Studies in Quantitative Philosophizing 9783110319613, 9783110319200

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Nicholas Rescher Studies in Quantitative Philosophizing

For Patrick Grim In cordial friendship

Nicholas Rescher

Studies in Quantitative Philosophizing

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Quantitative Philosophizing

CONTENTS Preface Chapter 1: On the Epistemology of Plato’s Divided Line

1

Chapter 2: Aristotle’s Golden Mean and the Epistemology of Ethical Understanding

51

Chapter 3: Ockham’s Razor and Ontological Economy

79

Chapter 4: Pascal’s Wager in Religion

99

Chapter 5: Leibniz on Coordinating Epistemology and Ontology

131

Chapter 6: Ethical Quantities

161

Preface

M

athematics is so powerful and useful a thought-tool that virtually from the start of the subject philosophers have occasionally throughout the years been tempted to use it in the course of their work—with varying but almost always interesting results. The present book brings together several case studies, dealing with relevant facets of the work of some of philosophy’s all-time greats. The subject-matter topic being addressed differs significantly, but in each case there is an attempt to apply mathematical methods and perspectives to the solution of a key philosophical issue in a way that throws instructive light upon it. On this basis it emerges that the question “Are mathematical methods useful in philosophy?” finds a suggestive response in the fact that over two millennia key figures in the history of the subject have indeed thought so. And they have substantiated this view not so much by abstract argumentation on the basis of general principles, but by making this point through actual practice. Plato is reported as insisting that the good philosopher must be competent in mathematics. And as these studies show that some of the most accomplished philosophers since his day proceeded in their own work in a way that indicates emphatic agreement. The first three chapters (on Plato, Aristotle, and Ockham) appear here for the first time. The final three have previously appeared in article form. (Detailed references are given in the footnotes.) I am very grateful to Estelle Burris for her help in putting this material into publishable form.

Nicholas Rescher Pittsburgh, Pennsylvania October 2009

Chapter 1 ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE 1. THE DIVIDED LINE AND ITS DIVISIONS

T

he principal contentions of this discussion are four: (1) that no interpretation of Books VI and VII of the Republic should be deemed adequate that fails to integrate its philosophical content with the mathematical detail Plato uses in his description of man’s cognitive situation, but that some plausible account must be given of why and how it is that those quantitative relations—the proportionalities and not mere analogies of the Divided Line—should hold; (2) that for a variety of reasons over 100 years of commentary has failed to meet this demand in a plausible way, a (3) that to remedy this situation it is instructive to take the Line’s narrative as literally as possible, and then look to its emplacement within the larger issues in Plato’s epistemology; and (4) that rather than dealing with different sorts of objects, the Line discussion deals with different modes (grades) of knowing (or, better, cognition).

* * * In Book VI of his classic dialogue, The Republic, Plato contemplated four factors at issue in inquiry and cognition: ideals or ideas (such as perfect beauty, justice, or goodness);1 mathematical idealizations (such as triangles, circles, or spheres); mundane, visible objects made by nature or man; and mere images, such as shadows and reflections. For abbreviative convenience we shall refer to these Platonic types as ideas (or forms), mathematicals, sensibles, and images, respectively. With this classification in view, Plato proceeded to envision our knowledge about the world in terms of an arrangement whose situation stands as follows: E D

C

B

A

In setting this out he proceeded as follows:

Nicholas Rescher • Quantitative Philosophizing

Suppose you take a line [EA], cut it into two unequal parts [at C] to represent, in proportion, the worlds of things seen [EC] and that of things thought [CA], and then cut each part in the same proportion [at D and B]. Your two parts in the world of things seen [ED and DC] will differ in degree of clearness and dimness, and one part [ED] will contain mere [sensory] images such as, first of all shadows, then reflections in water then surfaces which are of a close texture, smooth and shiny, and everything of that kind, if you understand.2

The realm of ideas is generated and organized under the aegis of a supreme agency, the Idea of the Good. In the lead-up to the discussion of the Divided Line in book VI of the Republic, Plato (or, rather, his protagonist Socrates) acknowledges (506d–e) his incapacity to expound the Idea of the Good itself, instead stressing its role in accounting for certain consequences, its “offspring” (ekgonos) and the “highest studies” (mathêmata megista, 504A) that provide a pathway towards it. And this path, so he maintains, can be illustrated by means of that diagramatic line. Plato’s Socrates then goes on to explain that in moving along a line from the mundane to the ideal we confront the following situation: In the first part [EC] the soul in its search is compelled to use the images of the things being imitated [that lie in DC] … In the second part [CA], the soul passes from an assumption to a first principle free from assumption, without the help of images which the other part [EC] uses, and makes its path of enquiry amongst idealizations themselves by means of them alone. (510B)

Plato correspondingly distinguished between the visible “things of the eye” (things seen, horata) and the intelligible “things of the mind” (things thought, noêta). Preeminent in the later category are the “ideas” or “forms” (ideai) that provide the model or prototype (paradeigma) conformity to which constitutes things as the kind of thing they are. Yet not these ideas alone, but also the mathematical idealizations have a paramount role in the realm of intelligibles: When geometers use visible figures and discuss about them, they are not thinking of these that they can see but rather the ideas that these resemble; a square in itself is what they speak of, and a diameter in itself, not the one they are drawing … What they seek is to see those ideas which can be seen only by the mind. (510D)

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ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE

Plato accordingly divided his line of cognition into two parts, respectively representing the intelligible and the visible realms, and then divides each of these into two parts into a higher and lower, each dealing with a correlative sort of cognition, as follows: I. “Intelligibles” 1. Higher: ideas (AB) 2. Lower: mathematicals (BC) II. “Visibles” 1. Higher: sensibles (CD) 2. Lower: images (DE) The cognitive landscape is mutually dualistic, contemplating two realms, the changeable and the unchangeable. However, the overall epistemology is quadratic, contemplating higher and lower modes of knowledge with respect to either category. Accordingly, as Plato saw it, what is pivotal with each of these four cognitive capacities in their relation to spatio-temporal issues can be indicated on the lines of Display 1.3 The four modes of cognition at issue thus differ in standing and status. At the top of the scale stand the Ideas—the timeless ultimates of Platonic concern. As G. W. Leibniz was to put it: The Platonists were not far wrong in recognizing four kinds of cognition of the mind … conjecture, experience, demonstration, and [finally] pure intuition which looks into the connections of truth by a single act of the mind and belongs to God in all things but is given to us in simple matters only.4

At the very bottom of the scale stand the “images” (eikones) at issue in suppositions based on the fleeting and superficial seemings of things: “shadows, reflections in pools and hard, smooth and polished surfaces, and everything of that sort” (510A).5 The formal deliberations of ratiocination and the concrete observations that ground our convictions about the world’s objects fall in between. As regards the mathematicals, there is an instructive passage in a critique of Plato in Aristotle’s Metaphysics.

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Display 1 HOW CAPACITY CONCERNS DIFFER Capacity

Mode of Cognition

Concerns

Temporal Aspect

(Mundane SpatioPhysical Aspect)

aisthesis

eikasia (supposition conjecture or imagination)

Images (eikones)

Fleeting

Present

aisthesi

pistis (observation-based conviction or belief)

Sensibles (aisthêta)

Transitory

Present

logos

logos dianoia (rationcination or discursive thought)

Mathematicals (mathêmatika)

Unchanging

Representable*

epistêmê (rational insight or reason)

Ideas (ideai)

Timeless

nous

Sensible Domain

Intelligible Domain Absent

*NOTE: What is here called mathematicals may encompass symbolically mediated thought in general. While physical objects such as diagrams and counters (“calculi”) can represent mathematicals, the physical world’s objects only “participate” in ideals and cannot represent them. Participation reaches across a wider gap than does representation.

Besides the Sensibles (aisthêta) and the Forms (ideai) he says that there are mathematicals (mathêmatika). These, so he says, are intermediate (metaxa) differing from the Sensibles in being eternal and immutable and from the Forms in that there are many like instances whereas the form itself is in each case unique. (Metaphysics 987b 14–18).

We thus have it that an individual Idea/Form is a single unique unit, despite there being a plurality of concrete particulars that participate in it. But a geometrical shape, for example a circle, has many abstract representations (differing in diameter, say), which are not concrete—though admitting of concrete participants in their turn.6 In summarizing the Divided Line discussion, the Republic stipulates that one should: Accept the four response-capacities (pathêmata) of the soul as corresponding to those four sectors: rational insight (noêsis) as the highest, ratiocination (dianoia) as the second, conviction (pistis) as the third, and supposition (eikasia) as the last; and arrange them proportionately, considering that they involve clarity (saphêneia) to the extent that the objects involve actual truth (alêtheia). (511E)

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Display 2 DESIGNATION FOR THE PLATONIC CAPACITIES epitêmê/logos/pistis//eikasia rational insight//ratiocination//conviction//supposition (Rescher 2009) intuition//demonstration//belief//conjecture (Whewell 1860) intelligence//thinking//belief//imagining (Cornford 1945) reason//understanding//belief//imagination (Wedberg, 1955) reason//understanding//belief//conjecture (Rouse 1956) intelligence//understanding//faith//conjecture (Malcolm 1962) intelligence//thinking//belief//illusion (Cross and Woozley 1964) intelligence//thought//conviction//conjecture (Robinson 1984) understanding//thought//confidence//imagination (Fine 1990) understanding//thought//belief//imagination (Grube 1974) intellect//thought//trust//fancy (Denyer 2007)

As Display 2 indicates, Plato’s translators have used a wide variety of terms for rendering the four Platonic faculties. While I believe my own translations come closest to what Plato has in view, I think that the time has passed for every discussant to introduce his own terminology. And so while I myself believe that the best nomenclature would be: Rational Insight//Ratiocination//Conviction//Supposition nevertheless, in the interests of impartiality, I think that we can live with the majority-rules reading of: Intellect//Thought//Belief//Imagination

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On this basis, every polled interpreter gets to have something their own way excepting—alas!—myself. Still, for the present I shall sink my own preferences in deference to the common good. Be the issue of terminology as it may, the fact remains that a definite four-rung ladder is at issue here, which conjointly characterizes both a type of knowing and a grade of knowledge. In ascending order these four are: superficial inspection (eikasia), observation (pistis), mathematically informed understanding (dianoia), and rational insight (epistêmê). Here mindmanaged dianoia, formal reasoning based on mathematics and logic, is seen as a more powerful cognitive instrumentality than anything that the senses have to offer us. But at the very top of the scale stands epistêmê, the authentic rational knowledge characterized by Plato as unerring (anmarêton: 477A), access to which is possible through dialectical reasoning alone. And what renders dianoia/mathematics inferior to noêsis/ideatics is that mathematical reasoning still relies on images (diagrams) and hypotheses while the methods of dialectic involve no such “contaminating” compromises with an inferior resource. Those four Platonic capacities are not different stages of learning, let alone “stages of mental development.” Nor do they address different kinds of existents of variantly inferior and superior nature, but rather different and variantly meritorious modes of cognition regarding existence: they deal not in degrees of reality but differently adequate degrees of insight into reality. And in just this way one recent interpreter speaks very sensibly of “les quatre degrés de conaissance.”7 What we have here is, in effect, four grades of knowing: superficial inspecting, close examination, quantitative measurement, and synoptic analysis. They represent different modes of knowing that offer increasingly more accurate insight into the nature of True Reality.8 Accordingly, the question “Does the Divided Line discussion deal with process (modes of cognition) or with product (objects of cognition): does it deal with ontology or with epistemology?” has to be answered by accentuating the latter. On the perspective at issue here, the crux lies in different modes of knowing, all addressed to one selfsame object, Reality, but dealing with it in different cognitive ways having very different degrees of clarity and adequacy.9 Along just these lines, Henry Jackson wrote “Now if the object of the inferior intellectual method is to the object of the superior as an image or reflection is to the thing itself … it would seem that the objects of the two sorts of intellectual methods are not distinct existence, but the same existences viewed [differently—] in the one case indirectly and in the other

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ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE

case directly.”10 To be sure, Stocks 1911 maintained that Plato subscribed “an old assumption, prevailed among the Greeks, [namely] that differences of apprehension must be due to differences of the apprehended.”11 There is, however, no reason to saddle Plato with the idea that different capacities must deal with different sorts of objects, but only that they can do so. In specific, those “higher” capacities need not deal with a higher class of objects: it is just that they can do so on occasion. The key point, as I see it, is not so much—or not saliently—that different sorts of things—different kinds of existing things are at issue here, but rather that we deal with different features of one single kind of thing—Reality—which can figure in cognition with very varying degrees of illumination. The issue, in sum, is a matter of dealing with things differently rather than one of dealing with different things. (See Display 3.) In her illuminating 1990 paper, Gail Fine contests what she calls the two-world theory according to which there is the world of sense and the world of intelligible forms, the first accessible only to mere belief but the second accessible to actual knowledge. The present approach takes this rejection one step further. It rejects not only the idea that different cognitive faculties address different “worlds” but also that they address different or dimensions of Reality. Instead, those different faculties address one object (Reality as it were) but with a very different yield in point of informative adequacy—though what even the lowest and most imperfect of them provides is not entirely useless. And the rationale of this view of the matter is in the final analysis that it best and most smoothly accommodates the comparabilities on which the entire Divided Line discussion is predicated. And this view of the matter is nowise contradicted by the discussion at the end of Book V where Plato stresses the different powers (dunameis) and different missions or functions of the former facilities. The contention heterô ara heteron ti dunanenê hekatera autôn pephuken ti. (Republic, 478) has indeed been translated: “Each of them, since it has a different power, is related to a different object” (Shorey-Loeb). But it would actually be more helpful—and more accurate—here to read product rather than object, seeing that this would alternate the suggestion that some distinctive type of thing is at issue. Viewed from this angle, the discussion of the cognitive faculties at the end of Book 5 is seen to hold that they deal with different takes on the real, and so not with different kinds of existents but with different ways of gaining a cognitive grip on what exists. It is certainly possible to argue for that variant interpretation, but the governing analogy of clarity of vision and il-

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Nicholas Rescher • Quantitative Philosophizing

lumination militates against this. The person who sees clearly, the person who sees poorly, and the person who is near-blind do not see different objects but rather all see rather differently and take what is seen to have very different features—only some few of which are authentic. Display 3 PLATO’S VIEW OF COGNITIVE PROCESSES AND THEIR OBJECTS Cognitive Resource or Capacity I.

KNOWING

(nous or gnôsis)

Process of Cognition I. INSIGHT (noêsis)

Resources of Cognition I. INTELLIGIBLE THOUGHTS (noêta)

1. Rational insight (epistêmê)

1. Intuitive grasps (epistasis)

2. Ratiocination (dianoia)

2. Formal reasoning (dianoêsis) 2. Mathematical Conceptions (mathêmata)

II. OPINING (doxa) [SENSING]

II. SENSORY APPREHENSIONS

1. Ideals and ideas, “Forms” (ideai, gnôsta)

II. SENSE JUDGMENTS (doxasta) or

(aisthêta) 1. Conviction (pistis)

1. Observation (horasis) and more generally perception (aesthesis)

1.Observed Features (horata)

2. Conjecture and seeming (eikasia)

2. Imaging (hêmoiôsis)

2. Casual Appearances or “Images” (phantsmata or eikona)

And so, notwithstanding the inclination of interpreters to have it that Plato holds that different faculties address different sorts or kinds or classes (Wedberg 1955, p. 108) of objects, the prospect is not only open but actually inviting of seeing what is at issue is a matter of different features or aspects of reality, differentiated with regard to the extent of the accuracy, authenticity of the information being furnished. So that, for example, those “mere appearances” do not reflect a clear grasp of a murky (or shadowy) object, but rather the confused, fuzzy product of a poor vision of reality.12 It is a salient feature of the Divided Line narrative that a certain proportionality obtains uniformly throughout these divisions, as represented by the dual proportions:

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ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE

I : II :: I1 : I2 :: II1 : II2 Thus overall, all of the following ratios (proportions) are all to be identical. • opinion : knowledge (EC : CA) • mathematical idealizations : ideal realities (CB : BA) • appearances : perceptions (ED : DC) Operative throughout is the crucial contrast between deep understanding (gnôsis) and mere superficial belief (doxa). The resultant situation is encapsulated in the line elaboration of Display 4. Scholars have worried—and of course disagreed—about whether the line is horizontal of vertical or diagonal.13 But this worry overlooks the clear lesson of Greek geometry that the orientation of a diagram just does not matter when the internal relations of a figure is at issue. Against this background, the present discussion will implement a certain definite perspective and procedure. It proposes to take the Divided Line narrative seriously as it stands literally and not more than minimally figurative or metaphorical. And it then asks where this leads in regard to the larger issues of Plato’s epistemology. So where most discussants have asked what Plato’s epistemology means for the Divided Line, the present discussion proposes to reverse this interpretative strategy. 2. WHAT DO THOSE PROPORTIONS REPRESENT? A helpful starting point for considering in Plato’s account here is the idea of a relational comparison or analogy based on the pattern: • Even as X is to Y in point of φ so also Z is to W in point of φ. On this basis, for example, the “ship of state” analogy would emerge roughly as follows: • Even as a ship’s people (crew and passengers) live under the aegis of a directive power (the captain) that is ultimately responsible for their well-being, so also do the people of a country live under the aegis of

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Nicholas Rescher • Quantitative Philosophizing

Display 4 PLATO’S DIVIDED LINE A MORE BRIGHTNESS (Greater Illumination)

INTELLECT (noesis) Ideas and Ideals (eidê)

a KNOWLEDGE Domain of the Good (Realm of Reason and Thought) (Authentic Knowledge: nous, episteme grosis)

B THOUGHT (dianoia) Mathematical Idealizations (mathematika)

b C

BELIEF (pistis) Objects of Authentic Vision (Observation) (horata)

c OPINION Domain of Vision (Realm of Sight and Sense) (Mere Opinion: doxa)

D IMAGINATION (eikasia) Images and Shadows (Appearance) (eikones)

d

MORE DARKNESS (Lesser Illumination)

E

a directive power (the government) that is ultimately responsible for their well-being. What is at issue in all such cases is an analogizing proportionality of the format: X : Y :: Z : W

10

in point of φ

ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE

Now whenever φ happens to be a feature that is quantifiable, then we are in a position to transmute the analogy at issue into an outright mathematical proportionality: X Z = Y W The ruling idea of Plato’s Divided Line is to exploit just this prospect of transmuting descriptive analogies into mathematical proportionalities. Plato’s Divided Line narrative transmutes what is a mere analogy (in our present sense) into a quantitative equation, an analogon in Aristotle’s technical sense of “an equating (isotês) of ratios or proportions (logoi)”.14 In its analytical role, the Idea of the Good mirrors the dual function of the sun in both providing the warmth that sensations organic life and the light by which existing things can be cognitively apprehended. On the cognitive side we reach the basic proportionality on which this process rests is: Light : Objects of sight :: the Good : Ideas

in point of φ

But what is φ to be in the Divided Line context? It must evidently be something that is quantifiable in order to provide for what can function as an outright proportionality-equation as per: (Sun)-Light The Good = Sight-objects Ideas And this, so Plato tells us, is illumination in its generally cognitive rather than merely visual sense. Light, of course, contrasts with darkness. At 478C–D we are introduced to yet another factor: ignorance (agnioia), and told that “opinion (doxa) is darker than knowledge (gnosis) and brighter than ignorance.” So ignorance (utter darkness), is at the bottom of the scale—“off the chart” so to speak. (And perhaps the Good is to be located similarly at the other end.) Both the general context of the discussion, and the Cave Allegory in particular, make it clear that those Divided Line segments are intended to correlate with the cognate power to give insight, to make intelligible, to illuminate. The crux of the matter is how much the information of a certain sort contributes to a proper understanding of the nature of reality and our

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Nicholas Rescher • Quantitative Philosophizing

place in it. Length is to reflect the comparative cognitive significance or importance in the wider setting of our knowledge of reality. Overall, in fact, Divided Line narrative presents us with a trio of proportionalities since we are told that: c a+b a = d = c+d b And we are told that these proportions are to reflect a differentiation in respect to truth and untruth (dihêrêsthai alêtheia te kai mê [510A]). And Plato has it that this is to be illustrative—preeminently daylight, the light of the sun. What is at issue here with illumination is increasing clarity of vision, be it by ocular sight or mental insight—the sort of thing inclined with the locution “Ah, it is now clear to me!” And as the discussion of the role of the sun at 507A–509B makes clear, the role of sunlight in apprehension is to mediate between the mind and its object. Just as the sun provides the power of visibility (ta to horasai dunamis) [509B], so the Good provides the power of intellection (ta to noêsei dunamis). Those proportions at issue are thus to reflect the comparative extent to which we are given significantly informative insights from the resources afforded by the mode of cognition at issue. The basic idea is that just as—and to the same extent that— sunlight makes sight-objects accessible to the mind through vision (horasis) so the Good makes ideas accessible to the mind through reason (noêsis). One writer has it that “One can easily argue that the length of the Line really does not matter. What matters are the ratios of its sections.”15 But this calls for qualification. All of those specified ratios and proportions are satisfied when those four segments are of equal length—something that clearly goes against Plato’s intentions and indeed has text which explicitly says that the line segments are unequal (anisa, 509D).16 Not only those ratios and propositions but also the (comparative) length of its segments are critical to the Line account. 3. THE ANALOGY OF LIGHT The divided line with its pinnacle of knowledge regarding the Ideas is joined to the simile of the Sun, that offspring (ekgonos) and resembler of the Idea of the Good (506E). And what both have in common is of course the illumination that constitutes a requisite for seeing things, be it with the

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ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE

eye of the body or the mind’s eye. Plato apparently holds that even as sunlight both reveals actual things and produces their shadows, so the intellect both reveals the Ideas and engenders their representations. The Divided Line is seen to provide a conjoint illustration of a cluster of proportions that implement the analogy of light. For explaining the proportions at issue, Plato tells us that the length of each segment measures the comparative “clarity and obscurity” (saphêneia kai asapheia) or “intelligibility” (alêtheia)17 of what is at issue—i.e., its comparative contribution to knowledge and understanding. To be sure there are many cognitive virtues: probability, informativeness, reliability, accuracy, detail, clarity. But none of these quite fill the bill. Instead, what seems paramount here is inherent in the simile of light: lucidity, illumination, insight, enlightenment. The model is the capacity for being seen that sunlight provides (ta tou horasthai dunamis [509B]). Something like profundity of understanding seems to be the issue—illumination or enlightenment (phanos) in short. Just this, we may suppose, is what Plato had in mind in speaking of “clarity and obscurity.” And just as the sum is the cause (aitia) of visual observation so the Idea of the Good affords “the very brightest illumination of being,” (tountos to phanotaton [518D]) in the realm of thought. This circumstance—that the line orders those faculties in point of cognitive power, and that the size of its segments reflects the amount of illumination achieved in the correlative domain—has been pretty much agreed upon since antiquity.18 Thus what we are dealing with are here differentially adequate cognitive responses to one single reality—not things of a different nature but different and differentially adequate takes on the nature of things. At the top, the dazzling brightness of the Idea of the Good yields greater—but not infinitely greater—information than that of our mundane observation. And this gearing to illumination means that the different parts of the line will deal— at least in the first instance—not so much with different kinds of knowledge as with different grades of knowledge, and not with different objects but with different levels of insight. Just what is to be made of Plato’s idea of illumination? It is clear, from what we are told, that even the image-mongering of mere “conjecture” (eikasia) provides some illumination and has some positive contribution to make. Granted, the illumination of the Good-illuminated Truth is vastly greater than that of the shadow-realm of mere images, but even this latter domain yields some illumination, albeit of a magnitude that is proportionally limited.

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Plutarch somewhat perversely suggested that the Divided Line narrative puts matters into reverse. As he had it, shorter line segments would better reflect coherence and unity of thought, while segments of greater length would better represent observability, indefiniteness, and of more obscure and less perspicuous knowledge.19 But this just is not how Plato’s account does it: his segments measure light rather than darkness. And even on the face of it, Plutarch’s complaint that the Line should measure obscurity rather than illumination seems problematic. After all, with measurement of all sorts one accentuates the positive: one measures the weight of objects not their lightness, the duration of time and not its brevity, the height of persons not their shortness. The cognitive level of authentic Knowledge (epistêmê) at issue in segment AB will always involve not just a certain fact but an explanatory rationale in which this certainty is grounded.20 The best construal for what Plato has in view is what one would understand as a matter of informativeness: insight into the significant truth of things. And this sets the goldplated standard by which the rest of our cognition must be judged. And illumination is the crux here since the mission of knowledge is to illuminate our way through this world’s darkness to the conception of a good life as encapsulated in the Idea of the Good. Gail Fine concluded in her instructive study of Plato’s epistemology by insisting that “Plato does indeed explicate epistêmê in terms of explanation and interconnectedness, and not in terms of certainty or vision.”21 But this view of the matter is predicated on maintaining a sharp contrast between a discursive and an intuitive approach to cognition. But in taking this position one elides the prospect that the apprehension of explanatory interconnections is the fuel that energizes the interactive apprehension of certainties, so that the grasp of explanatory connections can create an illuminative basis for intuitive certainty. One fails, in sum, to appreciate that discursive reasoning may open the gate to intuitive insight—that a great deal of reasoning may precede that EUREKA experience. But it seems to be along just these lines that Plato saw the connection between illumination and inquiry. To be sure, it is not the formal structure of the line itself but the substance of the overall explanatory discussion that is going to be crucial. For it is clear that the proportions of the Platonic Section do not of and by themselves accomplish the job that the Divided Line account is supposed to achieve. After all, the specified proportionality conditions are all satisfied when a = b = c = d = 1. To this extent, at least, there is justice in W. D. Ross’ observation that “the line, being but a symbol, is inadequate to the

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ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE

whole truth which Plato meant to symbolize.”22 For clearly the idea that equal illumination is provided by Vision and by Reason is a non-starter for Plato who rejects prospect out from the outset (at 509D). Overall, in coming to terms with the Divided Line narrative one must accordingly recognize: 1.

What is at issue are not items of knowledge, nor yet bodies or branches of knowledge, but types of knowledge as defined by the method of acquisition at issue: respectively superficial inspection, sensory observation, ratiocination/calculation, and dialectically developed insight. The focus is this less the product known than the process—the method of cognition that is at work.

2.

What is at issue is not the substance or theme of the sort of knowledge in question, but its significance or value.

3.

What is crucial in this valuation is neither the utility or applicative efficacy of the sort of true knowledge in question, nor yet the extent of time and effort needed for the mastery, but its illuminative strength: the extent to which it throws light on the condition of man in reality’s scheme of things.

4. The highest form of knowledge is not thought, ratiocination and calculation, but rather the philosophical wisdom achieved by the method of rational dialectic. However, even the world of shadows affords some instruction and enlightenment. While this is doubtless precious modest, it cannot be set at nothing, even in a comparison to authentic epistêmê. It would appear that in insisting on the philosophical importance of a mathematical informed view of things, Plato was putting his money where his mouth is in setting out the Divided Line narrative. 4. A POINT OF CONTENTION: DID PLATO MEAN IT? (METAPHOR OR MODEL?)

Scholars have long debated whether the Divided Line narrative is a mere flourish of literary ornamentation making the broad point that the realm of thought is superior to and more significant than the realm of sense, or

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whether something more substantial and significant is going on. Specifically, do those mathematical details matter? Some commentators have little patience with this entire Platonic exercise in mathematical epistemology. Some interpreters refuse flat-out to take the proportionalities of the Divided Line seriously. Thus R. N. Murphy has it that really “we are not dealing with mathematical proportions.”23 And another recent discussant dismisses the fourfold division and its proportionalities with the breezy comment that they are “at best a framework on which to hang the comparison of mathematics and dialectic, [and] at worst an empty play with the idea of mathematical proportion.”24 (If this is critical elucidation, then what price obfuscation?) The present discussion is predicated on the idea—the working hypothesis, if you will—that the mathematical detail of Plato’s discussion is to be taken seriously. It will thus be supposed that we are dealing not with some merely metaphorical analogy, but with a full-fledgedly mathematical description of man’s cognitive situation. And we shall suppose that Plato, good geometer that he was, formed his account with intention aforethought—that it was not some random whim that philosophers should be geometricians (mêdeis ageômetrêtos eisitô),25 and that Book VII of the Republic required that the training period in geometry for guardians be longer (indeed twice as long) as that in dialectical theory. Accordingly, the present deliberations will take the line that it is one of the salient tasks of an adequate interpretation to give some plausible account of why—and how—those quantitative relations would obtain on the basis of Platonic principles. Approached from this angle, the pivotal problem becomes that of explaining just how it is that the various mathematical specifications that Plato incorporated into his Divided Line narrative function to inform his theory of knowledge and to account for its formative features. The interpreter of Republic VI–VII who leaves those proportions out of consideration is offering us Hamlet without the Ghost. 5. THE PLATONIC SECTION Proceeding in this direction, let us envision the idea of a Platonic Section based on a diagrammatic set-up of the format:

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ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE

ED d

C c

B b

A a

where, as already noted, the magnitudes at issue are subject to the following specified proportionalities:26 a c a+b b =d =c+d The design of the Platonic Section has an array of significant mathematical consequences. These include: • b=c a a b c • b =c =d =d The appendices provided below will examine this situation more closely. To be sure, W. D. Ross is right in saying that “the equality of DC to CE, though it follows from the ratios prescribed, is never [explicitly] mentioned.”27 Some see this equality as “an undesirable though unavoidable consequence of the condition, which Plato would have avoided if he had been able, and to which we should attach no significance”28 with another commentator dismissing it as “as embarrassing detail.”29 But it would surely be unwise—as well as unkind—to fail to credit a geometer as sophisticated as Plato with recognizing consequences of his claims that would be at the disposal of a clever schoolboy. Anders Wedberg characterizes the equality of DC with CE as “obviously an unintended feature of the mathematical symbolism to which no particular significance should be attached.”30 But one wonders who conducted the séance at which Plato informed Wedberg of its unintendedness? 6. WHY SHOULD IT BE THAT b = c? Republic 510A says that the Divided Line’s segments represent “a division in respect of reality and truth” and not “in respect of decreasing reality and truth.” Yet nothing about the proportionalities at issue conflicts with the prospect that various segments have equal length. (However, Plato does

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block this prospect by a specific stipulation ad hoc at 509D). Nevertheless, it follows from the proportionalities of the Line that b = c. (See Appendix 1.) Moreover, Plato’s text nowhere explicitly acknowledges that BC = CD (i.e., b = c), and some commentators therefore think that “it may be indeed that he himself failed to notice that it was a consequence”.31 But could he really have been oblivious to this? Surely not! As J. S. Morrison has rightly insisted, “Plato was too good a geometrician for that.”32 For Plato, mathematically informed reasoning (dianoia) constitutes a mode of cognition superior to sense-based observation (aisthesis/pistis), seeing that it appertains to the intelligible rather than visible realm. On this ground, interpreters have been perplexed by the equating of BC and CD— the respective illumination afforded by those two cognitive resources. And such commentators have accordingly found it puzzling that a lower faculty should have as much to offer by way of cognitive illumination as a higher one. Thus H. W. D. Joseph observes that “the second and third segments as equal: whereas if Plato had wished to set forth to prosper in four stages, he should have given us a continuous proportion in four terms.”33 And Denyer wonders how this “surprising equality” can be reconciled with Plato’s view that cogitation/dianoia (Denyer calls it thought) outranks sensebelief/pistis (Denyer calls it trust).34 So why should Plato hold that sensory inspection and mathematical reflection to be co-equal in point of illumination? However, such puzzlement fails to distinguish between process and product: a more powerful process need not necessarily yield a greater result; it could well provide a product of co-equal value more elegantly or effectively. After all, an electronic typewriter is a more powerful instrument than a pen, but whatever it can write can be written by a pen as well. An automobile is a more powerful means of transport than Shank’s mare, but wherever the former can take you, the latter also can (if you have the energy and time). Through running together the Line and Cave, David Gallop depicts the line’s parts as involving the distinction between waking and dreaming35: A. the noêsis of dialectic: (“waking”) B.

the dianoia of mathematic: (“dreaming”)

C.

the horata of the natural science: (“waking”)

D. the ekasis of the plain man’s observations: (“dreaming”)

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ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE

But this raises real problems. In specific, why should mathematical cognition be a mode of “dreaming”? And why should the “dreaming” of B yield every bit as much illumination as the “waking” of C? Most likely, rather different considerations are at work in the Line and the Cave accounts—as will emerge more clearly below. The circumstance that while his specifications require that b = c, Plato himself does not make anything of this exerts a strange fascination on his interpreters. It emboldens them to think that they know Plato’s thoughts better than the man himself. Thus J. E. Raven writes: As Plato’s failure to mention the fact suggests, it is an unfortunate and irrelevant accident [that b = c]. Although it is a geometrical impossibility at once to preserve the [specified] proportions, which are all important, and to make each segment longer than the one below it, this is what Plato had it been possible, would have wished to do (Raven 1965, p. 145, my italics.)

So quoth the Raven. But how can he possibly know? Julia Annas endeavors to solve the problem by declaring that “Plato is not interested in having each section of the Line illustrate an increase in clarity …; his interest lies … in internal studies of each [segment], not in the whole line that results.”36 Yet one cannot but wonder why, if the Line structure is indeed immaterial, Plato should go to considerable lengths to set it out. No sign of disinterest, that! Morrison 1977 maintains that “the contents of the two middle subsectors (i.e., BC and CD) are identical, in the lower subsection (CD) used as originals and in the upper subsection (BC) used as likenesses” (p. 224n.) But this looks decidedly problematic. The crude diagram of the geometryteacher at issue with visualization is surely not a likeness of the theoretical mathematician’s abstract figure, but a crude representation (eikon) of it. The concerns of dionoia are a step upward from the deliverances of mere vision towards the Ideas, not a retrogression from them towards the phantasms of eikasia. Nicholas Smith has also been among those who wring their hands over the b = c of coordination of quantitative sense and quantitative thought. He writes: Plato never explicitly calls our attention to this equality [of b and c], and if we do attend to it we are led to problems in our understanding of the relative merits of the two subsections [of the Line] and to conflicts of what Plato does

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explicitly say about them … I am tempted to think that Plato might have purposefully wove these flaws into the intricate fabric of his own image because he wished to avoid the sin of perfection [as Persian rug-makers do].37

This is a certainly ingenious interpretative trope, but must surely function as an instrumentality of last resort. One plausible reason for having b = c is that this assimilates the defining proportions of Plato’s line to the harmonious mean that has a prominent place in the Greek mathematics of proportionality. For with b = c we return to the tripatitive proportionality relations commonly treated in the Greek theory of proportions where a salient example here is afforded by such proportionality relations as α : β :: β : γ This alone would yield a good reason for equating b and c. And there are, moreover, also substantive reasons grounded in the epistemological coordination of form and content, structure and substance in empirical knowledge. Thus BC = CD might perhaps obtain because everything in the world has a dual aspect: both a mathematically characterizable shape and a sense-provided qualitative texture. This idea is favored by Paul Pritchard who writes: “This much is clear, the objects in DC are the same as those in BC but now they are used as images of something else.”38 We are, that is, dealing with the same items regarded from different systemic points of view. And this perspective might well be grounded in Plato’s Pythagorean inclinations. After all, Plato seems drawn to the Pythagorean precedent of holding that such cognitive grasp as we securely have upon the mundane actualities of the world is mediated by mathematics.39 Accordingly, the guiding thought would be that of this world can be regarded either from the standpoint of empirical observation or from that of geometric analysis and that these approaches are of co-equal significance because each is informatively impotent without the other. Thus, equating BC and CD might be the result of the view that observation yields reliable information (“insight”) only—but to exactly the same extent—that it is mathematically formalizable. In the end, then, it may be that the relationship should be as one of coordination and that here something of a Kantian perspective is called for: observation without theory is blind and theory without observation is empty. The data of sensory perception (aesthesis) are only illuminating where rigorous reasoning (dianoia) can make sense

20

ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE

of them, and conversely dianoia cannot do its illuminative work without having the materials of aesthesis to address. 7. WHY SHOULD IT BE THAT a = b2 = c2 (WHEN d = 1)? Analysis of the proportions at issue with the Divided Line indicates that d, c, b¸ a stand to one another as per d, kd, kd, k2d. We shall designate these correlations as the Whewell Relations because this situation was first noted and discussed by William Whewell in his 1860 Philosophy of Discovery.40 These relations have it that when we do our measuring in terms of d as a unit (so that d = 1), we are lead straightway to the result that a = b2 = c2. And this opens up some larger vistas. For it means that when we use d as our unit of measure, then the overall proportionalities of the Divided Line will stand as follows: E D 1 d

C c c

B c b

A c2 a

During 1860–80, the Whewell Relations were considered by several commentators,41 but misunderstood by them as having d, c, b, a be 1, c, c2, c3 rather than 1, c, c, c2 as the just-given diagram indicates. (This error was noted by Henry Jackson in his 1882 paper.) Still, after the brief debate among the 19th century interpreters, the Whewell Relations simply dropped from sight. As far as I can see, no 20th century commentator has touched on these relations, and the question of their rationale remains in limbo. They do, however, have interesting ramifications. Specifically, they mean when we measure length in d units (with d = 1), we have it that the overall length of AE is 1 + c + c + c2 = 1 + 2c + c2 = (1 + c)2. On this basis we can say that the total illumination available in AE is exactly the square of the mundane illumination provided by the senses (in CE). Accordingly, c alone—the measurement of illumination afforded by the senses—can be seen as the determinative factor for the illumination provided by cognition at large. Can anything be said about the relationship of magnitude as between a and d? Not really! For the ratio a : d is left wide open. Nothing about those Line proportionalities bears upon the size of the other segments in relation to DE. That c2 is vastly greater than d is an ab extra supplementation to the

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postulated proportionalities, for—as already noted—nothing in the Platonic proportionalities prevents the prospect that a = b = c = d. (Clearly, these proportions tell only a part of the story!) After all, the Divided Line narrative must be construed in such a way that c is larger than d, and that in virtue of this a, which aligns with c2, becomes really enormous. The illumination of mindsight is vastly greater than that of eyesight. In context—but only then—are those proportionalities are effectively designed to carry a significant lesson. And so, when Sidgwick cavalierly dismisses “the fourth segment as of no metaphysical importance,” he ignores the inconvenient circumstance that the ratio d : a, albeit doubtless small, is nevertheless not zero.42 Yet why should Plato hold that a = b2 = c2 (with length measured in d units)? Why should the illumination of Reason so greatly amplify descriptive deliverances of qualitative perception and quantitative conception? Presumably the insight here is that we do not really understand something until we have embedded it within a larger framework of “scientifically” organized systematization—that is until we comprehend its place and role in the larger scheme of things and are able to characterize it descriptively but to explain it “scientifically.” Only when we know how an item figures in the larger explanatory framework of environing fact we really do understand it. Our comprehension of things is not real knowledge until we understand them in their wider systemic context. In short, what Plato seems to have in view is that higher kind of knowledge which Spinoza characterized as “an adequate knowledge of the essence of things” (adaequata cognitio essentiae rerum). In sum, intellect—that topmost “scientifically informed cognition” if you will—vastly amplifies the illumination provided by sensory information. In this perspective, Divided Line marks Plato as a quintessential rationalist. Still, why should it be that a = b2 = c2? Why is it a matter of squaring— why not have a come to c3 or 1000c? Dialectics, so Plato tells us, calls for “a synoptic view (sunopsis) of facts studied in the special sciences and their relationships to one another and to the nature of things” (537C). And so this square root relationship should really not be seen as all that puzzling. After all, when n items are at issue there will be n stories to be told by way of individual description. But if systemic understanding demands grasping how these items are related to one another, then there will be n x n = n2 stories to be told.

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ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE

8. THE ALLEGORY OF THE CAVE (514A–521B) How is one to fit the Divided Line narrative of Republic VI into the wider framework of Plato’s epistemology and, in particular, to coordinate it with the Cave Allegory of Republic VII? Especially because the Cave Allegory lacks the formalization of the Divided Line narrative, commentators have expended much ink on the question of how the two are related.43 It is clear that the Cave story envisions three regions: (1) the cave wall with its fire-projected shadows and images, (2) the entire cavern with its fire-illuminated visible objects, and (3) the exterior with its sun-illuminated realities assembled to the Platonic Ideas. As regards these, a fundamental proportionality is contemplated, for Plato tells us that “the realm of sight is like the habitation in prison [i.e., the cave], and the firelight there is like the sunlight” (517B). So overall, we are presented with the dual proportionality: the Good : ideas :: the sun : worldly things :: worldly things : images Interestingly, Plato begs off (at 506b–507a) from dealing with the Good as such in favor of dealing with its “offspring.” In effect, he says “Don’t ask me what the Good is, ask me rather about what it does.” (William James would love that!) And his response is that what the Good does is to serve a dual role. For one must distinguish between two questions “What (ontologically) is it that makes p be the case?” and “What is it (epistemically) that makes us think that p is the case?”—that is, we can ask both about a fact’s (ontological) truth-makers and also about the (epistemological) truthmarkers that lead us to accept it. And Plato’s line here is that as far as the world’s facts are concerned one and the same potency plays both roles. For the idea of the Good is the basis both of the world’s realities and of their knowability. As N. P. White concisely puts it, the idea of the Good is regarded by Plato “as the cause both of the being of intelligible objects as well as of our knowledge of them!”44 Like the sun which enables living creatures both to exist and to be seen, the Idea of the Good is the basic source both of the knowable and of its knowability. Viewed from this angle, there is nothing all that strange about the fundamental idea of the Cave Allegory. The Platonic parallelism between eyesight and insight, between vision and understanding, between the light of the sun and the enlightenment of thought, is actually pretty much taken for

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Nicholas Rescher • Quantitative Philosophizing

Display 5 POSSIBLE CAVE-TO-LINE CORRESPONDENCES Cave

Redistributional Match-Up of line Segments

α

I a

II a

III a+b

IV a

β

b

b+c

c

c

γ

c+d

d

d

d

granted by everyone. The student who grasps a mathematical concept immediately says: “Now I see it.” We say that it was a “flash of insight” that led Archimedes to exclaim eureka! Even in everyday use, “illumination” is as much mental as visual. In the Cave Allegory, three relationships are thus analogized in terms of proportionality among three triads:45 The Good//Rational Insight//Ideas The Sun//Sight//Visible Objects The Fire//Supposition//Shadows & Images The guiding idea is that the light of the fire in relation to the objects of the cave is like the illustration of the sun in relation to the objects outside. And so the Cave narrative envisions the analogy: shadows : objects :: object : ideas But if we now adopt a mathematical perspective and shift from analogy to proportion some basic facts come more sharply into view. For looking at the situation in terms of a linear arrangement (something that the Cave Allegory invites but does not explicitly state) we would have:

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ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE

Display 6 FOUR COGNITIVE PERSPECTIVES UPON THREE SORTS FO OBJECTS SUPERFICIAL IMPRESSIONS

mere sense-impressions

Perception OBSERVATIONS mundane realities GEO-METRICAL FACTS Cognition IDEALIZATIONS

γ shadows

β

pure theorizing

α visible objects

ideas

with the result of the following proportionality: γ β = or equivalently αγ = β2 β α This relationship is, in effect, simply the well-known geometric section of a harmonic mean amply elaborated upon in antiquity in the treatises of Niomachus and of Pappus.46 So if we once more conduct our mensuration in terms of γ-units, (so that γ = 1), then we again have β = α . Mathematical proportionalities once more confront us. But now there arises the critical question of bringing books VI and VII of the Republic into unison: Can the Divided Line narrative be reconciled with the Cave Allegory? Robinson 1952 maintains that Plato’s characterization of the Cave situation “forbids us to put it in exact correspondence with his Line.” But other commentators have disagreed: for example, Gould 1955, Malcolm 1962, and Morrison 1977. One potential strategy of reconciliation would proceed by reconfiguring the line segments to achieve a correspondence. The possibilities available

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Nicholas Rescher • Quantitative Philosophizing

here are inventoried in Display 6. As just noted, the Cave Allegory requires that: γ β = or equivalently αγ = β2 β α However, the Divided Line narrative requires both that b = c, and further that a = b2 when d is 1. So in relation to those cases in Display 6 we now require the following equations for transposing the α-to-γ range into the ato-d range: Case I. b2(b + 1) = b2. Not possible unless b = −1 or b = 0. Case II. b2 = (2b)2. Not possible unless b = 0. Case III. b2 + b = b2. Not possible unless b = 0. Case IV. a = c2. No problem! [See section 7 above!] And so, unless b = 0 the only viable match-up between Line and Cave is represented by IV which, exactly as one would surmise, leaves the Divided Line mathematicals (here represented by b) entirely out of view. We thus have a choice: we can annihilate the mathematical, or we can ignore them and simply let them drop out of sight. For on such an approach, dionoia with its concern for mathematika is dismissed. It simply appears to have vanished from the Cave account.47 How can this be? It would be tempting to try to reconcile the Line and the Cave accounts by the speculation that what is at issue is a matter of four cognitive perspectives upon three sorts of objects—rather along the lines of Display 6. Substantially this approach to the matter is taken in Wieland 1982. As he sees it, the things at issue with segments b and c, the sensibles and the arithematicals, represent one selfsame group of items, the natural world’s concreta, viewed two different perspectives, one qualitatively as objects of perception (Gegenstände der Wahrnehmung) and once quantitatively as material representations of forms (Abbilder [der Ideen]). Since merely different approaches (Einstellungen) toward the same objects are at issue, the two representing segments should be equal. This all sounds plausible enough, but, as the analysis relating to Display 5 clearly shows, it just does not square with the treatment of the mathematicals in the Divided Line

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ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE

narrative. For the preceding analysis blocks this otherwise attractive prospect of amalgamating b and c. While the overall account insists on their being co-equal, it blocks the prospect of their fusion via the impracticability of alternative II above. Again, it might be tempting to conjoin dianoia and epistêmê, then consolidating the two higher cognitive faculties into one—a prospect raised and informatively treated in Wedberg 1955 (pp. 103-111). But the impracticability of alternative III creates difficulties here. By a bit or fanciful legerdemain I. M. Crombie revamps the Cave account into four “stages” of image-to-original relationship in a way that coordinates the Cave with the Line, claiming that “Plato intended us to suppose that in the parable of the Cave he was putting flesh upon the bones of the skeleton he set out in the Line.”48 In effect he resorts to the matchup: a b c d

α β (part) β (part) γ

This seemingly provides for a smooth coordination between the Line and the Cave representations. But when we look at the matter in the reverse direction (from Cave to Line) we return to Case II of Display 5 and fall back into unavoidability. Rosemary Desjardines goes yet further in having it that a = c + d.49 Not only does this fail to be endorsed by Plato,50 but in the context of the ratios that he explicitly specified it would have some strange consequences. For example, since a a+b b =c+d it would mean that a2 = b(a + b) Note now that if we constructed out measurements in terms of a as a unit (a = 1), then the ratio d : c : b : a would be .38, .62, .62, 1.0 which would be bizarre given the intended interpretation of the Line.

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Nicholas Rescher • Quantitative Philosophizing

W. D. Ross believes that further ratios are also needed “and that this is mathematically impossible is only an indication of the fact that the line, being a symbol, is inadequate to the whole truth which Plato wanted to symbolize.”51 And some commentators incline to think that the mathematician’s symbols are somehow akin to shadows of the Cave allegory.52 (After all, both leave substance and content aside and deal only with structure and thus suggests a coordination of dianoia and eikasia.) But once again, the requisite detail for such an approach simply cannot be implemented satisfactorily. Colin Strang (1986) analyzes the Cave/Line relationship in a somewhat eccentric manner. With respect to the Line, he views it as having five divisions: noêsis dianoia epistêmê doxa pistis eikasia This introduces doxa as a separate division on a level with the other standard four. And as regards the Cave, he sees all five of these as functioning outside (i.e., above and beyond) the Cave and its firelight in the outer realm of the sun. It is unclear, however, both how this makes good textual sense, let alone how it provides for a more cogent philosophical systematization. Accordingly, Strang is constrained to insist that Plato’s own contention involves a variety of “misdirections,” and maintains that “no interpretation … can hope to emerge unscathed from the text,” claiming that his account “makes better philosophical sense than its rivals and can more easily explain away the anomalies that remain.” The first part of this contention may well be true, but the second part looks to be adrift in a sea of troubles. In particular, Strang’s account simply ignores the whole manifold of mathematical proportionalities that lay the groundwork for the Divided Line. How is one to reconcile the mathematical treatment of knowledge within the Divided Line account with Plato’s view of mathematics itself? Where—on its own telling—does the Line discussion stand? As Plato sees it, mathematics affords a suboptimal form of cognition in its reliance in hypotheses and assumptions—and nothing about Line division argues against seeing it in just this light. But on the other hand mathematics is

28

ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE

seen by Plato as also affording a step towards dialectics, and while the information it affords is not dialectically constituted epistêmê, there is certainly nothing false or misleading about it. Thus viewed in the light of its own epistemology, there is no reason why the Line discussion should not be taken seriously. It should certainly not be consigned to the shadowrealm of mere eikasia. Plato himself was keenly aware that the Divided Line narrative leaves a great deal unsaid and that its adequate exposition would require a much further explanation. In Book VII of the Republic, he has Socrates say: “But let us not, dear Glaucon, go further into the proposition between the lines representing the opinionable (doxaston) and the intelligible (noêton) so as not to involve ourselves in any more discussions than we have had already” (534A). Here Plato is clearly not retracting the Divided Line narrative but simply noting that it need not be further elaborated for the limited purposes just then at hand. And as Wedberg rightly observes “it is merely about the object of mathematics that [further] information is being withheld” at this particular juncture.53 So, what is one to make of this? How would one possibly account for the disappearance of the mathematics-oriented noêsis of segment b in the transit from the Line to the Cave account? There is, it would seem, one plausible way to do it—one that involves a fundamental shift of perspective. Suppose that dianoia is not conjoined with epistêmê, and b combined with a as per Case III of Table 6, but rather it is that the operations of dionoia are folded into and absorbed into epistêmê, high-level so that b effectively vanishes and its function is now provided for from within. The point is that even if we refrain from seeing the objects of mathematizing dionoa as themselves being Ideas (or Form), nevertheless as abstract and unchanging realities, they will fall into the same generically sense-transcending domain. Mathematics is thus seen as one integral part of a complex effort to detach people from the realm of sense: “to turn the soul’s attention upwards from the sensible to the intelligible” as one recent commenter puts it.54 To achieve real understanding we must leave any and all experientially guided suppositions behind, abandoning mere hypotheses for the solid ground of rationally apprehended principles. Mathematics itself thus becomes transformed into a science not just of basic ratiocination (which must inevitably proceed from premises) but one of rational insight because the fully trained mathematician comes to see why those hypotheses (those fundamental definitions, axioms, and post-

29

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ulates) come to be just what they are.55 Mathematics is now effectively seen as part of dialectic, and mathematical training becomes an integral component of the paidea of the philosopher-king. That is, we effectively shift from ousia to paideia, from ontology to education. On this basis, mathematics is no longer to be seen as a distinct discipline with a subjectmatter realm of its own (the mathematica), but rather a methodology of thought-discipline that is an essential part of the training of the philosopher-kings.56 Mathematics is thus cast in the role of the training-ground for abstracting from the mundane details of the sensible world and ratiocination (dianoia) is comprehended within reason (epistêmê) and b is not a supplement to a, but a component part of it. In this regard, the present analysis gives full marks to Henry Jackson, who wrote well over a century ago: There is no place for the mathematika [in the cave account]. Plato, as I understand him, is here concerned not with mathematika as apposed to other noêta, but with mathematika as types of noêta.57

From this standpoint, the condition of dianoia (b) is like that of a conquered state that is neither annihilated by nor annexed to another, but rather bodily absorbed into it. So regarded, mathematics acquires a different status, not as a distinctive field of inquiry but as a characteristic methodology of thought—a circumstance that merits its substantial emphasis in the educational deliberation of Book VII. But something comparatively radical is clearly needed. No wonder, then, that with such a shift of perspective considerable confusion might arise. In some of the expositions of Books VI and VII of the Republic—Fine 1990 for example—the mathematical aspect of the Divided Line is a non-entity, with the detail of those proportions seen as philosophically irrelevant. Other accounts take note of such detail as the fact that c = d but do not venture into an explanatory rationale (Fogelin 1972, for example). But be this as it may, it should be stressed that no interpretation of Books VI and VII of the Republic deserves to be deemed adequate that does not integrate the philosophical views being articulated with the mathematical detail being used in their articulation. And, above all, the Whewell Relations cannot simply be dropped from view. Granted, the proportionalistic structure of the Divided Line, which, after all, is its very reason for being as such, is something that simply does not interest various commentators. No doubt, the tentative suggestions of the present discussion can and should be improved upon. But the overall 30

ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE

project of getting this sort of thing right would seem to be something from which Plato’s interpreters cannot consciously beg off, and the offhand dismissal of the whole project by various interpreters is something that does little credit to Platonic scholarship. A couple of generations ago, A. S. Ferguson wrote that: “The similes of the Sun, Line, and cave in the Republic remain a reproach to Platonic scholarship because there is not agreement about them.”58 This may be going a bit too far. It is simply too much to expect scholars to agree on what Plato meant. But that he meant something—and something sensible at that—ought not be to a bone of contention.59, 60 To be sure, Plato’s interpreters would have been spared a great deal of grief if he had specified his Divided Line somewhat differently, namely as follows (with D and C equated): D

C

E

B d

c=b

A a

The construction of this line would be specified as follows: First divide the line EA twice, at D(=C) and at B, ensuring that the proportion of the first segment (ED) to the second (DB) is the same as that of this second to the last (BA).

The overall result that d : c :: c : or da = c2 would then align readily with the deliberations of the Cave discussion. If only the claim of Republic 509D had run along these lines, the work of interpreters would have been peaches and cream. But alas, it just was not so. As matters stand there is more work to be done. Appendix 1 PROOF THAT b = c

a c a+b b = d = c + d is given as basis Now observe:

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Nicholas Rescher • Quantitative Philosophizing

b=

a (c + d ) ac + ad c ( a + b) ac + bc = a+b = = =c a+b a+b a+b

a b a c Note that with b = c, the Platonic Section’s b = d at once yield’s b = d , the defining relation for the classical Harmonic Section. Furthermore it follows that we shall also have: a+b a+c a+b = = c+d b+d b+d c a Given basic relations of the Platonic Section, specifically b = d , the fact that b = c means that if we do our measurement in terms of b units (b= 1) 1 we have a = d . There is an inverse complementarity universe relationship between a and d. And this means that if d is very small (as it is bound to be) then a must be very large. Appendix 2 2

2

PROOF THAT a = b = c WHEN d IS THE UNIT OF MEASUREMENT

Let the ratio of c to d be r, so that c = dr. Then since b = c, we also have b = dr. But since the ratio of a to b is also to be r, we have a = rb = dr2. Thus the quartet d, c, b, a will be as d, dr, dr, dr2. All this is exactly what Whewell noted in 1860—and so represents what may be referred to as the Whewell Relations. And they mean that the overall length of the line, λ, c must be d(1 + 2r + r2) = d(1 + r)2, with r = d . Thus if we make our measurements in terms of d units (d = 1), then λ = (1 + c)2. And if we were to make our measurements in terms of b units (so b = c = 1), then λ = d(1 + 1 2 d ) . Either way, the main point is clear. As long as d is very small in relation to the other qualities, the overall length of the line will be large, with a taking the lion’s share.

32

ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE

Appendix 3 OBSERVATIONS ON THE PLATONIC SECTION

The abstract proportionalities at issue with Plato’s Divided Line will not of themselves determine the relative size or magnitudes of the quantities involved. For consider once more those proportions superimposed upon the linear scheme d

c

b

a

namely a c a+b = = b d c+d Note that the specified relations are such that a could in theory be as small as d itself, seeing that a = b = c = d = 1 will satisfy all of these proportionalities. This seems to appeal to some interpreters. Thus to the proposition a c b a b = proposes to add that these quantities also come to so that b d c b =c = c c . This preserves a uniform proportionality throughout, so that if we let d d b+a c = k, then c = kd, b = k2d, a = k3d, But now the only way to have c+d = d is to have it that d(k2+k3) dk2(1+k) 2 d(1+k) = d(1+k) = k = k This means that k = 1 so that four of these line segments must now be equal. But this hardly squares with the circumstance that those segments represent comparative illumination. Plato’s own stipulations have the consequence that a can be larger than the rest by any desired quantity whatsoever. For let us once more proceed to measure length in terms of d-units (i.e., d = 1). Then a discrepancy in the size in the magnitudes at issue of any size whatsoever will be able to satisfy all those proportionalities—however large is may be—as long as:

33

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a=k b= k c = k d=1 with the magnitude of k left open. So as regards the potential disparity between b, c, d and a, the sky is the limit. An interesting perspective emerges when measurement is made in terms of b (b = 1). For d, c, b, a will now stand as d, 1, 1,

1 . Since we now have d

b = c = 1, a relationship of reciprocal complementarity between d and a = 1 must obtain. That is, with d a very small quantity a will be a very big d

one (and conversely). The dimness of those mundane reflections is in diametrical contrast with the brightness of sunshine. Appendix 4 IS THE GOLDEN SECTION RELEVANT?

Some interpreters try to bring the Golden Section into the analysis of the Divided Line.61 This is based on the setup: x

y

y x with x : y :: y : x + y or equivalently y = x+y Applying this Golden Section specification to the Divided Line quantities would mean: a b c d = and = b a+b d c+d However, the specified Divided Line relations have it that

34

ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE

c a+b = d c+d This leads to d = a + b. And this makes d bigger than either a or b, which is of course absurd in view of Plato’s interpretation of these quantities. A possible alternative way of bringing the Golden Section into it would be to stipulate that: c+d a+b = a+b a+b+c+d Or equivalently: 1 c+d z = 1 + 1+z where z = a+b c+d This means that z = a+b ≅ .4 Consequently c + d ≅ .4 (a + b) so that c + d ≅ .4a + .4b. But since b = c this means .6c + d = .4a or a ≅ 1.5c + 2.5d. Since d is very small in relation to a and c, this makes a substantially smaller than c (and for that matter b). This too is starkly inconsistent with Plato’s specified interpretation of these quantities. So the Golden Section and the Divided Line look to be irresolvable. As regards the idea of relating the Cave Allegory to the Golden Section, consider the following. The Cave situation is based on the set-up: γ shadows where

β visibles

α ideas

γ β = β α

Now let us superimpose upon this the perspective of Golden Ratio situation: x

y 35

Nicholas Rescher • Quantitative Philosophizing

where

y x = x+y y

There are two possibilities here No. 1: x = γ + β and y = α No. 2: x = γ and y = β + α With alternative No. I we would need to have

α γ+β = α+β+γ α Since γ must be very small in relation to α and β this yields

α β ≅ α α+β

But since

β γ = , this comes to α β

α γ ≅ α+β β However, γ is to be very small in relation to α and β. And this means that α must be very small in relation to α + β, or in other words that β must be very large in relation to α. But this is totally out of synch with the Platonic stipulations. With regard to possibility No. 2 we would need to have

γ β+α = β+α α+β+γ Since γ is very small in relation to α and β we have α + β + γ ≅ α + β. So now

36

ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE

β+α γ ≅ =1 α+β β+α But given that γ is very small in relation to α and β, this too is impossible. So in no case can the situation of the Cave Account be squared with the idea that the relationships involved come to a Golden Section. One had best leave the Golden Section out of it. NOTES 1

Contemporary discussants often call these forms. But a rose by any other name . . .

2

Republic, 509D. Henceforth otherwise unspecified references are to this dialogue.

3

Plato uses the term hexis, i.e., capacity or skill or facility involved with a certain practice, what translators often render as Facility (Greek dianamis) a terms which, in this context, awaits Aristotle. But for dianamis in the sense of power see 509B.

4

G. W. Leibniz: Philosophical Papers and Letters, ed. by L. E. Loemker (Dordrecht & Boston, 1969), p. 593. [Letter to M. G. Hansch on the Platonic Philosophy: 25 July 1707.]

5

For lucid accounts of eikasia see Paton 1921 and Hamlyn 1958. (With references of this format see the Bibliography.)

6

On the mathematika see Mittelstrass 1976. Admittedly, Myles Burnyeat is quite right, the Republic leaves the question of the ontological status of the mathematicals “tantalizingly open.” (Burnyeat 2000, p. 22). However, see also Gill 2004 and 2007.

7

Lefrance 1977, p. 429.

8

However, on this dogmatic view of the matter see Joseph 1948 who covers a wide range of opinion on the topic.

9

That what is at issue with those different “faculties” is not a matter of ontologically different thing-kinds but rather different ways of knowing is also maintained in Smith 1996, p. 38.

10

Jackson 1882, p. 135.

11

Stocks 1911, p. 76.

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NOTES 12

Note how well this accounts with the treatment of flawed judgment in the Theaetetus and the Sophists, whose “images” are seen not as a mode of false reality but rather as a falsification of reality. Error, as Plato sees it, is not a matter of faithful subscription to a false reality, but rather an unfaithful subsumption to reality.

13

Their discordant views are surveyed in Smith 1996, pp. 26–27.

14

Niomachean Ethics, 1131a31. On Plato’s handling of analogies see Ferguson 1934.

15

Balashon, 1994, p. 284.

16

It may be true that “Nos plus anciens manuscrits révelent déjà une hesitation entre deux lectures possible de 509d7, c’est-à-dire isa at anisa” (Lefrance 1977, p. 430.) But this is hardly problematic as long as degrees of illumination—of clarity and obscurity—are to be at issue, as the text explicitly stipulates.

17

See 509D and cf. 478C-D.

18

Proclus’ Commentaries on the *Republic* of Plato observes—as is Plutarch’s view (in his Platonic Questions)—that in order of volume/quantity of information (rather than quality) one would have to reverse the size of the segments. See also Section 5 below and Denyer 2007, p. 293.

19

Plutarch, Platonic Questions, 1001 d–e.

20

Meno 98a, Phaedo 76b, Republic 531c, 534b.

21

Fine 1990, p. 115.

22

Ross 1951, p. 46.

23

R. N. Murphy, The Interpretation of Plato’s Republic (Oxford: Clarendon Press, 1951), p. 159.

24

Robinson 1978, p. 193.

25

However anachronistic the latter of this observation, its spirit seems thoroughly Platonic.

26

38

a c a b DesJardines 1990 (p. 481) has it that not only b = d , but also that c+d = c . This second proportion looks to be without visible means of support. Compare Pritchard 1995, p 97.

ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE

NOTES 27

Ross 1951, p. 45.

28

Prichard 1995, p. 91. Here Pritchard does not speak in propia persona.

29

Gould 1955, p. 31.

30

Wedberg 1955, pp. 102–03.

31

Cross & Woozley 1964, p. 209. Raven [1965, p. 145] speaks of it as “an unfortunate and irrelevant accident.” And Prichard 1995 notes that various commentators view this as “an undesirable though unavoidable consequence of the [specific] conditions, which Plato would have answered if he had been able, and to which we are to attach no significance” (p. 91).

32

Morrison 1977, p. 213.

33

Joseph 1948, p. 32.

34

Denyer 2007, p. 295.

35

See Gallop 1915.

36

Ammos 1981, p. 248.

37

Smith 1996, p. 43.

38

Pritchard 1995, p. 92.

39

On Plato’s Pythagoreanism see Frank 1923.

40

William Whewell, The Philosophy of Discovery (London: Parker & Son, 1860), p. 444. See Appendix 2. There should be little wonder that Whewell had a firmer grip than most on the mathematics of the Divided Line, for alone among Platonic scholars he was a senior wrangler at Cambridge.

41

See Jowett 1894 and Sidgwick 1869.

42

Sidgwick 1869, p. 102. Plato’s widely discussed rejection of the “Third Man” argument in the Parmenides (131e–133a) and elsewhere (Republic, 597c and Timaeus, 31a–b) is crucial here. For if that ad infinitum regress worked and the realm of forms had an infinite complexity, then notwithstanding the Divided Line account

39

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NOTES

it could bear no fixed proportion to the realm of mundane cognition at issue with the inferior capacities. 43

Virtually every commentary cited in our bibliography has much to say on the subject.

44

White 1979, p. 180. It is striking, but not untypical, that White’s commentary leaves the mathematical proportionalities of the Divided Line out of consideration.

45

Scholars have disputed about just how many epistemic division are in play with the Cave Allegory. (See, for example, Robinson 1953 and Malcolm 1962.) The tripartite picture contemplated here looks to be not only the simplest but also the most natural reading of the text.

46

See T. L. Heath A History of Greek Mathematics, Vol. 1 (Oxford: Clarendon Press, 1921), p. 87.

47

That those accounts are irreconcilable is sometimes maintained. (See, for example, Robinson 1984, pp. 181–82.) However the simplicity of the reason for this—viz., the Cave’s indifferent to dionoia—has not been duly emphasized.

48

Crombie 1962, p. 116.

49

See Desjardines 1990, p. 491.

50

See Pritchard 1995, pp. 97–98.

51

Ross 1951, pp. 45–46.

52

See Ferguson 1921, p. 148.

53

See Wedberg 1955.

54

Meuler 1992, p. 189. Here one need not go quite as far as Frank 1923 and hold that “Plato die Idee noch rein quantitativ als die bloße mathematische ideale Form der Dinge, d. h. als Zahl gefasst hat” (p. 60).

55

Compare Burnyeat 1987 and 2000, as well as Gill 2004 and 2007.

56

On this issue see Robins 1995.

57

Jackson 1882, p. 141.

40

ON THE EPISTEMOLOGY OF PLATO’S DIVIDED LINE

NOTES 58

Ferguson 1934, p. 190.

59

I cannot forego the observation that with regard to the specific issue being investigated here—namely the proportionalities at work in the Divided Line narrative of Republic VI and the Cave Allegory of Republic VII—I find the 19th century Platonic commentators—and in specific Whewell, Jackson, and Ferguson—more helpful than their 20th century successors.

60

I am grateful to Paul Scade for his constructive comments.

61

On the relevant controversies see Smith 1996, p. 28 (p. 12).

41

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Brentlinger, J. A., “The Divided Line and Plato’s ‘Theory of Intermediates’,” Phronesis, vol. 8 (1963), pp. 146–166. Burnyeat, Myles F., “Platonism and Mathematics: A Prelude to Discussion,” in Andreas Graeser (ed.), Mathematics and Metaphysics in Aristotle (Bern & Stuttgart: Paul Haupt, 1987). Burnyeat, Myles F., “Plato on Why Mathematics is Good for the Soul,” in T. Smiley (ed.), Mathematics and Necessity: Essays in the History of Philosophy, Proceedings of the British Academy, vol. 103 (2000), pp. 1–81. Campbell, Lews, Plato’s Republic: The Greek Text, Edited with Notes and Essays by B. Jowett and L. Campbell (Oxford: Clarendon Press, 1894). See Vol. II, Essays, pp. 161–64. Calvo, Tomás, and Luc Brisson (eds.), Interpreting the Timaeus and the Critias (St. Augustine: Akademie Verlag, 1997). Carone, G. R., “The Ethical Function of Astronomy in Plato’s Timaeus,” in: Tomás Calvo Martínez; Luc Brisson (eds.), Interpreting the Timaeus and the Critias (St. Augustin: Academia Verlag, 1997). Carrive, Paulette, “Encore la caverne, ou 4 = 8,” Les Études philosophiques, vol. 30 (1975), pp. 387–397. Cooper, Neil, “The Importance of διαυοια in Plato’s Theory of Forms,” Classical Quarterly, N. S., vol. 16 (1966), pp. 65–69. Cornford, F. M., “Mathematics and Dialectic in the Republic VI–VIII,” Mind, vol. 41 (1932), pp. 37–52 (Part I) and pp. 173–90 (Part II). Reprinted in: R. E. Allen (ed.), Studies in Plato’s Metaphysics (London: Routledge & K. Paul, 1900), pp. 61–95. Cornford, F. M., Plato’s Theory of Knowledge (London, 1935). Cornford, F. M., The Republic of Plato (London: Oxford University Press, 1941). Crombie, I. M., An Examination of Plato’s Doctrines, 2 Vol. (London: Routledge & Kegan Paul, 1962 and 1963). Cross, R. C. and A. D. Woozley, Plato’s Republic: A Philosophical Commentary (Basingstoke: Macmillan, 1964). Davies, J. C., “Plato’s Dialectic: Some Thoughts on the Line,” Orpheus, vol. 14 (1967), pp. 3–11.

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Denyer, Nicholas, “Sun and Line: The ‘Role of the Good’,” in G. R. F. Ferrari (ed.), The Cambridge Companion To Plato's Republic (Cambridge: Cambridge University Press, 2007). Des Jardins, Gregory, “How to Divide the Divided Line,” The Review of Metaphysics, vol. 29 (1976), pp. 483–496. Desjardines, Rosemary, The Rational Enterprise: Logos in Plato’s Theaetetus (Albany, NY: State of New York University Press, 1990). De Strycker, Emile, “La distinction entre l'entendement (διαυοια) et I'intellect (υουç) dans la Republique de Platon,” in Estudios de historia de la filosofia en hominaje a1 Professor R. Mondalfo (Madrid, 1957). Dreher, John Paul, “The Driving Ratio in Plato's Divided Line,” Ancient Philosophy vol. 10 (1990), pp. 159–172. Elliot, R. K., “Socrates and Plato’s Cave,” Kant Studien, vol. 58 (1967), pp. 137–57. Ferguson, A. S., “Plato’s Simile of Light Part I: The Similes of the Sun and the Line,” Classical Quarterly, vol. 15 (1921), pp. 131–152. Ferguson, A. S., “Plato’s Simile of Light Part II,” Classical Quarterly, vol. 16 (1922), pp. 15–28. Ferguson, A. S., “Plato’s Simile of Light Again,” Classical Quarterly, vol. 28 (1934), pp. 190–210. Ferguson, John, “Sun, Line and Cave Again,” Classical Quarterly, vol. 13 (1963), pp. 188–93. Fine, Gail, “Knowledge and Belief in Republic V,” Archiv für Geschichte der Philosophie, vol. 60 (1978), pp. 121–39, reprinted in Fine 2003. Fine, Gail, “Knowledge and Belief in Republic VI–VII,” in Stephen Everson (ed.), Epistemology: Compass to Ancient Thought, Vol. I (Cambridge: Cambridge University Press, 1990), pp. 85–115, reprinted in Fine 2003. Fine, Gail, Plato on Knowledge and Form: Selected Essays (Oxford: Clarendon Press, 2003). Fogelin, Robert J., “Three Platonic Analogies,” The Philosophical Review, vol. 80 (1971), pp. 371–82. Reprinted in his Philosophical Interpretation (New York: Oxford University Press, 1992).

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Frank, Erich, Plato und die sogenannten Pythagoraer (Halle: Max Niemayer, 1923, 2nd ed., Darmstadt: Wissenschaftlishe Buchgesellschaft, 1962). Gallop, David, “Image and Reality in Plato’s Republic,” Archiv für Geschichte der Philosophie, vol. 47 (1965), pp. 113–31. Gallop, David, “Dreaming and Waking in Plato,” in J. P. Anton et. al. (eds.), Essays in Ancient Greek Philosophy, Vol. 2 (Albany, 1971), pp. 187–201. Gill, Christopher, “Plato, Ethics and Mathematic,” in: M. Migliori and D. DelForno (eds.) Plato Ethicus: Philosophy is Life: Proceedings of the International Colloquium Piacenza (St. Augustin: Academia Verlag, 2004), pp. 165–75. Gill, Christopher, “The Good and Mathematics,” in Douglas L. Cairns, Herrmann Fritz-Gregor, and Terry Penner (eds.), Pursuing the Good: Ethics and Metaphysics in Plato’s Republic (Edinburgh: Edinburgh University Press, 2007) pp. 251–74. Gould, John, The Development of Plato’s Ethics (Cambridge: Cambridge University Press, 1955). [See Chap. 13.] Grube, G. M. A. (ed.), Plato’s Republic (Indianapolis: Hackett, 1974). Hackforth, Reginald, “Plato’s Divided Line and Dialectic,” Classical Quarterly, vol. 36 (1942), pp. 1–9. Hahn, Robert, “A Note on Plato’s Divided Line,” Journal of the History of Philosophy, vol. 21 (1983), pp. 235–237. Hamlyn, D. W., “Eikasia in Plato’s Republic,” The Philosophical Quarterly, vol. 8 (1958), pp. 14–23. Hare, R. M., “Plato and the Mathematicians,” in Renford Bambrough (ed.), New Essays on Plato and Aristotle (London: Routlledge & Kegan Paul, 1965), pp. 21– 38. Jackson, Henry, “On Plato’s Republic VI, 509D sqq.” The Journal of Philology, vol. 10 (1882), pp. 132–50. Joseph, H. W. D., Knowledge and the Good in Plato’s Republic (London: University Press, 1948). Jowett, Benjamin, Plato’s Republic: Text, Translation and Commentary, 3 vol.’s (Oxford: Clarendon Press, 1894).

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Kraut, Richard (ed.), The Cambridge Companion to Plato (Cambridge: Cambridge University Press, 1992). LaFrance, Yvon, “Platon et la géométrie: la construction de la ligne en République, 509d–511e,” Dialogue, vol. 16 (1977), pp. 425–450. LaFrance, Yvon, Pour intelpréter Platon. La Ligne en République VI, 509d-511e. Bilan analytique des études (1804–1984) (Montreal and Paris, 1986). Larson, Raymond, The Republic (Arlington Heights, IL: AHM Publishing, 1979). Lee, H. D. P., Plato, The Republic, 2nd edn (Baltimore, 1974). Malcolm, John, “The Line and the Cave,” Phronesis, vol. 7 (1962), pp. 38–45. Mansion, S., “L’objet des mathematiques et l’objet de la dialectique selon Platon,” Revue Philosophique de Louvain, vol. 67 (1969), pp. 365–388. Martínez, Tomás Calvo and Luc Brisson (eds.), Interpreting the Timaeus and the Critias (St. Augustin: Academia Verlag, 1997). Meuller, Ian, “Ascending to Problems: Astronomy and Harmonies in Republic VII,” in: John Peter Anton (ed.), Science and the Sciences in Plato (New York: EIDOS, 1980), pp. 103-122. Meuller, Ian, “Peri tôn mathêmatôn: Essays on Greek Mathematics and its Later Development,” Apeiron, vol. 24 (1991), pp. 85–104. Meuller, Ian, “Mathematics and Education: Some Notes on the Platonist Programme,” in his 1991. Meuller, Ian, “Mathematical Method and Philosophical Truth,” in: Richard Kraut (ed.), The Cambridge Companion to Plato (Cambridge: Cambridge University Press, 1992), pp. 170–99. Mittelstrass, Jürgen, “Die Dialektik und ihre wissenschaftlichen Vorübungen,” in: O. Höffe (ed.), Platon: Politeia (Berlin: Akademie Verlag, 1997), pp. 229–49. Mohr, Richard, “The Divided Line and the Doctrine of Recollection in Plato,” Apeiron, vol. 8 (1984), pp. 34–41. Morrison, J. S., “Two Unresolved Difficulties in the Line and Cave,” Phronesis, vol. 2 (1977), pp. 212–31.

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Mourelatos, Alexander, “Plato’s ‘Real Astronomy’: Republic 527d–531d,” in: John Peter Anton (ed.), Science and the Sciences in Plato (New York: EIDOS, 1980). Murphy, Neville Richard, “The Simile of Light in Plato’s Republic,” Classic Quarterly, vol. 26 (1931), pp. 93–102. Murphy, Neville Richard, The Interpretation of Plato’s Republic (Oxford: Clarendon Press, 1951). [See Chap. 8.] Nettleship, R. L., Lectures on the Republic of Plato (London: Macmillan, 1898). Paton, H. J., “Plato’s Theory of Eikasia,” Proceedings of the Aristotelian Society, vol. 22 (1921/22), pp. 69–104. Philipousis, J., “La gnoséologie de Platon selon la République. Connaissance et dialectique”, La communication: Actes du XVe Comreès de l’Addociatoin des Societés de Philosophis de Longrie Francaise (Montreal: Montmorency, 1971, pp. 90–95). Pritchard, Paul, Plato’s Philosophy of Mathematics (St. Augustin: Academia Verlag, 1995). Raven, J. E., “Sun, Divided Line, and Cave,” Classical Quarterly, vol. 3 (1953), pp. 22–32. Raven, J. E., Plato’s Thought in the Making: A Study of the Development of his Metaphysics (Cambridge, Cambridge University Press, 1965). [See Chap. 9.] Richards, I. A., “Plato’s Republic (Cambridge: Cambridge University Press, 1966). Ringbom, Sixten, “Plato on Images,” Theoria, vol. 31 (1965), pp. 86–109. Robins, Ian, “Mathematics and the Conversion of Mind: Republic VII 522C1–531E3,” Ancient Philosophy, vol. 15 (1995), pp. 359–91. Robinson, Richard, “Hypothesis in the Republic,” in Gregory Vlastos (ed.), Plato: A Collection of Critical Essays (Notre Dame, IN: University of Notre Dame Pres, 1978). Robinson, Richard, Plato’s Earlier Dialectic (Oxford: Clarendon Press, 2nd ed. 1984). Rose, L. E., “Plato’s Divided Line,” Review of Metaphysics, vol. 17 (1963–64), pp. 425–435.

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Whewell, William, The Platonic Dialogues for English Readers, Vol. III The Republic and the Timaeus (London: Macmillan, 1861). White, Nicholas P., A Companion to Plato’s Republic (Indianapolis, Ind.: Hackett, 1979). Wieland, Wolfgang, Platon und die Formen des Wissens (Göttingen: Vandentrooek & Ruprecht, 1982). Wilson, J. R. S., “The Contents of the Cave,” in R. S. Shiner and John King-Farlow (eds.), New Essays on Plato and the Presocratics, (Guelph, Ont: Canadian Association for Publishing in Philosophy, 1976), pp. 117–27. Wilson, J. R. S., “The Argument of Republic IV, Philosophical Quarterly, vol. 26 (1976), pp. 111–14. Wright, J. H., “The Origin of Plato’s Cave,” Harvard Studies in Classical Philology, vol. 27 (1906), pp. 130–42. Wu, Joseph S., “A Note on the Third Section of the Divided Line,” New Scholasticism, vol. 43 (1969).

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Chapter 2 ARISTOTLE’S GOLDEN MEAN AND THE EPISTEMOLOGY OF ETHICAL UNDERSTANDING 1. THE ARISTOTELIAN MEAN

I

n the introductory remarks to Book I of the Nichomachean Ethics, Aristotle apologized for a lack of precision and clarity of his ethical deliberations regarding the good life: Our discussion will be adequate if it has as much clarity as the subject-matter admits of, for precision is not to be sought for alike in all discussions, any more than in all craft-productions. It is the mark of an educated man to look for precision in each class of things only insofar as the nature of the subject admits; it is evidently equally foolish to accept probable reasoning from a mathematician and to demand from a rhetorician scientific proofs. (NE 1094b 11–27.)

Why should such an apology be needed? Just what is there about ethics on Aristotle’s conception of the matter that mandates this rather guarded view of the field?1 Aristotle’s apologia may seem odd in the face of his efforts to present a decidedly systematic and indeed mathematically geared account of his ethical theory. For Aristotle taught that a guiding standard of human virtue or excellence is constituted by a proportionate intermediation between opposed extremes of insufficiency and surfeit. As he put it: [In matters of conduct] as in everything that is continuous and divisible, it is possible to take a lot or a little or a middling amount, and to do this either in the thing itself or relatively to as (pros hêmos), and the proper amount is something intermediate between excess and insufficiency. (1106a 26–29).

Even in regard to a potentially positive mode of human comportment there nevertheless is something that can be over- or under-done through its presence to a greater or lesser extent on more or less frequent occasions. As

Nicholas Rescher • Quantitative Philosophizing

Aristotle accordingly sees it, virtue pivots on getting the right balance between too little and too much. And he substantiated this view on the basis of numerous illustrations along the lines of Display 1, explaining this doctrine of what Horace was to call a golden mean (aurea mediocritas) in the following terms: Virtue (arête) is a mean (meson) between two vices, one of excess and one of deficiency, being an intermediary in relation to which the vices respectively fall short or exceed that which is right alike in reaction and in action, while virtue finds and chooses the [appropriate] intermediary (meson). (1107a3).

The man who fears nothing is foolhardy; the man who fears everything is a pusillanimous coward; but the man who fears on rare but appropriate occasions is courageous (1116b15). Neither the self-denial of asceticism nor the over-indulgence of licentiousness hits the happy medium of a healthy care for the body’s requirements (1118b29).2 The relativisation of these criterial means to a mode of comportment indicates that this way of acting is something that is not inherently bad or inherently virtuous. Its quantitative aspect as well as its quality is pivotal. Accordingly, Aristotle maintains that: Not every action or reaction admits of a mean. For some have names that already imply negativity, such as spite, shamelessness, envy, and in the case of actions, adultery, theft, and murder.3

Aristotle thus viewed virtue/excellence (arête) as a purposive or preferential disposition (hexis prohairetikê) in favor of the things of the middle way, where this intermediacy is to be located where the wise man (ho phronimos)—the man of practical wisdom—understands it to be.4 The sort of virtue at issue is thus a stable state of character concerned with choice in matters that are self-regarding (tê pros hêmos)5 calling for achieving a proper mean (mesotês) that is a suitable intermediary (meson) between two complementarily correlative vices of excess or insufficiency. And as that first column of Display 1 makes manifest, the reflexivity of self-development is a crucial core of Aristotle’s ethical project. One thing that this display brings clearly into view is that any sort of intermediation has to proceed in point of some particular mode of comportment (here represented by that initial column). And in each case of such Aristotelian examples, this variable parameter is some reflexive (selforiented) mode of personal comportment that ranges over a wide spectrum of degree or intensity. With the human virtues or excellences that stand at

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Display 1 VIRTUE AS INTERMEDIATION Mode of Comportment • SELF-PROJECTION • CARE FOR ASSETS • SELF-TREATMENT • SELF-INDULGENCE • CARE FOR SELF-STANDING • CARE FOR ONE’S SELF-IMAGE • SELF-REVELATION • SELF-RISK • SELF SHARING • SELF-INVOLVEMENT • SELF-ESTEEM • SELF-INTEREST • SELF-ASSERTIVENESS • VERBAL SELF-MANIFESTATION

Flawed Insufficiency

Mediating Excellence

Flawed Excess

Fearfulness/Cowardice Stinginess/Meanness Self-denial Asceticism Self-centeredness Self-abasement Secretiveness Over-caution Stinginess Callous Indifference Self-denigration Spendthrift Pusillanimity Dullness

Courage Liberality Temperance Moderation Proper Self-regard Self-respect Candor Prudence Generosity Neighborliness Seemly modesty Prudence Righteous Integrity Ready Wit

Rashness Profligacy Self-indulgence Licentiousness Self-heedlessness Self-aggrandisement Blabbelmouthiness Foolhartiness Profligacy Busibodyness Conceit Avarice Irascibility Buffoonery

the forefront of Aristotle’s concerns, there will always be some quantifiable feature of self-concern that can be present either with insufficiency or excess. And proper virtue lies in realizing this factor to the right and appropriate extent as between too little and too much along a generally continuous spectrum.6 For Aristotle, personal virtue is accordingly determined by a proportion (1107a1) and thereby requires a decidedly mathematical character.7 For us nowadays the crux of ethics is morality, and its object is specifically moral goodness, so that, as we see it, the crux of ethics is acting with due heed for the interests of others. By contrast, Aristotle takes a different and broader view. For him ethics is thus part of a larger triad, according as one’s concern is for the condition of people at large (politikê), people proximate to oneself (oikonomia) and one’s own self (ethikê). As Aristotle saw it, the object of the enterprise is not the (moral) goodness of man but rather the good for man more broadly construed to encompass the prime desiderata of justice, health, and affection (1099a26).8 His ethics aims at achieving a deservedly satisfyingly life of personal excellence and worth (1099a21), and its target is not so much specifically moral comportment as fine and proper behavior at large (1094b14). After all, if Aristotle’s para-

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mount interests had been in morality (as we understand it) he would not have stressed the intellectual virtues to the extent he actually does. Not moral rectitude but excellence in living is the objective of his ethics. Aristotle is a good deal closer to Samuel Smiles than to Prince Albert. His conception of what his commentators like to call “excellence of character” or “moral worth” is actually closer to what could be characterized a “satisfying lifestyle”—a way of living in which an intelligent being can have rational content. The question of whether what Aristotle viewed as virtuous life-praxis actually encompasses morality—of whether he construed virtue/excellence/nobility to include morality as we understand it—is certainly discussable.9 But, in any case, the two cannot be equated because Aristotelian excellence addresses an entire range of issues that goes above and beyond what is nowadays understood by morality. To make a convenient phrase of it, Aristotle’s concern is not for the specifically moral virtues (as we conceive of them) but more largely for the human virtues; its aim is not so much the good man as the true man. The excess or deficiency at issue is not something that is inherent in the virtue itself, but rather relates to the mode of comportment that those virtues characterize. To speak of excess (huperbolê) or of deficiency (ellipsis) in point of a virtue or vice as such “would imply that one could have a medium amount of excess on and of deficiency, that is, an excess of excess and a deficiency of deficiency” (1167a19). This would be quite inappropriate. Rather, a proper “sense of proportion” is the crux—not just a balanced diet, but a balanced lifestyle. The orthos logos of the Nicomachean Ethics is a ratio or proportion conceived of along the lines of the Pythagorean doctrine of harmony, duly extended from medicine to culture.10 For the very idea of preserving a due balance (“harmony”) of proportion in matters of personal comportment is deeply Pythagorean in its origins and there can be little doubt that Aristotle’s doctrine has roots in Plato.11 Such antecedents clearly prefigure aspects of Aristotelian pursuit of the “happy medium” in personality development.12 And over and above these Pythagoreanly Platonic resonances, the Aristotle theory of the mean also has analogues to Hippocratic medicine, where therapy is a matter of restoration of balance by “subtracting what is in excess and adding what is in deficiency” as the Hippocratic treatise On Breaths puts it in characterizing the Hippocratic conception of medicine as looking to a vis mediatrix naturae that works by marking a proper balance of opposites.13 And Aristotle is prepared to deploy the theory of the mean in the philosophy of nature as well as in the philosophy

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of man. He tells us, for example, that “it is in virtue of an intermediation (kata mesotêta) that the dry and the moist and the other opposites produce flesh and bone and the remaining compounds.”14 In ethics, Aristotle likens failures to achieve virtue to an archer’s failure to hit the bull’s eye (1106b29ff) or again to a geometer’s failure to pinpoint the center of a circle (1109a29ff). In both cases, to be sure, one can miss the mark in many directions. But in any given direction, one can only be off in one way or the other, hitting wide or over, as it were. And of course the center in each direction is the same, for “it is possible to fail in many ways” with the upshot that “men are good in but one way, but bad in many” (1106b29ff). As Aristotle puts it: It is possible to fail in many ways . . . while succeeding is possible in only one way, so that the former is easy but the later difficult, even as missing the mark is easy but hitting it difficult. In this way excess and defeat are characteristic of vice, and the mean of virtue. (1106b28).

The mean itself represents an “extreme,” a maximum, and deviations from it, be it by way of shortfall or excess, become increasingly negative. Aristotle accordingly maintains that virtue is limited but vice unlimited, and our diagram is designed to suggest that.15 His position prefigures Cicero’s thesis that the virtues are grounded in order and limit in everything that is done (modesty, self-control, and restraint for example). We must so comport ourselves ut neve maior neve minor cura et opera suscipiatur, quam causa postulet.16 But just how is insufficiency or excess to be conceived of? After all, when one encounters too much or too little of a good thing, this can be so in a great many different respects. Here Aristotle’s theory enjoys the advantage of considerable elbow room.17 For the relevant measure can proceed in point of: Frequency—ranging from too seldom to too often (from never to always) Degree—ranging from too mildly to too intensely (from not at all to overly) Duration—ranging from too short to too long (from numberless to gargantuan)

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Extent—ranging from too few to too many (from none to all) Amount—ranging from too little to too much (parity to infinite) The quantitative parameters that coordinate virtues and vices can function in various qualitative regards, and this gives Aristotle’s theory a good deal of flexibility. It is important, however, to note that all of these factors relate quantitative matters. Aristotle is emphatically not recommending it would be virtuous to settle for mediocrity in the quantitative spectrum across the range: too little // just right // too much The situation is not all that unlike the predicament of Goldilocks in the house of the three bears. However, Aristotle’s doctrine of the mean is not merely a series of pedestrian variations on the trite if ever-popular Delphic injunction to “moderation in all things” (mêden agan). It reaches well beyond this to a recognition that balance and proportion of one kind or another are pervasively ineliminable requisites to a life in which we humans can find rational contentment. That fundamental issues of balance and proportion are involved here are not a solution to the problem of how one is to live, but rather an instrument for thinking sensibly about how this problem is to be handled. (We must, after all, achieve clarity as to what the problem is before the issue of its solution is profitably addressed.) The extent of the negativity of those opposed vices will of course differ from case to case. Thus vices—even opposed vices—are not created equal. Undue modesty (self-abnegation) is not as vicious as its opposite, vaingloriousness, because, like stinginess, “they do not bring serious discredit since they are neither injurious to others not unduly disgraceful” (aschêrmones) NE 1128a31). For Aristotle, then, neither virtues nor vices have an equivalent status: facilitating or impending realization of the good to the same extent: some are graver than others. (Making the proper assessment here is the salient capacity of practical wisdom.) Aristotle’s illustrative example at 1106a33 looks to daily eating, as per the following set-up: MODE OF COMPORTMENT : Self-nourishment

Insufficiency/deficiency : abstemiousness (2 lb. or less)

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Proper intermediation : sensible eating (3 lbs.) Excess : gluttony (10 lbs. or more) This sort of case makes it clear that, as Aristotle explicitly maintains,18 what is called for in such matters of intermediation is certainly not the arithmetical midpoint of those extra qualities (i.e., roughly 3 lbs. and not 6!), but a different (and in this case lesser) quality that acknowledges that abstemiousness is less harmful than gluttony. Aristotle went to some length to argue that while there indeed must be a proper intermediation to represent the boundary (horos) between insufficiency and excess, this intermediate point need not be the actual halfwaypoint between the extremes.19 Thus Burnet 1900 (p. 70) very properly insisted on Aristotle “mean” (mesotês) does not “mean only or even principally the authentical mean [that is, the midpoint]” but rather is simply “the oldest word for a proportion of any kind, however determined.”20 With Aristotle, everything depends on the seriousness for us of what lies at those extremes. Vices are not created equal, and the vice lying at one of those extremes may be a more serious negativity than that which lies at the other. (For example, while cowardice and rashness are reciprocally opposed vices, the former personal failing represents a graver defect than the latter.) And so, the location of that “happy medium” will not be just the same for every virtue so that, for example, the proportionate distance from temperance to self-indulgence on the self-control scale would be greater than that from spendthriftiness to heedless squandering on the personal expenditure scale. When Aristotle denies (at 1106a35) that the mean he has in view is not the arithmetical midpoint (the meson kata tên arithemetikên analogian) he is not denying mathematical proportionality at large but mere arithmetical halfwayness. After all, the arithmetical mechanism of proportionality is always to be on hand to achieve a proper intermediacy, seeing that what is at issue is not a midpoint but a proportionalized mean. Thus distributive justice too is subject to the principle of giving people their due (suum cuique tribuere) and then neither too little nor too much, and justness, like other human excellences, is a matter of too much and too little.21 Locating the mean is not a matter of calculation but of proper judgmental assessment: it has to be placed where someone of sound practical judgment (ho

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phronêsis) would put it, though, to be sure, precision (akribeia) can be achieved to only a limited extent in these matters.22 Aristotle is very emphatic that ethics is not a matter of precision (akribeia) in reasoning and calculation, but rather of good judgment.23 Anyone can be a clear reasoner, but only the person of good judgment can think rightly about ethical matters.24 Clearly, a person’s conditions and circumstances will have to dictate the proper proportionality. The poor woman who gives 1% of her money to charity is generous (“the widow’s mite”), and one who gives 50% will be an imprudent spendthrift. But the rich man who gives but 1% is stingy and only with 50% or more does he achieve proper generosity—while spendthriftiness might be reached only with 99.99%.25 The pros hêmos usrelativity of all those ethical concerns (1106a29) applies both collectively and individually.26 In theory, the idea of pros hêmos, FOR US, admits of three distinct construals, namely (1) for us universally, that is for us humans at large, as a species, (2) for us sortally, people of a certain sort (farmers, say, or athletes, or philosophers), or (3) for us individually, as the specific individual person one happens to be. Clearly how one construes this will make a difference. Aristotle’s own examples generally point to the via media of that second, itself intermediated resolution. After all, the virtues at issue in Aristotle’s ethic are neither of the universalist, one-size fits all type, nor yet of the rampant individualism that makes each agent a law unto itself, but rather construes ethical appropriateness as a type-coordinated conception, relating to what is generally appropriateness for doctors, farmers, wrestlers, etc. The good, after all, comes in two nowise identical forms: the generic and the particular. On the one hand there is the good for people-in-general, and on the other the good for X the farmer and Y the carpenter. And the idea of what is a pros hemos cuts both ways as generalities must be tailored to particular cases. (1443b5). However, Aristotle insisted that the matter is not one of one-size-fits-all, but requires adjustment to individualized conditions of personal circumstances, much as in eating the proper balance for an athletic is one thing and that for an ordinary person another. Accordingly, the mean (mesotês) sets the ratio (logos) that determines the balancing point that gives neither of those approach vices an undue advantage but rather keeps them in mutually offsetting suspension. And the pathê that Aristotle invokes to characterize appropriate intermediacy (at 1105b20) is not so much feeling, emotion or passion as simply

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response. (The proper dietary intermediation between insufficiency and excess—Aristotle’s own example—is not a matter of emotion.) In the context of the Nicomachean Ethics, pathê is often best translated not as feeling/passion (has been the all too usual practice) let alone emotion (as per Urmson 1973), but rather as reaction or response. Any inclination to conceive of pathê as a matter of emotion or affective sentiment would distort what Aristotle is after in this regard. For what is at issue is not just the emotions that people feel but how they respond in thought and action, something broader in scope than the merely affective through encompassing not just feeling, but also judgment and—above all—action.27 And the terms response or reaction are particularly suitable here because they open a natural pathway to the ideas of under- and over-reaction. Some interpreters take the view that “Aristotle must have felt . . . difficulties when he tried to fit courage into the theory of the mean” because “Aristotelian courage makes two distinct feelings if indeed fear and confidence are two distinct feelings.”28 But once one sees the issue in the light of reactions rather than mere feelings, the difficulty dissolves. The rash man underreacts in his response to dangers (he dismisses them and rushes forward); the coward overreacts (he sees danger everywhere and shrinks at shadows). The courageous man reacts on target: he recognizes those dangers when they are real and treats them with due respect—he “reacts and acts as is appropriate and as reason requires” (1115b20). All those problems posed by the affective/emotional/feeling dimension now simply fall aside. In seeing virtue as intermediate between vices Aristotle is explicit that the intermediacy is not fixed at the midpoint, the arithmetical mean. Nor need it be some other specified instance of the various means studied in Greek mathematics.29 It is, rather, something that will have to vary from case to case according to circumstances. To see how this works, consider the set-up of Display 2. And here let Wd be the negativity-weight—i.e., seriousness—of the vice of deficiency Vd and similarly let We be the negativity weight of the vice of excess Ve. We must then expect that the segments at issue will reflect these values: The two negative extremes are locked into a position of interactive complementarity: we can only distance ourselves from the one at the cost of appointing the other. The degree of negativity at issue with those complementarily correlative vices—its seriousness or gravity as a vice—will of course differ, for example over-caution is less harmful than foolhardiness.

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Display 2 ARISTOTLE’S ETHICAL INTERMEDIATION Weight of the Vice of Deficiency W1d

Weight of the Vice of Excess W2e

d1

Δ

d2

Here d represents the deficiency of the “too little” side and e that of surfeit on the “too much” side of the scale. And W measures the seriousness or magnitude of the vice at issue. For negativity minimization we require W1/W2 = d1/d2.

But now if the distance (D) reflects remoteness from a vice and its weight (W) is index of its seriousness, then our exposure to the negativity at issue is W/D, W divided by D, a quantity which clearly it increases with weight and decreases with distance. On this basis, we must distance ourselves from the extremes in line with the seriousness at issue with the correlative vices, so that w1/w2 = d1/d2. It is indeed a matter of proportion, the Greek term for this being analogia which Aristotle defines (at 1131a31) as an identity of ratios.30 The object is to avoid Scylla as much as possible without succumbing to Charybodis. Aristotle’s theory of the mean contrasts interestingly with Archimedes’ theory of the lever. Archimedean balance requires a due distancing of weight, Aristotelian intermediation requires a due distancing of negativity. With Archimedes, balance requires that W x D = const, so when the weight is increased, the distances for the fulcrum must be decreased (and conversely) if balance is to be maintained. With Aristotle’s theory of the mean, proper mediation requires that the distance must be proportionate to the “weight” (“gravity,” seriousness of negativity). Thus if a vice’s seriousness is increased, the distance one should keep from it must be increased correspondingly so that a W~D proportionality obtains and therefore W/D = const. In effect, then, the Aristotelian scheme inverts that of Archimedes. But an underlying analogy is at work. Where Archimedes balances out the weight of conflictingly opposed material objects, Aristotle balances out the seriousness of conflictingly opposed vices.

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2. THE PROPER RANGE OF THE POSITION Aristotle holds that ethical virtue, while positioned as a mean, is nevertheless an extreme “with regard to what is best and right” (1107a26). There is no denying that one cannot have too much of a good thing or too little of a bad (1107a9). Take, for example, temperance, reliability, trustworthiness, honesty, or good judgment. Clearly these desiderata are not quantitatively comparative virtues which admit of too much or too little. But they themselves are mean-determinations exactly as per the point of maximality at issue in Display 3. As noted above, appropriate intermediation between those extremes of excess and deficiency will depend on the specific situation of individuals. The extent of self-aggrandizement that is acceptable in a general who must inspire his troops with confidence in his leadership may not be appropriate for a greengrocer. Locating that proper mean will depend on the specific condition of people—not just in general, but even in particular. And so, while the make-up of virtue lies in intermediation, a moderation between extremes, there would not be any modesty holding back in the display of virtue itself. Here the issue of excess or deficiency IN POINT OF SOMETHING becomes crucial, this being in relation to some sort of selfcomportment very distinct from the exercise or virtue itself. And so there now comes to the fore the already stressed distinction between the moral virtues nowadays of prime concern to ourselves and the existential virtues paramount in Aristotle’s ethic. Granted, such specifically moral virtues as honesty or reliability are not a matter of an intermediation between what can be over- or under-done. Aristotle could readily concede that some modes of action can be inherently good (reliability might be an example) and some (such as adultery, theft, and murder) inherently bad. (1107a9) without any reference to degree or extent.31 These modes of action, however, just are not the sort of existentially self-developmental virtues that form the target range of the doctrine of the mean. As Aristotle’s discussion makes clear, his theory of the mean is designed to treat specifically the one virtue which is self-regarding and selforiented (pros hêmos). His point is that with self-regarding pros hêmos virtues affecting “the inner man” (so to speak) are qualitatively intermediate; it is not that all virtues are so, including those that are other-oriented and of a nature that is more decidedly moral as we nowadays see it.32 The excel-

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Display 3 ON DEVIATIONS FROM THE MEAN Point of Maximality Extent of Positivity Vice of Deficiency

The Proper Mean

Vice of Surfeit

NOTE: This diagram modifies that of Oats 1936, p. 390.

lences of self-development rather than the benevolences of morality are at issue here—the sort of thing that makes one good as a man rather than a good man. The crux is the achievement of human excellence rather than what makes one moral goodness as such.33 Self-developmentally individual character rather than other-oriented morality is the proper object of Aristotle’s enterprise. What he has in view is best described as strengths or as sets of personality. And so, to reemphasize, Aristotle’s ethical concern is with laudability rather than moral goodness as we understand it—with achieving personal excellences rather than being a morally “good person.” This readily explains why Aristotle does not focus his theory of the mean upon morality in specific but upon human excellences in general “in the gymnastic or the medical arts, in building or navigation, and in any sort of action cognitive or other, skilled or unskilled.”34 The Nicomachean Ethics is not really a treatise on moral philosophy as nowadays understood. For, strange though it may seem to us, what we see at the moral virtues just are not the objects of focal concern in Aristotle’s theory of the mean. And so while it must be granted that Aristotle’s ethics, taken overall, is also mindful of some of the specifically moral virtues as contradistinguished from the personal excellences, it should be realized that this involves subsidiary issues and that different considerations are focally at issue in Aristotle’s ethic of intermediation.35 The pivotal contrast here is that between existential and moral virtue, between excellence and goodness or more specifically between being good

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as a man and being a good man. And the critical fact here is that while for us nowadays ethics is in effect morality and aims at propriety, for Aristotle it is, first and foremost, a matter of aiming at excellence in self-development. Aristotelian excellence, then, is not so much specifically moral (in our sense) as role-geared: excellence, say as an athlete or as physician—or even more broadly as a person. 3. THE PROBLEM OF MENSURATION The mathematical aspect of Aristotle’s doctrine of the mean has various significant ramifications. Thanks to its involvement with Pythagorean harmonies, Aristotelian ethics is going to encounter exactly the same problem that beset its Pythagorean predecessors. For even where there is a theoretical proportion this might nevertheless not be realizable in matters of practical implementation. The original Pythagorean idea was to see all quantification as a matter of proportion and to treat numbers in general as proportionately rational. When this prospect was unraveled by the great discovery of Hippasus of Metapontum (ca. 500 BC) that some qualities would not be represented as proportions of integers—that there were qualities beyond the reach of proportioning—that earlier Pythagorean vision reached a dead end.36 And Aristotle’s ethical program runs into something of the same difficulty. To see the matter more clearly, let us take a detour through another mode of proportionality celebrated in ancient Greece, namely the idea of a Golden Section. Consider again the basic set-up of the diagram

d

Δ

e

And now have it be that that division-point Δ is to be so placed that the smaller is to the larger-segment as that larger segment itself is to the whole: d e = e d+e For the sake of simplicity, let us take as our unit of measurement simply that magnitude d itself, so that d = 1. Then

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e 1 = or equivalently 1 + e = e2 1+ e e

Solving this equation for e yields an irrational number close to 1.62, which means that e is going to be incommensurable with d itself. No yardstick whatsoever will enable us to measure out such a proportion. Getting this right is a matter not just of calculation but of judgment. And just as the Pythagorean program of geometric proportionalism foundered in the wake of the discovery of the immeasurability of the diagonal of a square with its sides, so the deployment in practice of ethical proportionalism is going to collide with the prospect of incommensurable proportions in ethics. And so the mensuration problem faced by Aristotle’s proportionality approach to mean-determination can prove insurmountable. In Aristotle’s ethics merit view in a matter of proportionality and just this—proportionality—is something that can leave us in the lurch by failing to be determinable when the measurement involved is operationally intractable. In consequence the exactitude of calculation is not going to be practicable here: locating—fixing upon—the exact mean appropriate specifically circumstanced particular individual is not something that can be achieved by abstract calculation—it will, on occasion, have to be a matter of approximation. His theory of the mean is a clear indication that Aristotle’s ethics of self-constitution is destined to be an inexact rather than an exact science. And what is needed to close the gap between the generalities of ethics and its particularized application to concrete human affairs is not calculation or retrocination but sound judgment. Moreover, Aristotle’s doctrine of the mean runs up against the paradoxes projected by his contemporary Eubulidean of Megara (born ca. 390 B.C.), the notorious paradoxer. His “Paradox of the Heap (sorites) is posed by the following account: A single grain of sand is certainly not a heap. Nor is the addition of a single grain of sand enough to transform a non-heap into a heap: when we have a collection of grains of sand that is not a heap, then adding but one single grain will not create a heap. And so by adding successive grains, moving from 1 to 2 to 3 and so on, we will never arrive at a heap. And yet we know full well that a collection of 1,000,000 grains of sand is a heap, even if not an enormous one.37

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Display 4 THE CONTRAST BETWEEN VIRILITY AND MORALITY VIRILITY

MORALITY

• • • • •

• • • • •

personal positivities strength of character nobility self-development the real or true man

moral virtues moral propriety decency benevolence the morally good or virtuous man

This opens the door to a whole list of what might be called “boundary line puzzles”—at what point is a bald man bald or a rich man rich? Exactly this issue of fixing the ethically decisive point of intermediation at which there are too little of a good or too much of a bad will clearly afflict the problem of locating that “golden mean” of Aristotle’s ethics. Given his wide-ranging interests and sources of information, Aristotle could not possibly have been unaware of the work of his contemporary Eubulides. And one of the salient ideas of his thought relates to the impossibility of establishing precise boundaries between opposites: between a heap of sand-grains and a non-heap, or between a bald man and one who is non-bald. There is simply no way for us to say just how few sand-grains will fail to make a heap and just how many are a superfluity; and the same for the hairs on a man’s head in relation to baldness. And, of course, just exactly the same story holds those proportionate means of Aristotelian ethics that defines, for example, where being sensibly cautious ends and being pusillanimous begins. 4. THE NATURE OF THE FIELD As Aristotle sees it we are, in ethics, dealing with shadowy penumbral regions that preclude any prospect of mathematical exactness. And it was doubtless for this very reason that Aristotle put that monitory warning at the outset of the Nicomachean Ethics that we must not ask for more precision than the nature of the subject will bear. But this very concession marks the fact that the field is not an exact science but at best one that rest not on exact principles but on inexact, for

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the most part rules of thumb. As one recent commentator has very rightly emphasized, it is Aristotle’s clear teaching “that ethical knowledge is only approximate . . . [and] lacks mathematical precision (akribeia).”38 Like Kant long after him, Aristotle had his reservations about the prospect of mathematical methods in philosophy. As he saw it, they have their use but also their limits. Which way are we to go here? Is the approach of Urmson’s “Doctrine of the Mean”39 intended as a serious model for a formalized theory of human virtue? Or is it, as per Hurtshouse,40 a mere (and often misleading) expository device—a “figure of speech” that should be replaced by more serious and accurate explanations? My inclination is for a middle position here (no pun intended). We have to deal with an analogy which, like all analogies, does not envision a literal identity, but rather a substantial similarity. My vending of it is that Aristotle’s division of quantitative intermediation should be taken seriously, subject to acknowledging the analogical nature of the case. Aristotle has it that: The [correct and proper] actions, which ethical science investigates, admit of much variety and fluctuation of opinion, so that they may even seem to exist only by convention, and not by nature . . . We must be content, then, in speaking of such subjects and with such premises to indicate the truth roughly and in outline, and in speaking about things which are only for the most part true and with premises of the same kind to reach conclusions that are no better. (NE 1088b 14–23.)

What we are going to have here is an inexact science—an applied science where the imposition of theoretical templates upon a complexly variegated reality requires approximations and compromises. But not total abandonment! There seems no doubt that Aristotle means his instructive analogies to be taken seriously albeit with caution. And Hurtshouse speaks with the flippant dogmatism of a skeptical era in rejecting the idea of “something right in what Aristotle says . . . as, to be blunt, simply whacky.”41 In this regard, Aristotle’s view of the nature of science becomes critical. As he saw it, there are three different levels of consideration in point of generality/particularity. • strict universality: holding good throughout an entire genus.

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• limited universality: holding good in general, i.e., for most (but not all) species of a genus. • mere particularity: holding good sometimes but not universally (in either mode). In regions of strict universality we have exact science. But in those of limited universality we have an inexact science. And when there is mere particularity and no universality—be it unrestricted or limited—an achievable, scientific knowledge of any sort is infeasible. Science, in sum, stands coordinate with the realization of generality, and thus has two versions: hard science and soft science, as it were.42 And so, as Aristotle sees it, the very nature of its subject-matter means that ethics just is not going to be an exact science. After all, the issue of mean-appropriateness hinges on the particular condition of particular individuals. And just this is going to be crucial for our present concerns. For Aristotle science (epistêmê) encompasses not just the exact sciences, which, like geometry, hold good for claims that obtain always and everywhere, but also of the inexact sciences whose generalizations hold only for the most part (epi to polu).43 The applicative implementation of such a science in relation to particular cases is always something of an art (technê) because following cases under generalizations is now a matter of good judgment rather than automatic deductive subsumption. All this means that Aristotle’s ethical mediation is—as he himself sees it—a process whose concrete application is going to be a matter of approximation.44 Aristotle’s belief is that exact science pivots on deduction—of fitting particular cases into the purview of general rules. What is there about ethics that prevents it from being an exact science in this sense? Aristotle is acutely aware of the difficult dialectic that connects generality and specificity. He writes: We must, however, not only make a general statement, but also apply it to specifics. For among statements about matters of practice, those which are general apply more widely, but those which are particular are more informative [or truth-like], since conduct has to do with the particular cases with which generalities must harmonize (NE, 1107a27ff).45

Now as regards the relation of general rules to particular cases there are, in principle, three alternatives:

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1.

The rules are definitive: they decide the cases: those cases are always to be resolved by subsumption under general rules.

2.

The rules are useless: cases must be decided individually on the specific merits of the prevailing circumstances. The so-called rules are simply statistical summaries of how things stand “for the most part” with regard to cases.

3.

The rules are helpful instrumentalities: they are rules of thumb: they are guidelines for resolving cases, telling us how to resolve them provided that “other things are equal”—which they are often not.

As regards Aristotle’s position here, scholars agree that alternative 1 is out. But at this point there arises a pervasive disagreement. Woods 1986, Gadamer 1990, and Dunne 1993 opt for alternative 2. Nussbaum 1978 and 1986 tries to have it both ways as regards alternatives 2 and 3. However, as the present deliberations conspire to indicate, my own preference is decidedly for alternative 3. Time and again Aristotle insists that abstract theorizing cannot establish particular ethical assessments. He tells us that “it is not easy to detect how and with whom and for what and for how long one ought to be angry,” and that “it is not easy to delimit this by a rational function.” 46 Subsuming particular circumstance under general principles is seldom all that straightforward. There will indeed be such principles in ethics, but the application of such principles to particular concrete cases is always an art. Those principles can indeed help and guide, but they cannot resolve. Determining the ethically appropriate mean that is case-specifically appropriate in particular applications is not a matter of calculations or ratiocination on the basis of governed principles but requires the sound judgment of the man of practical wisdom (ho phronimos). On this basis, Aristotle appears to suppose a moral sense whose operation mirrors that of our bodily sense, seeing that: Each of our senses is a sort of balance (mesotês) between sensory opposites, and through this we make sensory judgments. The balance-point (to meson) is what distinguishes (to kritikon).47

There are indeed general principles at work in the ethics of character, but they no more determine proceeding in particular cases than do the guidelines of theoretical medicine determine particular courses of treatment, or

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the principles of political theory determine particular laws.48 Central though it is in Aristotle’s ethical doctrine, the theory of the mean neither is nor is intended to make ethics into an exact science. The complex variability of the phenomena it addresses means that its relation to the actual management of life ethics—like medicine—would ultimately have to be developed as a judgmental art rather than as an inferential science on the model envisioned in the Posterior Analytic. In sum, Aristotle sees ethical deliberation as a matter of phronesis (proper judgment) rather than one of reasoning (noêsis), and its application in concrete situations is an art (technê)— albeit a cognitive one—rather than a science (epistêmê). However, Aristotle’s ethics is emphatically not Kantian. In Kant’s ethics the focus is on specifically moral virtue and its pivot is the good will toward others; with Aristotle it is on excellence of one’s own character and its pivot is good judgment. The salient lesson of the golden mean is that it takes practical wisdom to figure out the proper mediation between extremes, so that cognition plays a pivotal role and ethically appropriate action is a thought and not just will. Aristotle does not insist that the good life for man is one of thought alone, but he does hold that thought alone can discern what the good life actually is. To be sure, to live a good life is to do as the man of practical wisdom would counsel, just as living a healthy life is to do as the sagacious physician directs. In neither case is it indispensably necessary that the expertise at issue must emanate from the agent himself. But of course then such practice roots in understanding rather than trust, its foundation is all the more secure—and more credible. *** In closing, then, a brief summary. For Aristotle, ethics is not moral philosophy, but a philosophical examination of the good for man. And here the good is always a matter of intermediacy between too little and too much, between deficiency and excess. However, the appropriate intermediacy will depend of the agent’s taxonomic placement in relevant regards: is he an athlete or a thinker, a youth or a person of years. And determining the appropriate proportion is an art that requires good judgment in estimation and not a science that admits precise calculation in line with universal generalities. Viewed in a present-day perspective, then, Aristotle’s project in the Nicomedean Ethics is rather a guidebook for life-management counseling than a how-to manual for moral reformation.

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NOTES 1

Seen in wider perspective, the theory of the golden mean has a substantial preAristotelian history and scholars have had little difficulty in finding precedents for Aristotle’s teaching (see especially Kalchreuter 1911). It has deep risks in Greek medical theory, but also makes its appearance elsewhere. Plato in his Statesman, praises “that which is moderate (metrion) proper (prepon) appropriate (kairon) and as it should be (deon) and whatever avoids the extremes in favor of the mean (mason).” Plato, The Statesman, 284E.

2

Contrary to various interpreters, I propose understanding what here termed “modes of comportment” as types of reaction (pathê) generally, and not merely or even principally as emotional reactions (for the contrary view see Terzis 1995).

3

NE 107a9ff. Compare EE 1221b20ff. Aristotle might have added that some modes of action are inherently good, such as honesty or trustworthiness.

4

NE 1107b34.

5

NE 1106b35. It is clear from the examples he offers that when Aristotle speaks of appropriateness “to us” (pros hêmos) he has in view not just the human species at large, but such subgroups as the practitioners of a craft, a sport, or profession. But this is clearly no invitation to extent such subcategorization to the point of individualism.

6

Compare Joachim 1955, pp. 89–90.

7

W. D. Ross—and many others—render hôrismenê logô (at 1107a1) as being determined by a “rational principle,” but it would just as well be rendered as “definite ratio” or “fixed proportion”—as I would certainly prefer.

8

Unless otherwise noted, references are to the Nichomachean Ethics. Aristotle’s ethical project is something very different from that of our contemporaries.

9

See, for example, the elaborate treatment of the issue in Cooper 1975.

10

See Burnet 1914, p. 7.

11

The idea of a mean has a long an notable history in Greek thought, duly depicted in Kalchrentner 1911. See also Schilling 1930 and Dühring 1966, p. 448–49.

12

On the essentials of Greek medicine, and in particular on the Hippocratic theory of health visa the harmonious valance of humors, see M. R. Cohen and I. E. Drabkin, A Source Book on Greek Science (New York: McGraw Hill, 1948), pp. 467–529 (see esp. 486–90).

13

Aristotle himself is perfectly explicit on the parallelism between determining the right proportion in ethical matters and in matters of medicine (1138b 21–32). Compare also Topics 139b21 and 145b7, and The Generation of Animals 767a20, and The Parts of Animals 652b17–20. The parallelism between Aristotelian ethics and Greek medicine is rightly stressed in Clark 1975, p. 84ff.

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NOTES 14

Aristotle, On Generation and Corruption, 334b29–31.

15

On this issue of limits see Grant 1857).

16

Cicero, De officiis, I, xxxix. (Cicero connects this position with the Stoic eutaxia i.e., orderliness.)

17

This point was emphasized in Loxin 1787.

18

NE 1106b35ff.

19

On the role of horos (limit) in Aristotle see Peterson 1998.

20

On this issue see Tracy 1969, especially pp. 340–42.

21

NE 1137b7.

22

NE 1106b36ff. See Dunne 1953, pp. 301–13.

23

NE 1103b35.

24

NE 1144b27. Broadie 1991, pp. 95–96 and 101–102, thinks that Aristotle’s sound judgment (lio phrnisics) is caught between the standards of his own judgment and that of the community. But of course there should be no problem when both are properly and soundly formed.

25

On this issue see the dissent of Hardie 1980. But Aristotle simply foreshadows Cicero’s thesis pros hêmos that duty and virtue can root not only in our universal nature via features we share with everyone, but also in those things that become incumbent on us owing to specific aspects of our personal situation. (De officiis I, xxx–xxxi.)

26

On pros hêmos see Leighton 1995.

27

Compare Hursthouse 1980/81.

28

Pears 1980, p. 172.

29

On this point see Hardie 1980, p. 135.

30

isotês logon. Aristotle contrasts this with the “arithmetical proportion” at issue with establishing distributive justice by transfer, from A’s amount a to B’s amount to an equalizing quantity so that a – z = b + z. Compare 1137b30. On these issues see Hardie 1980.

31

Compare also the Eudemian Ethics at 1221b20.

32

On the rather complicated issues involved here see Brown 1997.

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NOTES 33

In addressing the aretê of a horse (NE 1106b20), Aristotle certainly does not have its morals in view!

34

Eudemian Ethics, 1220b22ff.

35

Anthony Kenny held that, “Aristotle’s aim is not to show that wisdom is a moral virtue, but to stress that there can be an application of the mean in the case of wisdom” (Kenny 1978) This looks to be right on target. For if wisdom were a moral virtue in our sense of the term, rather than a personal asset in the Aristotelian sense, then (contrary to fact) it would not be subject to a triadic analysis as an intermediation between the know-nothing and the know-it-all.

36

On Pythagorean arithmetic see T. L. Heath, Greek Mathematics, Vol. I (Oxford: Clarendon Press, 1921), pp. 65–117.

37

On this paradox and its ramifications see Chapter 2 of R. M. Sainsbury, Paradoxes (2nd. ed., Cambridge: Cambridge University Press, 1995), pp. 23–51. Originally the paradox also had a somewhat different form, as follows: Clearly 1 is a small number. And if n is a small number so is n + 1. But this leads straightway to having to say that an obviously large number (say a zillion billion) is a small number. (See Carl Prantl, Geschichte der Logik im Abendlande, Vol. I, p. 54.)

38

Hardie 1980, p. 142. Compare NE 1198a26 and 1094b11.

39

See Urmson 1973 and also Curzer 1996.

40

See Hurtshouse 1980/81.

41

Hurthouse 200x, p. 99.

42

Regarding Aristotle’s view of soft science and “on the whole” (epi to poli) generalization see discussed in Lindsay Judson, “Chance and Always or For the Most Part,” in idem (ed.), Aristotle’s Physics: A Collection of Essays (Oxford: Clarendon Press, 1991), pp. 73–99.

43

See Gisela Striker, “Notwendigkeit mit Lücken,” Neue Hefte der Philosophie, vol. 24/25 (1985), pp. 146–64.

44

See Greenwood 1909, p. 132.

45

On the complementarity trade-off between generality and informativeness see N. Rescher, Epistemetrics (Cambridge: Cambridge University Press, 2005).

46

NE 1109b14. Compare 1126a31 and 1104a8.

47

De anima, II, xi, 17.

48

NE 1181a36.

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REFERENCES Ackrill, J. L., “Aristotle on Eudaimonia,” in: A. Rorty (ed.), Essays on Aristotle’s Ethics (Berkeley & Los Angeles: University of California Press, 1980), pp. 15–34. Bosley, Richard, “Aristotle’s View of the Theory of the Mean,” in: R. Bosley, R. Shiner, and J. Session (eds.), Aristotle, Virtue and the Mean: Aperion, vol. XXV (1995), pp. 36-66. Broadie, Sarah, Ethics with Aristotle (New York: Oxford University Press, 1991). Brown, Leslie, “What is *The Mean in Relation to Us* in Aristotle’s Ethics?” Phronesis, vol. 42 (1997), pp. 77–93. Burnet, John, The Ethics of Aristotle, Edited with an Introduction and Notes (London: Methuen, 1900). Burnet, John, “On the Meaning of LOGOS in Aristotle’s Ethics,” Classical Review, vol. 28 (1914), pp, 6–7. Burnyeat, M. F., Platonism and Mathematics in Aristotle,” in: A. Graeser (ed.) Mathematics and Metaphysics in Aristotle: Proceedings if the 10th Symposium Aristotelicum (Bern & Stuttgart: Haupt, 1978), pp. 213–40. Burnyeat, M. F., “Aristotle on Learning to be Good,” in: A. O. Rorty (ed.), Essays on Aristotle’s Ethics (Berkeley, CA: University of California Press, 1980), pp. 69– 92. Clark, Stephen C., Aristotle’s Man: Speculation upon Aristotelian Anthropology (Oxford: Clarendon Press, 1975). Cook-Wilson, J., “On the Meaning of λóyoç in Certain Passages in Aristotle’s Nicomachean Ethics,” Classical Review, vol. 27 (1913), pp. 113–17. Cooper, John M., “The Magna Moralia and Aristotle’s Moral Philosophy,” American Journal of Philosophy, vol. 94 (1973), pp. 327–49. Cooper, John M., Reason and Human Good in Aristotle (Cambridge, MA: Harvard University Press, 1975). Curzer, H. J., “A Defense of Aristotle’s Doctrine that Virtue is a Mean,” Ancient Philosophy, vol. 16 (1996), pp. 129–38. Dirlmeier, Franz, Eudemische Ethik (Berlin: Akademie Verlag, 1963).

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Dunne, Joseph, Back to the Rough Ground: Phronêsis and Technê in Modern Philosophy and in Aristotle (Notre Dame: University of Notre Dame Press, 1953). Düring, Ingemar, Aristoteles: Darstellung und Interpretation seines Denkens (Heidelberg: Carl Winter, 1966). Engberg-Pedersen, Troels, Aristotle’s Theory of Moral Insight (Oxford: Oxford University Press, 1983). Festenbaugh, W. W., “Aristotle and the Questionable Mean-Disposition,” TAPA, vol. 99 (1963), pp. 203–31. Gadamer, Hans-Georg, Truth and Method, 2nd revised edition (New York: The Crossroad Publishing Company, 1990). Garver, Eugene, Confronting Aristotle’s Ethics (Chicago: University of Chicago Press, 2006). Gigon, O., “Phronesis und Sophia in der Nicomachaeischen, Ethik des Aristotles,” in: J. Mansfeld and L. M. Rijk (eds.), Studies, Offered to Professor C. J. de Vogel (Assen: Van Gorcum, 1975), pp. 91–104. Grant, Alexander, The Ethics of Aristotle, 2 vols. (4th ed.: London: Longmans, Green, 1866). Greenwood, L. H. G., Aristotle: Nichomachean Ethics, Book 6 (Cambridge: Cambridge University Press, 1909). Hardier, W. F. R., “Aristotle’s Doctrine of Virtue as a ‘Mean’,” Proceedings of the Aristotelian Society, vol. 65 (1964–65), pp. 185–204. Hardie, W. F. R., Aristotle’s Ethical Theory (Oxford, Clarendon Press, 1968; 2nd ed. 1980). Hursthouse, Rosalind, “A False Doctrine of the Mean,” Proceedings of the Aristotelian Society, vol. 81 (1980-81), pp. 57–72. Hursthouse, Rosalind, “the Central Doctrine of the Mean,” in: Richard Kraut (ed.), Aristotle’s Nicomachean Ethics (Oxford: Blackwell, 200x), pp. 96–118. Hutchinson, D. S., “Doctrines of the Mean and the Debate Concerning Skills in Fourth-Century Medicine, Rhetoric and Ethics,” in R. J. Hankinson (ed.), Apeiron, vol. 4 “Method, Medicine, and Metaphysics” (1988), pp. 17–52.

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Irwin, Terence H., “First Principles in Aristotle’s Ethics,” Midwest Studies in Philosophy, vol. 3 (1978), pp. 252–72. Irwin, Terence H., “Aristotle’s Method of Ethics,” in Dominic J. O'Meara Studies in Aristotle (Washington, D.C.: Catholic University of America Press, 1981). Irwin, Terence H., “Aristotle’s Conception if Morality,” Proceedings of the Boston Area Colloquium in Ancient Philosophy, vol. 1 (1985), pp. 115–43. Joachim, H. H., On Aristotle: The Nichomachean Ethics—A Commentary (ed. by D. A. Rees; Oxford: Clarendon Press, 1955). Jowett, Benjamin, Aristotle’s Politics (Oxford: Clarendon Press, 1905). Kalchreuter, Hermann, Die mesotês bei and vor Aristoteles (Tübingen: H. Laupp Jr., 1911). Kenny, Anthony, The Aristotelian Ethics (Oxford: Clarendon Press, 1978). Kraut, Richard, Aristotle on the Human Good (Princeton, N. J: Princeton University Press, 1989). Leighton, Stephen, “Relativizing Moral Excellence in Aristotle,” Apeiron, vol. 25 (1992), pp. 49–66. Leighton, Stephen, “The Mean Relative to Us,” in: R. Bosley, R. Shiner, and J. Session (eds.), Aristotle, Virtue and the Mean: Aperion, vol. XXV (1995). Lloyd, G. E. R., “The Role of Medical and Biological Analogies in Aristotle,” Phronesis, vol. 13 (1968), 68–83. Losin, Peter, “Aristotle, Doctrine of the Mean,” History of Philosophy Quarterly, vol. 4 (1987), pp. 329–41. Nussbaum, Martha Craven, The Fragility of Goodness (Cambridge: Cambridge University Press, 1986). Nussbaum, Martha Craven, “Non-Relative Virtues: An Aristotelian Approach,” Midwest Studies in Philosophy, vol. 13 (1988), pp, 32–53. McCullagh, Mark, “Mediality and Rationality in Aristotle’s Account of Excellences and Character,” in: R. Bosley, R. Shiner, and J. Session (eds.), Aristotle, Virtue and the Mean: Aperion, vol. XXV (1995), pp. 155–74.

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Müller, A. W., “Aristotle’s Conception of Ethical and Natural Virtue: How the Unity Thesis Sheds Lights in the Doctrine of the Mean,” in: J. Szaf and M. LutzBachman (eds.), What is Good in Human Beings (New York: Walter de Gruyter, 2004), pp 18–53. Oates, Whitney J., “The Doctrine of the Mean,” The Philosophical Review, vol. 45 (1936), pp. 382–98. Pears, David, “Courage as a Mean,” in: A. P. Rorty (ed.), Essays on Aristotle’s Ethics (Berkeley/Los Angeles/London, 1980), pp. 171–187. Peterson, Sandra, “Horos (Limit) in Aristotle’s Nicomachean Ethics,” Phronesis, vol. 33 (1988), pp. 233–49. Peterson, Sandra, “Apparent Circularity in Aristotle’s Account of Right of Action in the Nicomachean Ethics,” Apeiron, vol. 25 (1992), pp. 83–107. Reeve, C. D. C., Practices of Reason. Aristotle’s Nicomachean Ethics (Oxford: Oxford University Press, 1992). Ross, W. D., Aristotle (London: Methuen, 1923). Rowe, C. J., The Eudemain and Nicomachean Ethics: A Study in the Development of Aristotle’s Thought (Cambridge: Cambridge University Press, 1971). Schilling, H., Die Ethos des Mesotês (Tübingen: J. C. B., Mohr, 1930; Heidelberger Abhandlungen zur Philosophie und ihren Geschichte, No. 22). Stewart, John Alexander, Notes on Nichomachean Ethics of Aristotle (2 vol.’s: Oxford: Clarendon Press, 1892). Stocks, J. L., “On the Aristotelian use of λóyoç: A Reply,” Classical Quarterly, vol. 8 (1914), pp. 9–12. Stocks, J. L., “The Golden Mean,” in: J. L. Stocks and D. Z. Phillips (eds.), Morality and Purpose (London; Routledge & Kegan Paul, 1969), pp. 82–98. Terzis, George N., “Homeostasis and the Mean in Aristotle’s Ethics,” in: R. Bosley, R. Shiner, and J. Session (eds.), Aristotle, Virtue and the Mean: Aperion, vol. XXV (1995), pp. 175–89. Tiles, J. E., “The Practical Import of Aristotle’s Doctrine of the Mean,” in: R. Bosley, R. Shiner, and J. Session (eds.), Aristotle, Virtue and the Mean: Aperion, vol. XXV (1995), pp. 1–14.

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Tracy, Theodore James, Physiological Theory and the Doctrine of the Mean in Plato and Aristotle (Chicago: Loyola University Press, 1969). Tuozzo, Thomas M., “Contemplation, the Noble, and the Mean: The Standard of Moral: The Standard of Moral Virtue in Aristotle’s Ethics,” in: R. Bosley, R. Shiner, and J. Session (eds.), Aristotle, Virtue and the Mean: Aperion, vol. XXV (1995), pp. 129–54. Urmson, J. O., “Aristotle’s Doctrine of the Mean,” American Philosophical Quarterly, vol. 10 (1973), pp. 231–38. Reprinted in: A. Rorty (ed.), Essays in Aristotle’s Ethics (Berkeley: University of California Press, 1980). Welton, William A. and Ronald Polansky, “The Virtuality of Virtue in the Mean,” in: R. Bosley, R. Shiner, and J. Session (eds.), Aristotle, Virtue and the Mean: Aperion, vol. XXV (1995), pp. 79–102. Woods, Michael, Aristotle’s Eudemian Ethics, I, II, and VII (Oxford: Clarendon Press, 1982). Woods, Michael, “Intuition and Perception in Aristotle’s Ethics,” Oxford Studies in Ancient Philosophy, vol. 4 (1986), pp. 145–66. Young, Charles, “Aristotle on Temperance,” The Philosophical Review, vol. 97 (1988), pp. 521–42.

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Chapter 3 OCKHAM’S RAZOR AND ONTOLOGICAL ECONOMY 1. ONTOLOGICAL ECONOMY AND THE LAW OF SMALL NUMBERS

D

eep-rooted in the philosophical tradition of the West is the idea that Nature does nothing in vain, and does what it does efficiently and economically, getting maximal effect at minimum expenditure. And over the centuries this idea has found its expression in various different forms. Already the Presocratics insisted that a small number of basic elements (archai) make up the vast manifold of things we can observe about us. Archimedes—Isaac Newton’s very model of what a natural philosopher should be—provided an account for a vast array of phenomena on the basis of a modest number of principles. In the era from Leibniz and Maupertuis to Maxwell and beyond the tradition of rational mechanics accounted for a sizable manifold of physical laws on the basis of a single overarchingly determinative Principle of Least Action. Then too, in 1940 Paul K. Zipf1 developed a theory of natural economy in human affairs which gives rise to what could be called a Law of Small Numbers which in effect runs as follows: Whenever a process IN THEORY requires items of a certain genus for the realization of its product, the number of the species of this genus actually involved IN PRACTICE is contextually minimal—that is, is as small as is consonant with the effective realization of this product under the circumstances at issue.

To be sure, Zipf did not formulate the law in quite this form. The principle that lay at the core of his deliberations stipulated that the N-th most frequently occurring item of a class occurs with a frequency proportional to 1/N. But this means that a small handful of top performers will always account for a disproportionalely large portion of the whole. And we see this principle at work in innumerable circumstances:

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• A mere five vowels account for 60 percent of the letters used in English-language texts. • The fifty most commonly used words in a book will account for over half of the total text. • The ten most prosperous towns of a country will make up over half of its population. • Among the various causes of mortality across U.S. residents the top 5 account for 90% of all deaths. • In any hierarchical organization (and thereby in pretty well any organization), the functioning of the whole is determined via the operation of a few of its operatives. • While the total literature in any branch of science or learning is immense, a large fraction of the whole mass can be grasped by mastering a small fraction.2 • In any branch of science or learning the great bulk of really creative work is done by a small handful of unusually gifted contributors. In a wide variety of circumstances and contexts, nature is parsimonious and makes a little goes a long way. It is often maintained that such diverse manifestations of economy in the operations of Nature and of Man are to be accounted for on the basis of the fundamental principle of rational economy widely acknowledged and celebrated under the name of Ockham’s Razor. The aim of the present discussion is to explain how and why it is that this idea is deeply problematic and inappropriate. 2. OCKHAM’S RAZOR

The form in which Ockham’s Razor is traditionally presented stands as follows: • Entia non sunt multiplicanda sine necessitate. (“Entities are not to be multiplied without necessity.”)

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In this format the maxim was, as best we can tell, not medieval at all but originated ca. 1639 to the 17th Century Scotist commentator John Ponce of Cork. And it recurs in the Elementa philosophiæ seu ontosophia of Johann Clauberg (Groningen, 1647) as well as in his later Logica vetus et nova (Amsterdam 1654). This very coincidence suggests an older, Scotist source. However, Ockham’s extant writings fail to indicate that this is exactly how he himself formulated the principle at issue. Rather, we find there such injunctions as: • Pluralitas non est ponenda sine necessitate (“Pluralatives should never be assumed without necessity.”) • Frustra fit per plura quod potest fieri per pauciora (“It is pointless to do with more that which one can accomplish with fewer.”) • Paucitas est ponenda, ubi pluralitas non est necessaria (“Assume the fewer where the more are not needed.”) • Principia non sunt cumulanda (“Principles must not be proliferated.”) It would perhaps be best to rephrase the Ockhamite principle to read “beyond warrant” rather than “beyond necessity” with the idea that warrant can cover a considerable range. Ockham himself has it that “nothing should be posited without a given reason—be it self-evidence or experience or the authority of Sacred Scriptures.”3 He clearly even issued a variety of “necessities.” Ockham himself never spoke to shaving (rasere) or used the term razor (novacula)—or anything like it—in connection with his principle.4 But Étienne Bonnot de Condillac (1715–80) spoke of a “rasoir des nominaux,” giving rise to the latinate novaculun nominalium. And it was the Scottish philosopher Sir William Hamilton (1788–1856) who first used the expression “Ockham’s razor” to establish what he called a “Law of Parsimony.”5 He wrote: There exists a primary presumption in philosophy. This is the Law of Parsimony (sic): which prohibits, without a proven necessity, the multiplication of entities, powers, principles, or cures … We are therefore entitled to apply “Occam’s Razor” . . . a doctrine . . . [disallowing something] if what it would explain can be explained on less onerous conditions.6

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The leading idea at work here is deeply embedded in the Aristotelian tradition of Western philosophy—well before Ockham. Thus in his (ca. 1250) commentary on the Sentences, Odo Rigaldus stipulated the just-mentioned injunction that: Frustra fit per plura quod potest fieri per unum.

And Aristotle himself had already maintained in his Physics that: A single principle will not suffice [for according for Nature’s phenomena], because opposition (enantiosis) is an ultimate constituent in Nature, and opposition involves duality. Nor can the factors of Nature be unlimited, or Nature could not be made the object of knowledge. Now, as far as opposition goes, two principles would be enough, for every defined class comes under one general opposition and the whole sum of “things that exist in Nature,” opposition as such, forms a defined class. Since, then, one opposition will suffice, it is better not to go beyond it, for the more limited, if adequate, is always preferable.7

Aristotle was clearly convinced that the sensible thing is to adopt the smallest body of explanatory principles (archai) that will serve our explanatory needs. 3. OCKHAM’S RAZOR A MATTER OF METHODOLOGICAL ECONOMY

All of those versions of Ockham’s Razor cited above are procedural injunctions. And these are not accidents. The history of the topic clearly shows that from Aristotle to Ockham and then to Ernest Mach and beyond, quantitative parsimony was seen as a methodological requisite of explanatory procedure. It was taken to represent a principle of cognitive (not physical) process that is not necessarily linked to a conformable ontological (rather than methodological) product—a principle of the economy of thought rather than the economy of nature. Unlike (say) the Principle of Least Action, Ockham’s razor was not initially seen as applicable directly to nature, but rather as applicable to our ways of theorizing about nature. However, the Law of Small Numbers and Principle of Least Action are explicitly ontological and (nature-descriptive). Their claim upon us for acceptance is not grounded in general principles of rational economy but rather in empirical-observational considerations that result for scientific inquiry. In its original conception Ockham’s

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Razor was nothing like that. Throughout its traditional Aristotelian setting Ockham’s razor was thus seen as a principle of procedure whose status is methodological.8 Methodological economy of procedure is something different from structural economy of composition or operative efficiency. For while it too is antithetical to inefficiency and wastefulness, this now relates not to nature’s operations but to our own. The features we ascribe to reality need not mirror those of our proceedings. A sober theory of inebriation is perfectly possible. We find this situation encapsulated in the thesis of Ernst Mach that: The simplest and most economical conceptualized expression of the fact is the goal of natural science. (“Den sparsamsten, einfachsten begrifflichen Ausdruck der Tatsachen erkennt sie [d. h., die Naturwissenschaft] als ihr Ziel.”).9

Mach very rightly speaks of the goal of rational science, not the goal of nature. And philosophical theoretician agree in the methodological grounding of cognitive economy—even in contexts that deal with ontological issues. Thus W. V. O. Quine maintains that: Our acceptance of an ontology is, I think, similar in principle to our acceptance of a scientific theory . . . [where] we adopt, at least insofar as we are reasonable, the simplest conceptual scheme into which . . . [observations] experience can be fitted and arranges.

Economy, so regarded, is clearly a matter of rational practice in matters of theorizing.10 4. OCKHAM’S OWN USE OF THE “RAZOR”

Ockham’s own resort to the principle at issue did indeed direct methodological thrust in the direction of ontology. When one speaks in these contexts of having recourse to “more entities” (entia) or “multiplicities” (pluralitates), or to “more things” (plura), it is, of course, not individual particulars that are at issue, but rather types or kinds or species of things. The Ockhamite principle of economy does not envision a depopulated world without a multitude of concrete particulars— indeed quite the reverse is mandated by his nominalism. For him and his

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congeners what is at issue is a principle of rational economy in matters of taxonomic machinery. Early on, Ockham had espoused the theory of “objective existence” according to which there exist not only real objects (trees, dogs, rocks, etc.), but also imaginary thought-objects, such as a gold mountain or the winged horse Bellerophon, which do not exist as such, but only mind-correlatively as thought-objects. But by the time Ockham wrote his Quodlibeta, controversy had convinced him that such a view is erroneous. Instead of a twotier theory of existence (viz. existence-in-reality and existence-in-thought) he now advocated a one-tier theory which accepted only real objects.11 The grounding idea here is that those “thought-objects” are objects in name only: they do not have any sort of objectivity, let alone any form of being or existence. His reasoning was straightforward. It is simply that we do not need such objects: the work they are supposed to do can be achieved by noting that what does exist—and exist as perfectly real objects—are thoughts or ideas or beliefs regarding such things. Accordingly, we can get by with a one-tier theory of objects. And this is preferable and appropriate because “entitles are not to be multiplied beyond necessity.” And Ockham extended this theory that the real particulars have being or existence to accommodate not only thought-things like Bellerophon, but also universals. Only the concrete actualities of the real world have existence: universals like colors or shapes simply do not exist—not even as thought-objects. All that really exists are the ideas of thoughts of such universals. As Ockham thus saw it, there are neither imaginary nor universal entities: only concrete reals and their manifest features should be acknowledged as existing. All other needful items can and should be handled by their means. In the end, it should be clear that what does the probative work here is not ontology as such but methodology. There is no point in postulating more machinery than we need: “Entities are not to be multiplied beyond necessity.” In sum, this methodological principle gave expression to and provided a justifacatory rationale for Ockham’s rejection of “thoughtobjects”—the characteristic feature of his nominalism. In Ockham and the medievals we are not dealing with a ontological principle along the lines of “Nature does nothing in vain: Reality is economical in operation”)—nor yet the idea that, in Sir William Hamilton’s formulation: “Nature never works by more complex instruments than are

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necessary.”12 The thrust of the substantive dictum that natura nihil facit frustra—“Nature does nothing in vain”—moves in quite a different direction from that of the Ockhamite principle of methodology and finds no grounds for validation there. Still, the step from methodology to ontology seemed natural, and already St. Bonaventure wrote: Dicendum est quod ominis ratio et natura concordat—quod non fiat per plures quod potest sufficientissime fieri per unum; alioquin est ibi superfluum.13 (“It must be said that reason and nature both agree not to do by several that which can be done by one, since otherwise there would be waste.”) 5. THE METHODOLOGICAL RATIONALE OF COGNITIVE ECONOMY

It has long been recognized that simplicity must play a prominent role in rational inquiry. Considerations of rational economy and convenience of operation obviously militate for inductive systematicity. Seeing that the simplest answer is (eo ipso) the most economical one to work with, rationality creates natural pressure towards economy—towards simplicity insofar as other things are equal. Of course, we do not take this line because we know a priori that this simplest resolution will prove to be correct. (How could we?) Rather, when other things are anything like equal we adopt the simplest answer—provisionally at least—just exactly because it is the least cumbersome and most economical way of providing a resolution that does justice to the facts and demands of the situation. Recognizing that other possibilities of resolution exist we ignore them until further notice, exactly because there is no cogent reason for giving them favorable treatment at this stage. (After all, once we leave the safe harbor of simplicity behind we always encounter multiple possibilities for complexification, and generally lack sufficient guidance for moving one way rather than another.) With respect to the philosophy of nature, Henri Poincaré has observed that: [Even] those who do not believe that natural laws must be simple, are still often obliged to act as if they did believe it. They cannot entirely dispense with this necessity without making all generalization, and therefore all science, impossible. It is clear that any fact can be generalised in an infinite number of ways, and it is a question of choice. The choice can only be guided by considerations of simplicity. . . . To sum up, in most cases every law is held to be simple until the contrary is proved.14

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These observations are entirely in the right spirit. In science, we avoid needless complications whenever possible, because this is the course of an economy of effort. It is the general practice in scientific theory construction, to give preference to • one-dimensional rather than multidimensional modes of description, • quantitative rather than qualitative characterizations, • lower- rather than higher-order polynomials, • linear rather than nonlinear differential equations. As cognitive possibilities proliferate in the course of theory-projecting inquiry, a principle of choice and selection becomes requisite. And here economy—with its other systematic congeners, simplicity, and uniformity, and the like—are the natural guideposts. We subscribe to the inductive presumption in favor of simplicity, uniformity, normality,15 etc., not because we are convinced that matters always stand on a basis that is simple, uniform, normal, etc.—surely we know no such thing!—but because it is on this basis alone that we can conduct our cognitive business in the most advantageous, the most economical way. We effect our problem resolutions along the lines of least resistance, seeking to economize our cognitive effort by using the most direct workable means to our ends. Whenever possible, we analogize the present case to other similar ones, because the introduction of new patterns complicates our cognitive repertoire. We use the least cumbersome viable formulations because they are easier to remember and more convenient to use. Insofar as possible, we try to ease the burdens we pose for our memory (for information storage and retrieval) and for our intellect (for information processing and calculation). In sum, we favor uniformity, analogy, and the other aspects of simplicity because they ease our cognitive labor. At the dawn of modern science, Galileo wrote: “When therefore I observe a stone initially at rest falling from a considerable height and gradually acquiring new increases of speed, why should I not believe that such increments come about in the simplest, the most plausible way?”16 Why not indeed? Subsequent findings may, of course, render this simplest position untenable. But this recognition only reinforces our stance that simplic-

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ity is not an inevitable hallmark of truth (simplex sigillum veri), but merely a methodological tool of inquiry—a guidepost of procedure. Throughout the domain of rational inquiry we seek to provide a descriptive and explanatory account that provides the optimal way of accommodating the data that experience puts at our disposal. And when something simple accomplishes the cognitive tasks in hand equally well as some more complex alternative then it is foolish to adopt the latter. We need not certainly presuppose that the world somehow is systematic (simple, uniform, and the like) to validate our penchant for the systematicity of our cognitive commitments. The overall lesson is clear. All those manifestations of economy in the proceedings of Nature do not find their support in the Ockhamite principle of cognitive parsimony. The root idea is not that Nature herself is lacking in complications, but that we ourselves should be as sparing in this regard as we can manage to get away with. It may of course turn out to be true as scientific inquiry unfolds that Nature functions economically, and that Sir William Hamilton was right when he maintained that Law of Parsimony characterizes nature’s modus operandi. But, as J. S. Mill very sensibly objected that Hamiltonian principle is a purely [methodo-] logical precept. It is folly to complicate research by multiplying the object of inquiry; but we know too little of the ultimate constitution of the universe, to assume that it cannot be far more complex than it seems, or than we have any actual [i.e., current] reason for surprise.17

And in this light, William Thorburn is entirely right endorsing a construal of the Ockhamite principle that “has turned a precept of Methodology into a Metaphysical Dogma.”18 Accordingly, J. S. Mill rightly asks for the evidential credentials of this contention. Such a thesis belongs to empirical science and not to the theory of rational inquiry as such. Rather, we are dealing with a methodological rule of procedure—an operational injunction along the lines of: “Do not postulate mechanisms that are dispensable for your explanatory deliberations.” In matters of conjecture, assumption, postulation, and the like, fewer is better. Isaac Newton tells us that Dicunt utique philosophi, natura nihil agit frustra (“The philosophers certainly say that Nature does nothing in vain).19 But those philosophers who say this are clearly nature-describers and not mere methodologists. Hans Reichenbach declared: “Actually in cases of inductive simplicity it is not economy which determines our

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choice . . . We make the assumption that the simplest theory furnishes the best predictions. This assumption cannot be justified by convenience; it has a truth character and demands a justification within the theory of probability and induction.”20 This substantive perspective is gravely misleading. What sort of consideration could possibly justify the supposition that “the simplest theory furnishes the best prediction”? Any such supposition is surely unwarranted and inappropriate. Properly construed, the principle of simplicity belongs to the theory of scientific inquiry and not to the theory of nature. Even when driven towards greater complexity, we nevertheless try to keep things as simple as possible, with the emphasis on that as is possible in the circumstances. But just herein lies the rub. The penchant “Use the cheapest instrument that will do the job” may well lead to substantial expenditure under the pressure of that italicized proviso. The crucial proviso praeter necessitate (unnecessarily) deserves due heed. First of all there is the issue of how hard the idea of necessity is to be pressed. In daily life there are many things that are not absolutely necessary but nevertheless are highly desirable. What argues for them is not the harsh necessity of life-maintenance but the gentler desirability of lifepleasantness. And the same of course occurs in cognitive matters. There are many things we do not have to accept or postulate to make cognitive rationality possible, but whose acceptance or postulation greatly enhances the transaction of our cognitive business. We do not have to believe the deliverances of our senses but nevertheless the percept “Accept them unless and until there are counter-indication” is a sound rational principle. (In its absence there is, in fact, very little we can do). Analogously, the dictum “Make cognitive complications pay for themselves by affording some signified advantage in process or product” is sensible and sound. But this is a procedural injunction that does not prejudge any substantive results. Over the years, various theorists have construed Ockham’s Razor as an automatic pathway to ontological economy. Proceeding in this spirit, various schools of ontological minimalism go about posting signposts that put all risk of engaging larger issues OFF LIMITS. Such theorists turn Ockham’s razor into Robespierre’s guillotine. Their tumbrils carry off a wide variety of victims: • sets in the philosophy of mathematics, • abstracta in semantics,

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• unobservable entities in the philosophy of physics, • dispositional theses in the philosophy of language, • obligations that reach beyond the requisites of prudence in moral theory, etc. etc. And so the principle involved has been used as a supporting pillar in such contexts as: • nominalism in its controversy with a realism of universals, • phenomemalistic idealism in its controversy with materialistic realism, • scientific conventionalism in its controversy with theoretical-entity realism, • mathematical constructionism in its controversy with “Plutonic” realism. The fundamental reasoning is the same throughout such substantive appeals to economy, the idea being that Nature functions parsimoniously. 6. THE GROWTH OF COMPLEXITY

After all, what will it be that justifies this widespread insistence on economy in Nature’s modus operandi? It certainly is not the actual history of the sciences as we know it. For the history of science is in fact a story of everunfolding complexity. It provides no grounds to postulate simplicity of “Nature” and endorse the dictum that “Simplex sigillum veri.” The simplicity of Nature is certainly not an outcome that can be based on general principles of procedural rationality. (There is no proper transit from methodology to ontology.) Nor is it something we learn from science itself. Induction with respect to the history of science itself—a constant series of errors of oversimplification—soon undermines our confidence that nature operates in the way we would deem the simpler. On the contrary, the

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history of science is an endlessly repetitive story of simple theories giving way to more complicated and sophisticated ones. The Greeks had four elements; in the nineteenth century Mendeleev had some sixty; by the 1900s this had gone to eighty, and nowadays we have a vast series of elemental stability states. Aristotle’s cosmos had only spheres; Ptolemy’s added epicycles; ours has a virtually endless proliferation of complex orbits that only supercomputers can approximate. Greek science was contained on a single shelf of books; that of the Newtonian age required a roomful; ours requires vast storage structures filled not only with books and journals but with photographs, tapes, floppy disks, and so on. Of the quantities currently recognized as the fundamental constants of physics, only one was contemplated in Newton’s physics: the universal gravitational constant. A second was added in the nineteenth century, Avogadro’s constant. The remaining six are all creatures of twentieth century physics: the speed of light (the velocity of electromagnetic radiation in free space), the elementary charge, the rest mass of the electron, the rest mass of the proton, Planck’s constant, and Boltzmann’s constant.21 It would be naive— and quite wrong—to think that the course of scientific progress is one of increasing simplicity. The very reverse is the case: scientific progress is a matter of complexification because over-simple theories invariably prove untenable in a complex world. The natural dialectic of scientific inquiry ongoingly impels us into ever deeper levels of sophistication.22 In this regard our commitment to simplicity and systematicity, though methodologically necessary, is ontologically unavailing. And more sophisticated searches invariably engender changes of mind moving in the direction of an ever more complex picture of the world. Our methodological commitment to simplicity should not and does not preclude the substantive discovery of complexity. The explosive growth of information of itself countervails against its exploitation for the sake of knowledge-enhancement. The problem of coping with the proliferation of printed material affords a striking example of this phenomenon. One is forced to ever higher levels of aggregation, compression, and abstraction. In seeking for the needle in the haystack we must push our search processes to ever greater depths. And this ongoing refinement in the division of cognitive labor that an information explosion necessitates issues in a literal dis-integration of knowledge. The “progress of knowledge” is marked by an ever continuing proliferation of ever more restructured specialties marked by the unavoidable circumstance that any given specialty cell cannot know exactly what is

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going on even next door—let alone at the significant remove. Our understanding of matters outside one’s immediate bailiwick is bound to become superficial. At home base one knows the details, nearby one has an understanding of generalities, but at a greater remove one can be no more than an informed amateur. This disintegration of knowledge is also manifolded vividly in the fact that out cognitive taxonomies are bursting at the seams. Consider the example of taxonomic structure of physics. In the 11th (1911) edition of the Encyclopedia Britannica, physics is described as a discipline composed of 9 constituent branches (e.g., “Acoustics” or “Electricity and Magnetism”) which were themselves partitioned into 20 further specialties (e.g., “Thermo-electricity: of “Celestial Mechanics”). The 15th (1974) version of the Britannica divides physics into 12 branches whose subfields are— seemingly—too numerous for listing. (However the 14th 1960’s edition carried a special article entitled “Physics, Articles” on which surveyed more than 130 special topics in the field.) When the National Science Foundation launched its inventory of physical specialties with the National Register of Scientific and Technical Personnel in 1954, it divided physics into 12 areas with 90 specialties. By 1970 these figures had increased to 16 and 210, respectively. And the process continues unabated to the point where people are increasingly reluctant to embark on this classifying project at all. Substantially the same story can be told for every field of science. The emergence of new disciplines, branches, and specialties is manifest everywhere. And as though to negate this tendency and maintain unity, one finds an ongoing evolution of interdisciplinary syntheses—physical chemistry, astrophysics, biochemistry, etc. The very attempt to counteract fragmentation produces new fragments. Indeed, the phenomenology of this domain is nowadays so complex that some writers urge that the idea of a “natural taxonomy of science” must be abandoned altogether.23 The expansion of the scientific literature so great that natural science has in recent years been disintegrating before our very eyes. An ever larger number of ever more refined specialties has made it ever more difficult for experts in a given branch of science to achieve a thorough understanding about what is going on ever in the specialty next door. It is, of course, possible that the development of physics may eventually carry us to theoretical unification where everything that we class among the “laws of nature” belongs to one grand unified theory—one allencompassing deductive systematization integrated even more tightly than

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that Newton’s Principia Mathematica.24 But the covers of this elegantly contrived “book of nature” will have to encompass a mass of every more elaborate diversity and variety. Like a tricky mathematical series, it will have to generate ever more dissimilar constituents which, despite their abstract linkage are concretely as different as can be. And the integration at issue at the principle of a pyramid will cover further down an endlessly expansive range and encompassing the most variegated components. It will be an abstract unity uniting a concrete mishmash of incredible variety and diversity. The “unity of science” to which many theorists aspire may indeed come to be realized at the level of concepts and theories shared between different sciences—that is, at the level of ideational overlaps. But for every conceptual commonality and shared element there will emerge a dozen differentiations. The increasing complexity of our world picture is a striking phenomenon throughout the development of modern science. The lesson of such considerations is clear. The history of science presents us with a scene of ever-increasing complexity. It is thus fair to say that modern science confronts us with a cognitive manifold that involves an ever more extensive specialization and division of labor. The years of apprenticeship that separate master from novice grow ever greater. A science that moves continually from an over-simple picture of the world to one that is more complex calls for ever more elaborate processes for its effective cultivation. And as the scientific enterprise itself grows more extensive, the greater elaborateness of its productions requires an ever more intricate intellectual structure for its accommodation. The complexification of science betokens our ever more complex realization of complexity in Nature. 7. COGNITIVE ECONOMY IN SCIENCE

Our conception of Nature as developed over the historical course of scientific inquiry does not betoken simplicity, economy or parsimony as such. Instead, the so-called “Principle of Simplicity” is really a principle of complexity-management in relation to our cognitive affairs: Feel free to introduce complexity in your efforts to describe and explain nature’s ways. But only when and where it is really needed. Insofar as possible “keep it simple!” Only introduce as much complexity as you really need for your scientific purposes of description, explanation, prediction, and control.

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Such an approach is eminently sensible. But such a principle is no more than a methodological rule of procedure for managing our cognitive affairs. Nothing entitles us to transmute this methodological precept into a descriptive/ontological claim to the effect that nature is simple—let alone of finite complexity. It is indeed economy and convenience that determine our regulative predilection for simplicity and systematicity in general. Our prime motivation is to get by with a minimum of complication, to adopt strategies of question-resolution that enable us among other things: (1) to continue with existing solutions unless and until the epistemic circumstances compel us to introduce changes (uniformity); (2) to make the same processes insofar as possible (generality), and (3) to keep to the simplest process that will do the job (simplicity). Such a perspective combines the commonsensical precept, “Try the simplest thing first,” with a principle of burden of proof: “Maintain your cognitive commitments until there is good reason to abandon them.” From this perspective, systematicity-preference emerges as a matter of a procedural simplification of labor, a matter of the “intellectual economy” of the cognitive venture. Why use a more complex solution where a simple one will do as well? Why depart from uniformity—why use a new, different solution where an existing one will serve? And so it is accordingly important to distinguish between substantive and methodological considerations and separate economy of means from substantive economy. For process is one thing and product another. Simple tools or methods can, suitably used, create complicated results. A simple cognitive method, such as trial and error, can ultimately yield complex answers to difficult questions. Conversely, simple results are sometimes brought about in complicated ways. A complicated method of inquiry or problem solving might yield easy and uncomplicated problem solutions. Our commitment to simplicity in scientific inquiry accordingly does not, in the end, prevent us from discovering whatever complexities are actually there. And our striving for cognitive systematicity in its various forms persists even in the face of complex phenomena. For such reasons, the ontological systematicity of nature is ultimately irrelevant for our procedurally regulative concerns: the commitment to inductive systematicity in our account of the world remains a methodological desideratum regardless of how complex or untidy the world may turn out to be.

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The basic principles of procedural rationality should certainly govern our choice of a procedural methodology. In specific, it is clear that our procedural modus operandi — should be ontologically neutral. Ontological conclusions should not result from substantive inquiry—from our methodological practices and processes as such—but only from their application. — should not be reductive. It should not yield reductive analytical conclusions—eliminate possibilities on general principles. — should not be such as to impede our discovery—something which, in theory, could well be the case. Charles Sanders Peirce advocated the appropriate injunction: “Never bar the path of inquiry”—that is, never to adopt a procedural practice or policy that would automatically punctuate discovering something that might actually be the case. And any over-zealous substantive application of an Ockham’s Razor would certainly fall afoul of this rule. 8. CONCLUSION

The history of the matter makes it clear that two decidedly different versions of Ockham’s Razor are at work. One is the wholly unproblematic rational procedural/methodological injunction not to introduce such descriptive of explanatory instrumentalities as distinctions, classifications, assumptions, etc. that do not “pay for themselves” by making a significant contribution to our understanding of the relevant issues. The other is the more ambitious substantive thesis that nature’s modus operandi is economical and efficient, the “nature does nothing in vain” and that its functioning accords with such operative “Principles of Parsimony” as the Law of Least Action or of a Law of Small Numbers. Accordingly, two very different principles are at issue: the one a procedural impetus regarding the transaction of our cognitive business, the other a fact-purporting thesis about the modus-operandi of Nature. And what deserves stress here is that in Ockham himself and in most of the Aristotelian tradition to which his own work belongs, it is the former, methodological version of the prospect that stands at the forefront. And here the crux was not that nature/reality exhibits a simplicity tropism but

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rather that the basic principles of procedural rationality in inquiry speak loudly on behalf of simplicity preference. The lesson of these deliberations is clear. The manifestations of functional simplicity and economy in Nature’s modus operandi along the lines of a Principle of Least Action or of a Law of Small Numbers is not something that can be accounted for on Ockhamite principles of rational economy of method. Insofar as they are valid their validation remains a matter of the same sort of empirical systematization used in the substantiation of scientific theories at large. And to all appearances the jury is still out on this factual issue. In the end, then, it would conduce greatly to clarity and accuracy to effect the clear terminological distinction achieved by (1) characterizing as Ockham’s Razor the traditional principle of rational economy as stressed by Ockham and the medieval schoolmen, and then (2) contradistinguishing this by adopting William Hamilton’s expression, Law of Parsimony for the factually descriptive (and thereby purportedly scientific) thesis of economy in the operations of Nature along the lines of the Principle of Least Action or Zipf’s Law of Small Numbers. NOTES 1

Paul K. Zipf, Human Behavior and the Principle of Least Effort (Cambridge, MA: Addison Wesley, 1649).

2

See the author’s Epistemetrics (Cambridge: Cambridge University Press, 2006).

3

William of Ockham, Commentary on the Sentences, Pt. 1, dist. 30, q. 1.

4

The first traceable appearance of the standard formula was in Sir William Hamilton’s 1852 Discussion in Philosophy, Literature and Education (London: Longman, Brown, Green and Longmans, 1852), p. 590 (“On Causality”). In the second edition it was cited on pages 616 and 629.

5

The history of Ockham’s Razor is the subject of a magisterial and painstakingly scholarly study by William Thorburn, “The Myth of Occam’s Razor,” Mind, vol. 27 (1918), pp. 345–53. The present historical account is based upon it.

6

William Hamilton, Discussion on Philosophy and Literature, Education, and University Reform (Edinburgh and London: William Blackwood & Sons, 1858), p. 61112. (The discussion at issue first appeared in 1852.) Hamilton also dedicates a long footnote to antecedent of the principle on pp. 624–28.

7

Aristotle, Physics 189a14–18. Italics supplied.

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NOTES 8

Thus in his dissertation on Nizolius, Leibniz very properly spoke not of a principle but of a rule (regula). (See his Opera Omina, ed. L. Dutens, Vol. IV, p. 36.)

9

Ernst Mach, “Die ökonomische Natur der physikalischen Forschung” in his Populär-Wissenschaftliche Vorlesungen (Leipzig: Johann Ambroscus Berthm, 1903), pp. 215–242 (see p. 236).

10

Quoted in J. Kim and E. Sosa (eds.), Metaphysics: an Anthology (Oxford: Blackwell, 1999), p. 10.

11

Marilyn McCord Adams, William Ockham, 2 vol.’s (Notre Dame, IN: University of Notre Dame Press, 1978), Vol. I., pp. 103–105.

12

William Hamilton, op. cit, p. 617.

13

St. Bonaventure, Commentary on the Sentences, Bk. I, d. 10, art. 1, ad 4. In his Opera Omnia, Vol. I (Quarrachi, 1882), p. 196.

14

Henri Poincaré, Science and Hypothesis (New York: Dover Press, 1914), pp. 145– 146.

15

Note that in explaining the behavior of people we always presume normalcy and rationality on their part—a presumption that is, to be sure, defeasible and only holds “until proven otherwise.”

16

Galileo Galilei, Dialogues Concerning Two New Sciences tr. by H. Crew and A. de Salvo (Evanston: University of Illinois Press, 1914), p. 154.

17

J. S. Mill, An Examination of Sir William Hamilton’s Philosophy (London: Longman, Green, Longman, Roberts & Green, 1865), Chap. 24, p. 542, 4th ed.

18

William Thorburn, op. cit., p. 352.

19

Isaac Newton, Principia mathematica, 3rd ed. (1721), “De mundi systemate,” LII, p. 387.

20

Hans Reichenbach, Experience and Prediction (Chicago and London, 1938), p. 376. Compare: “Imagine that a physicist … wants to draw a curve which passes through [points on a graph that represent [the data observed. It is well known that the physicist chooses the simplest curve; this is not to be regarded as a matter of convenience … [For different] curves correspond as to the measurements observed, but they differ as to future measurements; hence they signify different predictions based on the same observational material. The choice of the simplest curve, consequently, depends on an inductive assumption: we believe that the simplest curve

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NOTES

gives the best predictions. … If in such cases the question of simplicity plays a certain role for our decision, it is because we make the assumptions that the simplest theory furnishes the best predictions.” (Ibid., pp. 375–376.) 21

See B. W. Petley, The Fundamental Physical Constants and the Frontiers of Measurement (Bristol: Hilger, 1985).

22

On dialectical reasoning see the author’s Dialectics (Albany NY: State University of New York Press, 1977), and for the analogous role of such reasoning in philosophy see The Strife of Systems (Pittsburgh: University of Pittsburgh Press, 1985).

23

See John Dupré, The Disorder of Things: Metaphysical Foundations of the Disunity of Science (Cambridge MA: Harvard University Press, 1993).

24

See Steven Weinberg, Dreams of a Formal Theory (New York: Pantheon, 1992). See also Edoardo Amaldi, “The Unity of Physics,” Physics Today, vol. 261 (September, 1973), pp. 23–29. Compare also C. F. von Weizsäcker, “The Unity of Physics” in Ted Bastin (ed.) Quantum Theory and Beyond (Cambridge: Cambridge University Press, 1971).

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Chapter 4 PASCAL’S WAGER IN RELIGION

P

ascal’s Thoughts (Pensées) is a large assemblage of brief notes and drafts that he jotted down during the years 1657–62 in preparation for writing a projected Apology for the Christian Religion. The passage which concerns us here is a minuscule part of this whole. It consists of two leaves, covered on both sides by handwriting overlapping in several directions, full of erasures and emendations.1 This brief text endeavors to present a new line of thought in support of Christian religious faith. Pascal himself was deeply persuaded of the importance and novelty of these deliberations. Toward the end of the text, which takes the form of a dialogue between him and an unbelieving friend, his interlocutor exclaims: “Oh, your discourse delights me, carries me away!” However, to judge by the tenor of the large literature to which this small passage has given rise, Pascal’s imaginary friend is very much in the minority in this regard. Pascal’s Wager argument can hardly be said to have set the world on fire. From its inception to the present day, when philosophers and theologians deign to mention it at all, they tend to do so by way of scornful dismissal. It is no exaggeration to say, with one nineteenth-century student of Pascal, that the argument “has been a scandal even to some of his greatest admirers.”2 Theologians have treated it with lofty disdain, and “philosophers feel it somehow as a professional obligation not to accept its cogency.”3 And yet, as this discussion will try to show, the charges of moral insensitivity and other such high-minded complaints often launched against the argument are based on a total misunderstanding of its aim and import, and reflect a callous refusal to accept the argument on its own terms. Only by failing to recognize the job that Pascal’s argument is designed to accomplish—by seeing it as attempting a task altogether different from its actual probative aim—can one support the facile recriminations all too often thrown its way. And this is hardly reasonable: surely any argument can be made to look silly if reconstrued to aim at conclusions never even remotely intended for it. In actual fact, Pascal’s Wager argument contains the core of a deeply insightful innovation. Its ground-breaking idea of a theological use of prac-

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tical reason marks an important departure which, in the hands of such subsequent masters as Immanuel Kant and William James, has come to make a substantial impact on the philosophy of religion. Let us have a closer look at what is involved. 2. THE SKEPTICAL BACKGROUND OF THE WAGER ARGUMENT The Wager argument passage opens with the following prologue: Infini-rien; infinity-nothing. Our soul has been cast into the body, where it finds number, time, dimension. Thereupon it reasons and calls this nature or necessity, and can believe nothing else. Yet unity added to infinity adds nothing to it, any more than one foot added to an infinite length. The finite is annihilated in presence of the infinite, and becomes pure nothingness. Even thus does our intellect stand to God. (Pensées, 343/233)

The close connection with Descrates’ Fourth Meditation is striking: I am a kind of intermediate between God and nothingness, between the Supreme Being and non-being (non ens). My nature is such that, in so far as I am a creature of the Supreme Being, I have nothing in me to deceive me or lead me astray. Yet in so far as I also participate somehow in nothingness, non-being—that is, in so far as I am not myself the Supreme Being, and am lacking in no end of things—it is not surprising that I am deceived.4

To be sure, Pascal himself will hear nothing of intermediation: in relation to God, man stands squarely on the side of nothingness. But the similarities outweigh the differences. Descartes and Pascal alike see man’s utter inadequacy as paramount. Both stress man’s utter dependency on God for whatever goods he may enjoy—cognitive goods prominently included. The matter of justification in the face of skeptical doubts is part of the common problem-heritage of these thinkers. And the starting point of Pascal’s Wager is clearly Descartes: its drama is played out on the stage of Cartesian skepticism. Moreover, the strategy of Pascal’s Wager argument in philosophical theology can fruitfully be viewed against the background of the Cartesian revolution in philosophy. Descartes put man as knower at the center of the stage. Instead of addressing questions regarding the nature of reality directly, the issue of our knowledge became the pivot point. Beginning with the question How do I really know?, Descartes assigned to God the pivotal role

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of guarantor of our knowledge of the world. And he maintained that man can come to know about the existence and nature of God through the light of God-given reason alone. God, our maker, has implanted an idea of himself in our minds somewhat as a silversmith impresses his hallmark into his product. This idea provides the basis on which we humans, relying wholly on our innate intellectual resources, can come to knowledge of God’s nature and existence through the clear and distinct intuitions of our mind in much the same way that we come to know the truths of mathematics and metaphysics. As far as the mechanism of our knowledge is concerned, the necessity of God is, for Descartes, on the same cognitive plane as the necessity of the Pythagorean theorem or that of the non-vacuity of space. Pascal retained Descartes’s skeptically inspired preoccupation with the processes and products of the human intellect. But he abandoned Descartes’s reliance on cognitive operation of human reason and thus his orientation toward demonstrative knowledge. “The metaphysical proofs of God are so remote from human reasoning, and so complicated, that they make little impression. If some find them profitable, it is only at the moment when they grasp them; an hour later they fear they have been mistaken.”5 In place of Descartes’s Theomistic/scholastic concern for demonstrating the existence of God, Pascal substitutes an Augustinian concern for the validation of belief in God who is beyond the reach of the unaided human intellect and outside the grasp of feeble human reason. Indeed, knowledge as such would not really serve our need in this domain: “There is a great difference between the knowledge and the love of God.”6 God lies beyond the reach of our ordinary cognitive resources: “If there is a God, he is infinitely incomprehensible, since, having neither parts nor limits, he has no affinity to us. We are incapable of knowing either what He is or if He is. God is deus absconditus, Dieu caché, a hidden God.”7 As Pascal saw it, reasoning can only impel the outsider into the fold of believers by a circuitous route: what we can establish by reasoning is not the direct conclusion that there is a God, but only that oblique results that belief in God is warranted. The task is not to deploy the processes of rational demonstration to “prove” to skeptical outsiders that God exists (in Pascal’s opinion a hopeless endeavor—as the skeptics have established), but to show uncommitted indifferentists that belief in God is rationally legitimate. Pascal holds, with the Renaissance skeptics, that our human resources for securing knowledge by inquiring reason are wholly inadequate to the demands of apologetics. For Pascal the pivotal question is thus no longer How can one demonstrate that a God of such-and -such a charter

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exists? But rather: How can one validate having faith in such a God? He writes: It is a remarkable fact that none of the canonical writers ever employed nature to prove God. They all endeavor to instil belief in him. David, Solomon, and the rest never said: “There is no void, therefore there is a God.” They must have had more knowledge than the most learned men who came after them, all of whom used such an argument. This is highly significant. (Pensées, 19/243)

Pascal shifts the issue from the demonstration of facts to the justification of faith. And he presses this question in the face of the profound skepticism regarding the capacity of reason in the theological sphere—one deeper than that of Descartes. Pascal turns to faith rather than theoretical reason precisely because of substantial doubts regarding the capabilities of the latter. The pivotal issue with respect to belief is not one of rational feasibility alone; desirability also enters in. Would we really rest content with a God whose existence is a matter of demonstration? Is the God who emerges at the end of a syllogism (or some yet more complex course of demonstration) the sort of God we need or want? Pascal thinks not. He anticipates Kiekegaard’s view that insufficiency in point of proof and evidence is advantageous, indeed even necessary to a viable faith. Accordingly, a transposition is at issue here that might be called “Pascal’s shift in theological argumentation”—a shift away from theoretical arguments that purport to argue probatively for the existence of God (in the manner of Aquinas’ five ways) to a different style of practical argumentation, geared not to a theoretical demonstration of the existence of God as an ontological fact but to a practical resolution regarding what we ought to believe. The salient feature of the argument is thus its recourse to praxis— and to prudence. Notwithstanding its methodological modernity as a course of argumentation cast in the mold of decision theory, the spirit of Pascal’s Wager is thus profoundly conservative in its substantive message. It turns its back on a medieval scholasticism that had left its deep marks as recently as Descartes’s unacknowledged borrowings from Suarez and returns to the perspectives of the Church Fathers. Its noncognitivism, its practicalism/voluntarism, and its Augustinian fideism are all throwbacks to an earlier era of Christian religious thought. Yet despite its backward-looking aspect, the approach taken by Pascal breaks significant new ground in philosophical theology. Renaissance

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Christian skeptics from Nicholas of Cusa to Erasmus had emphasized the inadequacy of human reason in theological matters and urged the sufficiency of simple piety in religious practice unsupported by any warrant from doctrines and dogmas—i.e., without the rationalization of a justificatory basis of accepted theses. Luther had rejected such a noncognitivist approach as unworthy of a rational creature, holding that piety without rational conviction is not just rationally unsatisfying but even hypocritical. Pascal sought for a middle way between Erasmus’ corrosive skepticism regarding reason and Luther’s dogmatic insistence on the evidential legitimacy of belief through scripture and reason. In Montaigne, who greatly influenced Pascal, we are offered a stark choice between a religion based on cognitive reason and evidence and one resting on faith alone, based not on our capacities but solely on God’s grace.8 Pascal sought to find an intermediate route via the mitigated skepticism of the middle Academy by arguing that belief can be validated on other than strictly evidential grounds, namely, on grounds of practical reason—as an instrumentality of action rather than theoretical cognition. As he saw it, considerations of praxis geared to the interest of man can provide the crucial linking tertium quid between the two problematic extremes of skepticism and dogmatism. Pascal’s shift away from the rational conviction secured by demonstrative reason has important consequences. To accept something, to believe in something, and to place one’s trust in something can all be seen as actions of a certain sort, things that one can decide on and do, or else decide against and refrain from. Accordingly, the level of the discussion is shifted from establishing a fact to justifying an action. Such a shift from the cognitive (or theoretical) to the practical use of human reasoning puts the discussion on an entirely different basis; it puts the practical issue of deciding what we should do at the forefront. The basic issue now is not “Does God exist?” as such, but “Should we accept that he exists—is it appropriate to endorse this proposition?” The skeptic is right in maintaining that theorizing reason leaves the issue undecided. Nevertheless, in practical matters indecision yields an inactivity tantamount to negation: to suspend judgment is, in its practical effect, to resolve the issue in the negative. And, so Pascal insists, we cannot afford the comfort of suspended judgment: “A bet must be laid. There is no option: You have joined the game.”

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3. THE WAGER ARGUMENT For convenience in analyzing its line of thought the continuous discussion of Pascal’s Wager argument can be divided into six component sectors: 1. Who then can blame Christians for not being able to give a logical reason for their belief, professing as they do a religion which they cannot explain by reason? They declare, when expounding it to the world, that it is a foolishness, stultitia. And then you complain that they do not prove it! If they proved it, they would give the lie to their own words; it is in lacking proofs that they do not lack sense. 2. Let us examine this point and declare: “Either God exists, or He does not.” To which view shall we incline? Reason cannot decide for us one way or the other: we are separated by an infinite gulf. At the extremity of this infinite distance a game is in progress, where either heads or tails may turn up. What will you wager? According to reason you cannot bet either way; according to reason you can defend neither proposition . . . “Both are wrong. The right thing is not to wager at all.” Yes, but a bet must be laid. There is no option: you have joined the game. 3. Which will you chose, then? Since a choice has to be made, let us see . . . Your reason suffers no more violence in choosing one rather than the other, since you must of necessity make a choice. That is one point cleared up. But what about your happiness? Let us weigh the gain and the loss involved in wagering that God exists. Let us estimate these two possibilities; if you win, you win all; if you lose, you lose nothing. Wager then, without hesitation, that he does exist. 4. “That is all very fine. Yes I must wager, but maybe I am wagering too much.” Let us see. When there is an equal risk of winning and of losing, if you had only two lives to win, you might still wager; but if there were three lives to win, you would still have to play (since you are under the necessity of playing); and being thus obliged to play, you would be imprudent not to risk your life to win three in a game where there is an equal chance of winning and losing. But there is an eternity of life and happiness. That being so, if there were an infinity of chances of which only one was in your favor, you would still do

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right to stake one to win two, and you would act unwisely in refusing to play one life against three, in a game where you had only one chance out of an infinite number, if there were an infinity of an infinitely happy life to win. But here there is an infinity of infinitely happy life to win, one chance of winning against a finite number of chances of losing, and what you stake is finite. That removes all doubt as to choice; wherever the infinite is to be won, and there is not an infinity of chances of loss against the chance of winning, there are no two ways about it: you must risk it all. 5. Now there is no use alleging the uncertainty of winning and the certainty of risk, or to say that the infinite distance between the certainty of what one risks and the uncertainty of what one will win equals that between that finite good, which one certainly risks, and the infinite, which is uncertain. This is not so; every player risks a certainty to win an uncertainty, and yet he risks a finite certainty to win a finite uncertainty, without offending reason . . . For the uncertainty of winning is proportionate to the chances of gain and loss. Hence, if there are as many chances on one side as on the other, the right course is to play even; and then the certainty of risk is equal to the uncertainty of the gain, so far are they from being infinitely distant. Thus our proposition is of infinite force when there is the infinite at stake in a game where there are equal chances of winning and losing,9 but the infinite is to be won. This is conclusive, and if men are capable of truth at all, there it is. 6. . . . “But I am so made that I cannot believe. What then do you wish me to do?” . . . That is true. But understand at least that your inability to believe is the result of your passions; for although reason (now) inclines you to believe, you cannot do so. Try therefore to convince yourself, not by piling up proofs of God, but by subduing you passions. . . . You desire to attain faith, but you do not know the way. You would like to cure yourself of unbelief, and you ask for remedies. Learn from those who were once bound and gagged like you, and who now stake all that they possess. They are men who know the road that you desire to follow, and who have been cured of a sickness of which you desire to be cured. Follow the way by which they set out, acting as if they already believed, taking holy water, having masses said, etc. Even this will naturally cause you to believe and

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blunt your cleverness. “But that is what I fear.” Why? What have you to lose? So reads Pascal’s discussion. 4. THE WAGER ARGUMENT ANALYZED Let us trace the flow of the argument point by point. It runs roughly as follows: 1. There is a strong streak of skepticism in Pascal—but not of irrationalism. Pascal is not concerned to downgrade reason per se—nor even to deny it a role in theology. Nor did he espouse the heresy of Fideism and maintain that proofs for the existence of God have no place whatsoever in the theological scheme of things.10 But one can only reason effectively from conceded premises, and in this apologetic context we can expect no substantial concessions. Theological reasoning is thus inadequate to the needs of apologetics—it cannot reach those who move on a purely mundane level. From the apologetic point of view those preuves de Dieu Métaphysiques, Pascal tells us in sect. 381/543 of the Pensées, are simply useless: they are too complicated and too remote from the way people ordinarily reason.11 He is not a misologist. (How could so fine a mathematician and scientist be?!) He simply thinks that there are important tasks that theoretical reason cannot accomplish satisfactorily—that of demonstrating the fundamentals of the Christian religion to skeptical nonbelievers among them. He does not want us to abandon theoretical reason but simply to recognize that it has limits.12 Reason itself requires this recognition of us: “The highest achievement of reason is to recognize that there is an infinity of things beyond its grasp” (Pensées, 373/267). 2. The question of inaugurating belief in God should thus be approached from a different angle—not that of constraining conviction but of motivating a decision. While theoretical, probative reason fails us in this apologetic context, practical reason can achieve a great deal. (Skepticism thus accomplishes something positive and useful in clearing the way for the crucial shift from theoretical to practical reason.) The issue is to be seen as one of choice under uncertainty, of a

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gamble with regard to belief or nonbelief. We return to the pragmatic justification of belief mooted by the praxis-oriented thinkers of the Middle Academy. Even as the exigencies of action force physicians and engineers to judgments going beyond the limits of a firm theoretical footing, so with other pressing issues in life we must resolve our questions in ways that outrun the resources of abstract cognition. Here we are, emplaced in this world in medias res—like it or not. The decision whether or not to accept that God exists, and that life extends beyond the grave, confronts us unavoidably. Our disadvantaged place in the world’s dispensation, and the importance and urgency the issue has for us, leaves us no alternative here as regards “playing the game.” We have to opt one way or the other; the issue is simply too pressing to be relegated to the limbo of suspended judgments—it represents an option which (in William James’s terms) is living and momentous. Moreover, it is forced as well; we cannot assume indifference—to be indifferent is effectively to come down on one side: suspension of belief is tantamount to disbelief.13 3. In the choice between accepting and rejecting God reason as such is unveiling: “Reason cannot make you choose either; reason cannot prove either wrong.”14 Since the question does not admit of effective resolution in the province of evidential reason, we are entitled to resolve it in the province of interest. And here Pascal has it that a clear advantage lies with belief. This line of deliberation is helpfully approached by considering a dominance argument. The options stand roughly as follows: Returns to the chooser Options

if God exists

if God doesn’t exist

Bet on God (cost B)

+∞

0

Bet against God (cost 0)

little or nothing (perhaps even something negative)

0

Betting on God clearly affords the better option here, for comparison of the two columns shows that it “dominates” over its rival in that its returns are in some cases better and in none worse than those of the alternative. If the choice stood on this basis, its resolution would be very straightforward: if

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you win, you win all; if you lose, you lose nothing—“Si vous gagnez, vous gagnez tout; si vous perdez, vous ne perdez rien.”15 But this argument, as it stands, has a serious flaw. It fails to take account of the cost (B) of “betting on God,” the price we pay for ordering our lives on religious principles—saying all those prayers, performing all those good works, etc. And here, in accepting that God exists and acting as he commands, the libertin realizes that he pays a price in giving up his selfindulgent ways. “I may perhaps wager too much.” But how much is too much? Consider an arbitrary gamble the following lines: Returns to the chooser Options

if outcome is favorable (probability p)

if outcome is unfavorable (probability 1 – p)

Bet

X

-Y

Do not bet

0

0

The expected value of these two alternatives stands as follows: Bet: P(X) + (1 – p) (–Y) = p(X + Y) –Y Do not bet: 0 Now if making a bet itself costs B dollars, then this simply decreases the expected value of that (top) alternative by just exactly the amount B. But if X is large enough, this will not affect matters significantly. And if the value of a favorable result is relatively infinite (X = +∞), as is presumably so in the theological case, then the expected value of the “Bet” alternative is also +∞ as long as p is nonzero. No finite price whatsoever is now too high. “Wager, then, without hesitation.” The Wager argument thus unfolds in two stages, the second of which removes a limitation or oversimplification of its predecessor. The first step is a dominance argument. The second, by taking account of costs, shifts the issue from one of straightforward dominance to one of preponderant expectations. At this point Pascal’s argument is effectively complete. Its job is done. The succeeding part of the discussion simply tries to show that this line of reasoning does indeed conform to the standard principles of probabilistic

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decision under conditions of uncertainty: the wager is simply an instance of the general strategy of probabilistic decision. 4. The next phase of deliberation proceeds in terms of the model of a lottery. However, it envisages a sequence of cases, and not just one. In each case you “stake your life” as the price of entry (the price of a ticket) and stand to win additional lives. The cases run as follows: Case 1. Win: gain one extra life—have two lives Lose: lose your life—have zero lives If there are two tickets (equal-risk case), the expected value stands as follows: 1

1

EV = 2 (2) + 2 (0) = 1 Since the expected return just equals the entry fee, “you might still play or you might not.” Case 2. Win: gain two extra lives—have three lives Lose: lose your life—have zero lives Again, if there are two tickets (equal-risk case), the expected value stands as follows: 1

1

1

EV = 2 (3) + 2 (0) = 1 – 2

Since the expectation exceeds the entry fee, “you would be imprudent” not to accept this gamble. But consider now the case of an infinitely great return:

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Display 1 A THEISTIC LOTTERY

God exists

God does not exist

1 (probability: n+1 )

n (probability: {n+1 )

Bet on God

+∞

0

Bet against God

1

1

____________________________________________________________ Case 3. Win: gain eternal life (in heaven)—have infinitely many ( ) lives Lose: lose your life—have zero lives As long as there is a finite number (n) of tickets (“a finite number of chances of loss”), the expected value stands as follows: 1

n-1

EV = n (+∞) + n (0) = +∞ Provided there is some finite chance of winning—however small—the expected return infinitely exceeds the entry fee, and the gamble should be run. What is thus contemplated is a lottery of the sort given in Display 1 with the returns indicated in “life units”:16 Since the top alternative in this display has an infinite expectation (regardless of the magnitude of n), while its rival has the uniform expectation of a single unit, betting on God is clearly the best plan. “There is no need to hesitate; you must risk all.” The lesson is simple. The reasoning at issue in the Wager argument is merely an instance of the standard process of probabilistic decision on the basis of expected-value calculations. Q. E. D. Pascal’s reference to the prospect of gaining many lives is in one way misleading. The difference in value relates not merely to temporal extent (finite vs. infinite lifespan), but to the “quality of life” reflected in the great gulf which separates travail in this “vale of tears” from infinite bliss. (The

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Display 2 THE STRUCTURE OF CHOICE IN PASCAL’S WAGER ARGUMENT Wager that God exists, and put a finite amount B at stake

p

God exists

+∞ -B

1-p

God does not exist

-B

p

God exists

Little or nothing (or even something negative)

1-p

God does not exist

0

Wager that God does not exist (at cost 0)

value of infinitely extended mundane life might well not be infinite— repetition is boring.) Critics of Pascal are often misled by his supposition of “equal chances of winning and losing”. They rightly protest against any (clearly fanciful) use of the principle of indifference to arrive at one-half as representing the chance of God’s existence.17 But this objection is entirely misdirected against Pascal. His reliance on the 50:50-case is only illustrative and methodological. It plays no substantive role whatsoever in the actual argument. Nowhere does Pascal say that the probability of God’s existence is one-half, and nothing in his argumentation requires it. All that matters for his reasoning is that this probability be nonzero. As long as there is a finite chance of God’s existence—no matter how small—the expectation of the “bet-and-believe” alternative outweighs that of its rival. The salient fact, then, is that Pascal’s discussion at this fourth stage is simply concerned to make the point that the reasoning of the Wager can easily and smoothly be accommodated by the standard mechanisms of probabilistic decision theory.18 In its full generality the choice situation contemplated in Pascal’s Wager argument takes the form set out in Display 2. His reasoning pivots on the fact that the expected value of the top alternative—the sum of the several prospective gains/losses proportioned by the probability of their realization—is going to be +∞ in any event. As long as the probability P of God’s existence is nonzero, and the size of the stake is B finite, the top alternative is bound to prevail in a comparison of

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expectations: regardless of B’s specific size it produces the maximum net expected gain. This recourse of expectations brings the central role of uncertainty to the fore. Pascal’s reasoning is only in a position to persuade someone who believes that God may exist.19 If the issue could be settled one way or another by theoretical argumentation (so that we know that P = 0 or P = 1), then this entire process of probabilistic argumentation would become unnecessary and pointless. The viability of this whole recourse to decisiontheoretic argumentation rests on a certain incapacity, a certain ignorance. And just this ignorance, so Pascal insists, is an inherent aspect of the human condition. 5. But what can be said of the objection that in taking the gamble at issue in the Wager argument we risk a certain loss (the comforts of a self-centered and worldly life) against an uncertain gain (divine rewards). The answer is that this is an inevitable and pervasive feature of all gambling situations (lotteries, insurance, etc.). Here we standardly use mathematical expectations as our guide. And accordingly, Pascal reasons, as long as we are guided by the established rational gambler’s standard of expected returns, we are bound to bet on God’s existence. For a gamble is advantageous on the basis of this expected-value standard whenever: chance of winning cost of stake potential loss > = chance of losing size of prize potential gain And if the potential gain is infinite, this standard favors the gamble as long as the chance of winning is nonzero. To judge matters on this basis is—so Pascal insists—entirely fitting and proper. It simply applies to the particular case at hand the general principles of probabilistic decision theory that govern the rational resolution of practical choices in conditions of uncertainty. 6. The argument, of course, only shows what the rational man should do—that he would be well-advised (self-interestedly speaking) to become a believer. But what if one cannot manage it? What if rationality alone does not suffice to move someone? Then he can simply take various practical steps to induce himself to go whither reason points. The inducing of belief itself is a technical matter in the psychology

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of self-persuasion and not necessarily a matter of reasoning at all. The Wager argument as such—that is, as an argument—does not touch it. The job of the Wager argument is simply to establish that belief is rationally warranted. We not step outside the province of the argument as such into the region of effective praxis—of practical steps needed to give effect to its deliberations. At this point we must supplement reasoned reflection with the psychology of self-management, for a person is not only a thinking being but an animal as well, and thus (as Descartes teaches) has a mechanical side.20 Habit can lead the mind where it is unwilling to go of itself: a reluctant person can be made to go where reason dictates, can be trained and rendered docile (“vous abêtira”). Pascal realized perfectly well that belief as such is something that lies beyond volition and cannot be constrained. But he stresses that once we recognize that belief is rationally warranted, we can and should set out to induce it. “I am so made that I cannot believe (i.e. cannot constrain belief). What then would you have me do?” The hypothetical respondent here concedes the weight of the Wager argument and grants its conclusion that belief is the reasonable thing. He says, “I admit that’s what I ought to do, but I really don’t think I can get myself to do it.” Pascal’s reply here is the recommendation to submit, that you can cure unbelief as others have done, by acting as if you believe until you finally come to do so: “Follow the way by which they begin . . . taking holy water, having masses, etc.”—and, above all, by joining a community of believers.21 Custom is man’s very nature and can lead him to belief where all else fails. This point is important. It is often objected against Pascal that it makes no sense to wager with belief. For, as philosophers from Hume onwards insist,22 belief is not something at our disposal: whether or not we actually believe something does not lie within our control. Is this not fatal to Pascal’s conclusion? In response to this abjection it is sometimes replied that while we cannot control what we believe, we can control what we try to believe or to come to believe (or else what we accept as a basis for action.)23 But this inherently plausible line is beside the point. For while the objection that belief as such does not lie in our control is perfectly true, it is entirely irrelevant to the thrust of Pascal’s argument. The argument does not issue in the injunction “Believe P!” but in the recommendatory finding “It would be well-advised for you to believe P.” Its concern is not directed at belief

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per se at all. The pivotal issue is: What is rational for someone to believe? The argument simply maintains that a certain belief is reasonable (appropriate, rationally warranted). The fact that someone might (through foolishness or weakness of will) fail to believe something which he recognizes as rationally deserving of credence is entirely irrelevant and immaterial to the argument as such. In sum, it is not the task of the Wager argument to constrain actual belief (no mere argument could achieve that!). Nor is it to motivate a course of action “as if” one believed. If practice rather than faith constituted the sum total of our religiosity, belief would be beside the point and hypocrisy would rule the roost. (Merely to act “as if” a religion were true is, quite obviously, not really to accept that religion at all.) Rather, the Wager argument’s task is to show that belief is rationally warranted—in the specifically prudential (not evidential!) mode of rational warrant. For the Wager argument unabashedly portrays faith as a means to an end. The end we hope to attain is represented by the Christian idea of salvation in an afterlife. The means necessary for attaining this end, at least according to that conception of the God which is a “live option” for us, involves a life of Christian piety and religious practice. Since faith is an integral component in such a life, its pragmatic justification follows. 5. LIMITATIONS OF THE WAGER ARGUMENT It emerges from this account of the matter that the Wager argument is subject to certain limitations (of which Pascal himself was doubtless well aware). 1. It will certainly fail to touch the convinced atheist. Someone who sets the probability of God’s existence at zero will obviously not arrive at the argument’s conclusion.24 2. Nor does the argument reach the all-out hedonist who lives for the pleasure of the moment alone. Someone who lets the future look after itself or who is prepared to dismiss future benefits entirely—who (extending the lines of Daniel Bernoulli’s resolution of the St. Petersburg paradox) is prepared to set the goods and evils of the next world at nought—would also remain untouched by the argument.

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3. The argument will also have no impact on the all trusting disbeliever. It is bound to leave untouched the person who, while believing that God does not exist, thinks that he is bound to be all-forgiving if (contra factum) he nevertheless did—and is therefore convinced that God, did he exist, would respond to disbelief and disobedience no differently from their opposites and would thus recompense disbelief no less amply than belief. Belief cannot be recommended on the basis of interest to someone who concedes it no possibility of advantage. 4. The argument does not deal with the radical skepticism of Pyrrhonian philosophy: the theorist who denies not only knowledge (epistêmê) but reasonable conviction (to pithanon) as well. For, like any other argument, it proceeds from premises and is accordingly impotent to enjoin its conclusion on someone who does not accept them. Unless we have some views about the nature of God (for example, believe that, should he exist, he will preferentially reward those who believe in him), Pascal’s reasonings will leave us untouched. It is a grave mistake to think of the argument as addressed to the all-out philosophical skeptic. Pascal’s skepticism clearly has its limits in this context. Precisely because he thinks that cognitive, theoretical reasons cannot yield knowledge that is useful for apologetic purposes, he is prepared to fall back on practical, interestoriented reason. Pascal’s skepticism is on the sort envisioned in the Middle Academy: the fact that theoretical knowledge is unavailable being offset by the availability of reasonable conviction of just the sort by which we manage the affairs of everyday life.25 5. Again, the argument carries no weight with someone who disdains the whole process of expected-value calculation and rejects the idea of letting the probability of profit afford “a guide to life.” The argument will only reach the prudently self-interested man who is prepared to proceed “calculatingly” in matters of self-interest in line with standard decision-theoretic principles. 6. Finally, the argument is addressed not to those who incline to rival theologies (Zoroastrian, Buddhist, etc.) that have very different ideas about the rewards of belief, but to the ordinary indifferent, noncaring Christian-in-name-only (if that). It is only in a position to reach

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someone who conceives of God in a particular way, and it is not addressed to those who have nonstandard ideas about the nature of God. The Wager argument, like any other, rests on particular premises—in its case, certain suppositions about the nature of God and his possibility. And it lies in the very nature of things that no argument, however cogent, can exert rational constraint on those who do not accept its premises. This is simply a fact of life and nowise a defect or limitation of the Wager argument. ***** Since the days of Lucretius the following sort of reasoning has been advanced by antitheists: Disbelief is actually beneficial. For rejection of god, and supernatural beings in general, squarely puts the responsibility for our human concerns on us humans, forcing us to come to grips with our problems.

We have here an interesting inversion of Pascal’s line of thought—a theological argument from interest, to be sure, but on the side of disbelief rather than belief. In principle this sort of table-turning is fair enough. But this particular argument turns out to be specious—against Christianity at any rate. For the Christian God neither meddles in human affairs in the manner of Greek gods on Mount Olympus not manages them altogether in the manner of the fate decreed by an oriental-style God-potentate. He neither creates our problems nor relieves us of the task of grappling with them. Of course, where the God at issue is conceived of in a way substantially different from that of standard Christianity, the Wager argument can make no impact. Here it is necessary to bear the apologetic purposes of the Wager argument in mind. Pascal’s discussion is directed at l’homme moyen sensuel, the ordinary, self-centered “man of the world” preoccupied with his own well-being and his own prudential interests. Pascal does not address the already converted, but the glib worldly cynic—the free-thinking libertin of his day, the sort of persons who populated the social circle in which Pascal himself moved prior to his conversion.26 The format of the discussion is that of a dialogue with just such a person. And it is part of the tacit ground rules of the discussion that there is to be no appeal to faith, to religious experiences, to authority—to any evidence that goes beyond man’s “natural

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light.” Pascal’s concern is thus with the rational justification of an action through an appeal to self-interest. The aim of Pascal’s Wager argument is one of apologetics and not of theological theorizing. And, even here, it is a special-purpose instrument with a limited and special mission—to stiffen the backbone of the slack and worldly Christian. The convinced atheist, the radical philosophical skeptic, the all-out hedonist, the all-trusting disbeliever, and those otherwise predisposed toward alien theological systems will not (and are not intended to) be reached by this mode of argumentation. These other battles of apologetics must be fought on other fronts with other weapons. In view of its status as an appeal to self-interest, many commentators have felt that Pascal’s Wager argument is altogether beneath the dignity of serious religiosity and unworthy of a religious thinker deserving of this name. But this line of objection loses sight of the job that Pascal actually intends the argument to do. His deliberations are intended to motivate l’homme moyen sensuel into making a start at religious faith, and his pivotal question is this: What line of thought could prove effective in bringing the nominally Christian but actually slack, indifferent, and worldly outsider into the fold of believers? If we fail to reckon with the apologetic purpose of the argument as delineated in these terms, we will not be in a position to evaluate it appropriately. The standard supporting pillars of the edifice of Christian belief are much as Display 3 indicates. Most of these have serious drawbacks from the standpoint of apologetics. Revelation generally speaks only to the already converted. Most ordinary men do not have the right sort of experience: personal religious experience exerts its impetus only on the select. The church’s testimony leaves the “outsider” cold. Cultural habituation often points a contrary way. And while it might seem that rational demonstration is the best prospect given the universal cogency of its appeal, it too has its problems. For it is precisely when the argumentation is at its most tight and rigorous that it cannot lead beyond the reach of conceded premises—whose availability cannot be supposed in the present context. As Pascal saw it, practical reasoning prevails in apologetics faute de mieux, for want of a better alternative. Following Montaigne and the Renaissance tradition of Christian skepticism, Pascal; insists that the sorts of proof and demonstrations by which scholastic thinkers and their congeners sought to validate actual knowledge (epistêmê) in the best Platonic and Stoic manner are simply not available for apologetic purposes. Indeed, such demonstrative arguments are not

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Display 3 ROUTES TO THE LEGITIMATION OF FAITH REASON • Cognitive/evidential reasoning; rational demonstration (the reasonings of the Church Fathers, the scholastic doctors, Descartes) • Practical reasoning CUSTOM • Group traditions: the collective testimony of the church: the “cloud of witnesses” in the history of ongoing commitment (the attestation of the living church and the witness of the faithful) • Cultural habituation in education and acculturation EXPERIENCE • Personal experience (inspiration): the “call” of the prophets, the insight of the mystics, or the believers contact with God in prayer • Revelation (as transmitted vicariously through the prophets of scriptures)

even desirable, because the features of God accessible to demonstrative reason alone could not suffice to enable him to answer to the needs of the human condition. In the setting of apologetics we would not even want a demonstrable God if we could have one, because the “God of the philosophers” cannot speak to the condition of the plain man. This line of thought impels Pascal into giving his apologetics a pragmatic cast. For Pascal a leading positive attraction of the Wager argument is thus its negative aspect: its turning away from outright demonstration and its abandonment of any claims to purely rational compulsion. In his view an apologetically effective argument cannot and must not even seek to demonstrate. Rather, it should motivate in the practical order of things. And this shift from cognition to praxis is the pivot of the Wager argument. 6. THE IMPORT OF THE WAGER ARGUMENT: THE PROBABILISTIC TURN Theological argumentation from self-interest with a view to the prospect of gain or loss was of course not invented by Pascal. Homilists had been

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going on in this vein for centuries. After all, if even Paris vaut bien la messe, how can one balk at the prospect of attaining heaven? Again, the idea of prudent choice under uncertainty was not really new in this context. A century before Pascal, St. Thomas More told the anecdote of the courtier and the poor cleric: And it fareth betwene these two kynde of folke as it fared betwene a lewde galante and a pore frere. Whom when the galante sam going barefote in a great frost and snowe, he asked him why he did take such payn. And he aunswered that it was very little payn, if a man would remember hell. Ye frere, quoth the galante, but what and there be none hell? Then art thou a great foole. Ye maister, quoth the frere, but what and there be hell? Then is youre maistershyppe a much more foole?27

Already the Christian theologian Arnobius, who flourished in the reign of Diocletian (284–305), wrote in his apologetic tract Adversus gentes as follows: [T]here can be no proof of things still in the future. Since then, the nature of the future is such that it cannot be grasped and comprehended by any anticipation, is it not more rational, of two things uncertain and hanging in doubtful suspense, rather to believe that which carries with it some hopes (quod aliquas spec ferat) than that which brings none at all? For in the one case there is no dander if that which is said to be at hand should prove vain and groundless; in the other there is the greatest loss, even the loss of salvation, if, when the time has come, it eventuates that our expectation was erroneous.28

Prudentialism in the face of risk and uncertainty, then, is nothing new in religious discourse. But here is an important element in Pascal’s argumentation that was wholly new and original: the concept of a measured gamble, a probabilistically calculated risk. Such a deployment of probabilistic reasoning with its use of the decision-theorectic tactic of guiding action on “the balance of probabilities” was something quite novel. This innovative recourse to gambling, probability, and doctrine of chances—to that decision-theoretic employment of the calculus of probabilities which was Pascal’s own invention29—illustrates yet again the common phenomenon of bringing new intellectual tools to bear on old problems that are encountered in all departments of intellectual endeavor.

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In this regard John Locke’s reasoning goes badly awry. Locke argues in Pascal-reminiscent style that a life of virtue (rather than religiosity) is preferable under a bare possibility of an afterlife: He that will allow exquisite and endless happiness to be but the possible consequence of a good life here and the contrary state the possible reward of a bad one . . . must conclude [that a virtuous life is to be adopted]. I have foreborne to mention anything of the certainty of probability of a future state, designing here to show the wrong judgment that anyone must allow he makes . . . who prefers the short pleasures of a vicious life upon any consideration whilst he knows and cannot but be certain, that a future life is at least possible.30

As Locke sees it, the possibility of an infinite reward outweighs the possibility of a finite one, wholly apart from any probabilistic considerations. The view seems to be that if only the stake is big enough, one should go ahead and “take the chance and go for broke”—probabilities need not enter in. But whatever the merits (or demerits) of this view, it is not the position that is at issue in Pascal’s deliberations. For him the probabilistic aspect is crucial. Many treatments of the Wager argument evince discomfort about any recourse to mathematics—Pascal’s included. As one recent commentator puts it: By his introduction of mathematics into the argument Pascal added the subtilty and curiosity of the argumentation, but it may be questioned whether he has thereby added to its form and value. . . . [T]o many his mathematics can be a factor that detracts from the persuasive and hortatory character of the whole. The mathematicizing is too elaborate and too subtle to add strength to the appeal that the argument is designed to make.31

One wonders where this author got his information about “the appeal that the argument is designed to make.” Surely not from Pascal’s discussion itself, which makes it clear as crystal that the argument is not designed to be popular, to appeal to “ the ordinary common sense of common folk.” It is addressed precisely to the man who is clever and calculating, who fancies himself a shrewd person who stands above and beyond the reach of popular homiletics. The mathematical character of the argument as an exercise in decision theory is integral to the function that Pascal intends for it.

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Some critics object that Pascal’s argument runs into trouble at exactly this point of conjuring with probabilities. Thus Anthony Flew writes: Yet this apparent force [of the Wager argument] depends on a gigantic assumption, concealed and false. For it assumes that the tally of possible mutually exclusive, Hell-threatening systems is finite. . . . But this is parochial and unimaginative. . . . [And if there are infinitely many alternatives] we cannot make a reasoned bet. But with regard to the transcendent there is no limit to the range. No odds can be given. The whole idea of betting breaks down.32

But this objection hits wide of the mark. Pascal has no need for calculating the numerical value for the probability of God’s existence from some body of raw data regarding possible alternatives. His invocation of probability here is subjectivistic, not combinatory (and is, moreover, independent of specifically quantitative valuation). For the matter turns on the way in which the Wager argument’s addressee himself evaluates the prospect of God’s existence.33 Pascal’s argument is simply addressed to those who see the existence of the Christian God and a real possibility to which they are prepared to accord a nonzero probability (however “imponderable” they may deem this quantity to be in other regards). Apart from this the numerical status of this probability—even its having a definite and stable value— is quite irrelevant. And this fact has important ramifications. Hume held that we cannot accept the testimony of an interested witness, and Laplace decked this view out with mathematical argumentation. If the witness has a great personal stake on the side of the question his testimony supports, the temptation for him to lie is great—indeed asymptotically infinite. And so the biblical stories transmitted by the earliest Christians are doubtful in the extreme—they lack even a finite probability. Laplace saw this point as fatal to Pascal’s position34—and so indeed it might be if it established that the existence of the Christian God is of probability zero.35 But Laplace’s view is of questionable soundness. From the two premises • X claims that he has good evidence for maintaining P (e.g., that he saw this with his own eyes) and

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• X lies; he actually has no good evidence but spoke from interest alone nothing whatsoever follows regarding the probability of P itself. The probability of the substance of a claim is wholly unaffected by the fact that those who make the claim have not adequate warrant for it and are actually ignorant in the matter at hand. I ask Smith whether heads will come up on the next toss of coin. He replies brazenly, “I know that it won’t—I have infallible foresight in these matters.” Let it be granted that he is lying through his teeth. Does that make the probability of heads zero?36 The Laplacean argument turns on the equivocation of “lie” = speak falsely as to the FACT of one’s claims, on the one hand, and “lie” = speak falsely as to the JUSTIFICATION of one’s claims, on the other. The circumstance that an interested witness may well “lie” in this second sense may plausibly disincline us from accepting his testimony. But it does nothing whatever to establish the falsity—or improbability—of his claims. (We can get no further then the Scot’s verdict, “not proven.”) In arguing against Pascal’s wager, Laplace in effect casts the church in the role of a set of interested witnesses who say: “Pay the price P and I can promise you on God’s behalf that he will give you a great reward R.”37 Laplace now argues as follows. If we do as the witnesses say, we confront the pay-off situation The witnesses speak truly (probability p) R–P

The witnesses speak falsely (probability 1 – p) –P

The expected value of this action will now be EV = p(R – P) + (1 – p)(–P) = pR – P Pascal tries to assure that this is infinite by setting R at +∞. But this, Laplace now insists, is a mistake. For P and R should be presumed to be interdependent: the bigger the reward R the witnesses promise us, the less, so we are to suppose, is the likelihood that they speak truly. Suppose that, in fact,

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1

pR = constant = 2 P Then the expectation at issue will in fact be uniformly negative; it will be 1

– 2 P. Laplace’s counterargument is fine as far as it goes. But it does not really tell against Pascal’s position. For it pivots crucially on the “fish-story Principle”: the bigger the claim, the less likely its truth. Now this may be very well in its place, but its place is not in theology. It just won’t do to say that gods who promise less are ipso facto more likely. The apologetic context at issue will have to define the parameters of the operative god-conception, and it is this that must set the stage of our deliberations. Issues regarding the creditability of witnesses are beside the point. Pascal certainly does not base his view that the probability of God’s existence is nonzero on the testimony of the early Christians. On the contrary, he insists that such testimony is effectively unavailable for apologetic purposes. His own starting point is skepticism—we must suppose that nothing apologetically useful can be established one way or the other through evidence and factual deliberation. We must simply implement our subjective inclination to acknowledge that the possibility exists. We must, in the final analysis, come to terms with our own ideas about God. This aleatory aspect of Pascal’s innovation was no less shocking and offensive to contemporary ideology and received opinion than its prudential orientation. Cries of outrage had greeted Thomas Gataker’s 1616 essay “On the Nature and Use of Lots” which defended the lawfulness of lots when not used for divination, and which led to violent attacks on its author as an advocate for unsavory games of hazard.38 Even a century after the Wager argument was propounded, Voltaire (of all people!) objected to Pascal’s infini-rien discussion: “This article seems a little indecent and puerile; the idea of a game, and of loss and gain, does not benefit the gravity of the subject.”39 Gambling was deemed a distinctly disreputable activity in those times. Pascal’s friend, the clever but somewhat shady Chevalier de Méré, was deplored by the pious as a typical specimen of the wordly libertin: “brilliant talker, fearless freethinker, and inveterate gambler.”40 Yet it is exactly this sort of person that Pascal set out to address in the Wager argument. The choice of his intended audience represented a departure altogether characteristic of Pascal’s view of the human condition. As he saw it, man is not respectable—his motivation is base, his thought calculating. Effec-

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tive apologetics must begin by moving down to man’s own natural level; it must reach him in the base region that is his natural habitat. To do effectively the job that needs to be done, it must fight fire with fire. (To be sure, this is not the end of the matter; never for an instant did Pascal deem this appeal to interest more than the start of a much longer and ultimately very different story.) Pascal’s Wager is an appeal neither to man’s scientific prowess (to competence at factual inquiry) nor to his “better self” (to human idealism), but to rational self-interest. As he saw it, the religious journey will no doubt eventually lead to a certainty of higher principle but can (and for many must) begin in the moral certitude of man’s practical affairs. The pragmatic tradition in philosophical theology thus received a powerful impetus in Pascal’s turning from theoretical/demonstrative argumentation to practical reasoning. And the course of historical events that led from Kant through James to later pragmatism was to assure that this turning was not just a haphazard idiosyncrasy, but a permanent gain for philosophy. In its light we may see Pascal not as an isolated eccentric, but as a key link in a long chain of praxis-oriented thinking that runs from the mitigated skeptics of the Middle Academy through Kant and William James to parts of the amply flowing stream of contemporary thought in the pragmatic tradition.41 NOTES 1

The passage is numbered as section 343 in Lafuma’s Delmas edition of 1952 and is numbered 233 in Léon Brunschvicg’s earlier edition (Paris: Hachette, 1914). The original text is now in the Bibliothèque Nationale in Paris. A photograph of the MS is given in George Brunet, Le Pari de Pascal (Paris, 1956), in Lafuma’s 1962 tercentenary edition, and in Henri Gouhier, Blaise Pascal: Commentaires (Paris, 1971). There has been much indecisive speculation regarding the time and circumstances of this draft.

2

Principal John Tulloch, Pascal (Edinburgh and London, 1878), p. 192. He insists that “it is impossible to defend this essay on any principle of sound philosophy” and maintains that the argument “was hardly worthy of Pascal” (p. 193).

3

Terence Penelhum, God and Scepticism (Dordrecht, Boston, Lancaster; 1983). This book is virtually unique in treating the argument with respect.

4

René Descartes, Meditations on First Philosophy, no. 4. cf. Descartes: Philosophical Writings, tr. by E. Anscombe and P. T. Geach (London, 1954), p. 93.

5

Pensées 381/543. While this passage appears in all editions of the Pensées remote from the Wager discussion, it is one of four jottings written on the same paper

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NOTES

used by Pascal for the “infini-rien” fragment. See Georges Brunet, Le Pari de Pascal (Paris, 1956), pp. 48–51. 6

Pensées, 727/280.

7

Pensées, 343/233. To “incapable of knowing” here, we must add “by way of human inquiring reason.” Knowledge by way of experience is not to be precluded. After the miracle of the Holy Thorn in 1654, Pascal adopted the motto Scio cui credidi, “I (now) know him in whom I (heretofore merely) believe.” See H. F. Stewart, “Blaise Pascal,” Proceeding of the British Academy, vol. 28 (1942), pp. 196–215 (see p. 203).

8

Michel de Montaigne, “Apologic de Raimond Sebond” in P. Villey (ed.), Les Essais de Michel de Montaigne, tome 2 (Paris, 1922), pp. 367–71.

9

Pascal’s supposition of an equal chance of gain and loss is simply illustrative. It is emphatically not an invocation of the Laplacian principle of indifference to establish equiprobability between the two possibilities of God’s existence and God’s nonexistence. The particular value at issue is in fact entirely irrelevant to the outcome of the argument. (Compare notes 17, 18, and 29 below.)

10

The third session of the Vatican Council of 1869–70 was to reiterate this longstanding rejection in the canon De Revelatione: “If anyone says that one true God, our creator and Lord, cannot be known for certain by the natural light of human reason through the things which are made: let him be cast out of the Church.” (Denzinger, sect. 1806)

11

Indeed, reason is man’s noblest feature; though but a reed he is a thinking reed. Pensées, 391/347. Man’s dignity and greatness consists in thought (ibid., 126/146 and 232–33/365–66).

12

Compare Pensées 9/276. For Pascal’s many-faceted conception of rational demonstration see chap. 1, “Proof and Proofs,” in Hugh M. Davidson, The Origins of Certainty: Means and Meanings in Pascal’s Pensées (Chicago and London, 1979). Regarding that section of the theological tradition which urged not that reason is dispensable, but (with Pascal) that faith must bring grist to reason’s mill before reason can accomplish any useful work in theology, see Richard H. Popkin, The History of Skepticism from Erasmus to Spinoza (Berkeley and Los Angeles, 1979).

13

See William James, The Will to Believe and Other Essays in Popular Philosophy (New York and London, 1896), pp. 2–3. The issues that arise here are usefully canvassed in Marcus G. Singer, “The Pragmatic Use of Language and the Will to

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NOTES

Believe,” American Philosophical Quarterly, vol. 8 (1971), pp. 24–34, and in Richard M. Gale, “Williams James and the Ethics of Belief,” ibid., vol. 17 (1980), pp. 1–14. 14

Pensées, sect. 343/233. This is the core of Pascal’s skepticism: in this apologetic domain we are in a terrain where theoretical, inquiring reason (Kant’s “speculative” reason) cannot resolve our questions by providing properly authenticated knowledge.

15

This idea is foreshadowed in the Mémorial Pascal wrote concerning the inspirational religious experience he had on the night of November 2, 1654: “Total submission to Jesus Christ and my director/Everlasting joy in return for one day’s effort on the earth.” (The passage may, however, be a later addendum.)

16

This model reflects the situation as described in the notes on pp. 147–50 of L. Brunschvicg’s edition of the Pensées.

17

“But then we are brought back to our original and persistent question: Are the existence and the non-existence of God equally balanced in probability as far as our knowledge goes—like the two sides of a newly minted coin?” Monroe and Elizabeth Beardsley, Philosophical Thinking: An Introduction (New York, 1965), p. 140.

18

On this point our analysis differs decisively from that of Ian Hacking, who sees this section of the infini-rien discussion as developing a further “argument for dominating expectation,” because the earlier argument somehow depends on the obviously problematic supposition that the probability of God’s existence is onehalf. See his essay “The Logic of Pascal’s Wager,” American Philosophical Quarterly, vol. 9 (1972), pp. 186–92 (see p. 189). The same discussion recurs in The Emergence of Probability (Cambridge, 1975), pp. 63–72. (Compare notes 11 and 30.)

19

This is what nonzero subjective probability comes to here: accepting with substantial confidence that God may possibly exist. It is not a matter of accepting with low confidence that God does in fact exist.

20

“[W]e must not lose sight of the fact that we are automatic as well as intelligent beings. . . . Proofs only convince the mind. Custom is for us the strongest and most readily accepted proof: it sways the automatic, which bears the unthinking mind along with it.” (Pensées, 7/252). On Pascal’s view of the “automatic” dimension of belief and in the aspect of man as “la machine” see chap. 3, “Fixation of Belief” of Hugh M. Davidson, The Origins of Certainty (Chicago and London, 1979).

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NOTES 21

Pensées, 343/233. Terence Penelhum makes an important point in this connection. The task of those religious practices Pascal recommends is to engender faith. He is not under the illusion that they constitute it: Pascal does not say that the practices which one enters in pursuit of faith themselves constitute faith. They are merely those activities that someone who has faith engages in, and he proposes that someone who wishes to be of their numbers should engage in them as well in order to acquire it. The enterprise he proposes is only successful if real faith actually ensues. Believing as he does that faith is a gift of grace, he implies that someone seeking to acquire it may be granted it if he makes efforts to remove some of its obstacles. It seems to me theologically unacceptable for Christian thinkers to insist against him that someone wanting faith, even for prudential reasons, would be denied it by God. (God and Scepticism [Dordrecht, Boston, Lancaster, 1983], pp. 72–77.)

22

David Hume, A Treatise of Human Nature, appendix: p. 624 in the edition by L. A. Selby-Bigge (Oxford, 1888).

23

R. G. Swinburne, “The Christian Wager,” Religious Studies, vol. 4 (1969), pp. 217–28 (see p. 222). That our beliefs are not within our control is argued at length in Bernard Williams, “Deciding to Believe,” in Problems of the Self (Cambridge, 1973), pp. 136–51. That we can embark on courses of action that are likely to induce certain beliefs is persuasively maintained H. H. Price, Belief and Will,” Proceedings of the Aristotelian Society, Supplementary Volume 28 (London, 1954), pp. 1–26.

24

Many writers object that the Wager argument is defective because it supposedly requires Pascal to have recourse to a Laplacian principle of indifference in establishing that the existence of God is a possibility of nonzero probability. See, for example: James Cargile, “Pascal’s Wager,” Philosophy, vol. 41 (1966), pp. 250– 57 (see p. 250). Peter C. Dalton, “Pascal’s Wager,” The Southern Journal of Philosophy, vol. 13 (975), pp. 31–46 (see p. 40). But this criticism misfires. The argument neither can nor endeavors to convince those who do not see God’s existence as a real possibility. Someone for whom the Christian God is a dead hypothesis (in William Jame’s sense) is going to be untouched by Pascal’s Wager.

25

These issues are treated in greater detail in the author’s Scepticism (Oxford, 1980).

26

See M. L. Goldman, “Le Pari, est-il ecrit ‘Pour le libertin’?” in Blaise Pascal: L’Homme et l’oeuvre (Paris, 1956; Cahiers de Royaumont), sect. 4. Here it is maintained that the Wager argument is aimed not at the skeptic who is satisfied with life in this world, but at those who are alive to the misery of man’s lot and the

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NOTES

anguish of the human condition. As Pascal sees it, however, this includes virtually everybody, since however smug and self-sufficient we may be, everyone has some share of existential Angst, of deep-rooted doubt about the ultimate value of what life in this world offers to man, seeing that the inevitable terminus of all our efforts and strivings is the grave. 27

Thomas More, “The Supplication of Soules” in The Workes of Sir Thomas More (London, 1557), p. 329; quoted in John K. Ryan, “The Argument of the Wager in Pascal and Others,” The New Scholasticism, vol. 19 (1945), pp. 233–350 (see p. 328).

28

Arnobius, Against the Heathen (Adversus gentes), II, 3–4; Ante-Niocine Fathers, tr. by H. Bryce and H. Campbell (New York, 1907), vol. 5, p. 434. Pierre Bayle already remarked on the kinship of Pascal’s line of thought with that of Arnobius. For a discussion of similar paralles see Jules Lachelier, “Notes sur le poari de Pascal” appended to his Du Fondement de l’induction (Paris, 1907), pp. 175–208, and Louis Blanchet, “L’Attitude réligieuse de Jesuites et les sources du pari de Pascal,” Revue de Métaphysique et de morale, vol. 24 (1919), pp. 477–516 and 617– 47. Precursors of Pascal are also discussed in George Brunet, Le Pari de Pascal (Paris, 1956), p. 62. For precedents in Islamic philosophical theology see Miguel Asin Palocios, Los Precedentes Musulmanes del Pari de Pascal (Stantander, 1921). Such precedents, however, only relate to the general (and old) idea of using prudence as a prop of faith. The characteristic features of the Wager argument in its reliance on probability and decision theory are altogether new. Thus Antony Flew is quite wrong in saying apropos of Asin Palacios that “Scholarship has now apparently traced the origin of the (Wager) argument back to the Islamic apologist Algazel” (Is Pascal’s Wager the Only Safe Bet?” The Rationalist Annual, vol. 76 (1960), pp. 21–25 [see p. 21]).

29

On Pascal’s contribution into the theory of probability see Isaac Todhunter, A History of Mathematical Theory of Probability (London and Cambridge, 1865) and Oystein Ore, “Pascal and the Origination of Probability theory,” American Mathmatical Monthly, vol. 67 (1960), pp. 409–19. See also Jean Mesnard, Pascal et les Roannez (Bruges, 1965). Todhunter and Ore do not, however, take any notice of the Wager argument. But it is treated at length in Ian Hacking, The Emergence of Probability (Cambridge, 1975).

30

John Locke, Essay Concerning Human Understanding, bk. II, chap. 21, sect. 70.

31

John K. Ryan, “The argument of the Wager is Pascal and Others,” New Scholasticism, vol. 19 (1945), pp. 233–50 (see p. 247).

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NOTES 32

Antony Flew, “Is Pascal’s Wager the Only Safe Bet? The Rationalist Annual, vol. 76 (1960), pp. 21–25 (see pp. 24–25); reprinted in revised form in God, Freedom, and Immortality (Buffalo, 1984), pp. 61–68 (see p. 68).

33

For an informative account of the basic issues regarding various modes of probability and their uses see Wesley C. Salmon, The Foundations of Scientific Inference (Pittsburgh, 1967).

34

P. S. de Laplace, Essaie Philosophique sur les probabilités, in Oeuvres, vol. 11 (Paris, 1886); A Philosophical Essay on Probabilities, tr. by E.W. Truscott and F. L. Emory (New York, 1948), see chap. 11, pp. 109–25.

35

This disagrees with the view of Ian Hacking, who dismissively comments: “Curiously, Laplace take . . . (his argument) to refute Pascal.” (“The Logic of Pascal’s Wager,” American Philosophical Quarterly, vol. 9 (1972), pp. 186–92 (see p. 192).) If Laplace’s point were correct and his analysis did indeed establish that P, the probability that the Christian God exists, were zero, then Pascal’s argument would indeed be in difficulty.

36

Its being reported by a highly unreliable source tells us effectively nothing about the probability of the claimed fact itself. An unreliable source is of course not one that is contra-indicative (i.e., that reliably speaks falsehood), but one that is irresponsible—that speaks truth and falsehood more or less randomly. For such a course we only know that the proportion of (relevantly comparable) cases where it “knows what it is talking about” (k) is small. Accordingly, the a posteriori probability that a statement P (of the relevant class) that is attested by our source is true is: k • 1 + (1-k) • pr(P) + k • pr(not-P). (We suppose the source is right within the range of its knowledge, and only meet with randomly average success elsewhere.) From the fact that k is small we can thus infer no more than our course’s inability to augment the a priori probability of its statements by any substantial amount.

37

P. S. de Laplace, A Philosophical Essay on Probabilities, (op. cit.), chap. 11, pp. 109–25 (see esp. pp. 120–22).

38

For Gataker (1574–1654) see the DNB, vol. 21 (London, 1919; 2nd ed., 1927).

39

F. M. A. de Voltaire, “Remarques sur ls Pensées de M. Pascal.” Letter XXV (1728) of Lettres Philosophiques par M. de V*** (Amsterdam, 1734). For a reprint and critical study see Gustave Larson (ed.), Voltaire: Lettres Philosophiques, 2 vols. (Paris, 1916 and 1917 [3rd ed. 1929]).

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NOTES 40

H. F. Stewart, “Blaise Pascal,” Proceedings of the British Acadamy, vol. 28 (1942), pp. 196–215 (see p. 204). For Méré’s role in the origins of probability theory see Ian Hacking, The Emergence of Probability (Cambridge, 1975).

41

This essay is a revised version of the opening chapter of my Pascal’s Wager (University of Notre Dame Press, 1985).

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Chapter 5 LEIBNIZ ON COORDINATING EPISTEMOLOGY AND ONTOLOGY 1. LINGUISTIC FINITUDE

I

n 1693 Leibniz launched into a series of studies of issues of eternal recurrence with a draft that was submitted to the Académie des Sciences in Paris and sent to its president, the abbé Bignon. Under the heading of “palingenesis” or apokatastasis1 these studies afford a highly instructive insight about Leibniz’s view of the human condition in its cognitive aspects, and specifically with regard to the limits of human knowledge.2 For Leibniz, it is a salient aspect of cognitively capable beings that they are symbol users and that their knowledge of matters of fact (in contrast to their perceptual phenomenology and their performatory, how-to knowledge) is unavoidably mediated by language. Whatever we factually know is—or can be—put into words. And what is put into words can be put into print. This circumstance reflects—and imposes—certain crucial limitations. As Leibniz put it: Or puisque toutes les connaissances humaines se peuvent exprimer par les letters de l’Alphabet, et qu’on peut dire que celuy qui entend parfaitement l’usage de l’alphabet, sçait tout; il s’en suit, qu’on pourra calculer le nombre des verités dont les hommes sont capables et qu’on peut determiner la grandeur d’un ouvrage qui contiendroit toutes les connaissances humaines possibles. So since all human knowledge can be expressed by letters of the alphabet, and one can say that one who understands the use of the alphabet perfectly knows everything, it follows that it would be possible to compute the number of truths accessible to us, and thereby determine the size of a work that could contain all possible human knowledge.3

After all, only a limited amount of material—number of letters—can be presented per page. So even someone who reads a goodly number of pages per day (Leibniz supposes 100), and who does so daily for a very long time

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(Leibniz assumed a Methuselah with 1,000 years!), will take on board only a limited amount of material (365 x 105) pages. And with a limited number of statements per page (strangely, Leibniz fixes this at 1) our avid reader will take on no more than a fixed number of informative contentions in his very long lifetime.4 Since any alphabet devisable by man will have only a limited number of letters (Leibniz here supposes 24), it transpires that even if we allow a word to become very long indeed (Leibniz supposes 32 letters) there will be only a limited number of words that can possibly be formed (namely 24 exp 32). And so if we suppose a maximum to the number of words that an intelligible statement can contain (say 100) then there will be a limit to the number of potential “statements” that can possibly be made, namely 100 exp (24 exp 32).5 Even with an array of basic symbols different from those of the Latin alphabet the situation is changed in detail but not in structure. And this remains the case of the symbols at work at those of mathematics, where Descartes’ translation of geometrically pictorial propositions into algebraically articulated form stood before Leibniz’s mind, to say nothing of his own project of a universal language and a calculus ratiocinator.6 With an alphabet of 24 letters, there are 24 exp n words of exactly n letters. Accordingly, the total number of “words” with up to (and including) n letters will be 24 + 242 + 243 + . . . + 24n This formula was effectively proposed by Fr. Paul Guldin in his 1622 study de rerum combinationibus7 It yields a sum that comes to (24n – 1) x (24 ÷ 23). For all practical purposes we can take this to be 24n, seeing that that the sum’s big final term will turn out to preponderate. A language whose average sentence is W words long and which has w words at its disposal will offer (at best) some w exp W sentence-candidates. Most of this astronomical number of such symbolic agglomerations will of course be meaningless—and most of the remainder false. But this does not alter the salient and fundamental fact that—astronomical though it may be—the amount of correct information that can be encoded in language is finite.8 And so, on the combinatorial approach projected by Leibniz, the number of books will be finite—albeit very large. Thus let it be—for example—that a book has 1000 pages of 100 lines each of which has 100 letters. Then such a megabook will have room for 107 letters. With 24 possibilities for each of

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them, there will be at best 24 exp (10 exp 7) possible books.9 No doubt it would take a vast amount of room to accommodate a library of this size.10 But it would clearly not require a space of Euclidean infinitude.11 Accordingly, as Leibniz sees it, we arrive at the crucial conclusion that as long as people manage their thinking in language—broadly understood to encompass diverse symbolic devices—the thoughts that they can have— and a fortiori the things that they possibly can know—will be limited in number.12 2. THE LIMITS OF LANGUAGE In his official duties in Hanover and Wolffenbuettel, Leibniz was a librarian (as indeed Kant was for a time). Books were a matter of lifelong interest to him. And even in the seventeenth century when—as we ourselves see it—the impact of Gutenberg’s great innovation had scarcely begun to make itself felt, Leibniz spoke of “cette horrible masse des livres qui va tousjours en augmentant”,13 lamenting “une infinité de mauvais livres qui etoufferont enfin les bons et nous ramèneront à la barbarie”.14 The relation of the world of print to the real world was of great interest to Leibniz. Given that our thought proceeds by means of language, the limitedness of what can be said enjoins corresponding limits upon what can be thought. Thus let it be that the thought-life of people consists of the propositions that they consider. Then let us suppose that people consider propositions at about the same speed at which they read, say 100 pages per hour, with each page consisting of 20 sentences. Now supposing a thought-span of 16 waking hours on average, it will then transpire that in the course of a year a person entertains a number of thought equal to: 365 x 16 x 100 x 20 ≅ 12 x 106 Subject to the hypotheses at issue, this is how much material we would need to restate the thought life of a person for an entire year, and a lifetime might encompass some 80 times that. But there is only so much thinking that a person can do. And these limits of language mean that there are only so many thoughts to go around. Once again we are in the grip of finitude. Moreover, this encompasses fiction as well—our knowledge of possibility is also finite and fiction is for us just as much language limited as is the domain of fact.

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3. COGNITIVE FINITUDE IN AN ONTOLOGICALLY INFINITE WORLD Our verbal characterizations at the level of theoretical knowledge are unable to keep pace with the world’s actual detail—and indeed even merely not only its actual detail but even its observed detail. After all, even a world with only 100 locations for objects each of which can manifest (or not) merely any one of 10 properties will run off into big numbers. For with 210 object-descriptions one of which can answer to each of 100 locations we will have (2 exp 10) exp 100 possibilities, that is 21000 ≅ 10300 of them. Even this actually modest level of phenomenological complexity is problematically large. Reality, as Leibniz sees it, is infinitely complex and endlessly detailed in its make-up. And so is our reality-attuned experience of the real, albeit only at the unconscious, subliminal level. For the level of conscious awareness the situation is quite different. Explicit cognition—actual conscious awareness—is always verbal and correspondingly discretized. We view the color of the egg, but merely see that it is “brownish” with refined detail of shading etc. beneath the level of conscious apprehension. The actual make-up of any fact of observed reality is endless in its detail, but conscious discrimination—which, after all, is for us human language-bound— can go only so far. As Leibniz puts it: Sense-based truths that rest not in pure reason but in whole or part in experience can vary infinitely, even without becoming more prolix. Accordingly, they can provide ever new material through growing in quantity of information or implication. The reason for this is that sensations consist in confused perception which can vary in infinitely many ways without sacrifice of concision, there being infinitely many sorts of awarenesses, feelings, and sensations. But matters stand differently with truths that admit of adequate or complete demonstration, which can be explained in words and accordingly are numerally limited in quantity.15

We cognize (broadly speaking) more than we do—or ever can—know in verbalized form. There are, after all, those unconscious, petites perceptions of which Leibniz makes much. But consciously, realized knowledge— factual knowledge—must be explicit, “doctrinal”,16 and verbally formulated. And because of this, so Leibniz has it, our language-bound, propositional knowledge of factual matters cannot even keep up with the details of phenomenological experience let alone with that of ontology reality.17

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4. LANGUAGE AND LIFE It also becomes a real prospect that language cannot keep up with people and their doings. Suppose that the Detailed Biography of a person is a minute-by-minute account of their doings, allocating (say) 10 printed lines to each minute and so roughly 15,000 lines per day to make up a hefty volume of 300 fifty-line pages. So if a paradigmatic individual lives 100 years we will need 365 x 100 or roughly 36,500 such hefty tomes to provide a comprehensive blow-by-blow account of his life. But, as we have seen, the number of such tomes, though vast, is limited. In consequence, there are only so many Detailed Biographies to go around, so that it transpires that the number of Detailed Biographies that is available is also finite. This, of course, means that: If the duration of the species were long enough, then there would have to be some people with exactly the same Detailed Biography. Moreover, it also means that: If in the vastness of the universe there exists a sufficiently large number of intelligent beings then there would have to be some individuals with exactly the same Detailed Biography. On this basis, even a universe that is massively sizeable in its spatio-temporal extent can accommodate only so much descriptive variety. Eventual repetitions became inevitable. And, viewed in reverse, this poses an interesting question, namely: How much would the spatio-temporal extent of the world have to be limited in order to avert descriptive redundancy: to allow, for example, each intelligent creature to have its own unique Detailed Biography with its personalized life history. This of course would depend crucially on the amount of descriptive detail that is inserted into those Detailed Biographies. If the descriptive detail is infinitely refined then there need be no limitation at all. The needed differentiation would be provided for by the Leibnizian Identity of Indiscernables (the differentness of the diverse.) And moving on from Biographies (or Diaries) to public Annals we find much the same general situation. Thus suppose that (as Leibniz has it) the world’s population is one hundred million (that is 108) and that each generation lives (on average) for 50 years, then in the 6,000 years during which (we may suppose) civilized man has existed, there have existed some 1.2 x 1010 people—or some 1010 of them if we suppose smaller generations in earlier times.18 Recall now the above-mentioned idea of 36,500 hefty tomes needed to characterize in detail the life of an individual. It then follows that we would

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need some 36.5 x 1013 of them for a complete history of the species. To be sure, we thus obtain an astronomically vast number of possible overall annals for mankind as a whole. But though vast, this number will nevertheless be finite. Moreover, if the history of the race is sufficiently long, then some part of its extensive history will have to repeat itself in full with a parfaite repetition mot pour mot since there are only so many possible accounts of a given day (or week or year). For once again there are only a finite number of possibilities to go around and somewhere along the line it will transpire that men will act anew during their whole life acting in the same way in which others have already (ut homines novi eadem ad sensum penitus tota vita agerent, quae alii jam egerunt).19 To be sure, it would be rather unrealistic to suppose that humankind will last forever. But of course the universe is vast, and it would be far less unrealistic to suppose that intelligent life will always find a foothold somewhere in its limitless vastness: “Denique etiamsi non semper duraturum sit quale nunc est genus humanum; modo tamen semper ponamus existere mentes veritatem cognoscentes et indagantes.”20 But even if two symbol complexes—two books of annals, say— are identical, can they not always be differentiated by their wider context? Yes and no. It depends on how large those contexts are going to be. If we go just one year out, that triennium is just one more fixed timespan whose history can be told in a trilogy of annals that is exactly in the same position as the original: annal trilogies are just as limited in number as are annals. It is only when we allow their context to become indefinite (i.e., potentially infinite) in size that we escape the bounds of contextual finitude. (And indeed at this point we take the step into transdenumerability.21) 5. THE CONTRAST WITH REALITY The unbridgeable gulf between language and reality was already noted by Aristotle: It is impossible in a discussion to bring in the actual things discussed: we use their names as symbols instead, and we suppose that what obtains in the names, obtains in the things as well . . . But the two cases are not alike. For names are finite and so are their combinations, while things are infinite in number. Inevitably, then, the same words and a single name will have a number of meanings.22

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This is a theme on which Leibniz envisioned many sophisticated variations. While the thought of finite intelligent creatures is restricted by the limited discreteness of language with its limited access to detail the continuous structure, a very different situation obtains as regards natural reality itself: Even when many substances have already attained great perfection because their divisibility continues in infinitum, there will always remain in the depth of things heretofore insensible parts (components) that are now stimulated to greater and better, and so to speak, more sophisticated states. Accordingly, progress will never reach an end.23

The world as such has infinite detail, and this is something crucial for God’s knowledge of the sufficient reason for things. As Leibniz puts it in the Monadology: But a sufficient reason must also be present in contingent truths or truths of fact, that is to say, in the sequence of things dispersed through the universe of created beings. Here the resolution into particular reasons can go on into endless detail, because of the immense variety of things in nature and the ad infinitum division of bodies. There is an infinity of shapes and motions, present and past, that enter into the efficient cause of my present writing, and there is an infinity of minute inclinations and dispositions of my soul, present and past, that enter into its final cause. And as all this detail only contains other prior, or yet more detailed contingents each of which also requires a similar analysis to provide its reason, one is no further ahead. The sufficient or final reason must lie outside of the sequence or series of this detail of contingencies, however infinite it may be. And so the ultimate reason of things must be in a necessary substance in which the detail of the changes is present only eminently, as in its source. It is this that we call God.24

Reality, according to Leibniz, is infinitely complex: it is such that “la descente au détail est in principe sans fin”.25 Our verbally formulated propositional knowledge cannot even keep up with the detail that our senses afford (the veritates sensibiles), let alone with the infinitely vaster actualities of the real. It admits no limit to its descriptive detail: here refinement can go on and on and Leibniz’s Principle of Continuity comes into play. Accordingly, the epistemic situation of a being whose cognition is a matter of finite knowledge of a domain of infinitely refined detail is automatically disadvantaged. We cannot perceive things in their infinite detail but only sub-

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ject to some degree of “confusion.” Moreover, our limited discourse cannot even descend to the level of detail that perception does put at our disposal. For us there is no question of perceiving things in their ontologically infinite detail. And this has many important consequences. 6. THE PRICE OF FINITUDE: (1) OVERSIMPLIFICATION As Leibniz sees it, finite intellects can only come to terms with an infinitely complex reality through oversimplification, approximation, and the blurring of detail. And this circumstance has critical ramifications for our knowledge that are best conveyed by some (over)simple examples. Consider the following situation. Reality, so we will now suppose, consists of a pair of dots, one small and one large, cycling through those four compartments at an even pace as per: Reality

Appearance

A B

A+B

D C

D+C

And let it be that a certain phenomenon obtains only where the large dot occupies compartment C. Accordingly, the pattern of dot occurrence is given by the following law of succession: A → B → C → D → A → B etc. But now suppose that to the blurred sight of an imperfect knower compartments A and B look to be just one single unified compartment, and the same with C and D; and moreover suppose that those two dots look to be just alike, appearing as simply one middle sized dot. Then our dynamic system’s succession law will simply take on the static format of that “Appearance” illustration. The observable situation before us affords us no explanatory grip on the occurrence of the phenomenon at issue. There are two lessons here: • That to an observer who is oblivious to various details of reality, things may well appear simpler and subject to a cruder lawful order than is the case. And—

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• Thanks to such nomic (over)-simplification certain phenomena can become inexplicable. As such examples show, the element of confusion that is pretty well inevitable in our perceptual knowledge of the real is going to spill over into the range of our conceptual knowledge as well. Leibniz holds that just this sort of imperfection characterizes the cognitive situation of man. Any real object in the world, even a fly, is vastly more complex than any object of pure mathematics. To specify the fly’s make-up and to deduce from this its modus operandi would outrun the resources of even the most sophisticated theoretical science: When one considers a fly as the subject of a science—much as when the subject is a circle—it becomes clear that the definition of “fly” is something whose structure is immensely complicated and involves a great deal more than the definition of a circle does. And the demonstration of fly-appertaining theorems is something a great deal more elaborate [than geometry], and all that more so with particular species of flies, not to speak of individuals.26

To characterize reality in terms of its elementary components is simply beyond our human powers: “I think it might require a book as big as the terrestrial globe itself to explain fully the relations that a sensible body can bear to the prime elements, if indeed they can be truly known at all” (GPI 335). In sum one price that a cognitively finite being pays for its inability to come to full grips with Reality’s details is an oversimplification whose result is that one’s knowledge is not just incomplete but imperfect—that is, in certain ways incorrect albeit not so gravely as to become useless “for practical purposes.” 7. THE PRICE OF FINITUDE: (2) INDUCTIVE UNCERTAINTY Let us turn to Leibniz’s theory as to how general truths can be extracted from experience—that is, his theory of induction. Leibniz discusses this question of the methodology of inductive inference briefly in his relatively early (1670) Preface to his edition of On the True Principles of Philosophy of Marius Nizolius, and he returns to the issue at greater length in the draft Elementa physicae written ca. 1682–84.27 The deciphering of a cryptogram is Leibniz’s favorite illustration of the workings of this method of hypothesis-formation utilization.

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A hypothesis of this kind is like the key to a cryptograph, and the simpler it is, and the greater the number of events that can be explained by it, the more probable it is. But just as it is possible to write a letter intentionally so that it can be understood by means of several different keys, of which only one is the true one, so the same effect can have several causes. Hence no firm demonstration can be made from the success of hypotheses.28

The inductive search for nature’s laws is for Leibniz, a matter of trying to decrypt the message encoded into Nature by Nature’s God. It is a search for overall regularities in series that we see only in part. The method is conjectural because of its crucial reliance on hypotheses, and a priori because of its reliance on fundamental principles whose establishment lies beyond the reach of induction.29 The aprioricity at issue here indicates not the dispensibility of the empirical, but its incompleteness and insufficiency to the task of actually establishing and demonstrating the conclusion. As Leibniz writes: Perfectly universal propositions can never be established on this basis [demonstration] because you are never certain in induction that all individuals have been considered. You must always stop at the proposition that all the cases which I have experienced are so. But since, then, no true universality is possible, it will always remain possible that countless other cases which you have not examined are different.30

One point clearly emerges from this remarkably Humean passage: there is simply no question of us men ever being in a position to demonstrate any general truths of contingent fact. When dealing with the series of nature’s occurrences, the proportion of what is seen to what is not is as the finite to the infinite. Exhaustive verification thus becomes impossible so that no absolute certainty can be attained with respect to universal theses in the domain of contingent fact. As Leibniz sees it, induction relies crucially on the conjectural stipulation of theses whose full content lies beyond the reach of realizable experience. Induction itself is never able to afford any sort of “demonstration.” The fact that we rely on incomplete experience means that we can never quite step outside the realm of the probable or plausible in this contingent realm. In the absence of proof, the determinative considerations that underlie the attainment of moral certainty in this conjectural sphere are simply the standard epistemological parameters of inductive reasoning: comprehensiveness, generality, uniformity, coherence, simplicity, economy, etc. This last factor, economy, is the pivotal principle of induction. For 140

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The simpler a hypothesis is, the better it is. And in accounting for the causes of phenomena, that hypothesis is the most successful which makes the fewest gratuitous assumptions. Whoever acts differently by this very fact accuses nature, or rather God, its author, or an unfitting superfluity.31

Thus, while the method of hypotheses cannot attain demonstration or absolute certainty, it does enable us to attain to moral certainty when we place the experiential data of the induction into the rational framework of suitable principles of reason. For God’s penchant for the simplicity and systematicity provides a peg on which the possibility of induction pivots. The degree of probability that is attained through the conjectural method will hinge on the explanatory power of the conjectural hypotheses at which the method arrives: It must be admitted that a hypothesis becomes the more probable as it is simpler to understand and wider in force and power, that is, the greater the number of phenomena that can be explained by it, and the fewer the further assumptions. It may even turn out that a certain hypothesis can be accepted as physically certain [pro physice certa] if, namely, it completely satisfies all the phenomena which occur, as does the key to a cryptograph. Those hypotheses deserve the highest praise (next to truth), however, by whose aid predictions can be made, even about phenomena or observations which have not been tested before; for a hypothesis of this kind can be applied, in practice, in place of truth.32

We finite intelligences see reality as though through a glass, darkly. The definitive truth in contingent matters is known by God alone. However, we imperfect inquirers can, do, and should strive to discern those hypotheses which “deserve the highest praise” next to the truth and which we therefore may—and indeed must in practice, allow appropriately justified supposition to stand in its place. 8. THE PRICE OF FINITUDE: (3) INCREASING COMPLEXITY Given the inherently limited resources of language there will be corresponding limitation to what can be said. Let us construe the lexical complexity of a sentence as the product of the total number of its words by the number of letters in its longest word. (Thus the lexical complexity of the preceding sentence is 28 x 10 = 280.) It is clear that the number of sen-

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tences at any given level of lexical complexity will be finite. Given the natural penchant of humankind—and indeed any intelligent being—to address comparatively simpler issues first and only deal with more complex matters once the simpler have been exhausted—we are liable to concern ourselves in any given field of inquiry with simpler issues before addressing the more complex. Nevertheless, given the nature of language, the truths that can be discovered and articulated at any given level of complexity will be finite in number. And so the natural tendency of cognitive progress is to move from the simpler to the more complex. Thus we find that in geometry, for example, we will find that an ongoing tendency to more complexly articulated theorems: nova theoremata invenienda operteret crescere magnitudine in infinitum (Fichant, p. 74). Moreover, since the use of mathematics in a natural science such as physics affords the greater expressive power and economy of symbolic devices, this shifts the mode of communication from verbal to mathematical form: opus foret ad cognoscendam penitius naturam, veritates physicas reovcando ad mathematicas (Fichant, p. 74). Nature’s simplicity is the pivot-point of its intelligibility, but—as Leibniz will make clear, it is something that only goes so far. 9. THE PRICE OF FINITUDE: (4) VAGUENESS To illustrate the impact of oversimplification, consider someone whose visual myopia is such that he is incompetent with regard to telling 5 and 6 apart. As a result of such an inability to perceive 5 and 6 clearly, seeing both of them as a uniform blur Á, the individual may well through conflation envision 56 as ÁÁ. In consequence disorder (a random mix of 5’s and 6’s) can now appear as perfect order. Or again, the individual may through confusion envision 56 as 66. And now the opposite sort of problem can arise. For suppose that we are in reality dealing with the perfectly regular series R: 6 5 6 5 6 5 6 5 6 5 . . . but due to the occasional confusion of a mild cognitive myopia we may then actually “see” this (be it by way of observation or conceptualization) as M: 6 5 5 5 6 5 5 5 6 5 . . .

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The inability to distinguish has here effectively transmuted a lawful regularity into a random disorder. As even such crude examples serve to show, lawful order can unravel and be destroyed by the confusion engendered by an occasional inability to discern differences. And this situation has far-reaching implications. In specific, it means that if, even if the world is possessed of a highly lawful order, this feature of reality may well fail to be captured in even a mildly myopic representation of it. And this in turn means that the world-view presented in our world-modeling may well be no more than loosely coupled to the underlying reality of things. But does not the prospect of such a disconnection between appearance and reality lead straightaway to drastic skepticism? Not necessarily! For a fundamental feature of inquiry is represented by the consideration that: by constraining us to make vaguer judgments, the ignorance reflected in cognitive myopia enhances our access to correct information (albeit at the cost of less detail and precision).33 This state of affairs means that when the truth of our claims is critical we generally “play it safe” and make our commitments less definite and detailed. Vagueness blurring and imprecision effectively provide a protective shell to guard that statement against a charge of falsity. Irrespective of how matters might actually stand within a vast range of alternative circumstances and conditions, a vague statement can remain secure, its truth unaffected by the variation of detail that it leaves out of sight. In sum, the cognitive myopia of finite knowers does not constrain skepticism through blocking the access way to the realization of truth when we cease to operate at the level of precise detail. For it is not the truth per se, but the detailed truth that is bound to be inaccessible to beings afflicted by cognitive myopia. It is critical for Leibniz’s epistemology that the inevitable incompleteness of knowledge need not engender its pervasive incorrectness. 10. THE PRICE OF FINITUDE: (5) REPETITION In general, if the descriptive complexity of the objects we are dealing with is finite—if whatever complexity their meaningful description admits comes to an end—then there will only be a finite number of descriptively distinguishable objects of this kind. And so in a world whose scope is such

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as to accommodate a greater number of real objects there will have to be descriptively redundant recurrences of them. In a realm which is inherently finite in its elementary units such as (according to Leibniz) that of discourse, ongoing occurrence means eventual repetition. Given the finitude of the sayable—and thereby of the knowable—there immediately arises a troubling question that Leibniz does not hesitate to confront, namely that of an eventual recurrence in the cognitive domain. For let us suppose that the universe has a limitless future and that in this limitless future intelligent beings (not necessarily humans) will have a part. Now if the number of things that can be said and thought is finite, then a point of repetition must inevitably be reached. There will inevitably come a juncture when the saying that there is nothing new under the sun will be literally true within the realm of what can be thought and said. And so let us imagine a scriptorium of chroniclers who, ever assiduous in scribbling, put on record all of observed events and occurrences of the year. Here too repetitions must eventually set in. In due course the descriptive domain of world events must literally repeat itself, item by item. For there is, clearly, only a finite number of ways to fill in a text of a given length with letters. Thus (Leibniz observes) if a mega-book has 10,000 pages each of 100 lines each of which have 100 letters, then there will be room for 108 letters. And within a fixed number of letters at our disposal, say 25—a blank space included—there will be 25 exp (10 exp 8) possible mega-books. To symbolize this (finite) number—the number of possible mega-books of this scale—Leibniz adopts the letter N. He then turns to the idea of a universal public history—a history which, town by town, country by country, describes in detail the world’s public affairs during a given year. Such a history will of course be vast in scope. It will occupy many mega-books of the sort described in the preceding paragraph, say n of them. But clearly whatever be the size of the quantities at issue, N/n will be some definite finite number. And so, if the history of human kind is long enough, there will be no way to avert some recurrence or a repetition in its affairs beyond that number of years.34 Assuming that mankind survives sufficiently long in its present condition, and that a history of its public affairs can be produced, it will become necessary to re-tell parts of earlier history in exactly the same terms.35

Recall, to begin with, Leibniz’s calculation of the limited number of biographies or diaries that can be written in a language. And now assuming (with Leibniz) that the number of persons alive during a given year to be

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fixed at so-many individuals, each of whom comes equipped with his or her own determinate “Diary” for the year, there will be a grand overall total of personal diaries constituting (as it were) the annals of mankind for that year. This (astronomical) quantity Leibniz designates as Q. Accordingly, if the history of mankind is long enough then its years will eventually outrun Q and: There will eventually have to be a time when parts of written human history must repeat themselves.36 Accordingly, we do well to realize that there is a crucial parting of the ways between a finite mind like man’s and an infinite mind like that which theologians ascribe to God. Our thought is language-bound and our knowledge symbolically mediated. But God’s is not, being free from the alphabetic/symbolic/linearly linguistic modus operandi of our thought. Accordingly, the product of God’s thought is infinitely more complex than what one can manage to discern and describe about it in our human terms of reference. And in consequence, the reality that God’s creativity engenders— the realm of fact, if you will—is vastly more complex than anything that the finite intelligence of human knowledge is able to encompass and outruns whatever limits and limitations our discourse involves. Accordingly, while Wittgenstein may be right and the limits of our language may delineate the limits of OUR world—our thought-world, that is to say—they are not the limits of THE world. For there is more in heaven and on earth than is encompassed in the manifold of our thought and discourse. As Leibniz saw it, man as a cognitively finite creature is limited by his dependence on language and thereby he is restricted to finitude in regard to the horizons of what can be said and thought. But a being such as God is supposed to be is not comparably limited and is able to contemplate an unrestricted infinitude of possible worlds. And there is no limit to the detail, the inner complexity of those infinitely many and varied worlds that God envisions. There is thus radical disconnection between the analogue ontology of nature and the digital epistemology of man as a being whose cognition is recursive in structure. Those historical repetitions are accordingly only repetitions for us as finite knowers. They are artifacts of our cognitive limitations of our limited access to detail. The crux here is the imperfection of linguistic description— its inability to capture reality’s detail. For what repeats is described reality and not experienced reality. To illustrate: if all one tells about the weather is • above freezing vs. below freezing

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• rising barometer vs. falling barometer • sunny skies vs. cloudy skies then one only has eight alternatives for meteorological descriptions at one’s disposal. Over the course of nine days there is destined to be at least one repetition. But, of course, as one introduces more descriptive categories, the longer the prospect of repetition can be postponed. And as one moves on towards realty’s unending variability the prospect of repetition invariably vanishes. With Leibniz, order, simplicity, repetitiveness are substantially phenomena of oversimplification. Ontologically (for Reality) and theologically (for God) there is and can be no repetition. Everything that exists is unique. 11. THE PRICE OF FINITUDE: (6) MELIORATION COMPLICATIONS Leibniz also confronts the problem of reconciling his theory of apokatastasis with his optimistic doctrine of eternal improvement and cosmic melioration, which looked to a “progrès continuel et non interrompu à des plus grands biens”.37 At least two options offer themselves here. The first approaches the issue from the angle of time. Thus even with only two (eventually repetitive states A and B, of which the former is the more meritorious) we can have . . . (B x i) A . . . A B B B B A B B B A B B A B A A B A A A B A A A A . . . B (A x i) . . .

where x i indicates i repetitions. The ongoing temporal dominance of the superior state A clearly betokens an ever-improving condition of things. The second alternative takes resort to the slack between linguistic characterization and ontological constitution. It looks to the prospect that successive “recurrences” are not strictly identical but always arise in a somewhat improved state which, like a petite perception, beneath the threshold of linguistic characterization—with improvements that see ever-diminishing returns as per the series 1 2 , 13 , 1 4 , . . . The point is simply that Leibniz has more and better resources for averting contradiction here than his critics are prepared to credit to him.38 For the price that he has to pay for reconciling his apokatastasis analysis with his doctrine of melioration is not one of infeasibility but merely one of complication or sophistication.

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12. OVERBECK’S QUESTIONS AND THE UNTENABILITY OF CYCLICITY Reacting to Leibniz’s thesis that there is only a limited number of distinct accounts (be they personal biographies or public annals) Leibniz’s correspondent, the mathematician Adolph Theobaldus Overbeck (d. 1719) asks him four questions:39 (1) Must every history occur with the same period? The answer is clearly negative even if there were only two histories the pattern of the occurrence might be: (A always so far), A, B-once, A, B-twice, A, B-thrice, etc. Given only finitely many event-kinds, there must be repetitions: some segments must recur time and again. But there need be nothing periodic about this. (Otherwise—as Leibniz rightly remarks, there would be no irrational numbers.40) A full-fledged repetition of reality would be theologically objectionable: nunquam dantur regressus perfecti because dignitati naturae consentaneum non est, ut priora tantum repetantur.41 (2) Must any given segment of history recur? Again the answer is negative. And the previous example shows it for here the sequence ABA will never recur. No particular segment—no series of events—need ever recur. (3) Must SOME segment of a fixed size eventuality recur? The answer is affirmative because there is only a limited, finite number of different segments of this size, so that in continuing a series indefinitely there will have to be ongoing resort to one or another of them. (But not to any one in particular!) (4) Can there always be SOME segment [of a fixed size n] that has never occurred before? Here the answer is negative because (once again) there are only so many possibilities for a length-n segment. Since length-n segments are at issue—and there are only so many

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types of these—there must come a point when these finite possibilities are exhausted. And after this point a repetition cannot be avoided. The crucial point to emerge from Leibniz’s handling of Overbeck’s question is that on his approach to the issues there must indeed be some repetitions (redundancies) in personal or public chronicles—and massive repetitions at that—there need be no larger periodicities. Recurrence will indeed transpire, but no particular pattern of recurrence is mandated. And—above all—the finitude of constituents does not entail any sort of cyclic periodicity: repetition there must be, yet not only can it be partial, but it need not be circular or cyclic.42 Accordingly, the juxtaposition of points (2) and (3) are crucial. With finitely many basic symbols (“letters”) there is only a limited finite number of segments of a specified size (however large). Thus in an infinite series there must be (infinitely many) repetitions of some such segment. But not of any particular segment. And what is repeated can be a mere “space filler” in an evolving situation of ever-changing patterns. And so, what must be stressed here is that that fact that there is going to be some “eternal recurrence” in the overall series does not mean that any given segment is going to repeat.43 Some history must repeat itself, but not your history. It is a grave fallacy to think that finitude entails cyclically periodicity or unending repetition. The error at issue here is sufficiently prominent to deserve a name, and one might as well call it Nietzsche’s Fallacy, considering that he reasons as follows: The number of states, alterations, combinations, and developments of this [self-maintaining] force [in nature] is, to be sure, tremendously large and practically “immeasurable”, but in any case also determinate and not infinite . . . Consequently, the development of this very moment must be a repetition, and likewise the one that gave birth to it and the one that arises out of it and thus forward and backward further! Everything has been there countless times inasmuch as the total state of all forces always recurs.44

However, decisive considerations go counter to this perspective.45 The finitude of units—and thereby their recurrence in an ongoing process—does not entail the finitude of complexes. The finitude of types of events does not entail cyclic patterns of occurrence. Recurrence there must be, but not “total-state” recurrence. Thus consider again the sequence . . . (all A’s), B, A, B, B, A, B, B, B, A, B, B, B, B, A . . .

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There is infinite repetition here: A recurs infinitely often, as does AB. But there is no recurrence at all for certain other patterns (and infinitely many of them). Thus A B A occurs once but never again, as does A B B A. In a series of limitedly many units, a particular sequence of events need never occur again. Repetition does not entail cyclic periodicity, and the finitude of occurrence types does not necessitate eternal recurrence for all occurrence complexes. It is by no means the case that “everything has been (or will be) there countless times.” As his discussion makes clear, what Leibniz has in view with his apokatastasis is merely a restricted unending recurrence and not cyclicity. The crux, with Leibniz, is that some sector of history of any given length must ongoingly recur, and not that every sector of history of some given length must ongoingly recur. The latter, Nietzschean mode of eternal recurrence is not what Leibniz has in view. It is not the case that the events of the present era will repeat “if only one waits long enough.” Although his choice of the apokatastasis terminology was perhaps infelicitous, Leibniz was too good a mathematician to fall into the trap of Nietzsche’s fallacy of thinking that phenomenal limits entail cyclicity (Kreislauf) rather than merely partial repetitions. And not only is there no cyclicity but in reality there is no total repetition either. Repetition is no more than a merely phenomenal product of human finitude, of our inability to derive fine details beneath a certain level. Reality as such (ontologically speaking) is infinitely complex, infinitely differentiated when something seemingly recourse there is always a difference between the old and the new, albeit one that lies beneath the threshold of cosmic perception (and only open to those minute petites perceptions of which Leibniz is so fond). Only when we proceed at the cruder level of consciously noted and propositionally formulated will those ontological differences vanish amidst an epistemic blur. Recurrence there must be for Leibniz, but it is always only a seeming recurrence that figures at the level of the epistemology of finite intellects, and not at the level of ontology. Actual full-scale literal recurrence is ruled out by the Identity of Indiscernables. 13. COGNITIVE LIMITS The finitude of human knowledge spells inevitable limitations.

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In this regard, the limitations of language are akin to the limitations of counting. Let us adopt an abbreviation C(n) = Humans have counted the integers up to and including the integer n (Here “counting” is to be a matter of indicating an integer directly by name—e.g., as “thirteen” or “13”—rather than descriptively and obliquely, as per “the first prime after eleven”.46) We will now have it that • For any integer n, it is (in theory) possible for humans to succeed in counting that far: (∀n) ◊ C(n) Any particular number—however large—can in principle be reached by us through actually counting. But this contrasts decidedly with the clearly different—and clearly false—contention • It is possible for humans to count every integer. ◊(∀n)C(n) This last possibility is precluded by the realities of human finitude. The situation as regards the knowing of facts is substantially akin to that of the counting of integers. And this is so specifically in the following regards: 1. The manifold of integers is inexhaustible. We can never come to grips with all of them as specific individuals. In this regard, however 2. Progress is always possible: we can always go beyond whatever point we have so far managed to reach. In principle we can always go further than we have already gone. 3. Moving onward gets ever more cumbersome. In going further we must be ever more prolix and make use of ever more elaborate symbol complexes. Greater demands in time, effort, and resources are inevitable here.

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4. Thus in actual practice there will be only so much that we can effectively manage to do. The possibilities that obtain in principle can never be fully realized in practice. Exactly the same sort of situation characterizes the cognitive condition of finite intelligences whose cognitive operations have to proceed by a symbolic process—effectively by language. In inquiry, as in counting, we can go even further and do ever more and better. Though always and unavoidably limited, we can advance the frontiers and shift our horizon further onwards. Our present limit need not apply to the future.47 14. RETROSPECT The difference between a finite and an infinite knower is of far-reaching importance and requires careful elucidation. For an “infinite knower” should not be construed as an omniscient knower⎯one from whom nothing knowable is concealed (and so who knows, for example, who will be elected U.S. President in the year 2200). Rather, what is now at issue is a knower who can manage to know in individualized detail an infinite number of independent facts. Such a knower might, for example, be able to answer such a question as: “Will the decimal expansion of π always continue to agree at some future point with that of 2 for 100 decimal places?” (And of course the circumstance that an infinite knower can know some infinite set of independent facts does not mean that he can know every such set.) What is at issue here might be called Isaiah’s Law on the basis of the verse: “For as the heavens are higher than the earth, so are my ways higher than your ways, and my thoughts than your thoughts.” A fundamental law of epistemology is at work here—to wit, that a mind of lesser power is for this very reason unable to understand adequately the workings of a mind of greater power. To be sure, the weaker mind can doubtless realize that the strong can solve problems that the lesser cannot. But it cannot understand how it is able to do so. An intellect that can only just manage to do well at tic-tac-toe cannot possibly comprehend the ways of one that is expert at chess. It is not simply that a more powerful mind will know more facts than a less powerful one, but that its conceptual machinery is ampler in encompassing ideas and issues that lie altogether outside the conceptual horizon of its less powerful compeers.48

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Even though the thought and knowledge of finite beings is for this very reason destined to be ever finite, it nevertheless has no fixed and unalterable limits. The situation is analogous to counting. No matter how far out we go in counting integers, one never gets beyond the range of the finite. There is a limit beyond which we will never get. But there is no limit beyond which we can never get. The range of issues operative in these deliberations was mooted by Kant in the following terms: [I]n natural philosophy, human reason admits of limits (“excluding limits,” Schranken) but not of boundaries (“terminating limits,” Grenzen), namely, it admits that something indeed lies without it, at which it can never arrive, but not that it will at any point find completion in its internal progress . . . [T]he possibility of new discoveries is infinite: and the same is the case with the discovery of new properties of nature, of new powers and laws by continued experience and its rational combination . . .49

And just this difference is critical for Leibniz. Reality and the realm of fact it represents is inexhaustible, so that there are, in principle, no limits to our knowledge in the sense of boundaries (Kantian Grenzen). But practice is something else again—and here our condition in the cognitive scheme of things means that limitations in the sense of Kantian Schranken are inevitable. In practice our cognitive efforts can carry us finite creatures only so far. In the light of this distinction it is significant that Leibniz speaks of the HORIZON of human knowledge and not of its BOUNDARY. For a boundary is something definite, something fixed: you can go beyond it and look at it from the other side. A horizon, by contrast, is a variable limit. It moves on with the observer. There are, to be sure, matters that lie out of range— “over the horizon” as the expression goes. But while one can build a wall at a boundary, one cannot do this at the horizon. 15. CONCLUSION But back to Leibniz. As he saw it, the salient consideration lies in the essential finitude of human cognition and the continuity-infinitude of reality and God’s knowledge of it. The gap between the finite and the numberless infinite is as large as any gap can possibly be. And this gap is something we cannot possibly cross in operational fact but only in analogy. And the

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analogy at our disposal is that afforded by the differential and integral calculus. So what we have in Leibniz is a dualized, two sided situation. On the one hand we have the (human) epistemology of finite beings limited to the combinated resources of the universal characteristic and the calculus ratiocinatur. And on the other side we have the (divine) epistemology of omission with its cognitive realization of an endless feasible and complex reality. For with a God who creates an existent reality via the mediation of thought the difference between epistemology and metaphysics is obliterated. Thus in the final analysis the relation of human reason to diverse reasons is merely analogical so that here Leibniz’s standing as a philosopher of analogy comes vividly into the foreground once again. The world’s complexity—and indeed that of any possible world—is something that God alone can grasp in its full details. And the optimization problem at issue with the relation of the most perfect possible world is one that God also can solve. We finite creatures with our intellects can at best realize THAT reality-as-we-have-it represents the reality of this problem: we cannot grasp HOW it is so. But how, then, is it possible for us to achieve even this modest level of understanding? How can the finite come to terms with the infinite? The answer, for Leibniz, is that this can only be accomplished at the level of analogies. And here the sort of maximization process afforded us by the calculus is absolutely crucial and Leibniz the metaphysician is once again deeply in the debt of Leibniz the mathematician. Appendix PROBLEMS OF ETERNAL RECURRENCE Even very simple repetitive processes need not issue in cyclical recurrence. Consider three planets moving with uniform velocity in a circular orbit around a common center, as per the following diagram: A B C

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Let it be that A completes one orbit in 2 days, B one in two days, and C one in one day. Then B and C will recur to their indicted initial positions every N days whenever N is even. But at no time when N is even will A ever again be aligned with B and C in those positions. Notwithstanding the simple periodicities at issues, the indicted A-B-C configuration just cannot recur: Never again will those three uniformly moving planets recover their initial alignment.50, 51 NOTES 1

This term goes back to Plato’s idea of a great “cosmic year” for the positional recurrence of the heavenly bodies and figures in the pseudo-Platonic dialogue Axiochus (370B). The term is also biblical, occurring at Acts 3:21. In its later, theological sense, apokatastasis relates to Origen’s doctrine of the ultimate restoration of all men to friendship with God (a teaching sharply opposed by St. Augustine and ultimately declared anathema at the Council of Constantinople in 543). The theologian Johann Wilhelm Petersen (d. 1727) had published a treatise Mysterion apokatastaseôs pantôn which, as Leibniz wrote to Fabricius, he read soon after its publication with pleasure and profit (cum voluntate et fructu).[See Kortholt (Ravier Nr. 394), p. 116. (=Deutens, V, p. 278).] Leibniz contributed an extensive overview of the book to Eckhardt’s Monathlichen Auszug, in its issue of April 1701 (pp. 1–37). An abstract of this review is given in Guhrauer, Leibniz’ Deutsche Schriften, vol. II, pp. 342–47.

2

Leibniz’s tract Apokatastaseôs pantôn was originally published (and translated) by Max W. Ettlinger as an appendix to Leibniz als Geschichtsphilosoph (Munich: Koesel & Puslet, 1921). For an ampler treatment see G. W. Leibniz, De l’horizon de le doctrine humain, ed. by Michel Fichant (Paris: Vrin, 1991). This work assembles the relevant texts and provides valuable explanatory and bibliographic material. See also “Leibniz on the Limits of Human Knowledge,” ed. by Philip Beeley, The Leibniz Review, vol. 13 (December 2003), pp. 93–97.

3

L. Couturat, Opuscules et fragments inédits de Leibniz (Paris, Alcan, 1903), p. 532.

4

Leibniz, De l’horizon, pp. 51–52.

5

Ibid., p. 44. By a somewhat different route Leibniz arrives at 10 exp (73 x 10 exp 10) as an upper limit. (See COF 96). This perspective requires the decomposition of larger textual complexes into their constituted units. Leibniz is of course perfectly aware that individual sentences can always be combined into longer units by conjunction, so that an entire book can be recast as a single megasentence and indeed totius seculi Historia pro magno aliquo dicto haberi potest. (See LH IV, Vol. V, Sect. 9, folios 4 recto and 7 recto). But even though Leibniz contemplates the

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NOTES

prospect of superlong sentences (as per Huebener 1975, p. 59) the overall structure of the situation remains unchanged. 6

For details see Louis Couturat La Logique de Leibniz (Paris: Alcan, 1901).

7

In the generation before Leibniz, Fr. Paul Guldin had published (in 1622) a study de rerum combinationibus. Anticipating Leibniz, Guldin goes on to compute the number of different possible 1000 page books, the number of libraries needed to house them all, and the size of the terrain needed to hold all of these, which would exceed the whole of Europe. He states that the pages of these books would paper a path stretching not only around the earth but reaching far into the heavens: viam ostendimus qua itur ad Astras (Cited by M. Fichant in De l’horizon, pp. 136–38.)

8

See Ibid., p. 44.

9

Ibid., pp. 146–48. (Leibniz’s own sample calculation uses somewhat different numbers.) Leibniz further notes that with an alphabet of 100 letters the totality of such available statements would be 10 exp (73 x 10 exp 11), and comments that with 20,000 scribes it would require some 37 years to write out this number in full, if each scribe annually filled 1,000 pages with 10,000 digits each, and further that printing this number would take some two years with 1,000 printing presses each daily printing 1,000 pages. (Knobloch 1973, p. 88 [see note 12 for this reference].)

10

Overbeck (op. cit., p. 93) notes that the number of grains of sand in the world— which Archimedes had put at 10 exp 50—is very small potatoes in comparison. The Sand-Reckoner of Archimedes, the grandfather of all studies of large numbers, introduces the idea of successively large ordains of magnitude via the relationship en = 10 exp (8n). Archimedes sees the diameter of the sphere of the fixed stars no greater than 10 exp 10 stadia and, on this basis states that the cosmos would be filled by 1000 e7 = 1050 grains of sand. See T. L. Heath, The Works of Archimedes (Cambridge: Cambridge University Press, 1897).

11

Leibniz, De l’horizon, p. 61.

12

On Leibniz’s studies of combinational mathematics see Eberhard Knobloch, “The Mathematical Studies of G. W. Leibniz on Combinatorics,” Historica Mathematica, vol. 1 (1974), pp. 409–30, as well as his Die mathematischen Studien von G. W. Leibniz zur Kombinatorik, Studia Leibnitiana Supplementa, Vol. XI [Wiesbaden, Franz Steiner Verlag, 1973].

13

GP VII 160.

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NOTES 14

Klopp 10, 17. Leibniz characterizes the world of print as “a vast ocean in which . . . the curiosity of mankind will finally be drowned. For if people continue to print books with such facility, someday neither the houses nor the streets will suffice for libraries, and indeed hardly whole cities.” Annales imperii, Ann. 949 §16 [Ed. H. Pertz, Vol. II (1845), p. 575).

15

Leibniz, De l’horizon, p. 76.

16

See NE iii IX (Remnant-Bennett 335–38).

17

Following Leibniz, Baumgarten in his Aesthetics notes that aesthetic beauty often hinges on those subtle, virtually “indiscernible” differences that make aesthetic differentiations hard to put into words. As he has it, while systemically rational knowledge of various facts is not possible nevertheless there is also sensible knowledge on the basis of indications rather than reasons. Here processes such as examples, analogies, and induction come into play.

18

Leibniz, De l’horizon, p. 112.

19

Ibid., p. 54.

20

Ibid., p. 74.

21

The number of infinite sequences of finitely many symbols—say 0’s and 1’s—is of the cardinality of the reals. Such sequences cannot be completely enumerated because there will always be a missing entry that should be in the list but cannot be there because it differs from the i-th entry at the i-th place.

22

Sophistical Refutations, 165a5–13.

23

GP VII 308.

24

Monadology, §§ 36–38.

25

G, Grua, Leibniz: Textes inédits (Paris: Presses Universitaries de France, 1948), pp. 321.

26

Leibniz, De l’horizon, p. 76.

27

For these texts see GP IV 138–76 and Loemker 277–90 respectively. [When writing this paper I did not have access to the Latin text of the Elementa, and so quote from this source in Loemker’s translation.]

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NOTES 28

Loemker 283.

29

The method contrast with “they hypothetical method a posteriori, which proceeds from experiments, (and) rests for the most part upon analogies” (Loemker 284).

30

GP IV 161 (Loemker 129).

31

GP IV 158 (Loemker 128).

32

GP I 195–96 (Loemker 188 and cf. p. 128). On Leibniz’s views regarding probability in the sense of the calculus of probability. See Ian Hacking, “The Leibniz-Carnap Program for Inductive Logic,” Journal of Philosophy, vol. 68 (1971), pp. 597–610.

33

For example, if I have forgotten that Seattle is in Washington State then if “forced to guess” I might well erroneously locate it in Oregon. Nevertheless, my vague judgment that “Seattle is located in the Northwestern U.S.” is quite correct.

34

Leibniz, De l’horizon, pp. 67–68.

35

Ibid., p. 68. Compare “Si genus humanum in statu qualis nunc est per annos plus quam duret, necesse est redire prior annales Historias” (ibid., pp. 62–63). It might seem that Leibniz wishes his own discussion to illustrate the point.

36

Ibid., p. 71.

37

GP VII 308. Compare PNG, ad fin, as well as Phil VI 232; Theodicy, II § 195.

38

See, for example, Wolfgang Huebener, “Die notwendige Grenze des Erkenntnisfortschritts als Konsequenz der Aussagenkombinatorik nach Leibniz’ unveröffentlichem Traktat ‘De l’horizon de la doctrine humaine’,” Studia Leibnitiana Supplementa, vol. 15 (1975), pp. 55–71 (see pp. 62–63).

39

Leibniz, De l’horizon, pp. 89 ff.

40

Leibniz, De l’horizon, p. 89.

41

Huebner “Die notwendige Grenze,” p. 62.

42

Nietzsche commentators seem to loose sight on this qualification. See, for example, Karl Loewith’s Nietzche’s Philosophy of the Eternal Recurrence of the Same

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NOTES

(Berkeley and Los Angeles: University of California Press, 1997), pp. 89–91. See the Appendix. 43

To be sure, in a series of finitely many states driven by pure chance any particular segment of n states is effectively certain to recur sooner or later. But this of course has no bearing on Leibniz’s very different position where pure chance is not an issue. For a study of the relevant scientific issues see Abel Rey, Le retour éternel et la philosophie de la physique (Paris, Flammarion, 1921).

44

Nietzsche’s Werke, Grossoktav edition (Hamburg: Felix Meiner, 1986), Vol. XII, p. 51 (italics supplied). However, this argumentation would plausibly engender its intended conclusion only for a universe of finite variety and pure chance.

45

See the Appendix to this chapter.

46

This distinction between identifying numbers by name and doing so by description is crucial for resolving Richard’s “paradox”, which results from such specifications as, for example, “the largest number one can identify with fewer than 100 symbols” has in fact been identified with a good many fewer. Richard’s paper “Les principes de la mathématique et la problème des ensembles” was originally published in the Révue générale des sciences pures et appliquées in 1905. It is translated in Hiejenoort 1967, pp. 143–44. For a detailed discussion of this issue see Alonzo Church, “The Richard Paradox,” American Mathematical Monthly, vol. 61 (1934), pp. 356–61.

47

Leibniz wrote: Quaevis mens horizontem praesentis suae circa scientias capacitatis habet, nullum futurae (De l’horizon, p. 76). But this puts the matters too strongly. For while, on Leibnizian principles present limits need not confine the future, it is bound to have limits of its own.

48

On these issues see the author’s Epistemetrics (Cambridge: Cambridge University Press, 2006).

49

Prolegmena to any Future Metaphysics, sect. 57. Compare the following passage from Charles Sanders Peirce: “For my part, I cannot admit the proposition of Kant—that there are certain impassable bound to human knowledge . . . The history of science affords illustrations enough of the folly of saying that this, that, or the other can never be found out. Auguste Comte said that it was clearly impossible for man ever to learn anything of the chemical constitution of the fixed stars, but before his book had reached its readers the discovery which he had announced as impossible had been made. Legendre said of a certain proposition in the theory of numbers that, while it appeared to be true, it was most likely beyond the powers of

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NOTES

the human mind to prove it; yet the next writer on the subject gave six independent demonstrations of the theorem.” (Collected Papers, Vol. VI, sect. 6.556.) 50

A more general argument to this conclusion was initially presented by Georg Simmel, Schopenhauer und Nietzsche, Ein Vortragszyklus (Leipzig: Duncker & Humblot, 1967), pp. 250–51.

51

This chapter draws upon the author’s “Leibniz’s Quantitative Epistemology,” Studia Leibnitiana, vol. 37 (2006).

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Chapter 6 ETHICAL QUANTITIES

A

n ethical quantity as here construed is one whose mis-specification can prove to be not so much incorrect or inefficient as inappropriate or even wrong in a specifically ethical sense of these terms. For example, it is clear that if and insofar as the commandment to “honor thy father and thy mother” carries ethical weight, the number two, under the identifying description “the very least number of people one should honor,” will represent an ethical quantity. To mis-specify this quantity as “one” is to be mistaken in ethics, not in arithmetic. Thus consider the classic dictum of Sir William Blackstone to the effect that “it is better that ten guilty persons escape than that one innocent suffer.”1 Obviously, punishing the innocent is the wrong thing to do. But nevertheless Blackstone was a realist who doubtless realized full well that no system devised and operated by imperfect humans can be error free. And error in this sort of case comes in various kinds. No doubt we want to avoid false positives in designating innocent as the guilty. But improper negatives by way of unpunished malefactors are also undesirable. Yet how many otherwise escaping guilty does it take to justify one condemned innocent? Blackstone clearly tells us that 10 won’t do. But what of 1,000; what of 1,000,000; what of “all there are”? Here Blackstone was careful. It was surely not without thought that he picked 10 instead of one or another of those much larger available alternatives. After all, any workable system of criminal justice must strike a balance between two ethically geared desiderata: fairness and justice for individuals on the one hand, and on the other the larger public interest in maintaining a communally benign reign of law and order. Thus the occasional escape of the guilty from their just deserts can be tolerated in the interest of individual fairness, but such tolerance is justifiable only up to a point—a point which reflective consideration indicates as perhaps acceptable with Blackstone’s 10, discussable with 20, but pretty clearly out at 100. Yet why just exactly 10? It is, of course, sensible to say “Even one is one too many” in that it would be decidedly preferable to have none instead of even one. But it is not reasonable to say “Even one is too many” in such circumstances where there just is not actually on offer a flawless system of criminal justice that

Nicholas Rescher • Quantitative Philosophizing

avoids mishaps altogether, so that the only way to keep error free is to have no system at all. In many or most real-life situations such an “all or nothing” approach is not only unrealistic but—in the circumstances— undesirable. Here an unrealistic perfectionism is itself something decidedly imperfect. But if not all-or-nothing then what? Just why in such cases should it be that N is appropriate? What is there about N that enables it to capture sufficiency so that half N is too small but twice N too large? Too small or too large for what? The point surely is that in such matters an ethically oriented purpose is at stake. Specify that number improperly and you have a system that on the one side tolerates the unfair and unjust treatment of individuals and on the other side invites damaging important community interests. So the point is that with ethical qualities there is going to be an ethical stake: “too few” will always cash out to something like “too few for the general good,” “too few for public safety,” “too few for a viable system of justice,” or some such. “Too few” that is to say, indicates a shortfall in relation to realizing an ethically worthy good. Thus consider the somewhat more dramatic situation afforded by the splendid story of Abraham’s haggle with the Lord God in the book of Genesis: [25] And Abraham drew near, and said, Wilt thou also destroy the righteous with the wicked? [24] Peradventure there be fifty righteous within the city: wilt thou also destroy ad not spare the place for the fifty righteous that are therein? [25] That be far from thee to do after this manner, to slay the righteous with the wicked: and that the righteous should be as the wicked, that be far from thee: Shall not the Judge of all the earth do right? [26] And the Lord said, If I find in Sodom fifty righteous within the city, then I will spare all the place for their sakes. [27] And Abraham answered and said Behold now, I have taken upon me to speak unto the Lord, which am but dust and ashes: [28] Peradventure there shall lack five of the fifty righteous: wilt thou destroy all the city for lack of five? And he said, If I find there forty and five, I will not destroy it. [29] And he spoke upon him yet again, and said, Peradventure there shall be forty found there. And he said, I will not do it for forty’s sake. [30] And he said unto him, Oh let not the Lord be angry, and I will speak: Peradventure there shall thirty be found there. And he said, I will not do it, if I find thirty there. [31] And he said, Behold now, I have taken upon me to speak unto the Lord: Peradventure there shall be twenty found there. And he said, I will not destroy it for twenty’s sake. [32] And he said, Oh let not the Lord be angry, and I will speak yet but this once: Peradventure ten shall be

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found there. And he said, I will not destroy it for ten’s sake. [33] And the Lord went his way.

So ten righteous men would have done the trick. But why only ten? Why would not six have served? So, in the ethical approach of a society how many righteous people must there be in a town before God—or, rather more commonly, man with his missiles and bombs—should spare the wicked (or implacably hostile) from fire and brimstone? God apparently thought that as few as ten would do the trick, perhaps because in Judaic thought it takes 10 to constitute a viable community that deserves sustaining. (Man, of course, has often been far less generous, although even here it is not irrelevant to observe that Henry Stimson struck Kyoto off the target list for the atomic bomb because of its role as a religious center.) It is clear that the specification of ethical quantities is pervasive throughout the administrative management of our everyday offers. In many American jurisdictions, people can vote at the age of 18. But why not already at 9—or only at 36? Presumably because the former would put public affairs at the mercy of immature judgment, while the latter would disenfranchise otherwise qualified people from participating in the political process. Both shortfall and excess clearly run us into ethical problems here. The ethical aspect of the number-specification at issue in such matters roots in two considerations: (1) The quantities at issue are coordinate with an ethically valid goal and objective—a correlative ethically worthy aim or telos. (2) How the quantity at issue is specified can be facilitative or counterproductive with respect to the realization of this ethically appropriate goal. However, ethical quantities are in general inexact. “I say unto you,” said Jesus in the parable, “that joy shall be in heaven over one sinner that repenteth, more than over ninety and nine just persons, which need no repentance” (Luke 15:7). But what of nine hundred and ninety nine? Is the number here an amount that can be stretched ad infinitum? Surely not! Surely at some point numbers will tell even here. In a myriad ethically connected context in human affairs we must fix on a number. When is someone of age to marry, enter into contracts, drink alcoholic beverages, vote, be licensed to drive an automobile, serve as presi-

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dent of the USA? The lawmakers will have to decide. And sensible deliberation here has to recognize that some numbers are too big and others are too small relative to the ethically significant purposes at hand. The situation is somewhat akin to that of the ancient puzzle of the heap: “How many grains of sand does it take to make a heap?” Clearly three are too few and three hundred more than enough. But the border between sufficiency and insufficiency cannot be fixed with precision.2 But here the analogy ends and crucial disanalogy comes to the fore. For nothing moral—and certainly nothing ethical—turns on whether or not we qualify a certain rate of sandgrains as a heap. Or again “One swallow does not make a summer.” Fine! But just how many does it take? Well, it really just does not matter—from an ethical point of view, at least. On the other hand, the issue of how large an assemblage of rowdy people it takes to turn an unruly crowd into a riot—thus bringing in play the legal mechanism of a Riot Act or its equivalent—is an issue that is, in the circumstances, duly fraught with ethical ramifications. Of course some ethical qualities are (arguably) determinate. Noteworthy here—perhaps—is the specification of 2 as—in our society, at least—the number of individuals who can form a marital unit. (Perhaps in the evolving condition of things this may well eventually come up for discussion and debate.) But such determinacy is the exception to the rule. A “grey” area arises with ethical quantitatives has important implications. Throughout these cases the Goldilocks Principle applies: some numbers are clearly too little and others too big, while some appear to be right, perhaps, but yet not altogether precise and fixed. In such matters there is bound to be an indeterminate, indecisive region. And what is at issue here is not a matter of mere cognition: it is not that there really is an exact number that we just cannot manage to determine with precision. Rather, there is an inherently penumbral range of indefiniteness within which a precise determination is in principle impossible. Within a limited range there is no inherent priority but only what one might characterize as scope for “administrative efficiency.” It is important to distinguish between quantitative determination and quantitative specification. With determination there is an antecedently well-defined quality at issue, and one is endeavoring to measure it. How many 1-inch diameter spheres can one fit into a 1-foot sided cubic box? Clearly 5 is too few and 5000 too many. In-between there is going to be a number that is just right, and with a bit of mathematics one can determine its value. But just how many years suffice to yield an “age of consent” is

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something quite different. It is not that there are no limits here (six is clearly too young and sixty too old). But in fixing upon 16 (or 18)—or indeed anything in the 12–21 range—one is not discovering an antecedently defined quantity but rather making a specification for the purposes of the legal and administrative management of public affairs. There indeed are preexisting facts relevant to the determination (viz., those that render that 12–21 range plausibly discussable in this context). But a society’s fixing upon 16 (or 18) is a matter of specification, not determination. Here there is no preexisting numeric fact of the matter that leads us to 16 (or 18). Ethical numbers are bound to be problematic—and for good reason. Consider such generalities as: • “A crucial justice system is ethically unacceptable that punishes the innocent too frequently.” • “An educational system is unacceptable that does not permit persons of mature years a substantial latitude in shaping their studies.” These percepts are unproblematic, clearly acceptable, virtually tautologous. But then too they are vague and indefinite. For in such cases how many cases constitute “too often,” how many years does it take to be “mature,” how much latitude is “substantial?” We cannot ever begin to apply the precept until those unspecified quantities are given substance via specified amounts. Consider, for example, the idea of communal prayer and worship in the context of establishing religious group solidarity. But what of the size of the group? Jewish law stipulates that it take a quorum (minyan) of ten adult men to constitute a congregation sufficient for a valid religious service.3 The underlying idea in adopting this specification of ten seems to be that it takes ten to be authentically communal. The Talmud says that when two men pray together, God is with them (Ber. 62), thereby contemplating a sociable God, prepared to join in “where two or three are gathered together in my name …” (Matthew 18:20). Either way, however, the basic underlying idea is much the same, namely that to make things right with God—or if you prefer, to be ethical creatures in good standing as such—we must make them right with our fellows. But of course principles of this sort do not issue in specific numbers. In regard then to mandates of ethics and morality, zero is the favorite number and “thou shalt not” the favorite formula. (Even “Honor thy father

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and thy mother”—this one and only commandment that dispenses with “thou shalt not”—might be rephrased as “Never treat your father or mother disrespectfully.”) After all, there is no free pass in ethics—“Feel free to do this or that X times but no more” does not sound right in ethics: positive law does sometimes look at such questions in a different light, but in ethics there is no “three strikes and you’re out.” Philosophical deliberation regarding ethics is a matter of theory, of general principles. As standardly practiced, it deals in generalities and injunctions. And the Kantian universality that is sought after here insists on how things must be: they deal in all cases (“no exceptions”) rather than in “often” or “not more than 3 times.” Those ethical injunctions tell us what is to be done never or always (i.e., never omitted). So zero would seem to be the only number well known to ethics. Beyond this, ethical quantities will generally not be precise: we cannot pin them down to particular numbers. Is this good enough? Macaulay wrote in his essay on Machiavelli that “Nothing is so useless as a general maxim.” But this bit of self-refutation itself needs qualification and amendment. We have to add the proviso: “in the absence of further guidance for their implementation in concrete circumstances.” The general rules determinable on ethical principles alone are almost unavoidably numerally indefinite. “Share your wealth with those in need.” But to what extent? “Do not take more than your fair share!” But just what is that? To implement such precepts we must transmute generalities into specifics—and these specifics need to be aligned to the specific circumstances of the case. Something over and above honoring general principles is requisite to achieve these numeral specifics. And it is just here that a way-of-life utilitarianism can procure useful grist for its mill. The salient point here is that on the path downwards from general principles and governing procedure there is always a certain amount of slack. Strategic principles are one thing and tactical guidelines another; there is always some looseness in the linkage here, some lack of tightness of constraint that leaves room for variability.4 And in view of this, the circumstance that the general principles of ethical deliberation cannot constrain quantitative precision at the level of concrete procedural injunctions should occasion neither surprise nor discomfort. And the general principles of ethics cannot realize the quantitative detail that is requisite here. They can certainly detect flaws of egregious shortfall or excess. But quantitative precision is beyond their grasp. And so here as elsewhere we have to settle for the best we can possibly get. In ethical matters, there is need for good

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judgment as well as acute theorizing, and for common sense as well as for strict reason. NOTES 1

Commentaries on the Law of England, Vol. IV (Chicago: University of Chicago Press [Facsimile of the first edition of 1715–19]), p. 352.

2

On the so-called Sorties Paradox at issue here see the author’s Paradoxes (Chicago: Open Court, 2001), pp. 77–82.

3

See the Encyclopedia Judaica, art MINYAN.

4

On these issues see also chapter two of the author’s Moral Absolutes (New York: Peter Lang, 1989) and chapter three of Value Matters (Frankfurt: Ontos, 2004).

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Nicholas Rescher

Free Will An Extensive Bibliography With the Cooperation of Estelle Burris

Few philosophical issues have had as long and elaborate a history as the problem of free will, which has been contested at every stage of the history of the subject. The present work practices an extensive bibliography of this elaborate literature, listing some five thousand items ranging from classical antiquity to the present.

About the author Nicholas Rescher is University Professor of Philosophy at the University of Pittsburgh where he also served for many years as Director of the Center for Philosophy of Science. He is a former president of the Eastern Division of the American Philosophical Association, and has also served as President of the American Catholic Philosophical Association, the Americna Metaphysical Society, the American G. W. Leibniz Society, and the C. S. Peirce Society. An honorary member of Corpus Christi College, Oxford, he has been elected to membership in the European Academy of Arts and Sciences (Academia Europaea), the Institut International de Philosophie, and several other learned academies. Having held visiting lectureships at Oxford, Constance, Salamanca, Munich, and Marburg, Professor Rescher has received six honorary degrees from universities on three continents. Author of some hundred books ranging over many areas of philosophy, over a dozen of them translated into other languages, he was awarded the Alexander von Humboldt Prize for Humanistic Scholarship in 1984. In November 2007 Nicholas Rescher was awarded by the American Catholic Philosophical Association with the „Aquinas Medal“

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Frankfurt • Paris • Lancaster • New Brunswick 2009. 309pp. Format 14,8 x 21 cm Hardcover EUR 119,00 ISBN 13: 978-3-86838-058-3 Due December 2009

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Ontos

NicholasRescher

Nicholas Rescher

Collected Paper. 14 Volumes Nicholas Rescher is University Professor of Philosophy at the University of Pittsburgh where he also served for many years as Director of the Center for Philosophy of Science. He is a former president of the Eastern Division of the American Philosophical Association, and has also served as President of the American Catholic Philosophical Association, the American Metaphysical Society, the American G. W. Leibniz Society, and the C. S. Peirce Society. An honorary member of Corpus Christi College, Oxford, he has been elected to membership in the European Academy of Arts and Sciences (Academia Europaea), the Institut International de Philosophie, and several other learned academies. Having held visiting lectureships at Oxford, Constance, Salamanca, Munich, and Marburg, Professor Rescher has received seven honorary degrees from universities on three continents (2006 at the University of Helsinki). Author of some hundred books ranging over many areas of philosophy, over a dozen of them translated into other languages, he was awarded the Alexander von Humboldt Prize for Humanistic Scholarship in 1984. ontos verlag has published a series of collected papers of Nicholas Rescher in three parts with altogether fourteen volumes, each of which will contain roughly ten chapters/essays (some new and some previously published in scholarly journals). The fourteen volumes would cover the following range of topics: Volumes I - XIV STUDIES IN 20TH CENTURY PHILOSOPHY ISBN 3-937202-78-1 · 215 pp. Hardcover, EUR 75,00

STUDIES IN VALUE THEORY ISBN 3-938793-03-1 . 176 pp. Hardcover, EUR 79,00

STUDIES IN PRAGMATISM ISBN 3-937202-79-X · 178 pp. Hardcover, EUR 69,00

STUDIES IN METAPHILOSOPHY ISBN 3-938793-04-X . 221 pp. Hardcover, EUR 79,00

STUDIES IN IDEALISM ISBN 3-937202-80-3 · 191 pp. Hardcover, EUR 69,00

STUDIES IN THE HISTORY OF LOGIC ISBN 3-938793-19-8 . 178 pp. Hardcover, EUR 69,00

STUDIES IN PHILOSOPHICAL INQUIRY ISBN 3-937202-81-1 · 206 pp. Hardcover, EUR 79,00

STUDIES IN THE PHILOSOPHY OF SCIENCE ISBN 3-938793-20-1 . 273 pp. Hardcover, EUR 79,00

STUDIES IN COGNITIVE FINITUDE ISBN 3-938793-00-7 . 118 pp. Hardcover, EUR 69,00

STUDIES IN METAPHYSICAL OPTIMALISM ISBN 3-938793-21-X . 96 pp. Hardcover, EUR 49,00

STUDIES IN SOCIAL PHILOSOPHY ISBN 3-938793-01-5 . 195 pp. Hardcover, EUR 79,00

STUDIES IN LEIBNIZ'S COSMOLOGY ISBN 3-938793-22-8 . 229 pp. Hardcover, EUR 69,00

STUDIES IN PHILOSOPHICAL ANTHROPOLOGY ISBN 3-938793-02-3 . 165 pp. Hardcover, EUR 79,00

STUDIES IN EPISTEMOLOGY ISBN 3-938793-23-6 . 180 pp. Hardcover, EUR 69,00

ontos verlag Frankfurt • Paris • Lancaster • New Brunswick 2006. 14 Volumes, Approx. 2630 pages. Format 14,8 x 21 cm Hardcover EUR 798,00 ISBN 10: 3-938793-25-2 Due October 2006 Please order free review copy from the publisher Order form on the next page

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Nicholas Rescher

Epistemological Studies The present book continues Rescher’s longstanding practice of publishing occasional studies written for formal presentation and informal discussion with colleagues. They form part of a wider program of investigation of the scope and limits of rational inquiry in the pursuit of knowledge.

About the author Nicholas Rescher is University Professor of Philosophy at the University of Pittsburgh where he also served for many years as Director of the Center for Philosophy of Science. He is a former president of the Eastern Division of the American Philosophical Association, and has also served as President of the American Catholic Philosophical Association, the Americna Metaphysical Society, the American G. W. Leibniz Society, and the C. S. Peirce Society. An honorary member of Corpus Christi College, Oxford, he has been elected to membership in the European Academy of Arts and Sciences (Academia Europaea), the Institut International de Philosophie, and several other learned academies. Having held visiting lectureships at Oxford, Constance, Salamanca, Munich, and Marburg, Professor Rescher has received six honorary degrees from universities on three continents. Author of some hundred books ranging over many areas of philosophy, over a dozen of them translated into other languages, he was awarded the Alexander von Humboldt Prize for Humanistic Scholarship in 1984. In November 2007 Nicholas Rescher was awarded by the American Catholic Philosophical Association with the „Aquinas Medal“

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Frankfurt • Paris • Lancaster • New Brunswick 2009. 112pp. Format 14,8 x 21 cm Hardcover EUR 59,00 ISBN 13: 978-3-86838-048-4 Due July 2009 Please order free review copy from the publisher Order form on the next page

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ProcessThought 22

Nicholas Rescher

Ideas in Process A Study on the Development of Philosophical Concepts The book aims to provide a process-philosophical perspective philosophizing itself. It employs the perspectives of process philosophy for elucidating the historical development of philosophical ideas. The doctrine of historicism in the history of ideas has it that each era and perhaps even each thinker employs philosophical ideas in such a user-idiosyncratic way that there is no continuity and indeed no connectivity of public access across the divides of space, time, and culture. In opposition to such a view, the present processist deliberations see the development of ideas as a matter of generic processes that have ample room for connectivity and recurrence, permitting the very self-same conception to be shared by philosophers of different settings. Beyond arguing this histico-processism on general principles, the book presents a series of case studies of significant philosophical topics that illustrate and elaborate upon the developmental connectivities at issue.

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Frankfurt • Paris • Lancaster • New Brunswick 2009. 148pp. Format 14,8 x 21 cm Hardback EUR 59,00 ISBN: 978-3-86838-038-5 Due April 2009 Please order free review copy from the publisher Order form on the next page

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