Thinking Like a Computer: An Introduction to Digital Reality 9781645759270, 9781645759263, 9781645759287, 2020916187

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THINKING LIKE A COMPUTER An Introduction to Digital Reality

George Towner

AUSTIN MACAULEY PUBLISHERS

Thinking Like a Computer About the Author Copyright Information © Introduction 1. Digital Reality Theory Understanding the World Constructing Digital Realities DR Theory Trade-Offs DR Theory and Computing Verifying DR Theory 2. Understanding Existence Types of Understanding Time, Space, Pattern Using Ideals Sets and Digitization 3. Constructing Reality Digital Reality Types What Categorization Does Expanding Digital Reality The Power of Powersets The Structure of Knowledge 4. Social Realities Social Categorizations Worldviews Science and Religion Individual Liberty Consciousness 5. Personal Realities Personal Categorizations Natural Reality Formal Reality From Natural to Formal Spiritual Reality Beyond Ideals 6. Using DR Theory

Resolving Cartesian Dualism Clarifying Time and Space Generalizing Categorization Defining Statism and Individualism Explaining Nonlocality in Physics Rationalizing Epistemology Notes Afterword

About the Author The author, George Towner, studied logic and philosophy at Berkeley, then became assistant director of the Kaiser Foundation Research Institute, working on the biology of primitive organisms. When the computer revolution reached Silicon Valley, he switched to information technology and served 30 years on the senior technical staff at Apple. In his independent research, Towner analyzed how computers evolved from early number crunchers into today’s smart digital assistants. Thinking Like a Computer presents a compelling new explanation, based in set theory, of how both people and computers understand reality.

Copyright Information © George Towner (2020) All rights reserved. No part of this publication may be reproduced, distributed, or transmitted in any form or by any means, including photocopying, recording, or other electronic or mechanical methods, without the prior written permission of the publisher, except in the case of brief quotations embodied in critical reviews and certain other noncommercial uses permitted by copyright law. For permission requests, write to the publisher. Any person who commits any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages. Austin Macauley is committed to publishing works of quality and integrity. In this spirit, we are proud to offer this book to our readers; however, the story, the experiences, and the words are the author’s alone. Ordering Information:

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Towner, George Thinking Like a Computer ISBN 9781645759270 (Paperback) ISBN 9781645759263 (Hardback) ISBN 9781645759287 (ePub e-book) Library of Congress Control Number: 2020916187 www.austinmacauley.com/us First Published (2020) Austin Macauley Publishers LLC 40 Wall Street, 28th Floor New York, NY 10005 USA [email protected]

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Introduction Discoveries made during the last fifty years suggest a new approach to understanding how knowledge supports life. IF ANY OF MY GRANDCHILDREN grow up to be historians they will marvel at our present

age. Beginning in the 1980s, the widespread availability of computing power upended many traditional skills. As a teenager, I learned the rudiments of printing, bookkeeping, and photography. Today, most of what I learned is obsolete. Printing migrated from hot metal to desktop publishing, bookkeeping from paper to spreadsheets, photography from film cameras to telephones. All this and now my car wants to drive itself. To keep up with the times, I moved to Silicon Valley, learned programming, and joined the engineering staff at Apple. Pure luck gave me entree to the mosh pit that people began calling “the digital revolution.” During the next 30 years, I watched the revolution unfold and became aware that it affected more than just lifestyles and office work. At Berkeley, I had been trained in logic and philosophy, and afterward I had immersed myself in biology at the Kaiser Foundation Research Institute. At Apple, I was surrounded by rough-and-ready philosophers who were using logic to design machines that acted like living things. I was working with very bright guys who every day solved deeply theoretical problems of human knowledge that would have blown away the likes of Aristotle, Newton, and Kant. Other Silicon Valley enterprises contributed to this effort—SRI, Intel, PARC, NeXT, Google, Adobe—and it began to dawn on me that I was in the midst of something like a philosophical laboratory at work. Imaginative people, trying to make machines think, were experimentally challenging the foundations of traditional science. The upshot was that I gained a new understanding of how knowledge works at the nuts and bolts level. The idea was not that people are like computers. Rather, the idea was that computers were supposed to act like people. All those experiments with hardware and software, all those trials and errors, uncovered novel principles of human knowledge that made sense to me and that worked in machines. My key discovery was digitization, a complex machine technology developed in the twentieth century. You and I and our computers interact with the world around us in analog ways, yet we are capable of thinking and acting in digital terms. Digitization is not just peculiar to us or to Homo sapiens—it is baked into the nature

of life itself. In fact, analog-to-digital conversion is one of life’s primary skills. From this insight, Digital Reality Theory was born. The basics of DR Theory can be expressed in seven words: Life understands existence by constructing digital realities. A hundred years ago most people would have found this seven-word summary incomprehensible, yet today it makes sense. It took three intellectual developments during the nineteenth and twentieth centuries—evolution, set theory, and digitization —to achieve that change and to make DR Theory possible: During the second half of the nineteenth century, Darwin’s principle of evolution recast many ideas about life. Among them was the idea of fixed knowledge. “How does the world work?” was a question that tradition claimed had one answer—if only we knew how to find it. Thinkers such as Newton and Kant had searched for the principles behind the development of knowledge— Newton picked mathematics and Kant picked reasoning—but it never occurred to them that life itself evolved and that knowledge changed naturally with it. Set theory was invented in 1874. Mathematician Georg Cantor launched what is now called naïve set theory by showing how to construct sets of numbers as real mathematical objects. In the 1920s two logicians, Ernst Zermelo and Abraham Fraenkel, worked out the general rules for constructing sets of elements of any kind and for verifying that the sets were real objects. This became the tightly logical discipline known as axiomatic set theory. Digitization as an information technology originated in the twentieth century. While computers evolved from number crunchers to multimedia processors, their designers invented algorithms for converting analog data to digital form. The science of analog-to-digital conversion was born. The latter two developments laid the foundation for today’s smart computing devices. All such devices, from desktop computers to mobile telephones, are designed to construct sets of digital bits internally to represent external analog phenomena such as images, sounds, events, and even whole artificial realities. Smart devices do this for efficiency—digitization helps them sort out the essential from the trivial and adapt old solutions to new tasks. We and other living things digitize the world we live in for the same reasons. We construct digital realities inside to solve analog problems outside. DR Theory emerged when the principles of evolution, axiomatic set theory, and the science of digitization were added to traditional theories of knowledge. Books published during the last forty years have presented enough detail about the theory to make its messages clear. The present book summarizes the latest state of DR Theory in six chapters:

Chapter 1, “Digital Reality Theory,” outlines the theory’s basic ideas in plain language. It summarizes how DR theory explains knowledge and how it differs from older explanations. Chapter 2, “Understanding Existence,” analyzes in more detail how we and other living things grasp the world around us. This collection of mechanisms, developed during life’s evolution, can be described using set theory. Chapter 3, “Constructing Reality,” explains the processes by which we humans make our knowledge useful. It is primarily through digital categorization that our understanding of the world acquires its astonishing richness and complexity. Chapter 4, “Social Realities,” discusses the institutions and agreements that make human group behavior possible. It shows how these constructions, although artificially created, become real in our lives. Chapter 5, “Personal Realities,” summarizes the processes by which people create their natural, formal, and spiritual worlds. These internal digital realities, taken together, contain everything we know as individuals. Chapter 6, “Using DR Theory,” reviews some of the ways in which DR Theory can help bring the foundations of human knowledge up to date. One of the messages of DR Theory is that all knowledge is more or less iffy. Some ideas are pretty certain, but even what we think is our surest knowledge gets regularly overturned by better ideas. This book is no exception. The best theories make us reexamine what we think we know, for that is where we find new understandings. If DR Theory only helps with that task, it will have done its job.

1. Digital Reality Theory DR Theory can be outlined in seven words: “Life understands existence by constructing digital realities.” “LIFE UNDERSTANDING EXISTENCE” describes a familiar occupation. We do it all the

time. Here I use the word “understanding” in the sense of understanding a language or a game. When I say that I understand French or gin rummy, I’m claiming that in certain situations—such as being asked “ça va?” or being dealt a hand of cards—I know how to respond. I’m only a fraction of life as a whole, and a deck of cards is but a tiny bit of the existing universe, but the same principle scales up. A branch of philosophy—epistemology, or theory of knowledge—is devoted to explaining how we understand the world (or how we should be understanding it) and judging the results. So DR Theory should be classified as a broad theory of knowledge, because the last three words in its summary—constructing digital realities —describe what we and other living things actually do to understand existence. After reading this chapter you should be able to understand understanding, which sounds like a truly philosophical goal. DR Theory is based on existing information and explanations. It does not depend on any unpublished experiments or hitherto undiscovered truths. However, it does bring together ideas and trains of thinking that are not usually associated. It assembles a coherent and believable mental picture out of principles that have traditionally been scattered among various scholarly disciplines. Whether that picture is true to life is a matter of judgment in which you are the judge. The story of DR Theory can be told in ordinary language. Writing it down has not required made-up words or strings of arcane symbols. However, ordinary language is sometimes too broad for DR Theory. For example, one meaning in Webster’s definition of “behavior” attributes it to inanimate substances (“The behavior of various metals under heat”). DR Theory restricts behavior to living things. I have tried to flag instances where DR Theory uses ordinary words in limited ways so this book can stick with plain language—a small price to pay for avoiding jargon and scholarly ratiocination.

Understanding the World For starters, I’m going to try to define exactly what the subject of DR Theory

—life understanding existence—covers. “Life” includes all living things on planet Earth, from bacteria to rock stars, plus their footprints—all the artifacts and environmental changes they make. It also includes some semi-independent parts of living bodies (such as chloroplasts and mitochondria) and many groups of living things (such as species and societies). I mean by life every object or assembly in physical existence that manifests or is a consequence of the phenomenon of being alive. For example, an individual beaver is a part of life, but so is the genus Castor and so are the dams and lodges that beavers build. During its lifetime, each individual beaver constructs a digital reality within itself to help it build dams across one or more particular streams. Castor, the beaver genus, constructs a more general digital reality to help all beavers cope with all streams. The individual beaver stores its digital reality in its nervous system; Castor stores and transmits its reality in the beaver genome. Long ago the class Mammalia added to that genome a digital reality of urges and signals that helps Castor and other large animals reproduce; and so on. If you are a beaver, successfully behaving like one depends on having access both to the digital realities that you have constructed during your lifetime and to the genome that your species has constructed and corrected over millions of earlier beaver lives. As this book progresses, it will talk more about Homo sapiens and less about other kinds of life. That’s not because we’re exceptional—we’re just smarter, most of the time, and this book will explain why—but because we’re naturally interested in ourselves. “Understanding” is a defining process of life. In DR Theory, it’s something only living things do. Attributing understanding to living things goes far down the range of life, as when we say that ants understand what sugar is good for. In DR Theory, the result of understanding is called knowledge, even if the understanding is poor or incorrect. In DR Theory, knowledge is something that only living things can have: it shows up as a disposition to act one way and not another way. Many philosophical systems mix up knowledge with truth, calling untrue knowledge belief. This may work for humans, but it becomes awkward when applied to simpler creatures. For example, amoebas avoid direct sunlight because their genome knows that too much heat is usually unhealthy. Yet it would not add anything to our understanding of that fact to analyze it in terms of true or untrue beliefs. Consequently, most people have no trouble saying that amoebas know to stay out of the sun. “Existence,” in DR Theory, comprises everything. Yes, everything—ships, shoes, and sealing-wax, as well as my thoughts about ships and shoes plus the fact that seven is a prime number. It includes digital realities too, which living things construct as parts of their bodies, as well as all the knowledge they

contain. If a living thing wants to understand something, that something, by definition, must exist in some form. Contradictory things, such as the square circle of introductory logic, may exist only in someone’s imagination, but they must exist somewhere for us to try to understand them. That somewhere is digital reality. As Picasso once remarked, “Everything you can imagine is real.” The problem with existence, so defined, is that we don’t understand a lot of it. It is there, it affects us, we notice it, but we don’t understand it. To understand it we need to build a kind of reality narrative in which what we experience is explained in terms of distinct objects and events. Imagine that we are parts of a gigantic, complicated machine. We see and feel the machine as it pushes us around, and we know how to deal with parts of it because our ancestors wrote a book of instincts that we were born with. But much of the machine is what William James called “a great blooming, buzzing confusion,” which we cannot understand at first. Our solution is to study our experiences and write an operating manual for the machine—an addition to the book of instincts—using materials we find in our own thoughts and sensations. The manual we write explains the machine as a structure of levers and pulleys because those are things we can understand. As we grow up we constantly rewrite the manual, and we always consult it to know what to do. We are gratified to find that the machine usually responds in ways that the manual predicts. We congratulate ourselves on being “realistic” in our knowledge and actions when actually we are following our own description of something we couldn’t fathom in its original form. We are reacting to a constructed digital reality, something that is part of our existence now but was not an original part of our bodies or of the machine we are trying to understand. If we actually carried through the machine documentation exercise just described, we would soon discover that we needed to understand three quite different types of things: The personal experiences that help us know the machine; DR Theory calls them types of behavior; The machine as a physical thing; The abstract principles that make the machine work, which DR Theory calls ideals. A trio of adjectives—behavioral, physical, ideal—runs like a thread through all of DR Theory. The three appear at the outset as different ways we understand existence, and they appear at the finish as types of digital realities. Some philosophers have spent their careers trying to collapse the three into

one grand understanding; but, as DR Theory explains, the fact that there are more than one, added to the fact that they are intrinsically different, are what make human knowledge work.

Constructing Digital Realities The goal of DR Theory, therefore, is to explain how life on Earth, in all its forms and manifestations, accomplishes the task of understanding existence, which comprises both life and every object or event with which life interacts. The answer—constructing digital realities—sounds a bit mechanical. Isn’t life about imagination, hope, love, and a lot of other intangibles? Yes, it is, but these additions are all side effects of the knowledge that digital realities contain. DR Theory goes a level deeper and asks how knowledge, in general, is acquired and stored, as well as what its boundaries are and what its future might be like. To answer those questions, we need to interpret the last three words of the DR Theory summary, “constructing digital reality.” “Constructing,” as DR Theory uses the term, is another activity (like understanding) that only living things do. The theory further specifies that the result of every construction resulting from an attempt to understand existence is a set, as defined by axiomatic set theory. It is a kind of set categorized as a digital reality. What is a set? It is a group of things, called elements, identified together as a single object. A knife and fork taken together can be a set called a place setting. The knife can also be a set whose elements are a blade and a handle. Putting a blade and a handle together to make a knife is an act of construction, one that originated among humans more than 5,000 years ago. When you construct a set, you understand a collection of things as a single new thing. Curiously enough, this simple operation was not recognized as a basic unit of human thought until nearly the end of the nineteenth century. Today, it is deemed to be part of the foundational logic of mathematics and much more. Supported by a consistent set of axioms, set theory is as close as we have yet come to the ultimate basis for human thought. More practically, it is also an essential logical tool for computer programming. Every set is different in kind from any of its elements; this is the crucial fact it took logicians so long to recognize. A knife is not another kind of blade nor is it any sort of handle: it has its own unique reality. It also can belong to category sets in ways that blades and handles do not—it’s a tool, a weapon, a part of a carving set, and so on. By analyzing digital realities as sets, we bring axiomatic set theory on board as the logical foundation of DR Theory. This makes sense because the primitive ideas of DR Theory are easily described as sets or elements of sets. For example, parts of existence become the elements of digital reality sets and digital category sets become the building blocks of understanding. Set theory

helps us arrange our knowledge in ways that are both meaningful and useful. The primary reason for devising a list of axioms for set theory was to limit set operations to those that would not construct paradoxical or illogical sets. As you study the Zermelo-Fraenkel axioms, you find commonsense ways to combine things into single objects, e pluribus unum. Imagine, for example, that you have a heap of coins out of which you want to create some specific realities. You might sort the coins into three sets of pennies, nickels, and dimes. Or you could construct many sets that each add up to one dollar. Then from the dollar sets, you could assemble one set containing all the sets that contained no pennies. And so on, all using ZF axioms. The axioms would also help you prove truths about your coin sets, such as “if a dollar set contains one penny it contains at least five pennies.” Although the original heap of coins is not truly an analog continuum, you can see how sorting and grouping its elements into digital sets can yield knowledge not obvious from merely staring at the heap. “Digital” is contrasted to analog. In DR Theory, both terms denote a specific characteristic of the world around us. This characteristic is so basic that DR Theory uses it as the primary separator of reality from the rest of existence. Existence is an analog continuum, and we experience it—it affects us—but we don’t really know it. Reality is digital, and it’s what we know when we say we know the world. As a consequence, converting analog existence into digital reality is a constant preoccupation of living things. It is the behavior that produces our knowledge of the world, a necessity for us to stay alive. What makes raw analog existence and constructed digital reality different? One common answer is that a digital world is discrete while an analog world is continuous. The difference between the two worlds is often expressed in terms of how easy they are to describe. A discrete world can be described by listing the things it contains, all of which are more or less explicit. Describing a continuous world involves measuring it in many ways to get a picture of what’s there. We are limited to making quantitative measurements because a continuum contains no discernible parts or internal boundaries. For example, a jar of ice cubes can be described by counting the cubes. Melt them, and you have a jar of water. The weight and composition of the two jars may be identical, but it is easier and more explicit to count the cubes than to measure and analyze the water. To a scientist, water is not a continuum. It is made up of little assemblies called molecules, which are made up of smaller atom assemblies, and so on down. Most physicists believe the model of ever-smaller particles ends with discrete things that are not assemblies and not perfectly continuous. DR Theory, however, asserts that it ends at some form of existence that we are forced to treat as a continuum. We can measure its properties, such as its mass and energy, at various places but we can’t count any parts in it. For

convenience, DR Theory calls it analog existence. When our goal is understanding, the difference between analog and digital can be crucial. Understanding a region of analog existence (such as analyzing a jar of water) may become a never-ending process, whereas understanding a region of digital reality (such as counting a collection of ice cubes) can be quickly and certainly completed. Since life is mostly about making decisions about the world around it, this makes digital knowledge more efficient and useful. The efficiency of digital knowledge shows up in life’s ability to work with exceptional situations. Animal flight provides a good example. Over millions of years of evolution, bird species have learned to fly or soar efficiently through the Earth’s atmosphere, a medium that presents itself to the bird as a continuum. Much of the time, birds navigate the complex mechanics of lift and drag through analog responses to analog stimuli. Occasionally, however, birds must contend with the anomalies that human pilots call drafts, pockets, thermals, and so on. Although mostly invisible, these are digitally identifiable objects in a continuum otherwise suitable for level flight. Birds cope with them through digital counter-maneuvers, as human pilots learn to do. The difference between birds and humans, however, is that the flight school for birds was held for millions of years, its lessons were learned by the survival of its students, and its textbook was written in DNA. The crucial avian knowledge about these exceptions to normal flight is embodied in instinctive reflexes, coded digitally. Programmers would call them asynchronous interrupts. They are as important for each species to know as flight itself. “Realities” comprise the totality of what living things understand. We and other living things constantly interact with existence, but to understand those interactions we construct and know realities. For more convenient analysis, DR Theory talks about many separate realities, each of which is a repository of knowledge. For example, the theory recognizes that a beaver constructs and maintains a reality that represents the dams it builds, while a human living nearby will construct a somewhat different reality about the beaver’s handiwork. Each reality, constructed by life, is also a new part of existence. It’s a part that life can understand. Imagine a beaver confronting a flood that threatens to wash away its lodge. The beaver is not a hydrologist—it experiences the flood mainly in analog terms, as a set of existential threats. But the beaver is a builder, so it also understands how to meet those threats with a reality of digital sticks and branches. In effect, the beaver deals with existence by adding reality—an understandable construction of new existence—to it. That is life’s typical response. We humans are a bit more sophisticated when dealing with a flood. We start by measuring it. Measurement creates a digital reality—water so many feet

deep, an object we can understand. We can then figure, in digital terms, what we must do (if anything) to counteract the flood. The beaver’s innate measurement of the floodwater produces only a 1-or-0 binary number that measures too high or not too high. Ours produces a more complex number that measures how high, but the effect is the same. Visualizing reality construction. What is it like to understand existence by constructing a digital reality? For an illustration, let’s go back to cathode-ray tube (CRT) television sets as they existed before flat panel displays were introduced in the 1980s. In a CRT, the picture is painted by a moving dot of light—one dot for black-and-white, three dots for color. The dot, its intensity varying in response to an analog signal, zips back and forth like a tractor plowing a field. It draws a complete picture thirty times each second, and successive pictures convey motion by their frame-to-frame differences. To suppress motion blur, high-quality CRT phosphors usually have minimal persistence: the dot on the screen is lit only briefly at each position, while the dot-writing beam swings a millimeter or so to the next position. At any instant, what existed on the face of the early TV screen? A dot of light of smoothly varying intensity. Yet people focused on that little analog dot, enthralled. They saw the Beatles, they laughed at Lucille Ball, they watched a man walk on the moon. What they saw and understood was not a dimming and brightening dot roaming back and forth, but the digital objects that the TV cameras had captured—people, things, discrete events. Because human vision did natural analog-to-digital conversions, viewers understood an analog dot of light by constructing a reality of discrete visual objects in their minds. Today, CRT screens have been largely displaced by flat-panel displays, but they depend on the same optical principles. From analog color fragments presented or refreshed piecemeal, human vision constructs a world of discrete, moving objects. This works because the evolutionary goal of animal vision was never just to see light: its goal was to identify objects, using light and colors as clues. Often those clues were transitory—a tail disappearing into the trees or a glint in the underbrush. Hence, we became clever at digitizing the visual jumble of existence into knowable realities.

DR Theory Trade-Offs One way to characterize DR Theory holds that it tries to update our formal learning—theoretical knowledge taught in universities—by incorporating in it some of what we have learned during the twentieth century about multimedia digitization, biological evolution, and axiomatic set theory. Updating implies discarding some time-honored points of view, so DR Theory proposes a few changes in the ways we theorize about the world in which we live. You can think of them as philosophical trade-offs designed to broaden our knowledge

and help resolve some of its problems. Some important trade-offs are discussed below. Understanding vs. objectivity: Traditional theories aim for objectivity— we are supposed to “follow the facts wherever they may lead,” as the saying goes. In DR Theory, the existential facts-in-themselves are inaccessible to us. They are buried in the continuum of analog existence. The important thing is for us to recognize how we understand those facts when they impact us. It is our understandings that will determine what we do and how successful we will be. We can always seek a better understanding, which might include a different digitizing algorithm, but what we understand will still be a digitization of analog existence, not the existence itself. Philosopher Thomas Nagel addressed a similar problem in his classic essay “What is it like to be a bat?” (1974): I want to know what it is like for a bat to be a bat. Yet if I try to imagine this, I am restricted to the resources of my own mind, and those resources are inadequate to the task.1 This difficulty arises from the fact that digital realities—because we construct them—tend to respond to our needs as well as to the areas of existence they digitize. It is no easier for me to experience raw analog existence all by itself than to experience what it is like to be a bat. Science hallows objectivity as the gold standard of truth. Newton claimed that he did not make hypotheses, but in scientific research it is virtually impossible to discover a fact without first making a hypothesis. What Newton meant was that he discarded his hypotheses after he had used them in his search for objective knowledge. The crucial test for objectivity is that it appears to us whether we want it or not. I may imagine that I am fishing in a forest stream or sitting in a warm bath, conjuring up these experiences at will. I will usually realize that I am not actually in a forest or a bathtub; but, as Descartes argued, the fact of my imagining is itself real to me. It is also under my control; I can usually stop or start it at will. If someone sticks me with a pin, however, that experience will normally force itself upon me regardless of my wishes. The only way I can understand the pinprick is by hypostatizing an intractable objective world that is separate from my experience and my will. The slings and arrows of outrageous fortune in this world are a constant preoccupation for all living things. We humans may have a refuge from objective reality in our conscious imagination, but it appears that most other forms of life deal almost exclusively with the outside world. From pinpricks and all their ilk flows the conviction that at least part of what we know consists of an objective external world. Its objectivity makes it an actively pursued object of understanding. But we feel that the external

world does not exhaust the reality we know. In the case of humans, at least, a separate inner or subjective world, a world over which we have more control, is also part of everyone’s reality. To illustrate this difference, let us return to the physicist’s description of solid objects. Science has refined and limited our observations to the point where we believe that our instruments show us the characteristics of the smallest possible bits of matter, the subatomic particles. Every time we break up a solid object—wood, metal, or whatever—we observe the same bits. Moreover, we seem to be successful in extracting and isolating these bits. We can even rip them from one solid object and implant them in another, using a particle accelerator. When we do this, we are able to predict changes in the solid objects attributable to changes in their particle compositions. As early as 1919, for example, Rutherford transmuted nitrogen into oxygen by bombarding it with alpha particles. Finally, physicists can record movements of particles through films and cloud chambers, showing graphically how they enter and leave solid objects and interact in empty space. All this they accomplish with assurance and regularity, handling these tiny particles almost as easily as I handle the pencil lying on my desk. Surely, their picture of reality must be correct! But let us now leave the physics laboratory, where particles are cleanly isolated in evacuated chambers, and try to apply the physicist’s picture of reality to ordinary events. I pick up a pencil from my desk and it feels hard and smooth: how can I translate this observation into a statement about particles? The physicist will assert that such a statement might be very long and complicated, but “in principle” it can always be made. We start with the surface of the pencil in which quadrillions of electrically charged particles lie, each moving in a small orbit but tightly bound by electrostatic forces to particles farther inside the pencil. The surface of my finger is similarly composed of charged particles. As the two surfaces meet, the charges repel (being of the same “sign”). Because the particles in the pencil are more favorably distributed by its cellular structure than are those in my finger, the pencil remains rigid while my finger deforms. The deformation causes certain nerve endings in my finger to release electrically charged particles. These attach themselves to nearby atomic structures, causing further charged particles to be released farther away so that a chain of charged-particlereleasing events travels along a nerve to my brain. There I have learned to interpret the occurrence of such events as a message that my finger has encountered something hard. Consequently, the pencil feels hard to me. How good is this explanation? Suppose I have just been holding an ice cube before picking up the pencil so that my finger is numb and does not feel the hardness. The physicist will probably say that certain particles in my finger have decreased their motions enough to interrupt the passage of charges into my nerves. Suppose I have just been hypnotized to believe that the pencil

is a worm, and so feel that it is soft instead of hard. Here the physicist’s explanation may be less clear: perhaps some charged particles have migrated in my brain in such a way that they block those coming up the nerve from my finger. Suppose now I recall a dream in which I felt the hardness of a pencil when none was actually present. “Now we are getting into psychology,” the physicist will say; “that’s not my department.” But these are just the sorts of knowledge that are most useful to me: under what conditions the pencil feels hard, when the hardness is an illusion, and how the feeling of hardness relates to my handling of the pencil. The particle explanation may have some interest, but by the time I expand it to apply to these questions it has become exceedingly cumbersome and vague. Its applicability “in principle” has turned out to be largely an empty promise. Digital vs. analog: The world in which we live, including ourselves, operates on analog interactions. Before we can understand it, however, we have to digitize it. This is because we are living things, and life has evolved to regard digital objects and events as more significant than analog influences. What our knowledge knows are constructed digital realities, not analog source data. Modern computers work the way they do because their designers copied this idea from life, consciously or instinctively. During the last few decades, the words “analog” and “digital” have migrated from fairly specific terms in electronic technology to trendy labels for everything from photographs to lifestyles. When digital computers were first designed, they processed digital inputs—switches, keyboards, punched cards or tape, etc. When a key was pressed or a pattern of holes was read, the computer converted that discrete action into a set of binary bits in an electronic register designed to store binary numbers. For example, pressing the R key on a Teletype keyboard entered the five-digit binary number 01010 into the computer. This pattern of bits (as binary digits came to be called) represented the letter R in Baudot coding. After processing, the output might be rendered by punching holes into a paper tape, five rows across, which a Teletype printer could sense mechanically and type out as readable text. At the time, there also were analog electronic computers that accepted analog inputs. These inputs were usually varying electric voltages, but the voltages often represented other quantities—temperature, pressure, the level of liquid in a tank, etc. The inputs were called analog because the voltages varied as analogs of the quantities they measured—0.1 volt for every degree Fahrenheit above 0, for example. Digital processing quickly won out over analog processing, for several reasons: analog signals tend to degrade as they are carried in wires, they are difficult to store, they need regular recalibration, etc. A demand built up to convert quantities into digital bits at their source and then deal with the data digitally from there. But sensors, such as thermocouples and strain gauges, tend to be analog by nature; it is hard to design a device that converts

temperature directly into a number. It is easy to design devices that convert varying quantities into varying electric signals. Thus, the solution was to design sensors that converted temperature, pressure, etc., into varying electric signals and then feed the signals into an analog-to-digital (A/D) converter. The A/D converter periodically converted the varying electric signals into patterns of digital bits, which could be transmitted over long distances without change. Any computer engineer will tell you that A/D conversion is never fully complete nor absolutely accurate. A typical instance is bitmapping, used to digitize still images. A digital camera divides the image into pixels—tiny dots usually arranged in a rectilinear grid—and assigns a binary number to represent the color of each pixel. The more closely packed the pixels, the finer the definition of the digital rendering; the larger the range of color numbers, the more faithful its tones. The same is true of ocular vision in living things, which normally uses some form of bitmapping. Compared to that of some birds, our visual digitizing is second-rate. The fovea centralis of an eagle’s retina has five times as many light-sensing cones per millimeter as that of a human being, and pigeons can distinguish many more shades of color than we can. The image digitization in birds is what one might expect of animals that navigate largely by sight and view their world from a distance. Structures vs. measurements: Because knowledge is digital, it primarily knows the structures of things and events. Our senses and our instruments constantly metricize properties of the world, producing analog measurements, but then we digitize those measurements to help us construct the objects we suppose they represent. As a consequence, the best way for us to understand the realities we construct is through set theory, which is about objects, not through mathematics, which is about numbers. This approach is further justified by the fact that the foundations of mathematics can be explained by set theory but not vice-versa. When Newton looked through his telescope, he saw moving objects. His world was filled with things that attracted each other through gravity, moved when forced, and kept moving by virtue of inertia. But when it came to predicting these effects, Newton had problems. The mathematics of his day was good at calculating static properties, such as weights and forces, but it lacked a good way to describe similar properties when they were changing. The problem was generally understood in Newton’s day, so both he and the German mathematician Leibniz set out to create a new mathematical tool. It ultimately became called the infinitesimal calculus, and it was based on the idea that if you measured tiny enough bits of a changing event, you could pretend that each bit was stationary. You could then perform mathematical analyses of the bits without worrying that your underlying data was changing. To measure the entire event, you just added all your infinitesimal measurements together. The result was a static mathematical picture of changing reality.

Throughout the nineteenth century, mathematicians applied calculus to more and more physical phenomena. Experimenters found that electricity and magnetism created attractions and forces, much like gravity and motion, which could be analyzed using the same mathematical tools. When the twentieth century came, Einstein showed how space and time could be integrated with all these analog phenomena, wrapping everything up in a famous single formula that related energy and mass with the space-time speed of propagation of electromagnetic radiation. All these physical effects could be modeled using the concept of a field. In modern physics, a field may be defined mathematically as a set of quantities closed under a set of functions. The quantities are what we are capable of measuring physically in any part of the field: space-time location, mass, electric charge, etc. They can be treated as determiners of all possible physical occurrences in the field. The functions relate the quantities to each other in a compact and continuous way, so we can describe abstractly their mutual variations. Usually, we are specifically interested in how the other quantities vary with respect to space-time location; we collect the quantities into tensors and thereby assign to each point of space-time a bundle of measurements as if a tiny observer were reporting the potentials for physical action at that point. Note that the tensors are not understood to have a tangible physical existence in themselves; instead, they describe how the field acts. The very important requirement of closure disciplines this arrangement. It forces us to make sure that the field functions do not describe any measurements that could not actually be made. The proposition that field theorizing can explain all physical existence conflicts with DR Theory’s central thesis that we understand reality in digital terms, not analog. A discrete particle shows up in any field theory as a singularity—something other than a bundle of measurements. By its very existence, a singularity violates the closure of a field. Einstein’s attempts to formulate a unified field theory for physics were never concluded because he could not achieve a single closed representation of both gravitational and electromagnetic effects. Nevertheless, he was certain about the role of particles in field theorizing: What appears certain to me, however, is that, in the foundations of any consistent field theory, there shall not be, in addition to the concept of field, any concept concerning particles. The whole theory must be based solely on partial differential equations and their singularity-free solutions.2 DR Theory would characterize Einstein’s attempt to describe subatomic existence solely in field terms as trying to create a model of existence entirely

out of analog measurements. Such a model might be possible, and it might be mathematically consistent, but it would not help our digital understanding. Understanding and prediction. One way to characterize the progress of science over the last two centuries is by contrasting understanding with prediction. The two are interlinked: understanding how the world works helps us decide how to predict what will happen in the future. But prediction is where the money is, and predictions are usually based on measurements. Astronomy aided navigation by predicting the positions of celestial objects, physics helped engineers design more efficient engines by showing how heat produced pressure, geology helped prospectors find minerals by linking them to strata, and so on. While understanding what reality is may nourish our intellects, it’s through predictive procedures that we earn our bread. Hence, the tendency for science to value measurements over natural descriptions. Everyday concepts such as things and events are deeply ingrained in our common view of reality. To cast them aside in order to satisfy the needs of mathematics seems to frustrate the goal of understanding. The problem is succinctly summarized by computer scientist John F. Sowa: In modern physics, the fundamental laws of nature are expressed in continuous systems of partial differential equations. Yet the words and concepts that people use in talking and reasoning about cause and effect are expressed in discrete terms that have no direct relationship to the theories of physics. As a result, there is a sharp break between the way that physicists characterize the world and the way that people usually talk about it.3 In some versions, quantum field theory represents the subatomic world as space filled with overlapping fields. The fields are real and appear to us as particles at points where they are “excited.” DR Theory would call this an attempt to export a digital understanding into analog existence. Spreading a particle over all the space it affects does not digitize it—it only represents an attempt to justify our analog measurements.

DR Theory and Computing If you are familiar with computer science, DR Theory will ring a bell. Constructing digital realities is what modern computing systems do. The verb digitize was coined in the 1950s just to describe how data is prepared for entry into a computer. All the analog information that a smartphone or desktop computer processes is digitized into bits so the device can “understand” it. We could regard the bits inside a computer as forming a new kind of digital

reality, one which emulates the digital realities that living things construct. It is a reality devised for programmers, with its own characteristic objects of knowledge. At the same time, it is a reality constrained by existence because its results must work in the outside world. When we analyze software worlds in all their variations, we find a few common threads running through them. One thread is their division into disparate types of objects, which software designers are careful to work with separately. These types are commonly called data, programs (or code), and algorithms. We can draw an analogy between them and the physical, behavioral, and ideal types of digital reality that living things construct. Data consists of digital representations of “real-world” objects. A simple example of computer data is a bitmap, which embodies one of the ways that computing systems represent still images. To make a bitmap, a digital device such as a scanner or a camera measures the light from the source image at each point of a grid. There is no absolute way to convert colors to numbers, but several standards have been established. The most common, RGB, represents colors additively by the amounts of pure red, green, and blue they contain, as if we were mixing paint. For ordinary color rendering, such as in Web pages, the actual numbers stored or transmitted are usually three 8-bit binary values from 0 to 255. For more sophisticated color rendering, a palette may be defined for a group of images. The palette defines a manageable number of more accurately measured color mixes in specific areas, such as flesh tones or vegetation, up to the maximum choice of 262,144 color shades for each grid point. In a digital world, this is how the Mona Lisa becomes a bundle of data numbers. Programs convert data into other data. Ordinarily, the resulting bits are translated back into real-world numbers, sounds, images, and so on, by reversing the original digitizing process. Programs are time-driven, like organic behavior; they are incremented by the computer’s clock and act sequentially. Programmers think of program elements in temporal terms—do this, then do that. What do programs do to the digital data inside a computer? Mostly they group data bits into variables, attach identifiers to the groups, and use the identifiers to perform operations on the bits. Each variable usually has a type that tells what the bits inside it mean—whether they digitize a quantity or an image, for example. Programmers often assemble groups of bits into data structures, which are patterned as arrays or hierarchies. After all this grouping and labeling of data, the computer’s actual transformations on the bits boil down to a relatively small set of operations—performing arithmetic, applying “bit logic,” copying bits from one variable into another, and the like. Algorithms are general techniques for manipulating digital data in useful ways to produce other digital data. To help humans understand what they do, algorithms are frequently couched in lifelike terms; but when its

anthropomorphism is stripped away, any algorithm can be characterized as simply a recipe for converting sets of bits into new sets of bits. Algorithms do this by specifying patterns of arithmetic or logical operations to be performed on the bits. For example, several algorithms have been published for searching data structures to find specific patterns of bits. They have names, such as “binary search,” “linear search,” and “tree search.” Each algorithm is an ideal pattern that guides programmers in the task of writing software. Algorithms are independent of the programs that execute them. Most algorithms can be executed by programs written in a variety of programming languages, just as an idea can be expressed in various spoken languages. The differences among programs executing the same algorithm are more than just matters of expression because different programming languages often produce different instruction sequences in the machine code that is fed to the computer. Algorithms are expressible as ideals. They follow the logical and mathematical rules that govern numeric ideals, and they are abstract, enduring, and internally consistent. They are often the intellectual property in computer designs because they are the logical patterns for programs, which have commercial value. These three kinds of objects—data, programs, and algorithms—are as separate and different in the software world as patients, doctors, and treatments are in the medical world. Each kind of object is essential for a computer program to do meaningful work, but each can be developed independently. Data may be gathered before any program exists to process it; programs are written to process a variety of data; and computing algorithms are often devised abstractly, using tools such as flowcharting, before they are reduced to working software.

Verifying DR Theory How can DR Theory be verified? Many theories cite crucial experiments or broad-scale studies as proofs of their validity, but philosophers have pointed out how such demonstrations only show that a theory is useful, not that it is absolutely correct. DR Theory is in that position. Its confirmation depends on how well it helps us understand the world, not on how accurately it calculates measurements of physical quantities. DR Theory also bears the burden of being fundamentally dualistic. Ever since Descartes championed mind-body dualism in the seventeenth century (which could be reconciled with the deism of the time), philosophers have been spooked by every ontology that involves multiple divisions in reality. The problem with such systems boils down to answering the questions “In which of multiple worlds do we truly live?” and “How do we access another world from the world we are in?” To these questions, DR Theory offers simple and believable answers. We

truly live in an analog existence that affects us and other living things and with which we interact. When we want to understand and work with that world, however, we access one or more digital worlds that we have constructed, which DR Theory calls reality. Life has spent four billion years making our digital realities represent analog existence, having developed time, space, and categorization among our more basic tools. Understanding digital reality is the primary way we know anything. These answers may raise a further question: If converting analog data to digital reality is the secret of life and the key to understanding existence, why haven’t philosophers been talking about it during the last two thousand years? DR Theory’s answer is again simple: until the invention of multimedia computing, scarcely fifty years ago, no model has existed to show people how global analog-to-digital conversion might work. It took the creativity of computer engineers to demonstrate how digital realities might be constructed. Today we not only use digital realities in our daily lives, but we can also use axiomatic set theory to expose their underlying logic. We are no longer forced to depend on the mathematical analysis of analog quantities to understand the world in which we live. Finally, some may object that DR Theory replaces hard external objectivity with a potentially arbitrary reliance on mental modeling. The answer—again from the findings of computer technology—is that digital data is demonstrably more accurate, more durable, and more usable than analog data. That is why it has evolved as the choice of life and has become the legacy of mankind. Because we cannot find digital data in existence, we are justified in creating it for ourselves. A comprehensive theory about existence, such as DR Theory, is like a suit of clothing. It’s something nearly everybody has, but it adheres to no global standard. It is intended to accompany everyday activities, but it is seldom either essential or inimical to them. Whether it’s dhoti or denims, your clothing helps you feel comfortable with the rest of your lifestyle. Such is also the intention of DR Theory. By explaining how life works in terms of set theory and digital knowledge, it provides a relatively simple underlying structure for individual speculations about the world. Both science and spirituality can live under the DR tent, but understanding how they fit into the DR world does not require the specialized knowledge or terminology that scholarly disciplines demand. That’s why this introductory book about DR Theory more resembles a road map than a detailed tour guide. Astrophysicist and science popularizer Arthur Eddington once composed an image that evokes the overall message of DR Theory. “We have found a strange footprint on the shores of the unknown,” he wrote. “We have devised profound theories, one after another, to account for its origins. At last, we have succeeded in reconstructing the creature that made the footprint. And lo! It is our own.” 4

2. Understanding Existence We understand the world around us in three interconnected ways: behaviorally, physically, and ideally. LIFE AND UNDERSTANDING ARE COEXTENSIVE. Every living thing understands the world

around it, and understanding is something only living things do. A soil bacterium living on a plant root may not understand much; but without its limited grasp of the practicalities of soil chemistry, multiplied by that of trillions of its cousins, agriculture would disappear from our planet. At the other end of living evolution, human understanding ranges from knowing how to play hopscotch to finding antiderivatives in integral calculus. Though our understanding sometimes lacks depth, its variety is staggering.

Types of Understanding What do living things try to understand? Everything—i.e., all existence. Even my dog wants to understand the house and environment we live in, the things we do together, the rules I expect him to obey, the characteristics and activities of other animals in the neighborhood, his bodily functions, and so on. One might at first suppose that any attempt to catalog the things we understand would lead directly to the concept of a single total body of knowledge. If we start from the objects of everyday understanding, grouping them in ever-larger wholes, do we not finally arrive at the idea of the largest possible whole, containing everything we want to understand? The remarkable fact is that we do not. Consider the three examples, discussed below, of understanding a book, a chess game, and an image. Example 1: Understanding a book. On my desk is an ordinary book, which is part of the physical cosmos. If I move it about on my desk the resulting changes in gravitation will spread throughout the universe, making tiny but measurable changes everywhere. In at least this way, then, the book is objectively linked to other physical objects—to the earth and its atmosphere, to my body, to the other planets, and to the distant stars. But even if this were not the case, the book would still be linked to the physical cosmos as I understand it. By this, I mean that I expect the book will react upon other physical things. I can drive a nail with it; it will make a bruise if it falls on me;

it will burn in air. My natural understanding of reality places this book in a class with many other things, all of which react on one another in familiar ways and all of which add up to the physical cosmos. Once it is located in physical reality, I can provide as detailed a physical characterization of my book as I wish. It weighs such-and-such because it interacts with my scales in a certain way. It is hard, rectangular, and so forth because it interacts in certain ways with the appropriate instruments. I can further determine that it is flammable, floats in water, and so on, by bringing it into contact with other physical objects. By such procedures, I can eventually determine all the physical properties of the book in terms of its practical effects upon other physical things. By defining the book physically, do we eventually exhaust all our possible knowledge about it? There seems to be no natural limit to the detail with which its interactions with other physical objects could be cataloged. But I know that this book also happens to be a copy of Plato’s Dialogues. Surely such information has a place in human knowledge about this book; but where does it appear in our physical description? At first, it is tempting to say that the fact that the book is Plato’s Dialogues (and not, for instance, Scott’s Ivanhoe) is a subtle physical property. It is related to the distribution of ink on the book’s pages. By reflecting light from the pages into the eyes of human beings we can elicit the same sorts of reactions as those by which we determined that the book was hard, rectangular, and so on. They would characterize the book as Plato’s and not Scott’s. But now several complications ensue. The book on my desk happens to be translated into English, but suppose it were printed in the original Greek? Having been educated after 1900 I never learned Greek, and hence would probably fail to recognize Plato’s text or be able to distinguish it from a Greek translation of Ivanhoe. I would be blind to this property of the book, even though my eyes were receiving the proper light patterns reflected from its pages. To make this distinction about the book, then, we would have to show it to someone who reads Greek. As a physical property, the content of the book would have to be treated as something tested by special human instruments—namely an English reader for certain books, a Greek reader for other books, and so on. But then how are we to distinguish these instruments? There is no physical characteristic by which we could group them outside the fact that they identify certain classes of books. We would be led to the circularity that certain physical properties of Greek books, i.e. their contents, are only determinable by Greek readers, who are distinguished from other readers solely by the property that they recognize such properties in such books; and the same for English books, Arabic books, and so on. We can include in our physical knowledge all kinds of detail about light patterns reflected from the book’s pages—how the ink marks are shaped, what

variety of marks there are, the degree to which they occur in repeated sequences, and so on; but as soon as we try to extend our knowledge to connect these marks with such concepts as “subject,” “meaning,” “language,” and the like, our knowledge sinks in a quicksand of arbitrary distinctions. Taken physically, these properties of the book become functions of the properties of other objects—the living observers who distinguish them—who themselves cannot be distinguished physically. Does this mean we must abandon any effort to know the content of a book? Obviously not. Such efforts are impossible only when confined to our understanding of physical reality. Let us assume that its content is a property of the book but let us call it a behavioral property. By a procedure cognate to our locating the book in the physical cosmos we can now locate its contents in a behavioral cosmos. To do this we bring it into contact with other behavioral objects and observe the reactions. Just as we measured its weight on a scale, we now characterize its content as it relates to the thought processes of human beings. Using the content of the book as a starting point we can explore a new area of reality, behavioral reality, by understanding the ever-larger totalities of which it is a part. A wealth of new characteristics of the book now emerges. Besides the fact that it is in English and is a collection of dialogues, we find that it is philosophical rather than descriptive, more argumentative than narrative, and so forth. We can analyze its use of language (both in Greek and in English), its style of expression, and all such factors that literary critics discuss. None of these characteristics, so important to us, can be naturally included in any description of a purely physical book located in a physical cosmos. Moreover, the behavioral book—as we may roughly call the content of the physical book—is found to be part of a very large behavioral reality, just as the physical book was found to be part of a very large physical universe. The statements in the book are products of the thought processes of Socrates and Plato, which were in turn embedded in Greek culture of the fourth century BC. Behind them lay a tradition of Mediterranean and Middle Eastern cultures; afterward the writings of Plato were a persistent influence in Roman and European cultures. They helped shape institutions, establish moral values, and determine knowledge. We might compare the intellectual influence of Plato’s Dialogues moving through time to the gravitational influence of a physical object moving through space. Beyond the cultural effects just mentioned lies the whole of human behavior—drives, values, instincts, skills, and so on. These are further connected to living behavior as a whole, from viruses to primates; through the tree of evolution, we could trace the derivation of each behavioral tradition as it has been invented and perfected. Has the potential of this book as a starting point for exploring the totality of our understanding now been used up? The answer is no; there is at least one more universe with which it is associated. Let us turn to one of the dialogues,

called the Timaeus. It starts with a summary of part of the Republic, after which one of the persons of the dialogue, Critias, recounts the legend of Atlantis. Here there is no problem of understanding, even though as far as we presently know Atlantis never actually existed. The physical references—the size of the island, the earthquake and flood that destroyed it, the mud remaining where houses had been—are all comprehensible because they refer to the sorts of things we encounter in physical reality. The references to behavior—the bravery of her warriors, the magnanimity of her leaders—are similarly comprehensible in terms of the behavioral reality we understand. But then Timaeus starts unfolding an elaborate cosmogony, including a scheme for associating the elements of Empedocles (earth, water, air, and fire) with what are now known as the Platonic solids. The Greeks had already known about geometric solids bounded by identical regular polygons for a couple of centuries. Theatetus, Plato’s contemporary, described them and was said to have proved that there could be only five such. Euclid later made them famous by devoting the thirteenth book of his Geometry to them. Intrigued by the solids’ property of decomposing into one another under simple geometric transformations, Plato assigned four of them to what were then the traditional physical elements: the tetrahedron to fire, the cube to earth, the octahedron to air, and the icosahedron to water. The dodecahedron was taken to represent the whole cosmos. A geometric calculus could then be formulated in which the decomposition of each solid into sets of the others would parallel the transmutations that were thought to occur among the physical elements. All this is set forth in the dialogue. I mention this theory not for its intrinsic explanatory value, although it enjoyed a lengthy vogue during the Middle Ages. I mention it to illustrate this question: how do we understand the Platonic solids that it discusses? Are they part of physical reality or part of behavioral reality? Of course, it is easy to manufacture physical objects in geometric shapes—a cube of sugar, for example. But a sugar cube is not in any sense a geometric cube, because the sugar does not have any of the properties required of the geometer’s object. Its faces are not perfectly flat, its edges do not meet in exact points, and so on. When we prove a theorem about a geometric object we never refer to any physical thing; in fact, it is just as easy for us to prove theorems about shapes that can barely be represented physically at all, such as the tesseract. When we create physical things in geometric shapes as an aid to visualization, it is always clear that they are not perfect. Since perfect correspondence to description is a necessary property of anything subject to geometric proof, such things cannot be physical objects. A subtler explication for geometric objects is that they are figments of behavior. In this view Platonic solids, for instance, exist just to the extent that we think about them. All we know about them (and about all other entities of geometry, mathematics, and logic) we have learned through strictly mental

operations. The proof that there are only five possible regular convex solids does not require that we examine the shapes of all possible things, or indeed that we use our senses in any way. It follows from the axioms of geometry by logical processes. It is a truth we acquire by sitting quietly in a chair and thinking, the sort of knowledge some classical philosophers called a priori. Because the whole process begins and ends in behavior, it is natural to suppose that it refers only to more behavior—that Plato’s statements about the tetrahedron, for example, refer only to an idea that philosophers discussed. Those not in the field often fail to realize how extensive the disciplines of logic and mathematics are. Whole libraries are devoted to housing their conclusions. In an address delivered in 1900, mathematician David Hilbert set 23 fundamental problems as a background for twentieth-century mathematical research. Most have yet to be satisfactorily resolved, and some have yielded the remarkable conclusion that they are undecidable within present conceptualizations. It is clear that for each of Hilbert’s propositions, determining whether it is true, false, or undecidable on the basis of presently accepted premises is truly a search for knowledge. Yet it is not a search of behavior, for we have no control over its outcome. The only way we can influence the outcome is by changing our definitions and axioms, in which case it becomes a new and different search. Our understanding is enriched by such work, but what we end up understanding is neither physical nor behavioral. We come to know a third universe, a division of our understanding that contains objects which DR Theory calls ideals. Ideals in this sense are real objects of understanding. The objects mentioned in Plato’s Timaeus—tetrahedron, cube, and so on—are discovered by logical explorations that also yield a wealth of other ideal objects. Just as the physical book before me is a point of entry for understanding physical reality, and Plato’s Dialogues is a point of entry for understanding behavior, so the regular polyhedra provide a convenient (although arbitrary) starting place for understanding ideals. From them, we can branch in many directions into geometry, mathematics, logic, and beyond. The field of ideals is not limited to entities such as numbers and geometric shapes. In the twentieth century, the development of powerful general concepts in semantics and symbolic logic showed how most abstractions can be connected in our understanding. In particular, the concepts of form, essence, and universal that pervaded classical philosophy refer to what DR Theory calls ideals. It is now possible—through an understanding of relations, functions, and classes—to demonstrate the kinship of purely philosophical ideals to more rigorously described logical abstractions. They are all the same sort of thing in our understanding. Example 2: Playing chess. Discovering that the totality of ways we might understand things is divided into three different processes—physical, behavioral, and ideal—might sound fatal to DR Theory. Wasn’t the big

problem with Descartes’s famous Meditationes of 1641 his conclusion that the world of thought was completely disconnected from the world of material things? Ever since that time, philosophers have grumbled about how Cartesian dualism splits the world in two: isn’t DR Theory now proposing a digital triplism? Wouldn’t that be half again as bad as a dualism? The answer, of course, is that we’re presently talking about ways of understanding, not about tangible substances. Physical, behavioral, and ideal are like three different strategies in the “understanding game.” For example, let’s compare it with three ways that we might understand a position in a chess game: Physically: certain pieces are located spatially on certain squares of a chessboard; Behaviorally: certain sequences of moves are possible in future time, resulting in new positions; Ideally: the pieces on the board form certain abstract patterns. The physical representation of a chess position is familiar to anyone who has seen the game. It consists of 32 or fewer little wooden tokens placed on a 64-square checkerboard. We can recite a purely physical and spatial description of the chessboard and the layout of its playing pieces that can be understood by anyone, including someone who doesn’t know how to play the game. The behavioral representation, described in terms of possible future moves, is more complex. It is understandable only by someone who knows how to play the game. We say that a particular pawn guards a bishop or that a knight threatens a rook. This description treats the pieces as acting in time—their guarding or threatening characteristics depend on their being able to move in certain ways in the future. It is a time-based description of the potential behavior of the chess pieces. The ideal representation of a chess position is denoted by patterns. The pieces on the board make up configurations that only an experienced player will recognize. Here’s how a chess expert describes it: To a beginner, a position with 20 chessmen on the board may contain far more than 20 chunks of information, because the pieces can be placed in so many configurations. A grandmaster, however, may see one part of the position as “fianchettoed bishop in the castled kingside,” together with a “blockaded king’s-Indian-style pawn chain,” and thereby cram the entire position into perhaps five or six chunks. . . A grandmaster can retrieve any of these chunks from memory simply by looking at a chess position, in the same way that most native English speakers can recite the poem “Mary had a little lamb” after hearing just the first few words.5

In this way, a chess position may be understood as an ideal pattern made up of ideal sub-patterns. Which of these representations captures the real chess position? Is the position (and thus the entire chess game) a matter of physical piece distribution, temporal move behavior, or ideal patterns? Here are three equally valid descriptions of one thing, differing only by the types of understanding they exemplify. Yet their differences evoke divergent ways in which chess players understand and play the game. More importantly, they often determine who wins and who doesn’t. We can draw an analogy between components of the computer software world, described earlier, and the ways of describing a chess position. Data is like the physical configuration of a chessboard—a stored starting point for playing the game. A software program is like a sequence of chess moves, occurring step-by-step. And an algorithm is like a set of positional strategies for achieving some goal in the game. Example 3: Digitizing an image. Parallelisms exist between the ways that computers process information about the world outside them and the ways that living things accomplish similar tasks. This happens because, during the evolution of computer technology, designers tended to emulate in machinery the methods of human work with which they were familiar. Computers were designed to work like people do. Computer processing includes the technologies of analog-to-digital and digital-to-analog conversion, which translate between things in the real (noncomputer) world and sequences of bits that a computer can handle. These technologies let computers deal with text, images, sounds, and so on, none of which are made of bits in the noncomputer world. As an example, seeing how computers represent images as bits can help us realize why our understanding of existence is divided into three types. An image in the noncomputer world is a configuration of light on a surface. It can be detected by a matrix of photoreceptors, which generate electric currents that a computer can interpret as binary numbers. Conversely, binary numbers in a computer can modulate color cells in a flat panel display, producing a configuration of light on its surface. In both cases, the computer treats external images as continuous regions of light that blend into one another, not sets of discrete “light objects.” That is why images are called analog. The computer must represent such analog objects as bit sequences, a task that it can perform in several ways: As explained under “DR Theory and Computing” in Chapter 1. the computer can divide the image spatially into a bitmap, storing a measurement of the light present at each point of a grid. We might call this physical digitization because the computer’s bit sequences represent physical light occurring at specific locations in the image.

The computer can create a set of numerically coded instructions for drawing the image. This technique is most commonly used when the nature of the image lends itself to delineation—for instance, a block of text lettering. We might call this behavioral digitization because what is stored is a sequence of drawing instructions. The computer can analyze the image into geometric regions, each of which is defined in a file of equations for generating standard shapes. It then encodes the location, dimensions, color, and so on of each shape, thus representing the image as a collage of predefined geometric objects. We might call this kind of representation ideal digitization. Note that all of the digital representations are generically different from the analog original—regions of light on a surface are represented by sequences of bits. Also note that the different ways which computer engineers have devised to represent images digitally—as pixel maps, as drawing instructions, or as geometric decompositions—correspond to the types of understanding described earlier: physical, behavioral, and ideal. And finally, note that each of the three modes of digitization encodes the external analog image into a completely different sequence of bits inside the computer. What happens during image digitization? At one end, we have a real thing, a visual image, that the computer needs to “know.” At the other end, we have a choice of digital representations, each an ordered set of binary digits that the computer can digest. In the middle lies a process designed to generate digital numbers that represent the image. The computer can never know the image directly, for configurations of light are intrinsically incomprehensible to it. It can only process binary numbers, and it turns out that each image can be represented by at least three totally different sets of numbers—numbers that not only differ in value but also differ radically in the ways they are used. The physical pixel numbers are used to lay down points of color; the behavioral drawing numbers identify routines that outline and fill spaces; the ideal shape numbers are used to retrieve abstract graphic definitions from memory. Working on a single image, the three digitization processes not only produce different sets of numbers but also numbers that have different meanings. For the computer to treat the pixel representation as a set of drawing instructions, for instance, or as callouts from a library of standard shapes, would constitute a fatal processing error. Inside the computer, these different ways of knowing an image stay confined to separate processing tracks. This is analogous to the imperative that each type of digital reality— behavioral, physical, and ideal—has its own way of being understood. Although it is foreign to the computer, the image being digitized in our example exists independently. Any of three bit-sets in the computer’s digital reality—a physical pixel set, a behavioral instruction set, or an ideal shape set —can be factually bound to that analog image. Changing a single bit will

make the bit-set more or less faithful to the original. Veracity is a crucial property of each of the three digital representations of the image, even though their bit-sets are completely different. This truthfulness to nature is like the fact that analog existence engenders digital reality, even though digital reality may appear to us in a completely different form.

Time, Space, Pattern Living things survive in existence because they know how to extract energy from their surroundings and use it to build and develop themselves. The knowledge that makes this possible can be organized in various ways. DR Theory traces the evolution of three kinds of practical knowledge: How to organize behavior using time How to arrange physical objects using space How to recognize patterns using ideals DR Theory limits its definition of life to what is observable on Earth; it also accepts the current scientific view that Earth was formed about 4.5 billion years ago and that life arose about half a billion years later. Organizing behavior. DR Theory posits that time ordering is the skill on which terrestrial life was founded. What role did time play? First and most importantly, it enabled heredity. Some organisms developed behavior that increased their ability to survive and then—later in time—they managed, by fission, budding, or cloning, to pass their life skills to new organisms. Otherwise, they perished and their skills were lost. A digital time sequence of survival followed by reproduction was essential to this process and was also something new in existence. Its emergence in early organisms helped disengage life from raw analog existence. Time sequencing also enabled certain survival skills. Foremost among these skills must have been metabolism, which is based on repeatable sequences of chemical reactions. Energy transfer using adenosine triphosphate (ATP), which is universal in life today, must have appeared early. The Krebs (or citric acid) cycle used today by all aerobic organisms makes more than 30 ATP transfers, interleaved in a specific order with other reactions. Earlier forms of life may have used fewer. The point is that without temporal event sequencing, no process such as organic metabolism would have been possible. In DR Theory, time is the simplest form of digital set construction. It forms countable linear sets that accept new elements at only one end. You might visualize such an organization of set elements as like a bookshelf where you can add new books to the end but you can’t slip them among the books already shelved. Ordering algorithms. With the advent of genome-based reproduction, algorithms for ordering physical things in space may have evolved as an early

life skill. An organism that knows only a time dimension may reproduce itself by fission or cloning; but to establish an evolving species, the organism must create a set of coded instructions for building new creatures. Life’s crucial breakthrough was the development of the DNA molecule, which stores sequential instructions by representing time spatially. By reading codons— groups of amino acids strung spatially along the length of DNA genes—an organism can execute the time-sequential actions necessary to manufacture proteins. The positions of certain codons in one dimension of space represent the serial steps in time required to synthesize each specific molecule. A digital reality with one dimension of space is adequate for reading DNA and also for feeding oneself by random encounters with organic and inorganic molecules. But around three billion years ago, life on Earth achieved an evolutionary breakthrough in photosynthesis. At the time, a few phototrophic microbes could capture energy from sunlight, but they did so inefficiently and did not extract carbon from the environment. Solar photosynthesis ultimately evolved into a versatile chemical factory that uses the energy in sunlight to break carbon dioxide and water into their elements, while synthesizing a variety of useful organic molecules on the way and filling our atmosphere with oxygen. Life’s use of solar energy may be responsible for additional spatial dimensions in digital reality. Sunlight comes from a single distant source; it rewards organisms that can locate that source and collect the sunlight on large surfaces orthogonal to its vector of propagation. Why three spatial dimensions? If sunlight had been like lightning, striking Earth at single points, the organic structures that captured it could have been one-dimensional, recognizing only up and down. But sunlight is best captured by horizontal surfaces; it needs two non-vertical dimensions to define an effective collector. From a living organism’s viewpoint, moreover, sunshine falls not in successive lumps but continuously. Hence, its collection space cannot be linear—it must consist of multiple coexistent accumulators. To extend the bookshelf analogy, space must be like a library to which you can add books wherever you want, not just at the end of the time shelf. The evolution of space-time. Life could have evolved any of several combinations of time and space dimensions to organize its digital knowledge. But consider that life on the surface of the Earth is whirled about in a complex motion with respect to the sun—a cycloid curve compounded from our planet’s orbital speed of 67,000 mph and its rotational surface speed up to 1,040 mph. To maximize the ability of primitive photosynthetic organisms to track the life-giving sun from such an itinerant platform, the relation between time and space that life evolved had to be such that the four-dimensional vector of propagation of radiant energy remained constant as life moved around it. Einstein intuited the truth about this relationship between time and space—

equivalent to saying that the speed of light is the same regardless of the motion of its observer—but he regarded it as a given feature of existence, not as a product of biological evolution. He also intuited that gravitation can be explained by space-time curvature. In DR Theory, both the constancy of propagation of light energy and the proprioception of gravitation are features of life’s understanding of time and space. This understanding evolved over the billion or so years that life on earth was dominated by photosynthetic plants. At first, plant life was probably mere floating cells or surface scum. As species developed, they must have grown upward from the planet’s surface and competed to expose their leaves higher than their neighbors’ leaves. Success, for a plant, required both the ability to track the sun in the sky and constant knowledge of which way was up. Over a billion years, the understandings of time and space that plants developed became embedded in chloroplast DNA, from whence it was inherited by animal DNA and today is puzzled over by humans. One way to characterize space-time in DR Theory is that it represents the simplest ordering algorithm by which a variety of living things as simple as chloroplasts, located on Earth, can find and metabolize light energy from a distant source. Terrestrial life spent a billion and a half years developing this algorithm. Life in some other part of existence might come up with a totally different form of space-time. To cover this potentiality, Relativity Theory needs only a short extension: besides space-time being relative for observers in motion, it is also relative for observers with different evolutionary histories. Recognizing patterns. Philosophers do not usually include recognizing patterns as cognate to recognizing space and time. But we use patterns for the same reasons as we use space and time: to order objects in digital reality. We can place ideal objects within a pattern, just as we can locate physical objects in space and catenate behavioral objects in time. As with space and time, patterns help us separate and arrange objects algorithmically. It seems likely that the recognition of patterns emerged in the evolution of life as a result of predation. Prey could improve their chances of survival by recognizing patterns of hunting behavior in predators; conversely, predators could feed themselves more efficiently by recognizing patterns of avoidance in prey. In humans, pattern recognition has broadened out to spawn a whole intellectual world of ideals. From the days of Aristotle onward, our ability to order ideals by pattern has been a fundamental method for obtaining, preserving, and communicating the sophisticated knowledge that we possess. Many of the ideals that we understand would be useless to us were we not able to arrange them in lists or hierarchies. And our speech and writing would barely serve our needs if it were stripped of the analogies and other figures of speech that depend on people’s mutual recognition of patterns. Patterns take myriad forms. They include hierarchies, linearities, networks,

and all kinds of linkages. In mathematics, they are identified as fields (of numbers), manifolds (of points), functions (of variables), sets (of elements), groups (of transformations), and so on. In other disciplines, information is arranged in lists, arrays, trees, and so on. Although patterns are ideal, they often arrange physical or behavioral knowledge. Imagine a collection of physical color samples, of the kind used to sell paint. They may be arranged in a rack or on the pages of a book, but always in a pattern—for example, from red hues through the spectrum to violet, with each hue’s different tints and shades laid out next to it. We navigate through the color samples by understanding their pattern, making it easy to find the one we want. For a behavioral example, consider the hierarchical way the entries in a dictionary are arranged: first by each word’s orthography, then by its parts of speech, and finally by its different usages in various contexts. We find the word we want by matching its location in the dictionary’s hierarchy to the communication behavior we want to perform.

Using Ideals What DR Theory calls ideals are life’s latest evolutionary development. They are the universal patterns that philosophers call a priori plus the abstractions that make organisms clever as well as adaptable. In the language of set theory, ideals are found in the powersets of physical sets. A powerset is the set of all possible subsets of a set. In effect, it provides an inventory of all the ways the elements of a set can be associated with one another. When physical ordering in DNA provided life with a way to preserve recipes for survival, the powerset of physical ordering—ideals— emerged as a factor in evolution. By revealing all the possibilities of manipulating the physical world, ideals helped both species and individuals solve problems more efficiently than would have been achievable through trial and error. It turned DNA and neural tissues into laboratories for developing new ways to live. Nature interpreted through its patterns turns out to be not about the ordinary objects and events that we recognize in daily life. Our new explanations are couched in ideal terms. Understanding these explanations requires the formulation of definitions. Definitions, in turn, construct new objects in digital reality that are the properties of physical things and events. By formulating definitions, we create new objects in digital reality. For example, we might identify the source of some gray, solid, heavy sensations as a rock, a physical thing. A definition of weight lets us add a new object to our digital reality—the weight of the rock. We acquire new routines to measure the rock’s weight, by constructing a weighing machine and a unit of weight. The weight comes out as a number—an ideal—but for us, it is as valid as any sensation. The weight number, which we call a property of the rock,

joins its grayness, solidity, and heaviness in our understanding of that physical object. We discover that virtually every physical thing has a weight number. Even air can be weighed by using the right equipment. We can also measure familiar characteristics of our rock—its grayness (using a colorimeter), its hardness, density, and so on. But in so doing we would often be adding new property numbers to our digital reality, objects that we understand in terms of ideals, not replacing characteristics that we already know and understand in behavioral terms. The principles of modern science are all about properties: mass and energy, for starters, plus velocity, density, entropy, specific heat, and so on—a host of measurable properties that are required for scientific principles to work. For example, when I pick up a rock I have little idea what its magnetic permeability might be. Yet that property is as tangible as its shape or color. Certain natural principles are about magnetic permeability and are meaningless unless we define that property in physical things. Thus, the rock I hold in my hands is a more complex object than my perception tells me, and I might be surprised by the ways in which it would react when placed near a powerful magnet. At the simplest and oldest levels of living evolution, the actions of organisms can be understood in terms of time-ordered reflexes, urges, and needs. At the next higher level, physical objects ordered spatially become factors in life’s understandings—habitats, environments, ecosystems, etc., as well as human artifacts. At the top level, we must call on ideal patterns such as goals and strategies to understand how life survives as it does. DR Theory explains how the complexity of human life is attributable to our ability to be informed and guided by ideal categories. Ideals provide inventive alternatives to emotional reactions and physical pressures. Besides deepening our understanding, ideal patterns form the basis for much of our social and communal lives. They make us the most creative (and also the most troublesome) animals on Earth.

Sets and Digitization In 1623, Galileo analyzed the motions of heavenly objects using mathematics, and sixty-four years later Newton published what amounted to a conceptual proof of that idea, his Mathematical Principles of Natural Philosophy. Ever since that time, scientists have largely assumed that the primary way to understand physical nature is by measuring analog existence and then analyzing the resulting numbers. Numerical analysis of analog measurements has proved useful for making predictions. For example, by taking a few precise astronomical measurements we have been able to predict exactly when an eclipse of the sun or moon will

happen and where on the Earth’s surface it will be visible. The most compelling arguments for the validity of the physical sciences have come from their ability to predict future events. Yet numerical analysis has contributed very little to our understanding of the world. Knowing exactly when the sky will darken and the sun will be blotted out has told us very little about what is happening or why. As the physical sciences matured, it turned out that from a logical viewpoint nobody really knew what numbers were. Set theory. During the twentieth century, set theory came to the rescue. Logicians discovered that there were several ways to define numbers in terms of sets. The converse is not true; sets can be defined by axioms but not by numbers. What finally emerged, and is still a hot topic, was foundational or constructive mathematics—a logical discipline in which axiomatic set theory defines the basic ideas and procedures of traditional mathematics.6 Axiomatic set theory also provides the formal logic for DR Theory. Set theory has achieved a grip on human thought because it recognizes one of the basic operations that people use when constructing digital reality. This is the operation of recognizing an aggregation of objects as an object in its own right, different from and independent of any of its elements. A set is simply a collection of things of any kind, considered as a group. The power of this concept stems from the idea that the set is a new thing by itself, different from its elements. By forming a group, we create a new object. Among the things that might be collected into a set are other sets. For example, we might form a set of all the inhabitants of a village, plus a number of other sets each of which contained the members of one family in the village. Then we could form a set of all the families in the village. This set would be a completely different thing from the set of all the inhabitants, even though the ultimate real constituents of both sets—people—might be the same. The set of inhabitants would still be a set of people, but the set of families would be a set of other sets. Thus, set formation can be thought of as a tool for making new things—sets —that are considered to be as real as the ingredients from which they are made. In these terms, set formation symbolizes the construction of digital reality. DR Theory fills in the details as described below. Constructing a digital object is like forming a set out of some part of existence. The elements of a set do not need to be countable or discrete—for instance, they can be ranges of time or space. The principal requirement is that we be able to tell whether or not a given thing is or is not in the set. Hence our saying that an object can be represented by a set whose elements are parts of existence need not ascribe any properties to existence beyond the possibility of separating parts of it from other parts. “Object sets” can be formed simply by delimiting analog existence and giving what’s inside the limits a new identity as a set. Forming a digital category is like forming a set whose elements are sets.

New object sets, made by delimiting existence, typically become elements of existing category sets. These categories start out as sets of object sets that we associate in some way—for example, “red objects.” But as sets, categories may themselves be categorized; this is called objectifying categories. A category set becomes an object when it becomes an element of another set. The processes of set formation and the use of sets as elements of other sets thus result in hierarchies of sets—categories that explain other categories. The importance of categorization is that it lets us construct objects that are not formed directly from existence. In set theory, an apple might be treated as an object set formed from existence—a set containing the apple’s physical constituents. That object set could then become an element of the category set “red objects,” which would contain the apple and other object sets that we choose to group together while attempting to understand reality. Another category set, “colors,” could also be constructed in digital reality, containing the sets “red objects,” “blue objects,” “green objects,” and so on as its elements. Making the set “red objects” an element of “colors” would change it into an object in our digital reality. What had been only a grouping of apples and other objects that affected our vision in similar ways would become “red,” an object that we understand as a color because it is an element of the category set “colors.” Note that we have constructed an understanding of color by grouping objects in digital reality, not by delimiting parts of existence. Generalization is like forming a set using the logical operations of set theory. Logicians have extended “first-order logic” (by itself an extension of Aristotle’s syllogistic logic) to define ways of making new sets from existing sets. For example, the union operation forms a new set by combining the elements of multiple other sets, discarding duplicates; the intersection operation forms a new set from the elements common to several sets. These operations on sets are often illustrated by means of Venn diagrams. By combining them with other operations, any number of new sets may be formed around elements selected from other sets. Using Boolean algebra, we can express the selection criteria for forming these new sets to virtually any degree of complexity. Their set inclusions are determined by relating other sets, not just by aggregating elements of sets. Freedom of knowledge. Living things learn, and their species evolve, as a consequence of life’s freedom to form arbitrary objects and categories, right or wrong. This freedom appears in set theory as the axiom of choice. It asserts that from any collection of sets, a new set can be formed by selecting one element from each set in the collection. In terms of categorization, the axiom of choice guarantees that there is no natural limit to the ways we can categorize the objects of digital reality. The minimum requirement for our understanding any object in digital reality is that the object be an element of at least one category. Most objects belong to many categories and most categories contain many objects as

elements. When we encounter a problem with understanding one or more objects, a common solution is to place them into another category. The axiom of choice guarantees that we can always do that. It lets us take objects we don’t understand from the categories they are in and construct new categories around them that may help us understand them better. As Georg Cantor was developing what is today called naîve set theory, at the end of the nineteenth century, his guiding concepts arose from lifelike operations—grouping things, determining their number by counting them off (one-to-one correspondence), dividing groups into smaller groups, and so on. Although these concepts were subsequently formalized into rigorous axiomatic systems (including the addition of the axiom of choice in 1904), Cantor’s original concepts have survived as the core meaning of set theory. It seems evident that they formalize a deep-set human capability. Natural digitization. The behavior of forming a set is a natural act of digitization. What had been an ad hoc collection of elements becomes a new unitary object. You can find this kind of A/D conversion everywhere in living organisms if you know where to look. In animals, for example, the all-or-none law states that the strength of the response in a responsive system is normally uniform and independent of the strength of the system’s stimulus. Stimulation above a threshold level causes a full response; stimulation below that level causes no response at all. This one-bit analog-to-digital effect, which has been observed in both neural and muscular tissue and at all stages of biological data flow, filters out source noise and maintains information in purely digital form. Life’s non-neural chemical systems, particularly in plants, can be equally selective: cell membranes and molecular receptor sites pick and choose molecules as metabolism presents them, absorbing only the ones with useful digital characteristics. Natural categorization. In DR Theory, differences in set nesting can change our understanding of what we know without impairing its objectivity. Remember that a set and its elements are naturally independent. When you make one set an element of another set, you may change the way you understand the nested set but not its empirical source. During ideal categorization, for example, the original existential events that produced a physical or behavioral object remain in its history. This is how the knowledge produced by digital categorization retains its factual links to existence, and why it does not spawn dualisms of the Cartesian kind. The idea of a tripartite reality is not new with DR Theory—it is at least as old as Western science. In 1789, for example, Lavoisier wrote in his Traite Elementaire de Chimie: Every natural science always involves three things: the sequence of phenomena on which the science is based; the abstract concepts which call these phenomena to mind; and the words in which the concepts are

expressed. To call forth a concept a word is needed; to portray a phenomenon a concept is needed. All three mirror one and the same reality.7 In the language of DR Theory, Lavoisier’s phenomena are physical, his words are behavioral, and his abstract concepts are ideal. All three are present in our digital realities as different ways of representing analog existence. Example: Perception. Lavoisier’s chain of scientific understanding— from behavioral word to ideal concept to physical phenomenon—is more typical of formal reasoning, a topic discussed in Chapter 5. For a simpler example, consider the set operations that occur during an act of natural perception. Imagine that we are observing a cluster of red balloons. To understand what we are seeing, we must categorize our sensations of redness and roundness by grouping them in two-element sets, placing each set in the physical category of “tangible things.” The sets themselves we identify as “balloons.” In this way, we populate our digital reality with physical red balloons. To distinguish the balloons from one another, we assign each one a position in space. Our natural perception of physical space is not a Cartesian coordinate system; it is more like a large set of location subsets, each with a distinct identifier such as “left of the tree,” “above my head,” and so on. Identical-looking balloons can be distinguished from one another by tracing their inclusion in different location subsets in the space set. Now one of the balloons breaks loose and floats away. We have the same paired sensations of redness and roundness but with one pair the spatial location that we added to the red + round set changes at different stages along our behavioral timeline. We perceive a series of indistinguishable balloon objects, each in a different location. We explain this event by grouping the multiple physical objects into a new set in our behavior, identified as a change object. It’s like assembling the different frames of a movie into a single shot. As the balloon floats away, we categorize the new set in our perception behavior as a motion change, perhaps calling it “balloon floating.” If one of the balloons is punctured, on the other hand, we observe it in two states—big and round, then small and floppy. Our behavior combines the two perceptions into one physical set that we categorize as a transformation change. In both this case and the case of motion, what might otherwise be confusing in our behavioral perceptions is resolved as a change in physical reality. The end result is that we humans (like other living things) construct a workable digital reality filled with physically-categorized sets, each of which explains some set of sensations in our behavior. We perceive and understand things outside ourselves. As this example illustrates, in DR Theory sets are the ultimate units of

understanding. A set may contain elements of raw existence, but we and other living things have not evolved the ability to understand existence directly. What we understand are sets. If a set contains parts of existence, we digitize those parts; if it contains other sets, we combine their digital information. To understand a set, we trace its position as an element in other sets, which act as categories. This process of locating a digital set in a digital reality determines both what it is—by identifying its contents—and how we understand it—by analyzing the categories that include it. Categorization is discussed in more detail in the next chapter.

3. Constructing Reality From three types of understanding, we construct three types of digital reality, using each type to categorize the other two. TO SUMMARIZE THE PREVIOUS CHAPTER,

living things interpret analog existence by forming digital sets within a hierarchy of types of understanding—behavioral, physical, and ideal. As this chapter will explain, the physical sets are powersets of the behavioral and the ideal sets are powersets of the physical. All these sets and their elements constitute the evolving raw materials from which living things construct digital realities in their behavior.

Digital Reality Types The sets of raw materials from which living things construct digital realities do not arrive anonymously; they normally carry hallmarks of their origins and understandings with them. Here are the primary characteristics that digital realities derive from the sources on which they are built: Type of understanding: behavioral, physical, or ideal Set cardinality: aleph-null, aleph-one, aleph-two Element ordering: time, space, pattern Set packing: linear, dense, complex These characteristics define three basic digital reality types. They are associated as shown in the following table and are discussed in the next sections: Reality type Behavioral Physical Ideal

Cardinality Ordering Packing Aleph-null Aleph-one Aleph-two

Time Space Pattern

Linear Dense Complex

The three types of digital reality conform to our types of understanding. Sensations, thoughts, emotions, volitions, etc.—our internally understood experiences in general—become parts of our behavioral reality. What we understand as external objects and events, including those in our bodies, become parts of our physical reality. Universals and a priori truths become parts of our ideal reality. As they are categorized and understood better, reality sets of each type may become elements of sets of other types. This happens mainly as a result of theorizing, as explained later in this chapter. Cardinality refers to the ways that digital realities handle the concept of infinity. DR Theory includes it to link itself firmly with axiomatic set theory. In the Zermelo-Fraenkel axioms, the axiom of infinity guarantees the existence of the simplest possible infinite set (equivalent to the set of all integers). From that set, using other axioms, all infinite sets can either be generated or proved to exist. The existence of infinite sets is important in set theory because it distinguishes the concept of a set from the mere concept of multiple things. In founding set theory, Cantor postulated that one could determine that two infinite sets had the same number of elements just by showing how their elements could be counted in one-to-one correspondence, without specifying how the count was to terminate. Thus, for example, the set of all integers was the same size as the set of all even integers because the two can be counted side-by-side forever, despite the objection that the count would miss half the elements of one set. On the other hand, Cantor proved that the set of all algebraic numbers (such as numbers with decimal-point extensions) could not be counted one-for-one alongside the set of all integers. In a truly fundamental way, there were more algebraic numbers, and so the sets of all of the two kinds of numbers had logically different sizes—which came to be called their cardinalities. Finally, the two infinite sets had to be real things—apart from the elements they contained—because the sets by themselves had provably different cardinalities. Talking about sets had suddenly become legitimate. Discovering how to create a series of mathematical objects that were both real and generically different from one another, as Cantor’s transfinite sets were, ignited a firestorm among mathematicians and philosophers. They grumbled that Cantor was dabbling in metaphysics. The controversy did not simmer down until David Hilbert, the doyen of modern mathematics, dubbed set theory “Cantor’s paradise” in 1926. The cardinal numbers that represented the sizes of transfinite sets turned out to be strange beasts. They were larger than all the real numbers, yet they didn’t measure quantities in the normal sense. It appeared that sets came in noninterchangeable size types, like magnums and jeroboams of wine. Cantor adopted the Hebrew character aleph, with a subscript, to designate an ascending family of infinite set sizes, each of which was the powerset of its

predecessor. In Cantor’s numbering system, aleph-null is the cardinality of the set of all the integers (1, 2, 3…) and all its infinite subsets (such as the set of all even integers). Aleph-one is the cardinality of a continuum, such as the set of points on a line. An aleph-two set might the cardinality of the set of all functions of a continuum; think of it as every possible wiggly line you could draw in an infinite space. The crucial fact about two sets whose cardinalities have different subscripts is that you can never compare them element-by-element because the set with the higher cardinality is always larger in a truly fundamental way. DR Theory asks the trio of questions, “What are the limits of our inner behavioral experiences and responses, and of the external physical world, and of all the abstractions and truths that we might discover?” The instinctive human answer is, “The limits are infinite in all three cases.” But we are talking about digital reality, where all our knowledge is contained in sets, so the single answer “infinite” won’t do. If we want to use set theory to analyze digital reality, we must play by Cantor’s rules and assign transfinite cardinal size types to our sets of behavior, physical reality, and ideals As the table at the beginning of this chapter shows, the cardinality of behavioral sets is aleph-null, of physical sets aleph-one, and of ideals alephtwo. This is the key that admits DR Theory to Cantor’s paradise. It also helps resolve something that has bothered philosophers for centuries: why our inner (behavioral) worlds, the outer (physical) world, and Plato’s heaven of abstractions (ideals) feel so different to us and seem to be walled off from each other. It is because they are sets of different cardinalities. If I appear to be explaining one puzzle by an even deeper puzzle, perhaps the next sections on the ordering and packing of digital reality sets may shed some light. Ordering specifies the ways that contents are added to, or removed from, digital reality sets. Time orders behavioral sets, space orders physical sets, and pattern orders sets of ideals. One way to visualize how these three mechanisms work is by imagining a more mundane analogy, such as three different kinds of libraries. You access and return books in all three kinds, but each kind has its own rules. The “time library” is a single shelf of books. You can retrieve or return books only at the righthand end. The librarian is constantly adding books to the end, so you have a variety of literary fare; but when you want to read something you have to be satisfied with what is there. However, you can place a request slip at the end of the shelf and in most cases, the book you request will promptly be added. A simple aleph-null library, but a useful one. The “space library” is more like what we think of as a large public library. You can wander among its stacks, where books are grouped in various ways— some by title, some by color, some by size. To a limited extent you can add, remove, and rearrange the books; but if you want to read one you must go to

the conveniently located time library and request it. The aleph-one space library is built for storage and browsing. The “pattern library” is a daunting affair. People get lost in it. Once admitted, you can trek its endless halls, filled with books everywhere. Jorge Luis Borges called it the “Library of Babel.” For every book in the space library, the pattern library contains an entire room full of variations on it. Some variations are truer than the original, but most are not. Some variations seem to be nonsense until a reader comes along who understands their subtle messages. Others are truly nonsensical but widely read. Who can tell? It’s a fun aleph-two library, but one that must be used carefully. The patterns in the pattern library are as central to human life as space and time. They define the relations that make our thoughts and communications intelligent. Without them, our digital realities would be devoid of imagination, speculation, and nuance. They would be mere heaps of data, flat and undigested: when we tried to communicate their contents, our speech would be limited to the most basic pidgin. Packing denotes the structure of each digital reality—how each one stores the knowledge it contains. The library analogies described above foreshadow the terminology that DR theory uses, as described below. Behavior, ordered by time, has a linear structure. Every perception, thought, communication, volition—in short, every instance of a living thing’s behavior—is either before or after every other and hence can be counted in an orderly way. Moreover, past behavior cannot be changed and future behavior requires effort to make it happen. Our behavior marches endlessly in one direction, like the integers in Cantor’s set of cardinality aleph-null. Physical reality, ordered by space, has a dense structure. It is said to be infinitely specific, meaning that when physical things are dissected, they yield new details indefinitely. This feature also makes the parts of physical reality uncountable—between any two parts you can always find other parts. This suggests that space is a continuum, not a collection of points separated by something else, and hence that it forms a set of cardinality aleph-one. Ideals, ordered by their patterns, have a complex structure. One way to think of the world of ideals would be the set of every way to join together points in space to make lines and pictures, from the simplest wiggly scrawl to a detailed blueprint of the entire universe (as well as an infinity of blueprints that looked superficially the same but are different in some detail). Readers familiar with software structures may recognize in this list of packing types echoes of three levels of computer memory: queues, RAM, and databases. The similarities are described below. Linear structuring in software is typical of queues and buffers, most of which deal with timing. A queue usually lists instructions to be executed, messages to be sent, or data to be accessed, all in temporal order. The queue contains code or data in one dimension. It is usually set up so that elements

can be enqueued or dequeued at either end in a specific combination (first-infirst-out, first-in-last-out, etc.). Normally, the only way to locate a specific element is by its position in the queue, and queue elements usually cannot be randomly inserted or removed. Dense structuring is typical of random-access storage. Numerical addresses are assigned to memory elements (usually 8-bit bytes), and the location of any piece of code or data is defined by one or more address ranges. New code or data is normally stored simply by executing a write instruction to an address range, which overwrites whatever is already there. Storage is never empty; to clear it, the program writes in either zeros or random bytes. This more or less mimics physical space—when we want to store a thing, we simply move it to a location in space, where it displaces the air or whatever else had been there. Complex structuring distinguishes linked and relational databases, the most familiar of which is the World Wide Web. A single Web page may spawn thousands of other pages, all accessed through hypertext links. This concept is traceable back to an article written in 1945 by scientist Vannevar Bush.8 He proposed an information depository of linked microfilms called a “Memex,” organized by the logical relations among data. With the advent of personal computing power, visionary Ted Nelson cited Bush while conceiving the key software ideas that made hypermedia possible and which engineer Tim Berners-Lee implemented in working hardware in 1989. This history is worth tracing because it illustrates how the availability of new physical hardware (first microfilm, then microprocessors) led to innovations in physical reality’s powerset, ideals. The history of digital computer design is largely the history of inventing electronic devices that could handle data of the three kinds just described— linear, dense, and complex. Alan Turing showed the way in 1936 by envisioning a machine that could perform mathematical calculations by marking cells on a linear tape. The behavior of marking paper was emulated electrically during World War II by flip-flop circuits that used components such as relays and vacuum tubes. A flip-flop can be set to either of two states and retains that setting until it is set to the other state, making it function as a memory unit. In 1945 John von Neumann showed how the calculating routines on Turing’s tape could be executed by dense arrays of physical flipflops. In 1948 Claude Shannon identified flip-flops as the elementary units of information-handling machines, symbolizing their contents ideally as “bits.” This meant that every computer task, however complex, could be performed by manipulating sets of bits. Devising machinery to move, store, and interpret bit patterns became the general goal of computer design. The foregoing reflections of behavioral, physical, and ideal realities in computer technology are interesting for their own sake. In addition, they help us understand the intrinsic differences between our inner behavior, the external physical world, and the realm of ideals—something that has baffled

philosophers for more than two millennia. To a computer engineer, the differences between a data buffer, a file system, and the Internet are profound. They are not just differences in size or technical difficulty. Everything is different—the I/O and access protocols, the ways you expand or modify them, the kind of hardware you need. Yet the same information runs through them all, and to a computer user they appear to work seamlessly together. That’s one virtue both life and good software can claim.

What Categorization Does Aristotle introduced the idea of categorization in his short work Categoriae, which had a lasting effect on medieval and later thought. For him, categories were headings under which all the single things we could talk about were classified. He listed ten, all quite general: substance, quantity, quality, relation, place, time, position, state, action, and affection. Briefly, the sort of thing he intended by this scheme was to be able to specify that when we say (for instance) “the horse runs” we can analyze our statement further by saying that “the horse” is an example of substance and “runs” is an example of action. A set of categories thus gives us an overall view of how we think and talk about anything, by outlining the pigeonholes into which our terminology may be sorted. In his Critique of Pure Reason (1781), Kant developed a list of twelve “fundamental concepts of the pure understanding,” which he proposed as an absolute framework within which anything we can imagine must be cast. These categories, forming the cornerstone of his “Copernican revolution in philosophy,” were generated by an essentially logical process. As we would anticipate, they were even more abstract than Aristotle’s, featuring such headings as unity, plurality, causality, possibility, and so on. Kant speaks of Aristotle as having “merely picked [categories] up as they occurred to him,” whereas for Kant these entities represented the absolute forms of existence as we apprehend it, and hence could be defined through logical analysis alone. DR Theory would describe both Aristotle’s and Kant’s ideas of categorization as simple algorithms for set constructions. They both proposed to divide the ways of human understanding into enough gigantic sets that everything we wanted to understand would turn out to be an element of one and only one set. Aristotle used his encyclopedic knowledge to build his list of sets; Kant used his towering intelligence. Aristotle’s categories became outdated as the physical world became better known, but Kant’s ideal categories eventually contributed to some of the principles underlying the modern scientific method. DR Theory is able to add insights to the process of categorization that were unavailable in Kant’s day. Kant lived before Darwin was born and before set theory was invented. It never seriously occurred to him that categorization

might be an evolving set-construction process that living things had been working on for millions of years, nor did he realize that human logic as it was understood in the eighteenth century lacked some of the basic tools needed to understand how categories work. Descartes, who lived more than a century before Kant, had brought into focus the idea that there were two different types of human understanding— objective and subjective. It was obvious to Descartes that the two had to work together, but he was unsure how. Kant accordingly tried to link these types of understanding together, using what he called transcendental logic. In DR Theory, by contrast, life has naturally evolved the linkage that Descartes and Kant sought. Ideal categorization. Modern science maintains the most disciplined ideal categories for explaining digital reality. They are taught in schools and many people regard them as definitive. For example, consider a well-developed modern theory: say, the exposition of chemistry one learns in high school. On the first day of class, students are commonly told that the subject of chemistry comprises all physical matter and the transformations it undergoes. Typical instances are given: iron rusts, candles burn, cloth bleaches, sugar ferments. Iron, wax, smoke, bleach, alcohol—these are the familiar materials with which the theory deals. They form its subject. At first, the theory seems almost cosmological in scope; but it is soon evident that there are limitations on its subject. To start with, chemistry recognizes no transformations of matter below the atomic level; most events taking place in the sun, for example, are explained by physics, not chemistry. But more subtly, practical chemistry is also limited to relatively pure forms of matter. No chemist would undertake to analyze a whole housefly, because it is such a concatenation of compounds that overall analysis would hardly yield any meaningful information. It would be like trying to pursue botany by studying aerial photographs of forests. A chemist would assert that “in principle” a fly could be analyzed chemically, its matter becoming described as proportions of carbon, hydrogen, oxygen, nitrogen, and other atoms; but such figures would tell us very little. To understand a fly we must turn from chemistry to biology. Similarly, most other objects of everyday experience—earth, air, wood, cloth, etc.—are too mixed or contaminated to figure conveniently in chemical researches. Even water is usually distilled to remove minerals before it becomes an object of study. This does not mean that chemistry refuses to recognize such mixtures or cannot ultimately understand them; it’s just that practical chemistry displays an inherent tendency to set mixed matter aside as not being a fruitful area for inquiry. For most high school students, chemistry soon devolves into the study of relatively pure chemicals, i.e. materials purchased in bottles from chemical supply houses. The subject becomes esoteric, removed from common

experience; and later the students may be alarmed to learn that such chemicals are present in the food they eat. In this fashion, the discipline of chemistry retreats rapidly from its ostensive subject (non-subatomic matter in general) to knowledge of rather special materials under controlled conditions. It becomes the study of pure chemicals in laboratories. How does the modern theory of chemistry handle its subject? We say that it starts by categorizing physical matter. Of all possible purified materials that might be found on a chemist’s shelf, some ninety-odd are elements and the rest are compounds. Every compound is made of two or more elements. We can demonstrate this by subjecting a compound to various operations such as heating or electrolysis and noting that it eventually disappears, being replaced by an equal mass of elements. Alternately, we can often create the compound by bringing its elements together under the right conditions. For any given compound the ratio of elements by mass is always the same. This schema, enunciated more than two centuries ago by Dalton and J. L. Proust, forms the bedrock of modern chemistry. Since then, of course, more has been added to the theory. Compounds hold together because of bonds between elements—attractions with which we can associate definite amounts of energy. Elements have valence numbers that tend to predict the ways they will combine with other elements. Elements are composed of tiny identical atoms which make up identical molecules in compounds; this accounts for isomeric compounds, which contain the same elements in the same ratios but have different molecular structures. And so on. By categorizing matter (even the somewhat specialized materials on the chemist’s shelf) we alter our view of it. Iron and oxygen, although utterly different by everyday standards, are similar because they are both elements; rust, which is everywhere associated with iron in common experience, is different because it is a compound. Red, black, and brown rusts are similar because they are made of the same elements but different because their combining ratios are different. Materials as dissimilar as graphite and diamond are the same element (carbon); materials as apparently similar as the two colorless gases carbon dioxide and argon are fundamentally different, because one is a compound and the other is an element. The categories of modern chemistry, at least at the level of sophistication discussed here, are thus element, compound, bond, and so forth. But these headings are not physical things: there is no material object we can point out as element itself, pure bond, etc. In fact, they are ideals. Chemistry is a theory with a physical subject and ideal categories. As a consequence, familiar physical things are now treated as having ideal properties. For instance, substances such as iron (because they are elements) are regarded as inherently immutable during chemical transformations—not just usually immutable, or not hitherto transformed, but by their very nature not capable of being decomposed into anything else. If we start with an element in a closed

container, no matter what chemical operations we perform on it we shall still have exactly that much of the element. An element is thus like a Euclidean point or an arithmetic prime number: it has an inherent property that belongs to it by definition. We may discover that a material thought to be an element is not an element—as happened in 1894 with atmospheric nitrogen—but such a discovery does not affect the category “element.” It only changes the area of subject matter that we find fits the category. Similarly, a compound is matter that (when purified) always contains two or more elements in a constant ratio. We describe a compound by writing the symbols for its elements with subscripts indicating the combining ratio, e.g. Fe2O3. In chemical theory this totally defines the material (ignoring isomers); one physical sample will be identical in its properties to any other. The ratio is an inherent part of the compound. So chemical categories are abstract descriptions. To find the subject of modern chemistry we explore physical reality; to find its categories we turn to another type of reality, to ideals. The theory as it exists today began to take shape toward the end of the eighteenth century when a few thinkers began to conjecture that certain ideal concepts could be correlated with parts of the physical world. An ideal-physical fit had been conceived. Behavioral categorization. For a comparison with science, consider modern chemistry’s precursor, alchemy. Working with many of the same physical materials (often equally purified) it came up with an entirely different understanding. According to researchers Stephen Toulmin and June Goodfield, the categories of alchemical theory were behavioral: The first starting point for alchemical theory was Aristotle’s principle of development: the conception that all material things, unless interfered with, will naturally change and develop—turning, when properly fed, and nurtured, from an immature to a ripe or adult form. Rather than treating elementary matter as naturally inert and static, they thought of all things equally in a fundamentally physiological way.9 It had been believed for centuries that minerals grew organically in the earth. As a practical discipline, then, alchemy strove to reproduce the terrestrial womb in the laboratory, initiating and nourishing the gestation of one material into another, such as mercury and sulfur into gold. When considered from the alchemist’s viewpoint, it was a perfectly plausible idea. Were we to formalize alchemical theory, we would come up with categories such as seed, womb, and nourishment. The process of transmutation was one of preparing a proper womb (typically the carefully heated retort or alembic), infusing it with the correct seed (such as a portion of gold around which more gold was to grow), and adding nourishment over a period of months, much like cultivating a plant. The theory’s categories do not describe

these parts of physical reality by their abstract properties, but by what they do. The womb promotes growth, the seed grows, and the nourishment sustains the process. The same mercury that modern chemistry calls an immutable element alchemy characterized as a food that helps metals mature. This difference in categories is the difference between ideals and behavior, between reality that we understand through its description and reality that we understand through its lifelike action. Because they subscribe to disparate sets of understandings, the modern chemist and the medieval alchemist understand the same physical objects and events in radically different ways. Anthropologists sometimes call the alchemist’s worldview animism and disparage it as unscientific or superstitious. Nevertheless, it would be fair to say that animism is more generally used today, even in “advanced” cultures, than science. If I bake a cake, I think of the ingredients as having behavioral properties, not ideal properties. Flour, milk, eggs, and baking powder each “do” something to contribute to the finished product. A chemist might characterize baking powder in terms of the potential decomposition of sodium bicarbonate into sodium carbonate, water, and carbon dioxide gas, which proceeds at a certain rate in the presence of moisture and heat by virtue of ionization. I would say it simply “makes the cake rise.” To assure that the cake rises, I select a reputable brand of baking powder, thereby appealing to a behavioral agency (a manufacturer) who is supposed to compound the powder so that it works. Neither the chemist nor I will be very conscious of impersonal abstract laws embedded in the process. If the cake fails to rise I might blame a careless manufacturer. An instance of the cake falling flat will not contradict any beliefs I hold, at least not in the sense that the Michelson-Morley experiment was said to contradict Newtonian physics. It will just mean that something didn’t work. It is clear that this attitude is generally adopted in our everyday commerce with physical things. As I remarked earlier, the more we examine the actual uses of science the more we discover it is a discipline mainly confined to laboratories. In fact, it is clear that without a firm grasp of animistic thinking no human being, not even the most capable scientist, could long survive. When I put a bite of food in my mouth it is usually because I believe it will taste good, satisfy my hunger, and the like, not because it contains certain molecules or conforms to certain chemical specifications. When I take a step, I expect the floor will support me without my knowing its modulus of elasticity. Obviously, these beliefs may be wrong: I can get food poisoning, the floor can give way under me, and so on. But if science had never been devised or if I had never heard of calories, elasticity, and other idealizations, I would still be able to get along satisfactorily through my animistic conceptions of

physical reality. On the other hand, if I had no such conceptions—if, for instance, I could distinguish a potato from a rock only by measuring its carbohydrate content and dared not take a step until I had determined the engineering properties of the floor in front of me—I would quickly perish. While it may be argued that the ideal expressions of science yield advantages over behavioral animism, the fact remains that animistic theories are essential to human life whereas scientific theories are not. My everyday physical reality is categorized behaviorally, not ideally, and my usual everyday theories are animistic, not scientific.

Expanding Digital Reality The foregoing description of how digital realities are created has emphasized their origins in our behavioral experience of existence. This echoes Locke’s dictum that “no man’s knowledge here can go beyond his experience”10 —one of the cornerstone ideas of empiricism. But in fact, most people are quite adept at creating knowledge beyond their experience. They are like the composer who weaves harmonies and decorations around a simple theme. In surveying knowledge—the contents of digital realities—DR Theory finds three ways that life and living things broaden their realities beyond simple interactions with existence: The behavior of many living things, particularly humans, includes theorizing, which modifies and amplifies the realities we construct beyond our existing understandings; The physical aspect of life has evolved speciation, which creates new kinds of life that acquire new kinds of knowledge; Among ideals, powerset expansion breeds entirely new areas of digital reality filled with novel subjects of reasoning. These techniques are like computer algorithms—general procedures for performing a generic task. In this instance, the task is building new digital realities. The main ways that life does this are discussed below. Theorizing is something living individuals do. It is not confined to humans; in DR Theory, any living thing capable of learning must sometimes theorize. But as far as we can tell, theorizing plays only a secondary role when it appears in the lives of a few other species, whereas in Homo sapiens it can become an absorbing career. It may sound paradoxical to assert that one purpose of theorizing is to define error, but that is so. It is meaningless to ascribe error to natural facts— things are simply the way they are. DR Theory calls this lowest level of natural understanding common sense and formalizes it by saying that it explains each type of digital reality—behavioral, physical, or ideal—using

categories of the same type. When we theorize, on the other hand, we step outside our natural conceptual pathways; we explain one type of reality by using categories of a different type. DR Theory calls this maneuver crosscategorization. It is a common technique for expanding knowledge, which the rest of this book will explore at length. A principal goal of crosscategorization is to make us aware of errors or omissions in common sense. An example may help illustrate the difference between common sense and cross-categorization. Categorizing water as a liquid is commonsense because both the substance water and the category “liquids” are physical. On the other hand, categorizing water as a chemical compound is theoretical because the category “chemical compounds” is ideal while water is still physical. The difference between common sense and theoretical knowledge is not just a scholarly distinction. Although common sense forms the bedrock of knowledge, most advances in understanding have sprung from crosscategorization. Modern science, for example, is largely based on the ideal categorization of physical reality, as explained under “Formal Reality” in Chapter 5. Perceptual errors. Chapter 2 illustrated perception with an example of red balloons. All phases of the process described there involve choice and hence are subject to error. Grouping a sensation of redness with one of roundness under a single physical category could be a mistake; for example, the round thing might look red only through a trick of lighting. Adding a space location to the red + round categorical grouping might be wrong if we stood in a hall of mirrors. And separating the categorical grouping from the rest of physical reality, by objectifying it, could be erroneous if it turned out that we had unknowingly been watching a video. Every living organism struggles to identify the things in physical reality that correspond to their behavioral sensations. Among humans, the struggle is complicated by the fact that populating physical reality with things is only one of several ways in which we understand the world. Nevertheless, the struggle mostly succeeds. To deceive ourselves, we have to concoct deliberate illusions and virtual realities. For example, suppose a penny and a dime are on the table in front of me. I want to know which is larger. My physical common sense tells me the penny is larger; in the behavioral common sense of my sensations, the penny also appears larger. So far there is no problem and no need to theorize. But suppose the penny and dime are placed on a drawing of converging lines, creating an optical illusion in which the dime appears to be larger. My behavioral common sense (which is capable only of accepting my sensations uncritically) now tells me the dime is larger. Yet my physical common sense still claims that the dime is smaller. There now exists a conflict which can manifest itself in several ways: for instance, I might find my efforts to cover up the penny with the “larger” dime frustrated.

To resolve the conflict, I resort to a theory of perception. It starts by asserting that my thoughts of a certain kind are physical images; in other words, it applies physical categories to my thought behavior. Among these categories will be disc images (or images of flat things or of coins themselves, at various levels of categorization), which separate and identify the sensations I have of coins, and in particular of these two coins. The theory will also have categories identifying relative size, under which other parts of my thought behavior will fall. Using this theory, I will then be able to understand that one sensation I have refers to the physical penny and another refers to the physical dime, while the thought I have that the dime is larger than the penny refers to their relative physical size. Such a theory of perception (even in the rudimentary form sketched here) now provides me with a vital new piece of knowledge, for I already know from my commonsense grasp of physical reality that the penny is actually larger—therefore a conflict or error exists. Applying the theory further allows me to locate the source of the conflict: by moving the two coins around until the penny appears larger (i.e. the conflict disappears), I discover that the perceptual problem occurs only when they are on the drawing of converging lines. This may then become the starting point for enriching my common sense with an understanding of optical illusions, central to which will be a notion of erroneous perceptions. Speciation might be called the way that life as a whole theorizes. A unique feature of life is that when we compare individuals within a species we find them generally alike, but when we compare them between species we find them generally unlike. This result is not just an artifact of the way we understand living things; it is a feature constructed by life itself. Why? Competition between species lets life conduct trial-and-error experiments that improve its understanding of the physical and behavioral ecosystems within which it exists. The behavior of every living individual includes two major digital actions in physical reality: reproduction and dying. If the individual reproduces before dying, it creates one or more new physical individuals; if it dies first, it doesn’t. The mechanism for creating new individuals consists of a sequence of initiators for anabolic chemical reactions strung along physical DNA molecules. The anabolic reactions are chemically uniform, and hence ideal with respect to physical organisms. Once the groups of individuals that we know as species can generate new individuals that are like each other, but not like those of other groups, then groups can compete as a whole with one another. The payoff for group competition is that the species which succeed can evolve new kinds of life that embody a more effective knowledge of their environments. Computers are designed to emulate life. If we characterize life by its physical aspect, then it would make sense to place microprocessor-based

equipment somewhere within its hierarchy of species. We could name a new domain, at the same taxonomic level as the Eukaryota, to contain computers, integrated circuit chips, and smart devices. Hardware platforms might be classified like genera and operating systems like species. One reason for doing this would be to recognize that smart devices are becoming symbiotic with humans, at least economically. While the symbiosis has so far been largely mutualistic, parasitism is bound to emerge. Categorizing the actors and points of competition between our species and our machines would tend to give our species an advantage. Powerset expansion as a way to increase knowledge is a central feature DR Theory. Here is its formal statement: Sets of ideal knowledge are powersets of sets of physical knowledge and sets of physical knowledge are powersets of sets of behavioral knowledge The axioms of Zermelo-Fraenkel set theory define a powerset as the set of all subsets of another set, called its base set. To take a simple example, if a set contains three elements—x, y, and z—its powerset contains eight elements, all of which are sets. In set-theory notation, the powerset of the set {x, y, z} is written {{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}. This new set contains every set that can be made using the elements x, y, and z, including a set that contains none of them (called the null set, written {}) and the original base set {x, y, z}, which contains all of them.

The Power of Powersets David Hilbert may have been exercising his intuition when he called Cantor’s hierarchy of infinite sets a “paradise.” In many religions, paradise represents a place that grants cosmic wisdom to the blessed. By linking the behavioral, physical, and ideal forms of understanding, Cantor’s set theory supplies similar insight to living things as they construct digital realities. Constructing a set makes a new object of knowledge by combining other objects. Constructing a powerset—the set of all possible subsets of a set— creates a package containing all the new objects of knowledge that might have been made from the same materials. Consider the example of a powerset described above. If x and y and z are three digital objects of knowledge, then their base set, {x, y, z}, shows what happens when you put the three together. But their powerset contains all eight possible combinations of those objects. Each possible combination can help us understand more about the base set and its elements. For example, if x and z

have something in common, then finding them together among the powerset’s elements as the new, independent set {x, z}—without y—may tell us something we might not have realized by knowing x and z only when they were accompanied by y in the base set {x, y, z}. In this way, constructing a powerset actualizes, in digital reality, all the possibilities inherent in the construction of the base set. This is only half the story, however. In DR Theory, not only does the powerset of a set show us all its possibilities, but it does this in a new type of understanding. If a base set is composed of behavioral sensations, its powerset is filled with physical things. If the base set is physical, then its powerset is ideal. Besides showing us new possibilities, powerset expansion shows us those possibilities in a different and infinitely larger digital world. Let’s trace a very simplified example of how powerset expansion might work in an actual digital reality. Suppose we are looking at a red balloon. In our digital perception behavior, let us say, there is a base set of three sensations: {red, round, floating}. It’s a real-world instance of the set {x, y, z} in the example above. Since we are a clever animal, our neurological apparatus (our brain) is able to translate the behavioral set {red, round, floating} into a physical set {red thing, round thing, floating thing}. Congratulations. Our general knowledge of the world has told us that we can understand our perceptions in physical terms. At this point set theory intervenes and offers us further options. Using powerset expansion, we can break our physical understanding into parts. The physical statement of the new set of possibilities is longer, but it just follows the pattern of expansion of {x, y, z} written earlier in this chapter: {{}, {red thing}, {round thing}, {floating thing}, {red thing, round thing}, {red thing, floating thing}, {round thing, floating thing}, {red thing, round thing, floating thing}} Now the possibility of {x, z} without y has become an imagined physical object: a red and floating thing that is not round. We are encouraged to go out into the external physical world and build a kite (or an airplane) that is red and floats in the sky. Although the balloon—the only floating thing we can see—is round, roundness and floatability are not functionally related in the physical world. Inflating a round balloon may be quick and easy, but powerset expansion encourages us to try other ways to make things float. Besides feeding our imagination, powerset expansion accounts for the mental capabilities of abstraction and generalization which humans share with some other animals. For example, the three single perceptions now made physical—the sets {red thing}, {round thing}, and {floating thing}—can easily become category sets by adding other red, round, or floating elements to them. Powerset expansion fills our minds with prefabricated pigeonholes

into which we can store new or orphaned objects of knowledge. You might also think of powerset expansion this way: To limit our attention to meaningful knowledge, we living things have evolved to regard only sets as real. Powerset expansion serves this instinct by wrapping sets around every group of objects that might qualify as elements. (We know they qualify because they are already elements of another set). It’s a basic marketing strategy that our species uses to encourage human curiosity. In effect, the Powerset Emporium makes potential knowledge more attractive by wrapping it in convenient boxes and placing it on display.

The Structure of Knowledge In DR Theory, every category set belongs to one of the type sets of understanding discussed in Chapter 2: behavioral, physical, or ideal. Using categories to sort out our digital realities supports our understanding of them in several ways, described below. Our physical bodies perform A/D conversion on existence, turning it from an analog mass into specific objects and events. This is digitization, the first step in the process of constructing digital reality. The resulting sets form the lowest level of reality because they contain raw existence; all higher levels of reality are sets whose elements are other sets. How does digitization take place? Consider a typical example. Suppose my visual field contains a yellow, chalky sensation. If I were a child, or very ignorant, I might understand little or nothing about that sensation. But I am a chemist, so I categorize my sensation as “seeing sulfur.” The category I have just constructed is a behavioral set wrapped around an existential event that I do not or cannot understand. It is a new item in my digital reality. If I turn out to be mistaken (as will be discussed shortly), I may replace it with another category set; but if I care about the business of living, I cannot plausibly deny the existential event itself. I must categorize it somehow. Applying categories to an existing reality can enrich our understanding by showing us how parts of that reality are like or unlike other parts. Returning to the example of seeing sulfur, as a chemist I know how sulfur is categorized physically: it crumbles easily into a fine powder, it burns with a small blue flame, the odor of its burning is acrid, and so on. I can test the sample in front of me to verify that it belongs to these other categories, and hence it is sulfur and not something else. Applying alternate categories to the same reality can deepen my understanding by giving me a choice of ways to make sense of it. Behavioral categories such as looking yellow and physical categories such as making acrid smoke would identify sulfur equally well to a medieval alchemist and to a modern chemist. But the chemist’s digital reality also contains ideal categorizations that the alchemist would find hard to fathom. Sulfur as an

element can bond with hydrogen, oxygen, and water to form a bewildering array of polythionic acids, which some chemists have spent years studying and categorizing. Sulfur also bonds with metals, halides, and many organic substances, producing useful compounds such as saccharin, sulfa drugs, and penicillin. Achieving most of these discoveries has depended on applying ideal categories to physical chemicals. How can such a simple operation as set construction unlock so much new understanding? When we create a new set, we accomplish two goals: Veracity: by wrapping a set around other sets in a digital reality, making them its elements, we define a new digital unit in our understanding. At the same time, we establish the veracity of the new digital unit because each of its elements is traceable back to existence. Understandability: by inserting the new unit into a hierarchy of categories in a digital reality, we make it more and more familiar. It becomes an element in several overlapping sets, each of which helps us understand it. Any human individual’s knowledge consists of a mass of nested sets. Some sets have been constructed for veracity—they bore down through hierarchies of subsets to find their roots in digitizations of analog existence. Other sets have been constructed for understandability—they explore upward, through larger and larger categories, to seek enlightenment in the behavioral, physical, and ideal worlds of digital reality. DR Theory asserts that we humans know only the digital realities that we construct. Because they are digital, these realities can be described as sets whose elements are mostly other sets. If viewed as tree-like structures of nested sets, their trunks would be the three types of reality—behavioral, physical, and ideal—and their leaves would be sets whose elements are parts of analog existence. Had the human species never evolved, this simple description would be mostly true. Life on Earth would consist of creatures whose behavior followed a stimulus-response timeline with limited local adaptation, whose physical actions followed the electrochemical processes of their bodies, and whose ideal principles were encoded in their genomes. But we humans have a knack for innovation, and so we discovered cross-categorization. We learned how to understand sets in one type of reality by including them in sets of another type. The principles of naïve set construction permit cross-categorization, with far-reaching results. It’s as if different kinds of trees in the forest had decided to share their branches. The oaks started growing maple leaves while the elms made acorns. Mother nature cannot object, for it was she who set in motion the evolutionary processes that produced this capability. There is no guarantee

that intertwining the tree structures of real categories will not lead to logical paradoxes; on the other hand, we have no reason to believe that paradoxes within reality are truly bad. Emerson seems to have been trying to save us a world of trouble when he wrote “A foolish consistency is the hobgoblin of little minds.” It is usually more efficient to let our constructed and crosscategorized digital realities simply be what they are, paradoxes and all. The most striking consequence of cross-categorization in human behavior is the tangle of social realities in which most of us live. Helping us understand and define that tangle is a central task of DR Theory. Some preliminary results are discussed in the next chapter.

4. Social Realities Structures of cross-categorization in digital realities drive human group behavior. THE REMAINDER OF THIS BOOK

mainly examines how DR Theory may be used to

understand human behavior. In 1966 two sociologists, Peter Berger and Thomas Luckmann, published The Social Construction of Reality, a seminal book in which they treated social institutions as both objectively real and as humanly constructed: An institutional world, then, is experienced as an objective reality. It has a history that antedates the individual’s birth and is not accessible to his biological recollection. . . Since institutions exist as external reality, the individual cannot understand them by introspection. He must ‘go out’ and learn about them, just as he must learn about nature. . . It is important to keep in mind that the objectivity of the institutional world, however massive it may appear to the individual, is a humanly produced, constructed objectivity11 The term “constructed objectivity” raises ontological issues that the authors disown by understanding the world phenomenologically. DR Theory takes a different tack. It prefers to treat institutional worlds as existing externally in a fully objective sense, but as being understood internally through individually digitized realities. This makes it possible to analyze disagreements about institutional worlds as differences in digitization, instead of as errors in recognizing what they “really are.” We humans are social animals, and much of our success as a species is attributable to our successes in socialization. Human history tends to be centered around artifacts because that’s what researchers find in their diggings; but each new tool or shelter or hearth must have given the human socialization that produced it an advantage in its struggle for survival. Hence, while human discoveries in mechanics and technology are important, DR Theory gives equal weight to our advances in group organization. The three basic types of understanding described in Chapter 2—behavioral, physical, and ideal—permeate the ways in which we organize our social groups. This chapter describes the six ways that they can cross-categorize one another, plus the two ways that these types of understanding link together into complete societal worldviews. The results—distinctive social organizations such as families and

religions—are formed by cross-categorizing the behavioral, physical, and ideal realities of individual persons.

Social Categorizations In DR Theory’s analysis, cross-categorization in the behavior of individuals is the primary agent of socialization. There are six ways that each of the three types of digital reality can be categorized by one of the others, and each of these modes of categorization can be given a sociological name. Here they are: Source of categories Physical Behavioral Behavioral Ideal Ideal Physical

Reality categorized Behavioral Physical Ideal Behavioral Physical Ideal

Sociological name Communalism Authoritarianism Intellection Orthodoxy Legalism Collectivism

The same information can be rendered in a diagram:

This diagram shows the three types of digital reality at its three corners. The arrows represent modes of cross-categorization. The tail of each arrow comes from a source of categories and the arrow points to the reality categorized. In every case of cross-categorization, the categories give the realities they categorize positive or negative values or attributes they might not

otherwise possess. Positive values might include justification, sanction, acceptance, worthiness, necessity, etc., while negative values might include unfitness, immorality, wickedness, etc. For example, when we use physical reality to categorize human behavior in a social situation we are engaged in the kind of interaction that DR Theory calls Communalism. This is a social reality where behavior is characterized by physical issues affecting a social group. It encourages or sanctions individual behavior depending on how that behavior affects the physical success or health of the group. In effect, that social reality compares the group’s physical needs with individual trains of behavior and pronounces specific behavior either good or bad, right or wrong. Communalism is the most basic social reality. Human child rearing is an obvious example. Parent-child cooperation is also a principal point of entry for Communalism into individual personalities. We can appreciate this by considering Communalism from an infant’s viewpoint. To a neonate, certain physical situations are given. These include its own physiological needs, such potentially harmful situations as becoming cold, the mother’s breast as a source of nourishment, the cry as a means of signaling, etc. At the outset the infant is powerless to alter any of these physical factors by itself; they must be taken as ineluctable categories for its initial organization of responses, while the responses themselves must be directed toward another area. In fact, they are directed toward behavior, first that of the infant and then that of its mother. It is only through mother-child cooperation that the neonate’s survival is ultimately possible. Thus, our early and best-remembered responsive behavior takes as its subject matter, as its area of learning and manipulation, the interplay between a child and its mother; and it takes a portion of physical reality, that centered around our requirements for physical survival, as the categorical setting for our behavior. What behavior gains the breast? What new behavior then produces the milk? What to do when I am cold? Questions such as these fill the neonate’s first struggles with individual learning, which rapidly pass beyond the instinctive set of reflexes with which it was born. Many animals, other than man, exhibit Communalism. Most birds, for instance, display some sort of communal organization when rearing their young. Here again, the setting is physical—the need to nourish infant birds, the fact that they must stay in the nest until old enough to fly, the dangers from predators and the elements—and the solution is behavioral. The parents cooperate, often with risk and sacrifice for each one individually, until the physical setting has been resolved by the fledglings leaving the nest. Bird behavior while rearing young is quite different from that at other times, largely because of the emergence of this type of communal organization. A well-known example of a communal society was described by Margaret Mead in 1935. The Arapesh people of the Sepik River area in

northeastern New Guinea lived in an isolated and difficult land, protected from outside contacts by mountains so infertile that no neighbor envies them their possession, so inhospitable that no army could invade them and find food enough to survive, so precipitous that life among them can never be anything except difficult and exacting12 This provided the setting for their social organization, a setting composed primarily of severe physical problems. The Arapesh that Mead studied responded to this setting by adopting a nearly total dedication to cooperation. They tended each other’s gardens, built each other’s houses, shared the results of hunting, and helped care for each other’s children. Institutions that would reflect social organizations other than Communalism—such as political units, private property, competition, and even lines of authority within family clans —were largely absent. Mead’s account of the Arapesh provides a graphic description of what life in a truly communal society can be like. Modern societies occasionally develop a predominately communal organization. One immediately thinks of communes—small bands of individuals cooperating to maintain a common physical setting—which become popular from time to time. While these may approximate pure Communalism, they are more often mixed with other social organizations— Orthodoxy in the case of religious communes and Collectivism in the case of economically productive communes. Instances of truly communal societies are usually found only in reports of anthropologists. Human history shows that pure Communalism is not an enduring form of behavior for whole societies. Authoritarianism results when you turn the Communalism arrow around. Now behavior categorizes physical actions. This is a familiar concept in sociology; it is applied to groups where the will of a leader or small clique categorizes physical acts by individual members. The people whose behavior controls the group may include not only specific individuals (such as dictators) but also past tradition-makers and the diffuse but powerful behavioral consensus of the present. “The way our fathers did it” and “the way it is usually done” express Authoritarianism, even though they may not cite an active personal authority. The physical subject areas of Authoritarianism are as diverse as the interests of any group: who does which jobs, how goods are to be allocated, what individual actions are demanded or permitted or proscribed, even how individuals are to be punished when the authority is transgressed. The key to this social organization (and what separates it from Legalism) is that the basis for its dictates is a program of behavior, not a set of ideals. It springs from a group’s agreement to accept the will of a chief, the decisions of an oligarchy, or a history of hallowed traditional behavior as the basis for sorting out and

regulating the physical actions of its members. Authoritarianism is a common organization in human families, particularly in the subgroups containing young children. Once they pass the stage where they are wholly dependent on mother-child Communalism to satisfy their physical needs, children acquire an organization where they receive prescribed programs of behavior from their parents and in return are permitted individual manipulation of physical things. Parents, too, tend to treat these prescribed behavioral routines as intrinsic to the parent-child group even though they have the power to hold them in perspective, which the child does not. In other words, Authoritarianism arises in the family group through a common agreement that certain behavior programs are given, and its members (particularly the children) must deal with physical things in conformity with this behavioral setting. Studies by Piaget of children’s attitudes toward the rules of games illustrate Authoritarianism from their viewpoint. For several years after infancy children normally treat game behavior as utterly fixed: rules are regarded as sacred and untouchable, emanating from adults, and lasting forever. Every suggested alteration strikes the child as a transgression13 Despite this attitude, children are actually observed to play somewhat carelessly, randomly altering the physical configurations of their games. What is happening is that the child is learning the physical skill of playing (in this case, marbles) within a setting of behavior it regards as ineluctable. When asked to perform the physical game the child exhibits a range of trial-anderror learning; when asked to report the governing behavioral routine, it treats it as given by unquestionable authority. Any closely supervised work group tends to exhibit Authoritarianism. When a group achieves its goals through mere cooperation, of course, it is communal. But to the extent that its success depends on the members following behavioral directions from a leader, it is authoritarian. A relatively simple example is military discipline. Here the behavioral setting is clear and explicit; it is sometimes discipline just for the sake of discipline, and each individual series of acts is governed by the rigid organization of the group. On a larger scale, several functions of highly-regulated societies tend to be carried out by authoritarian groups. These may range from fire and police forces down to school traffic patrols. Usually, these groups display other social organizations as well, for pure Authoritarianism on a large scale seems despotic. The group may be guided by a book of abstract policy in addition to the established canons of behavior. But the principal organization emerges in the actions of each group member: each one performs physical acts in accordance with a group-sanctioned routine of behavior. If there can be no

appeal from the behavioral dictates, then it is pure Authoritarianism; if the prescribed behavior can be modified by reference to ideals, then it is Authoritarianism mixed with Legalism. Intellection introduces ideals into social reality. Many people may live their lives in a mixture of Communalism and Authoritarianism without paying much attention to ideals. But inevitably their consciousness will be raised by a book or speech in the kind of social interaction that DR Theory calls Intellection. Intellection defines ideals by categorizing them behaviorally. Words and inspiring actions call ideals to mind and implant them in our lives. We learn basic societal abstractions—truth, honesty, responsibility—by listening and watching. From there, our imagination multiplies these concepts in the theater of the mind: we discover principles and we grasp generalities instead of single perceptions. In such activities, the setting is thought behavior, the ability of human minds to conceptualize. The subjects of Intellection are ideals—not physical objects, not the behavior of other people, but pure abstractions. Socially, Intellection is promoted by writers, lecturers, academicians, and thinkers: this book, for instance, is primarily a product of Intellection. Among smaller groups, a good place to observe Intellection in a relatively pure form is in the classroom or seminar. Here the behavioral setting exceeds the thought processes of any one individual; the group as a whole agrees to join in a program of behavior designed to facilitate their mutual exploration of ideals. This program usually includes attempts to minimize physical distractions, an agreement to stick to the subject, a scheme of terminology (i.e. common language behavior), etc. Such classroom discipline is important, for it establishes much of the behavioral basis without which this social organization could not exist. Group Intellection of this type—education—is vital to industrialized societies. We can appreciate its importance from the fact that members of those societies typically devote a significant part of their lives to it. At one stage in European history, Intellection went underground, surviving mainly in behavioral settings where it could withstand the political Authoritarianism of the day. These settings included the monastic institutions that flourished between the dissolution of the Roman Empire and the rise of Protestantism. Although most of them also functioned as agents for the Orthodoxy of Catholicism, they comprised (at least at the beginning) the most effective sources of abstract learning in Europe. They preserved and communicated much of what had been previously known about ideals. The monastic life typically combined a regime of fixed behavior with the encouragement of individual insights into ideals. Orthodoxy is the flip side of Intellection, where ideals are used to categorize behavior. Our inner plans and urges, even our casual thoughts, become good or bad, honest or irresponsible. This is where codes of morals

and ethics come from. Perhaps the plainest examples of social Orthodoxy are established religions. A group ordains a set of ideals that are to be taken as categorical and not open to question; the members develop and adjust their behavioral habits to conform with these received ideals. Although they are both associated with religion, Orthodoxy (the regulation of behavior by ideal categories) must be distinguished from deism (the regulation of physical acts by behavioral commands). Deism hypostatizes a God or gods whose commands run the physical world. Orthodoxy replaces the concept of a behaving, willful God with that of an ideal divine order, and shifts its area of operation from controlling physical events to regulating human conduct. This change, from worshipping an Authoritarian God of commands and retributions to obeying abstract orthodox principles through conscience, is illustrated in Judeo-Christian religious history. The God of Moses was almost entirely Authoritarian; the divine guidance of modern Protestant sects is primarily Orthodox. Compare the opening line of the Pentateuch—“In the beginning, God created the heaven and the earth”—with the opening of the New Testament: “In the beginning was the Word.” Because they spring from different social organizations of behavior, these attitudes easily exist independently; for instance, authoritarian deism without Orthodoxy is found in some primitive nature god cults and Orthodoxy without deism in such belief systems as Confucianism. Somewhat less obvious examples of Orthodoxy can be found in human social classes. Sometimes these subgroups in complex societies have a common basis in physical reality, e.g. in their relationship to land or means of production. But their basic coherence is more often a product of a system of agreed values or principles. By their group acceptance of such ideals—each individual applying these ideals to everyday behavior—such classes tend to pull away from the rest of society and appear as distinct sociological entities. They can best be identified by uncovering the ideal systems that their members regard as categorical for various kinds of social behavior. Much has been written about the reasons for the stratification of modern societies into classes. Marx attributed it largely to physical factors—property, coercion, and physiological needs. But in many modern societies, group acceptance of ideal categories forms an equally potent separator. This can create a problem for schemes of class redistribution; a person’s economic or legal status can be changed by fiat, but the same person cannot be made to shift from one form of Orthodoxy to another without a difficult period of reeducation. Well-meaning social programs that seek to push individuals from one class to another by changing their physical environments may underestimate the importance of their categorical ideals, which often form the actual bases for class membership. People may associate their class membership with a particular set of moral or ethical theories. This style of “belief” or “faith” theorizing is typical

of Orthodoxy. In each instance, there is a presupposed set of ideal categories, more or less internally consistent, which is used to distinguish one train of behavior from another. The product of such theorizing is often a set of incontestable judgments that such-and-such kind of behavior is bad and should be avoided or prevented, while such-and-such kind of behavior is good and should be encouraged. Legalism is sometimes confused with Orthodoxy because they both rely on ideal categories. But Legalism is the kind of social interaction in which we use ideals to categorize physical actions instead of inner behavior. Orthodoxy regulates our motivations and emotions; Legalism regulates our physical acts. A typical instance of social Legalism is any system of legislatures, laws, courts, enforcement officials, and law-abiding citizens. As with Orthodoxy, the group adopts a set of ideal principles, which are taken by its members as fundamental and not to be questioned: these are the “principles of justice.” But unlike Orthodoxy, a legalistic social organization tries to regulate physical events rather than behavior. Legal systems often fail to control pure behavior (separate from physical manifestations) because enforcement officials find it hard to detect. Thus, laws banning “impure thoughts” or “unworthy motives” are technically alien to Legalism, although such sanctions are common in Orthodoxy. At first, it may be hard to realize that secular law is concerned only with distinguishing physical acts, not behavior. Is it not anti-social behavior that is proscribed and punished? But a careful examination of the theory of the law shows that it always tries to stick to tangible physical facts. When actual legal procedures depart from this policy they get into trouble. A properly drawn indictment, for instance, states that the accused performed certain physical acts at a certain time and place, such acts being proscribed by law. Where such behavioral factors as intent, motivation, or state of mind are relevant, a legal burden emerges to show by physical evidence (statements, actions, circumstances) that these behavioral factors must have been present. In some cases, physical evidence becomes converted by law into a substitute for intent, as when the possession of a weapon establishes an intent to use it. Much civilized social life is organized legalistically. Besides laws imposing physical punishments for physical transgressions, there are legal systems that define wealth, property, and political power. A monetary system, for instance, starts with a prescribed abstraction—monetary value—and uses it to measure many of the physical objects handled by citizens. Working within such a system, individuals accept the idea that financial worth applies to objects, and they manipulate the objects to exchange this worth as if it were a more tangible property such as weight or color. In a mature monetary system, financial worth may be attributed to all sorts of physically insignificant objects, such as the magnetic pattern on a bank’s ledger disk. The power of the legalistic organization is such that those

adopting it will accept this physical trifle as actually possessing the abstract financial properties assigned to it. Similarly, physical objects and land are associated with physical human beings in the legalistic relation of property. A society assumes the ideal concept of property rights as a basis for determining what things belong to which people. An accessory process is the granting of ideal qualities to physical legal instruments, such as deeds and securities. The democratic political election procedure arises from a sophisticated form of Legalism. The convention that major political decisions should be determined by tallying ballots and that each mature human body in a society should be allowed to mark just one ballot, is by no means self-evident. In fact, it smacks of mathematical elegance at the expense of practicality. Yet wars have been fought to preserve or export this arithmetic procedure. It is based on two abstract principles: that a numerical surplus of marked ballots should determine the course to be followed, and that the proper ballot markers are mature human individuals. Once these principles are adopted by the group, shifts of political power can be accomplished by an essentially mathematical process. The concept of social law suggests the concept of natural law. Legalism generates mechanistic theories that cover the whole range of physical science. In each such theory, certain ideal categories are adopted by the scientific community, whose members pry into reality using those categories. The social organization of Legalism posits that every physical fact conforms completely to a set of ideal descriptions; therefore, once we possess the proper ideal tools —a complete system of mathematics, for example—we will be able to find out all that can be known and predicted about physical reality. In view of this, it is not surprising that some of the most ardent supporters of mechanistic theorizing have been mathematicians. In 1796, for instance, Laplace wrote in his massive System of the World: In the midst of the infinite variety of phenomena which succeed one another continuously in the heavens and on the earth, one is led to recognize the small number of general laws which matter follows in its movements. Everything in nature obeys them; everything is derived from them as necessarily as the return of the seasons, and the curve described by the dust particle which the winds seem to carry by chance, is ruled in as certain a manner as the orbits of the planets14 In other words, physical reality is a gigantic machine driven by a few ideal principles. Mathematicians are the most adept at handling such principles, so it seems natural to them to suppose that the mysterious workings of physical events can be straightened out once we know how to relate them to the precisely ordered world of numbers and functions. Ever since Galileo wrote that the book of nature is written in the

language of mathematics,15 Western scientists have tended to assume that the key to understanding the world will be found in mathematics and logic. For example, a more detailed statement of Galileo’s view was written by Paul Dirac, Lukasian Professor of Mathematics at Cambridge, in 1963: It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better.16 To these views, DR Theory replies that their authors may be looking for knowledge in all the wrong places. God’s role, if any, is to goad us (through evolution) into improving our understanding of the universe, not to set increasingly difficult mathematical puzzles for us to solve. Collectivism is the sixth kind of social interaction, where physical reality categorizes ideals. In industrialized societies, group Collectivism is sometimes called socialism. A physical situation—the availability of land, a surplus of existing goods or productive facilities, etc.—supplies the categories; from this setting, the group selects ideal values or institutional principles appropriate to the given physical situation. In large groups, Collectivism may adopt a set of ideal guiding principles on the basis of physical (typically agricultural or industrial) facilities. When it is pursued independently of legalistic considerations, this kind of social organization tends to concentrate on defining ways for distributing goods, sometimes recasting the traditional legal concepts of money, property, and individual rights. In such “pure collectivism,” the unfailing availability of physical goods and facilities may be assumed categorically, to justify the adoption of socialistic ideals. In modern corporations, the topmost policy level often operates Collectivistically. Although externally the corporation is a creature of Legalism, internally it tends to create its own organization. At the policy level, its assets as a whole constitute the physical basis from which its employees do their work. Among those employees, the policy-makers are particularly charged to determine what ideals (principles, goals, guidelines, etc.) are appropriate to best exploit the assets. The topmost layer of corporate management thus develops abstract business formulas from a background of physical fact. Individual social realities. Carving human social realities into six types

of categorization should not be taken to imply that anybody has ever managed to construct a perfectly harmonious balance of these realities within a single personality. For example, Freud famously analyzed the typical child’s relationship with its parents and concluded that conflict was the norm. Today, much practical counseling is devoted to managing the alternations of Communalism and Authoritarianism that characterize modern marriages. The main goal of the sort of analysis that DR Theory supports is to uncover the linkages between the different types of social realities and the types of ontological realities from which they are built. For example, a truly authoritarian political regime will have chosen to organize itself between behavioral and physical realities, with little regard for ideals. Hence, such a regime is unlikely to respond to ideal arguments, as the world discovered in 1939. Similarly, orthodox religious zealots most value behavior and ideals, so tendering them physical threats or rewards seldom works; and so on. The social world may be a crazy place, but sometimes there are underlying methods in its madness. Using set theory and empirical data, DR Theory can help expose those methods and suggest how to work with them.

Worldviews In practice, most people use all six of the kinds of cross-categorization just described at one time or another during their daily lives. But to hold our lives together, each one of us typically constructs a worldview. In such a view, all three types of digital reality—behavioral, physical, and ideal—are crosscategorized and also used as categories for other types of reality, without loose ends or redundancy. There are only two workable ways to do this. DR Theory calls these two ways Individualism and Statism, They can be summarized in two simple diagrams:

The worldview that DR Theory calls Individualism starts with people acting Communally to establish a group physical environment. This leads some to make an Intellectual search for ideals that might govern that

environment. The ideals that are found are applied Legalistically, closing the ring. The worldview that DR Theory calls Statism starts with an existing physical environment that needs ideals for its Collective maintenance. The ideals that are adopted become an Orthodoxy that regulates people’s behavior and justifies Authoritarian control over people’s actions. The ring structure of each worldview insulates it from being questioned by the other worldview. All cross-categorizations in each ring are satisfied internally, and none overlap between the two. Every question that may arise in an Individualist or Statist world is answered by categorizations in that same world. Thus, each worldview delivers the snug feeling of being complete and certain. It provides mutual support for three of the six modes of social interaction and devalues the other three. To summarize, Individualism generally tends to support Communalism, Intellection, and Legalism, while actively discouraging societal Authoritarianism, Collectivism, and Orthodoxy. Statism is antithetical to Individualism: it generally encourages Authoritarianism, Collectivism, and Orthodoxy, while discouraging Communalism, Intellection, and Legalism. The effects of these worldviews are pervasive enough in the history of human behavior to warrant separate analyses of how they usually work in actual social realities. Individualism is so named because it tends to emphasize individual, rather than group, action toward physical reality. In the communalistic phase, a physical situation leads to cooperation among group members—an agreement to share work. This cooperative behavior next becomes the basis for exploring and establishing group ideals. The ideals then become the categorical setting for a system of laws governing group members’ actions toward the physical situation and one another. The intervening stage of abstract legal formulation shields individuals in the group from direct behavioral sanctions; they work within ideal guidelines, rather than obeying personal commands. The component of Legalism in this organization—instead of the Authoritarian phase of Statism—establishes rule by law instead of rule by individuals. It is worth noting that an individualistic worldview works better in groups whose physical situation is categorized more by opportunities than by dangers. It tends to be adopted by frontier societies, where individuals are encouraged to build or mine or plant; Statism tends to be adopted by threatened societies, where individuals are encouraged to unite and defend. The difference in effectiveness seems to stem from the different approaches of individuals to physical situations. In the Individualistic case, each person’s responses are abstractly categorized (by a legal system), and the individual is left to work out the details; in the Statist case, these responses are behaviorally dictated and judged. Individualism supports individual creativity while Statism emphasizes individual obedience to norms established by others. A basic manifestation of Individualism in human life is communication. Its

three types of social categorization are well adapted to making language work. Communalism categorizes verbal or writing behavior physically, giving words and expressions concrete meanings. Intellection categorizes ideals behaviorally so that the tenses and moods of a language can be expressed in verbal behavior. Intellection also allows the creation of rhetorical tropes and supports the verbal expression of abstractions. Legalism regulates the formation of linguistic structures so that what one person says or writes can be parsed and understood by another. The whole three-part cycle of categorization in human communication could be summed up by saying that physical things categorize the bits of language behavior that we agree refer to them; the language behavior then becomes the basis for adopting an ideal system of usage; and the ideal system regulates our actual generation of physical sounds or marks. Among animals other than man, communication consists mainly of one kind of physical reality—sounds or actions—modeling another. By chaining three cross-categorizations, we add an intervening phase of ideal systematization, giving our communication a modal character peculiar to human language. This increases the range of subjects that we can communicate so it equals the range of realities that we can understand. Philosophically, Individualism is materialistic as opposed to idealistic. It asks “What is behavior?” and answers, “It is the manifestation of physical events in living organisms.” It then asks “What are physical events?” and replies “They are realizations of the ideal laws of nature.” Finally it asks “What are ideals?” and concludes, “They are conceptions produced by human behavior.” This is the common, secular, down to earth worldview that underlies much of modern, practical knowledge. It evokes no world-spirit or ideal plan, but pins all its explanations on mundane, empirical, scientific concepts. By embracing all three primary organizations in one cycle of explanations, it appears to tie up loose ends more neatly than any of its component cross-categorizations can. However, Individualism displays a disquieting lack of absolutes. Everything is explained by something else, and the explanations never end. Because each of its three types of categorization is supported by another type, none of them seem to explain reality in its own right. Statism, discussed next, solves this problem by supporting the concept of an idealistic world-spirit that exists by definition and answers all questions. Statism is the opposite cycle of three cross-categorizations, where the subject area of each one provides categories for the next. In it, Collectivism classifies ideals on the basis of physical fact; Orthodoxy uses those ideals as the basis for regulating behavior; and Authoritarianism uses behavior to prescribe physical transactions, thereby closing the ring. The Statist worldview derives much of its coherence from this process of successive support among its component phases. An example can illustrate

how the three phases of Statism can contribute to the whole. Suppose a relatively complex society is threatened by incursions from a hostile neighbor. This physical situation provides categories for the development of a set of ideals, in the Collectivistic phase. The ideals would typically include militaristic concepts: the desirability of serving one’s country, the idea of war as an honorable profession, etc. These ideals now become the basis for an Orthodoxy, often creating a new social class—e.g., a warrior caste. The Orthodox phase defines certain behavior routines and imbues them with value: bravery, service, glory, and so on. Such behavior, finally, becomes the basis for the Authoritarian phase, which dictates what physical acts are to be performed. Arms and fortifications are created, citizens are impressed into service, war is waged. Note that the original physical problem (hostile incursions) is now being solved physically, by military retaliation. But this solution has been reached by a somewhat roundabout cycle of social categorizations, one that has significantly modified the society’s behavioral, physical, and ideal realities. The usefulness of the Statist worldview can be appreciated by considering a typical alternative. In the present example, another way of meeting the physical threat of hostile incursions would be to merge Communalism and Authoritarianism. Here the physical problem would lead to communal cooperation among the group members and adoption of a group behavior plan; this would then become the basis for authoritarian dictates of individual physical acts. The threat might be met by the group coalescing under a chief, who was invested with the authority to lead a war party. Such a response is common in smaller societies. By contrast, the statist organization yields a more complicated solution—but it is one that will ultimately prove more effective, especially in large groups. Note that the first Statist response to a physical problem is not behavioral, but ideal: instead of just forming ranks to take action, the group sets out to create an institution, a system of agreed ideals. The institution (not the physical problem) then becomes the setting for discriminating and regulating behavior. Only after this is accomplished is the group ready to take authoritarian action, to perform physical acts in the field designed to counter the original physical threat. The increased effectiveness of the statist worldview is also evident in smaller groups. A well-studied example is the growth of modern corporations out of simpler (usually authoritarian) businesses. Here the three implicit social phases (Collectivism, Orthodoxy, and Authoritarianism) are revealed by analyzing corporate behavior into three layers of management. As I mentioned earlier, the topmost policy-makers perform the collectivistic phase. They start from a predominately physical setting—the assets of the corporation—and strive to form a consistent set of general principles that will govern the exploitation of the assets. In the next layer, middle managers receive this set of ideals as a basis for defining specific patterns of behavior designed to

conform to its abstractions. They select employees, write job descriptions, issue general instructions, and monitor performance. Their job is an exercise in Orthodoxy, wherein the ideals prescribed by the policy-makers are translated into behavior to be pursued by the workers. In the lowest layer, workers and their supervisors adopt an authoritarian organization, manipulating physical things in accordance with the prescribed behavioral regime. Job instructions created by the middle managers become, in their hands, a setting for the sequences of physical acts that comprise the work done by the corporation. This work, in turn, alters the assets of the corporation, presenting new physical situations to the policy-makers. Thus, the management cycle forms a closed loop. Statism exhibits two characteristics that are not present with simpler organizations (such as Communalism-plus-Authoritarianism). First, the group adopting it acquires greater efficiency in solving chronic, long-term, or largescale problems. In the example of military Statism, it creates such things as permanent fortifications and a cadre of professional warriors, whereas with the simpler war-party organization the group may just band together, do the job, and then disperse. The second characteristic (related to the first) is that the things created by a statist organization—such as military establishments and corporate departments —are likely to acquire a life of their own, enduring after the original problem for which they were created has disappeared. This is because each phase of this worldview is instrumental to another phase and none are related directly to the problem. For instance, consider what happens when we question the validity of a military establishment. If we ask about the physical part—why does the group need troops, arms, and fortifications?—the answer will be that these are necessary to carry out the behavior of waging war. If we then ask, “why wage war?” the answer will be that such behavior supports necessary ideals: freedom, self-determination, perhaps also glory and destiny. If we finally ask where the ideals come from, the answer will be that they are appropriate to the society’s physical situation: the value of its natural resources, its strategic geographical location, even the physiological qualities of its members. Although the military establishment amounts to a physical response to a physical situation, the foregoing argument tends to explain it differently because of the intermediate behavioral and ideal factors in its generating organization. Such a roundabout explanation tends to shield Statist organizations from criticism. The physical part is maintained for behavioral reasons, the behavior is pursued for ideal reasons, and the ideals are held for physical reasons. All these reasons may appear individually sound, even when their totality is unreasonable or out-of-date. Thus, the same factors that make Statist organizations more effective in dealing with large, long-term problems may make them endure after such problems have ceased to exist. In effect, cyclic three-phase organizations such as Statism produce an

endless sequence of explanations. Physical reality is referred to behavior, behavior to ideals, and ideals to more physical reality. The process also shows up in theorizing, where the result might be called “general idealism.” When developed into a complete philosophy, it offers a chain of three explanations. What are ideals? They are the perfect forms of worldly things. What are worldly things? They are the creation of a behaving world-spirit. What moves the world-spirit? Its destiny unfolds according to an ideal plan. Such a threephase scheme lies behind much Eastern philosophy: in the European tradition, it is perhaps most closely represented by Hegelianism. It tends to be more complex and harder to grasp than the two-phase approaches discussed earlier, but it also provides a greater richness of conception because it touches on all three types of digital reality. It appears to cover more ground, in a consistent way, than simpler philosophical reasoning. Neither of the two worldviews discussed above offers a free lunch. Pure Individualism can be rife with hazards and inequities, but it supports the warmth of free Communalism and the stimulus of Intellection. Statism offers physical security but at the cost of Orthodoxy and Authoritarian controls. Trying to find a magic blend of the two seems to produce only conflict and wasted lives. A default worldview. We could regard the Statist construction of social reality as the “default” way of organizing groups, both human and animal, because it supports the fundamental ways that living things survive and evolve: Ideals categorize behavior through genetic inheritance. When we try to understand animal behavior, we frame our explanations largely in terms of the animal’s species. Individual behavior may be triggered by local stimuli, but the response is usually fashioned by an instinct or other inherited trait. Thus, one stimulus can trigger predictably different behavior in a dog and a cat; to understand what is going on, we refer to ideal models of canine and feline traits. Behavior categorizes physical reality through stimulus-response reactivity. Organisms react behaviorally to their physical environment, so it is behavior that explains an organism’s physical actions. Among plants and primitive animals, most reactions are determined by species and higher taxa; but in higher animals, they may come from individual learning. In humans, where we find widespread ideal categorization, most physical acts are still categorized by behavior. The effects of behavior on the physical environment constitute a hallmark of life. Physical reality categorizes ideals through natural selection. What determines the ideal techniques of life—the tissue structures, metabolic pathways, reflex patterns, and so on that are ultimately controlled by genetic and epigenetic codes in individual organisms? These codes are

passed from individual to individual through the physical reproduction of molecules such as DNA. In this way, the physical survival of the individual and its success in reproduction categorizes its codes as favorable. When an individual fails to reproduce, its codes are implicitly categorized as unfavorable. This was Darwin’s seminal insight; he compared the selection that took place in nature—what Spencer called “survival of the fittest”—with the selection that breeders of animals (such as pigeons) performed artificially. In effect, genetic codes are categorized by the physical survival and reproduction of the individual organisms that propagate them. Thus, the behavior by which organisms construct themselves and their digital realities links up in an overall Statist chain of categorizations, not Individualistically. We might expect this result because the primary way that life succeeds is by forming species—Collectivistic groups of organisms—that test ideal models against the environment, not by spawning individuals, each of whom struggles separately to survive.

Science and Religion The differences between Individualism and Statism are sometimes associated with a rivalry between science and religion. DR Theory endorses this connection, treating Individualism and Statism as alternative modes of social behavior with science and religion sometimes serving as their core beliefs. Thus, DR Theory treats science as a core belief system in many Individualistic societies, and religion as a core belief system in many Statist societies. Crosscategorizations account for their differences, as the following tables show.

Science: Source of Reality categories categorized Core belief Events Ideals Physical follow laws Human Behavioral Ideal spirit is free Life is Physical Behavioral materialistic

Religion: Source of Reality categories categorized

Behavioral Physical

Physical

Ideal

Ideals

Behavioral

Core belief God runs the world Human spirit is bound Faith is the goal of life

The differences between science and religion show up most clearly when digital realities are constructed. Since many of those constructions take place within a social context—that is, they take place with the help and within the sanctions of social groups—it can clarify the societal roles of science and religion to list those contexts:

Reality constructed

Social context

Core belief

Physical: Scientific Religious

Events Legalism follow laws Authoritarianism God runs the world Human Ideal: spirit is free Intellection Scientific Human Collectivism Religious spirit is bound Life is Behavioral: Communalism materialistic Scientific Orthodoxy Faith is the Religious goal of life

Individual Liberty Like the behavior of individual persons, the behavior of governments can be good or bad. A common measure of the “goodness” of a government (for example, by Freedom House, an American organization that rates governments) is the amount of individual liberty that is enjoyed by its citizens. Individual liberty, in this sense, is commonly defined as freedom from government coercion. In practice, it turns out that individual liberty is most often encouraged by governments whose behavioral, physical, and ideal powers are distributed among separate bodies. The eighteenth-century French philosopher Charles de Montesquieu is commonly credited with having first delineated the separation of powers on which many modern governments are based. In his 1748 work, De l’Esprit des Lois, Montesquieu identifies three functions of government —legislative, executive, and judicial—and concludes that combining any two in one body entails a loss of individual liberty: The political liberty of the subject is a tranquility of mind arising from the opinion each person has of his safety. In order to have this liberty, it is requisite [that] the government be so constituted as one man need not be afraid of another. When the legislative and executive powers are united in the same person, or in the same body of magistrates, there can be no liberty; because apprehensions may arise, lest the same monarch or senate should

enact tyrannical laws, to execute them in a tyrannical manner. Again, there is no liberty, if the judiciary power be not separated from the legislative and executive. Were it joined with the legislative, the life and liberty of the subject would be exposed to arbitrary control; for the judge would be then the legislator. Were it joined to the executive power, the judge might behave with violence and oppression. There would be an end of everything, were the same man or the same body, whether of the nobles or of the people, to exercise those three powers, that of enacting laws, that of executing the public resolutions, and of trying the causes of individuals. 17 Here we see cross-categorization laid bare in the construction of social reality. Governments affect their subjects physically by their executive power, which can arrest persons and seize property; their legislative power sets behavioral norms for their subjects; and their judicial power defines ideal values for them. As long as these three powers construct categories separately from one another, each category grouping objects in a different power, the whole machinery of cross-categorization comes into play. Each power in the society gives meaning to the other powers through categorizations that the other powers see as tentative and subject to error. In a typical government that separates its physical, behavioral, and ideal powers, the physical power has the mandate to construct physical objects that realize the laws passed by the behavioral power. It builds roads, forts, prisons, etc., and provides personnel to administer them. While so doing, the physical power must be careful not to violate the concepts of rights constructed by the ideal power. The behavioral power constructs laws based on the wants of the social group, but it must take care not to construct a law that the physical power either will not or cannot execute, nor one that the ideal power may categorize as illegal. The ideal power must try to abstract and express the universal patterns in social reality while rendering judgments that will neither incapacitate the physical power nor radically contradict the behavioral power. This balance of a government’s powers mirrors the balance of crosscategorizations within each member of the social group that it governs. In a group of individuals who tend toward Statism, the operation of the physical power of the group (in modern terms, the executive branch of the government) would usually be categorized behaviorally in an authoritarian style of social construction; with a group that tends toward Individualism, it would be categorized ideally in the style of Legalism. With Statism, similarly, the group’s behavioral legislative branch would be categorized ideally as an exercise in Orthodoxy; Individualism would categorize it physically as an instance of Communalism. And Statism would categorize the ideal judicial branch physically as a Collective, a body that controls material transactions,

whereas Individualism would categorize it behaviorally as an exercise in judicial Intellection. Individualism supports individual liberty within a group because it uses ideals to cross-categorize the physical actions of the group’s members, instead of using behavioral categories. In an Individualistic group, the ideals may be recorded in a constitution and laws; in a Statist group, the behavior that categorizes the members’ physical actions more often comes from a leader, a ruling class, or a tradition of conduct. My earlier discussion of Individualism versus Statism characterized Statism as the more natural way to organize human groups. There is a drag toward Statism that only individuals can counteract. But to pull a group toward Individualism requires that its members maintain a clear separation between the three types of reality; that was Montesquieu’s point. When categorizations overlap—for example, when behavioral dictates replace ideal laws— Individualism dissipates and individual liberty may become lost.

Consciousness DR Theory supports a subordinate theory of consciousness. While not essential to DR Theory itself, an understanding of consciousness can help us understand how knowledge is acquired and stored. Psychologists tend to group several distinguishable mental processes under the rubric of “consciousness.” For instance, Steven Pinker cites three areas of human experience that consciousness might denote: self-knowledge, access to information, and sentience.18 Of these, it is sentience that bedevils analysis. “Beats the heck out of me!” writes Pinker about sentience, and many other thinkers agree. The hard sciences, where objects of knowledge are supposed to sit still while we examine them, find it difficult to come to grips with the slippery, self-referential nature of the stream of consciousness. Doctor Johnson, citing Locke, defined it as “the perception of what passes in a man’s own mind” and William James pointed out that “the most immutable barrier in nature is between one man’s thoughts and another’s.” In the form of sentience, therefore, consciousness is a unique area of human digital reality: one that is personal, private, and—most importantly—known only to itself. In the history of human evolution, consciousness may be a late arrival. In his 1976 book, psychologist Julian Jaynes argues that human consciousness is a product of the evolution of society—something that developed during the second millennium BC, at least in the Middle East. In his account, earlier human mental states dealt mainly with physical reality. Most social reality was dominated by authoritarian dictates from rulers and gods, objectified within plebeian experience as inner voices. But as a consequence of having to

grapple with an increasingly complicated mix of civilizations and political institutions, people began to develop individual sentience. Jaynes concludes that consciousness is chiefly a cultural introduction, learned on the basis of language and taught to others, rather than any biological necessity.19 DR Theory would tend to agree. Mere awareness does not yield major benefits in dealing with the physical world. Reacting to physical reality means doing this to that; the further realization that I am doing this does not add efficiency or effectiveness to the process. But when one is dealing with social reality, the difference between knowing I choose to do this versus knowing that somebody else is choosing instead of me or this is happening despite my wishes is crucial. Consciousness can help make anyone an effective player in society. DR Theory suggests that this is the role of consciousness in society and its raison d’être in human life. Sandboxing. During the development of computer software, a programmer may create a sandbox to aid in testing new code. A sandbox is a functional area in a software system that can access system services but cannot interact with the rest of the system. Code under development can be safely tested in a sandbox; if it malfunctions or crashes the damage will be confined to the sandbox, which can be quickly rebuilt. Consciousness might be thought of as a sort of social sandbox, an area of digital reality in humans (and likely in other animals) where thoughts may be developed and plans matured without public disclosure. By walling itself off from the normal stimulus-response activities of life, such an area would be able to choose what to think about (“I’ll think about that tomorrow,” mused Scarlett O’Hara) and how to categorize it (“When I use a word, it means just what I choose it to mean,” said Humpty Dumpty). By manipulating knowledge as sets in a sandbox, we can interchange objects with categories and our imagination can build realities in ways that would be impractical in any other context. A theory of consciousness conforming to DR Theory could also import concepts from other areas of psychology to deepen our understanding of sentient reality. For example, Freud’s ideas of the id, ego, and superego might characterize sandbox proxies for certain types of physical, behavioral, and ideal components of individual personality. In DR Theory, the social sandbox of consciousness provides an essential portal between our social realities and the personal realities discussed in the next chapter. For example, suppose I am at work and spy a bowl of fruit on a colleague’s desk. I realize that I am hungry and in the bowl is an apple. If I had no knowledge of social mores, I might just walk across the room and seize the apple. But when I constructed the social reality in which I work, I

categorized things such as that apple as “someone else’s property.” The apple acquired a hierarchy of social attributes unrelated to my hunger. A resolution now emerges in the sandbox of my consciousness. I become aware of the conflict between my metabolic wants and the social constraints associated with the apple. I reflect on possible ways to obtain the apple and construct a scenario. I will walk over to my colleague and say, “What goodlooking fruit you brought in!” He, whom I know to be an obliging person, will probably reply, “Help yourself.” If so, I will accept and eat his apple, and tomorrow I will bring in my own fruit and offer to share it with him.

5. Personal Realities

We have evolved three ways of constructing digital realities in our everyday lives: Natural, Formal, and Spiritual. SURVEYING THE VARIED RESULTS of cross-categorization, DR Theory finds that some

construction techniques are more common than others. The construction of two big worldviews—Individualism and Statism—was described in the last chapter. Each one is made by linking three of the six kinds of cross-categorization into an endless cycle. This chapter describes three reality construction techniques that are more personal. Each of them results from combining two complementary kinds of crosscategorization. DR Theory calls the resulting realities Natural, Formal, and Spiritual. They are the kinds of realities that people construct and know on a daily basis, as integral parts of being human and alive. Because “everybody sees the world a bit differently,” they can serve as rough definitions of our individual personalities. For clarity in the following discussions, I retain the sociological names that I used in Chapter 4 to identify the types of cross-categorization. Although we construct digital realities for reasons beyond group behavior, we typically learn our construction techniques from others—and teach them to others—within our social groups. Personal realities are facets of our characters that we usually try to share with friends.

Personal Categorizations This chapter discusses three common ways of human life that DR Theory calls lifestyles. They are distinguishable through set-theory analysis by their dominant cross-categorizations. Each lifestyle can be further distinguished by the minima they recognize and the tokens they collect. These latter two concepts are merely simplified clues for identifying lifestyles: Minima are theoretical objects that each specific lifestyle posits as both real and irreducible in digital realities. Tokens are objects that people engaged in a specific lifestyle tend to

create and accumulate. Typical minima and tokens are shown in this table: Lifestyle Categorizations Behavioral and Natural Physical Physical and Formal Ideal Ideal and Spiritual Behavioral

Minima Tokens Causes

Possessions

Elements Truths Values

Acts

Here is a line-by-line summary of this table: Natural lifestyles cross-categorize behavioral reality with physical reality, each digital reality supplying categories to explain the other. People engaged in the Natural lifestyle tend to explain their worlds in terms of causes (“the wind made the apple fall”). Causal links are attractive explanations because the same object of knowledge can be categorized both behaviorally (A made B do something) and physically (both actors were material things). Tokens accumulated in the Natural lifestyle tend to be physical possessions that require behavioral energy to acquire and protect. Examples are wealth, property, and confirmations of recognition by others. These things are collected because they constitute physical evidence of successful behavior. Formal lifestyles cross-categorize physical reality with ideals. Many scientists and technologists favor this schema and see their worlds in terms of elements that are both corporeal and abstract. The subatomic particles of physics, for instance, are physical because they are the constituents of matter and ideal because they have absolutely identical properties. Formal tokens include “established truths” that are widely regarded as verified scientifically or logically. Acknowledging them tends to enhance the value of the rest of one’s constructed digital reality. Spiritual lifestyles cross-categorize behavior with ideals. Because physical reality is absent, people engaged in this lifestyle tend to act “other-worldly.” They often behave in accordance with their values, not their needs. They may regard the world as primarily a moral testing ground, not as a workshop for material prosperity. Spiritual tokens are often behavioral acts, such as virtuous deeds, that

exemplify higher values instead of material accomplishments. The lifestyles listed above are described in more detail below.

Natural Reality Communalism understands behavior from a physical basis; Authoritarianism regulates physical reality on a behavioral basis. Each type of crosscategorization provides categories for the other: Category type Physical Behavioral

Object type Behavioral Physical

Sociological type Communalism Authoritarianism

Applied together, Communalism and Authoritarianism help us construct whole realities that are the most prevalent in human life. The third type of reality, ideals, is absent. One could think of Natural reality as nonideological living. My discussion of Communalism and Authoritarianism in societies mentioned their importance in human families: Communalism is the approach by which infants are reared and Authoritarianism is the approach by which they are trained. As a lifestyle, they merge into what might be called family life. Physical conditions provide the setting for communalistic modifications of behavior, resulting in cooperation to achieve family goals; behavior is the setting for authoritarian regulation of physical acts by family members, producing effective results from a mixed group of children and adults. Although most typical of families, this organization appears in any relatively small group of people with common physical goals or problems requiring efficient group effort: tribes and clans, military units, labor details, project teams, exploring parties, etc. To the extent that they behave like families, such groups display this lifestyle. As a way of life, Communalism plus Authoritarianism predominates among “primitive” people, who may spend much of their time either maintaining lines of communal cooperation with one another or obeying the authoritarian dictates of leaders or traditions. But it is also a common paradigm in the everyday life of the members of “advanced” societies. Any human being whose behavior lacks this fundamental approach to life tends to become isolated and ineffectual. In theorizing, the two phases of this lifestyle first appear as notions of perception and animism. Perception, the most basic communalistic theory,

generates an understanding of our behavior in physical terms—this sensation comes from that object, this thought is about that event, and so on. Animism, the converse authoritarian theory, generates an understanding of physical things in behavioral terms, by the ways they act—one object moves, it affects another object, and so on. As they merge in Natural reality, the result is a concept of causation. In perception, physical things seem to force their qualities upon us; animism suggests that physical things must do the same to each other. Together they encourage us to interpret the world in terms of causes and effects, as a combined physical and behaving whole. Our notion of causation could be thought of as a basic worldview without a grasp of ideals. Things push one another, A results in B and leads to C, but nothing endures in the process, no principles are realized. It might as well happen entirely differently. Hume analyzed causation from a logical standpoint and concluded that it was irrational. Yet it is a deeply felt approach to understanding reality. This is because it embodies in theorizing a basic understanding that is common in everyday human life.

Formal Reality Legalism uses ideals to define and explain physical reality; Collectivism regulates ideals to conform to physical reality. Each phase of crosscategorization provides categories for the other: Category type Ideal Physical

Object type Physical Ideal

Sociological type Legalism Collectivism

The combination of Legalism and Collectivism helps us construct whole realities that we feel are objective and useful, but in which the behavioral type of reality is absent. One could think of Formal reality as absolute, objective reality, unaffected by human interference. Recall the Communal-Authoritarian lifestyle previously discussed, which associates physical reality with behavior. It produces social processes that are family-like and based on cooperation, tradition, and personal loyalty. When Legalism and Collectivism merge, they substitute for this behavioral association a reliance on abstract institutions. The collectivistic phase starts from a given physical basis—available goods and facilities such as land, livestock, tools, raw materials, etc.—and consists of a search for appropriate ideals to govern the uses of such things. The outcome is a system of advanced social concepts and institutions: private

property, transfer and inheritance, monetary units, and the whole edifice of the common law. These ideals then become the basis for the legalistic regulation of physical transactions in a society, through the establishment of courts and official means of enforcement. Merged into a lifestyle, the legalistic and collectivistic phases of this process display an intimate and continuous interplay. New ideal concepts and new physical transactions constantly generate each other. Thus, when a new physical situation arises that is not covered by the society’s current institutions (such as the introduction of mass automobile transportation or the emergence of the Internet), a search for new principles ensues. We say that we need new laws, and legislators set about drafting them. Enforcing the laws then tends to modify the physical situation, bringing it more into line with the new ideals. Legalistic-plus-Collectivistic theorizing, typical of modern physics, builds up a body of natural law instead of man-made law. A physical situation—a collection of data or a newly observed physical effect—suggests the need to formulate a new law of nature. The new law in turn suggests new researches into physical reality, often requiring the construction of novel machines or experimental methods that would not otherwise have been conceived. These researches turn up fresh data and the cycle continues. Knowledge of physical effects and an armory of descriptive abstractions grow side by side. In physics, this whole process might be called framework theorizing, because its system of abstractions constitutes a prescribed framework that is treated as underlying the things of physical reality. Theorizing in this lifestyle contrasts with the simpler explanations of Communalism-plusAuthoritarianism, which would prefer to explain the world in terms of dynamic chains of causal links.

From Natural to Formal People in industrialized countries today automatically move much of their daily working knowledge from Natural to Formal realities as they grow up. It is generally recognized that the realities known by adults must be different from the realities known by children. Parents and schools support this transition by inculcating children with “formal education.” Managing this change in personal lifestyles during the seventeenth to the twentieth centuries, through compulsory education, occupation licensing, work standards, etc., was one of the remarkable achievements of Western culture. As analyzed by DR Theory, the “pivot” around which this transition moves is physical categorization, as shown below: Category

Object

Sociological Personal

type Physical

type type reality Behavioral Communalism Natural Ideal Collectivism Formal

Such a transition is inherently materialistic because its categories are physical. In Natural reality, physical categories define personal behavior; in Formal reality they define personal ideals. Because behavior is typically overt and ideals are typically subjective, some psychologists refer to this transition in personal realities as an “internalization” of moral or ethical codes. When Thomas Hobbes published Leviathan in 1651, physical categories defined personal behavior for most European citizens. Their political attitudes were based on Natural realities. When Max Weber’s Economy and Society was published in 1922, physical categories were mostly used to define personal ideals, and the bulk of the citizenry based their lives on Formal realities. Sociologists have described this change in terms of a distinction made by Ferdinand Tönnies (1887) between a community (gemeinschaft) and a society (gesellschaft). Tönnies, a Hobbes scholar, described the feudal community that had dominated Europe before the eighteenth century. Weber, a student of the Protestant work ethic, described the modern society of free citizens under the rule of law that succeeded it. This is one of the most striking changes of personal realities to be found in the human record.

Spiritual Reality Intellection understands ideals through mental behavior; Orthodoxy regulates that mental behavior. Each phase of cross-categorization provides categories for the other: Category type Behavioral Ideal

Object type Ideal Behavioral

Sociological type Intellection Orthodoxy

Alternating Intellection and Orthodoxy helps us construct realities that are in some sense free of worldly influence because the physical type of digital reality is absent. One could think of Spiritual reality as nonmaterialistic or supersensory. How the two phases merge can be seen in the genesis of churches. An

individual—a prophet—exploring abstract principles comes up with a set of ideals that can be categorized behaviorally and that seems to have relevance for human life. If the prophet is successful (most are not) these ideals are picked up and promulgated by a group of followers, who establish a sect or a church. For the prophet, the ideals were principally the outcome of Intellection; for the followers, however, they become principally the basis of an Orthodoxy. The prophet sought knowledge; the followers seek to regulate behavior, to make people better. But for the church to survive these two approaches must blend into a single coherent process. Intellection alone is schismatic and leads to the church’s disintegration; Orthodoxy alone is dogmatic and leads to its overthrow. It takes a constant interplay between the two to satisfy human spiritual needs. In a successful church or sect, each new communicant is exposed to a comprehensive education in the ideal articles of faith, which is a process of Intellection; at the same time, these articles are consistently promulgated as an absolute setting for the definition and regulation of the behavior of the faithful. When this is done effectively, the communicant becomes convinced that the system of ideals and the church’s routines of human behavior naturally correspond to each other. Many industrialized societies today are progressively abandoning traditional churches, replacing them with an Intellection-plus-Orthodoxy lifestyle built around class membership. Here Intellection appears in school education and Orthodoxy in the maintenance of social class norms. Students in these societies, like the communicants of a church, undergo a long process of learning the ideals that will largely determine to which social class they belong. At the same time, the members of each class try to ensure that the education being given inculcates the Orthodox behavior they take to be necessary for social life. This combined organization shows up as a major paradigm in human behavior similar to that produced by churches, but now more ethical than spiritual. On an everyday level, theorizing within the Spiritual lifestyle appears as morality. Prevailing human behavior suggests a set of ideals by which it may be categorized; those categories then become the basis for calling behavior good or evil, moral or immoral. Actual societies may also acquire a “tone” that depends on the balance of their lifestyle. When Intellection predominates, we say a society is liberal or open, and when Orthodoxy predominates we say it is repressive or closed. On a more esoteric level, particularly in evangelical sects, the Spiritual lifestyle may tend toward mysticism. Ethical ideals, treated individually at first, may become interconnected in an abstract Absolute, separate from other digital realities, which is then taken to be directly accessible to human behavior. Here is how Evelyn Underhill describes the characteristic dual

conception of the mystic: he is able to perceive and react to reality under two modes. On the one hand he knows, and rests in, the eternal world of Pure Being, the ‘Sea Pacific’ of the Godhead, indubitably present to him in his ecstasies, attained by him in the union of love. On the other, he knows—and works in—that ‘stormy sea,’ the vital World of Becoming which is the expression of Its will. To the great mystic the ‘problem of the Absolute’ presents itself in terms of life, not in terms of dialectic. He solves it in terms of life: by a change or growth of consciousness which—thanks to his peculiar genius—enables him to apprehend that two-fold Vision of Reality which eludes the perceptive powers of other men.20 The “act of Divine Union,” known to the mystic, brings together these two factors, the ideal and the behavioral. It amounts to a decision to treat categories interchangeably: ordinary behavior becomes regarded as part of a Divine system, and Divine ideals become regarded as perfect forms of behavior. These ethical and spiritual patterns in human life tend to ignore physical reality. Because they arise by combining behavior with ideals, they do not include a grasp of physical states, and thus often treat physical reality as gross or something to be overcome. One of the first things taught by most church doctrines, moral systems, or mystical disciplines is that they embody truths outside human corporeal experience.

Beyond Ideals The scope of DR Theory extends only to the three types of digital reality discussed so far in this book—behavioral, physical, and ideal. Most people would agree that they cover the bulk of what is commonly regarded as real. Yet the analysis in this book began (in the Introduction) by suggesting that three modern innovations in knowledge needed to be more fully incorporated into our understandings of the world. These innovations are biological evolution, axiomatic set theory, and analog-to-digital conversion. While DR Theory may start this incorporation, it by no means finishes the job. So here are a few visions of where evolution, set theory, and digitization might take human knowledge in the future. Evolution is currently well understood as a driver of speciation and a determining factor in organic physiology. It is widely cited in explaining the configurations and behavior of living things besides Homo sapiens. With humanity, however, the role of evolution is assumed to have begun with the rise of hominids and ended in historic times. When we get to some of the

basic concepts of human science—such as time and space—it is hard to imagine that they might be adaptations evolved by life, not fixed features of existence. But why not? If we accept that reality is digitized existence, we are more likely to find ordering protocols such as time and space in the analog-todigital algorithm than in the analog source. Life as we know it evolved time first, then space. Suppose that another form of life had evolved space before time. Space for them would consist of a one-dimensional linear life-line, while time would be its multidimensional powerset. Creatures of that life form would spend their lives moving inexorably on a fixed track through space, but in every place where they visited they could move freely about in time, like visitors to a history museum. Now imagine our attempts to interact with creatures of that life form. They would appear to us to be always moving around but choosing to pop up at various disconnected times. We would appear to them to be separated into spatial groups that were always on the move, like wandering tribes. They would observe us as if they were traveling on a never-stopping spatial tour bus, taking occasional snapshots through the window of scattered herds of people in motion. We would see them as occasional visitors appearing from time to time, always on the way to somewhere else. If both we “timies” and our brethren “spacies” realized our situations, we might contrive to leave messages for each other; but it would be hard to arrange a place for them, and a time for us, where and when we could meet face to face. Set theory is widely used in practical computer programming. This fact is fortunate because one way our digital realities might expand would be through a fourth type of understanding, one that extends our familiar behavioral, physical, and ideal understandings. Set theory tells us that this fourth way of digitizing existence would be based on the powerset of the set of all ideals—i.e., on knowledge of all the ways that ideals could be usefully grouped together. Absorbing such a body of knowledge might well fall beyond human capacity. Disciplines such as metalogic—the study of logical systems—already explore the powersets of small areas of ideals. But showing how all ideals could be related by rendering their full powerset as a palpable digital reality might require a distributed network of many computers. Then it would take dedicated human thinkers of unusual ability to understand the contents of that reality. There is a modern tradition of accessing a fathomless view of raw existence through sudden inspirations. A good example is the kind of experience that theologian Emanuel Swedenborg called a “vastation.” Henry James, Sr., had such an experience, and his son, psychologist William James, documented the phenomenon in his classic Varieties of Religious Experience.

There he describes a faculty of human experience that generates a sense of reality, a feeling of objective presence, a perception of what we may call ‘something there,’ more deep and more general than any of the particular senses by which current psychology supposes existent realities to be originally revealed.21 The literature on this subject, written by those who report vastation-type experiences, is filled with references to “visions of absolute reality” and “the union with pure existence” that qualify as assertions of objectivity fully as much as the less passionate statements of empirical scientists. James writes of “a state of insight into truths unplumbed by the discursive intellect” and concludes that our normal waking consciousness, rational consciousness as we call it, is but one special type of consciousness, whilst all about it, parted from it by the filmiest of screens, there lie potential forms of consciousness entirely different. No account of the universe in its totality can be final which leaves these other forms of consciousness quite disregarded.22 As a purely practical matter, those who claim to have attained such consciousness behave differently from other people. They are the “enlightened,” the explorers of “the other side,” the “twice-born”; and they return, like travelers to an exotic world, not quite the same as when they left. It is not hard to believe that they have experienced something outside the boundaries of traditional behavioral, physical, and ideal knowledge. Digitization as a mechanical technology is relatively new and is mainly used by computers. As a life process, however, digitization is something living things do all the time. Life has evolved analog-to-digital conversion algorithms from its inception. That is the good news. The bad news is that life has evolved only those A/D algorithms that contribute to the pursuit of traditional living needs. In the case of humans, our needs may be widely varied but they are not exhaustive. Our bodies don’t come equipped with every possible A/D converter that we might want. An example was cited in Chapter 1. When we ask the question, “What is it like to be a bat?” we discover that bats have some senses, such as echolocation, that we don’t have. Because they depend on hunting in the dark, bats have evolved an ability to convert the existential space around them into a digital reality of solid objects without using ambient light; we can’t do this. We can try to emulate the bat’s skill by learning a sort of human echolocation formerly called “facial vision,” but it falls short of constructing the full digital reality that a bat enjoys. The foregoing considerations bear on some parts of the field of

parapsychology, including precognition, telepathy, clairvoyance, intuition, and extrasensory perception. Although commonly denounced as fakery or “violations of the laws of physics,” some of the phenomena studied in these areas could represent analog source data in existence that we are not biologically equipped to digitize. In cases where we conclude that paranormal sensing in humans reflects inadequate natural A/D conversion, we would have a choice of two resolutions: either try to sharpen our natural senses by adding intelligent learning, or design mechanical sensors to help us. The technology of sensor design is still young. For example, only recently has the field of machine olfaction begun to produce “electronic noses” that can not only smell noxious odors better than we can, but that can smell gases and objects that are odorless to us. In this way, computer technology is making it possible for us to understand new digital realities for the first time.

6. Using DR Theory

Digital Reality Theory helps bring the foundations of human knowledge up to date. IN OUR WESTERN CULTURE the traditional purpose of philosophy has been to support the

sciences, humanities, and common sense from which human understanding flows. It asks general questions about the assumptions, procedures, and concepts that become the starting points for analyzing and understanding our world. Philosophy is the avant-garde of knowledge. But like all avant-garde movements, philosophy often finds itself in conflict with established practices. It has flowed and ebbed several times in European history. Although philosophy is largely at an ebb today, I believe it is starting another flow. In Europe, explicit philosophy first appeared among the Greeks and Romans, for whom theorizing was a novel occupation. The dominant human reality at that time was what DR Theory calls Natural reality, the cross-categorization of physical reality with behavior. Philosophers such as Aristotle and Plato provided methodologies by which they and other thinkers could develop theories about human life and its relation to the material and spiritual worlds. A second wave of philosophy arose as Rome declined, this time about intellectual reality. Augustine’s City of God, a philosophical work written soon after the sack of Rome (410 AD), was a seminal example. It advanced practical arguments for adopting Christian monotheism, portraying the City of God as a place where human behavior and ideals supported and explained each other. In the City of Man, obedient to its Greco-Roman pantheon, the behavior of most people served mainly their physical needs. The City of God was an invitation to a new, spiritual, way of theorizing about life and behavior. By the seventeenth century, a third wave of philosophy managed to take hold. It laid the groundwork for theorizing about what DR Theory calls Formal reality, the cross-categorization of physical reality with ideals. Natural philosophers, such as Francis Bacon, Galileo, Descartes, and Boyle, defined the methods by which scientific theories could establish themselves. Before the end of the century, Newton had published his classic Mathematical Principles of Natural Philosophy, a proof of concept for explaining physical events through ideal categories. Today, scientific knowledge is firmly established in Western thought and has

become an integral part of daily life in the industrialized world. Its success has tended to preempt some traditional areas of philosophy. Metaphysics and ontology give way to physics, epistemology bows to experimental method, and so on. “Philosophy is dead,” wrote the late physicist Stephen Hawking, explaining that Philosophy has not kept up with modern developments in science, particularly physics. Scientists have become the bearers of the torch of discovery in our quest for knowledge.23 The successes of science notwithstanding, a fourth wave of philosophy began to emerge in the middle of the twentieth century. New openings for philosophical speculation have been created by the development of new information technologies, an event as transformational in our time as was the adoption of modern science in the seventeenth century or the spread of Christianity in the fifth. Although professional philosophers have been slow to realize it, the development of digital computers opened new vistas for epistemology. Computer architects were specifically tasked to design electronic systems that understood and worked with the world the way humans do. They responded by emulating in hardware many of the basic processes of life. The payoff for philosophers today is that we understand how these emulations work. We can experiment with them in ways impossible with human brains. By mining the decisions that resulted in successful computer technology, we can uncover philosophical truths about the workings of human knowledge. This book is offered as a contribution to the new wave of philosophy just described. Using DR Theory, the next sections address some perennial questions in human thought and propose some answers to them. These short summaries suggest how I believe DR Theory might help resolve a few age-old philosophical issues.

Resolving Cartesian Dualism The basic difference between mental realities and physical realities (epitomized as mind and matter) has always seemed obvious, yet it is baffling to analyze. By describing these realities as two kinds of constructed sets with different transfinite cardinalities, DR Theory shows both how they are different and why they are difficult to compare. Mind is behavioral, contained in a linear and countable set; matter is physical, contained in a dense and uncountable set. Philosophers sometimes divide mental reality into bits of experience called “qualia.” C. I. Lewis wrote in 1929: The quale is directly intuited, given, and is not the subject of any possible error because it is purely subjective.24

Typical qualia include the redness of a sunset and the pain of a headache. Experiencing a quale tells you pretty much all you can know about it. However, the physical concomitant—a sunset or a biological stress condition —may be merely the portal to a lengthy empirical search for the “origin” of the redness or pain in the intricacies of optics or physiology. Our inability to compare behavioral and physical events spawns the question of how they could possibly cause one another. DR Theory’s answer is that their relationship is not one of causation, but rather one of digitization. Sunsets and bodily stresses are analog happenings; seeing red and feeling pain are digital events. As living organisms, our bodies have evolved to convert the one into the other. Humans construct digital realities out of analog existence. When the sky or my body form hard-to-define analog configurations, I digitize them into easier-to-understand sunsets and headaches. A few generations ago, most people might have felt that the foregoing explanation consisted of explaining one mystery by posing another. But computer technologists today regularly earn their livings by performing the “magic” of analog-to-digital data conversion. In the jargon of engineering, it is well-understood. Programmers have worked out several ways to digitize the color red, and they are working on ways to digitize potential headache pain from analog measurements of body stresses.

Clarifying Time and Space Physics is forced to include metrics of time and space in its descriptions of phenomena without explaining what these parameters measure in themselves. DR Theory clarifies the adoption of time and space as responsive mechanisms that life evolved early in its history to make metabolism and photosynthesis work in the Earth’s environment. Time has been a perennial bugaboo for philosophers and scientists. It has been variously called an absolute feature of nature (Newton), a human “intuition” (Kant), or something entangled with space (Einstein). All this confusion results from trying to define time in physical terms. As something constructed by life, time is easy to understand: it orders behavior. Behavior has no inherent spatial location, but it does form sequential “trains” in which the ordering relations “before” and “after” are crucial. On the micro level, time is necessary to sequence metabolic reactions. On the macro level, before and after ordering is necessary to link behavioral events such as stimuli and responses, emotions and expressions, intentions and executions. Without time, living behavior wouldn’t work. Time’s role in behavior makes it likely that our perception of time evolved with the evolution of life. What about space? A big clue presents itself in the privileged position of the vector of propagation of light in space-time. As

Einstein intuited, this vector remains invariant with respect to all observers, even when the observers are moving uniformly with respect to the vector’s origin. This fact suggests that life’s perception of space evolved to facilitate and maintain the reception of light from a distant source while life was moving with respect to that source. Such a situation would be the case during the billion or so years that life on Earth was evolving photosynthesis. Those who seek theories that offer elegant explanations will be gratified to note that it makes more sense to treat time and space as evolved features of life’s processes of digitizing existence, instead of trying to understand why existence happened to provide just the space-time features that life needed to survive successfully on planet Earth.

Generalizing Categorization Traditional schemes of categorization (proposed by Aristotle, Kant, Lakoff, and others) have tended to assume that categories are mostly based on abstractions. In DR Theory, digital sets of any type may be used as categories. Allowing all types of sets to be used as categories resolves the “problem of universals.” It also accounts for the range and flexibility of human understanding and helps explain how human knowledge evolves. The metaphysical problem of universals has been a puzzle for philosophers since ancient Greek times. If all red things share the universal “redness,” does the universal exist by itself? DR Theory regards this ontological question as irrelevant; calling something a “universal” is just another way of identifying a category set. In set theory, a set is just as real as its members, so a universal property of several real things is just as real as the things that have that property. The supposed problem of universals is often the result of an innate philosophical tendency to demand that reality be one monolithic thing. The idea of natural divisions within reality scares up all kinds of theoretical bogeys. So, if universals are real, they must all be either behavioral, as phenomenologists and solipsists suppose; or physical, as materialists insist; or ideal, as Platonic idealists believe. DR Theory replaces the idea of universals with a reality of categories that are sets. Early in the dawn of its evolution, life discovered that sets could serve as generalizations. For example, any set of red things could serve the same function in knowledge as the hypothetical universal “redness.” It was a real thing that helped us understand what it meant to be red. Moreover, we could construct such a category set in our behavior (call it “seeing red”), in the physical reality that was the powerset of behavior (“being red”), or in physical reality’s powerset—“the abstract ideal of redness.” In the practical business of knowledge, each of philosophy’s definitions of a universal could be satisfied

by at least one of these sets. Categorization in DR Theory also explains how our knowledge acquired its remarkable range and flexibility. Human behavior—particularly consciousness —knows both sets that come from digitizing existence and sets that we create artificially through powerset operations. Unless we are psychotic, we generally distinguish the two, but we don’t assume the kinds of fundamental separations that philosophers try to make. Objects and categories tumble over each other in the sandbox of consciousness, generating ideas and speculations. These objects of knowledge are all just sets, so our mental behavior doesn’t need to follow special rules for handling categories.

Defining Statism and Individualism Among the ways that human groups organize themselves, Statism and Individualism are the most pervasive. Using its schema for social evolution, DR Theory explains how Statism and Individualism each justify themselves internally. It also explains why there can only be two such global organizing principles in human societies and why they tend to be incompatible. One can think of Statism and Individualism as coalitions of simpler social organizations that categorize one another in endless rings: Statism = Collectivism + Orthodoxy + Authoritarianism [Physical → Behavioral → Ideal → Physical] As social organizations, both Statism and Individualism are attractive to their adherents. Statism offers sufficiency, certainty, and stability. It is also the default way into which groups devolve naturally, as explained in Chapter 4. Individualism offers amity, freedom, and fairness. People who grow up in either one tend to prefer it over the other. As worldviews, both Statism and Individualism defend themselves internally. Statism takes the physical assets of society as given, using them to define ideal principles for their distribution or use. Those ideals prescribe the behavior of Statist citizens, which is enforced by physical means. Individualism assumes that physical assets must be created by appropriate behavior, which defines the ideals of work and productivity. These ideals define laws that regulate the physical actions of Individualist citizens, closing the loop.

Explaining Nonlocality in Physics Einstein’s “spooky action at a distance”—the mechanism by which particles

influence one another through empty space—is called “nonlocality.” It vexes modern physicists and has led to awkward concepts such as particle-wave duality. By separating reality from existence, DR Theory lets the appearance of nonlocality in the reality models of physics be treated as an artifact of digitization. Two and a half millennia ago, Democritus argued that atoms must exist because it would be impossible to keep dividing matter in half forever. At some point, you would have to stop, and what was left would be indivisible atoms. Aristotle rejected this chop-logic, holding that the four ancient elements—earth, water, air, and fire—were all obviously continuous. Physicists today, some of whom have spent their careers dividing matter into ever-smaller particles, laud Democritus as the “father of modern science.” Others call Aristotle the “father of Western philosophy.” DR Theory awards cigars to both fathers. It holds that existence is continuous and accessible only by analog measurements, whereas reality is discrete and known digitally. Analog existence ultimately determines our fates, but digital reality gives us the knowledge we need to influence those fates. Aristotle was right about the world as it is; Democritus was right about the world as we know it. The problem with populating existence with particles, whether they be atoms or quarks, is that existence contains more than just matter. When we digitize it into material objects, they interact and entangle one another in multiple ways. It strains credulity to imagine that all those particle interactions are the result of yet more particles traveling through a void between the particles we know about. Physicists try to map energy with fields, but even so, some fields in quantum mechanics are so full of probable values that they hardly qualify as reality. Chaos theorists take a somewhat different tack; they characterize existence as “scale-free” and digitize it by means of fractals. Describing existence as a continuum, while it may be discomforting to those looking for absolute objectivity, is as close as we have yet come to the truth. DR Theory supports that approach by showing how digital realities can contain multiple digitizations of a single physical continuum. For example, some digital sets may describe how masses act while others explain transfers of energy. The result would be a physics unified through interacting sets, instead of being divided by competing mathematical models.

Rationalizing Epistemology Traditional epistemology has tended to be empirical and pragmatic—more a compendium of kinds of knowledge than a system that analyzes knowledge in general. Attempts to formalize epistemology around reasoning (Kant), logic

(Russell), or language (Carnap) have not been durable. Using axiomatic set theory (Zermelo and Fraenkel), DR Theory offers a large and robust foundation for analyzing knowledge in terms of set transactions. Calling set theory a branch of mathematics is largely an accident of the fact that it was launched through the discovery of transfinite numbers. In its more general forms, set theory is a pragmatic discipline about general thought processes. Hence, it has much to offer a theory that attempts to define the relations between existence, life, and the reality we know. During the twentieth century, much traditional philosophy has been superseded by mathematical theories of physics and neuroscience. By redefining reality as the result of digitizing objective existence, explainable by set theory, DR Theory helps overcome some of the drawbacks of these numerical approaches. It encourages a more productive balance between philosophy and the analytical sciences.

Notes

1. Thomas Nagel, “What is it like to be a bat?” The Philosophical Review LXXXIII, 4 (October 1974), p. 440. 2. Albert Einstein, Out of My Later Years (New York, NY: Philosophical Library, 1950) 78. 3. John F. Sowa, “Processes and Causality,” at http://www.jfsowa.com/ontology/causal.htm (retrieved July 9, 2019). 4. Arthur Eddington, Space Time, and Gravitation (London, UK: Cambridge University Press, 1920), 182. 5. Philip E. Ross, “The Expert Mind” (New York, NY: Scientific American, July 2006). 6. See Laura Crosilla and Peter Schuster, From Sets and Types to Topology and Analysis (Oxford, UK: Clarendon Press 2005) 4. 7. Antoine Lavoisier, Elementary Treatise on Chemistry (Brussels: Cultures et Civilizations 1965). 8. Vannevar Bush, “As We May Think,” The Atlantic Monthly (July 1945). 9. Stephen Toulmin and June Goodfield, The Architecture of Matter (New York, NY: Harper Torchbooks 1962) 124. 10. John Locke, An Essay Concerning Human Understanding (London, 1689) Book II, Ch. 1, Sec. 19. 11. Peter L. Berger and Thomas Luckmann, The Social Construction of Reality (New York, NY: Anchor Books 1967) 60. 12. Margaret Mead, Sex and Temperament in Three Primitive Societies (New York, NY: Perennial 1963) 32. 13. Jean Piaget, The Moral Judgment of the Child (tr. Marjorie Gabain) (New York, NY: The Free Press 1965) 28. 14. Pierre-Simon Laplace, Exposition du Système du Monde (Paris, France 1796) Book III. 15. Galileo Galilei, Il Saggiatore [The Assayer] (Rome 1623). 16. Paul Dirac, “The Evolution of the Physicist’s Picture of Nature” (New York, NY: Scientific American, May 1963) 53. 17. Montesquieu, The Spirit of the Laws (Paris, France 1748) (tr. Thomas Nugent 1750) 11.6.

18. Steven Pinker, How the Mind Works (New York, NY: W. W. Norton 1997} 131 ff. 19. Julian Jaynes, The Origin of Consciousness in the Breakdown of the Bicameral Mind (Boston, MA: Houghton Mifflin Company 1976) 220. 20. Evelyn Underhill, Mysticism (New York, NY: Meridian Books 1955) 36—37. 21. William James, Varieties of Religious Experience (New York, NY: Modern Library 1936) 58. 22. William James, op. cit. 378—79. 23. Stephen Hawking, The Grand Design (New York, NY: Bantam Books 2010) 5. 24. C. I. Lewis, Mind and the World Order (New York, NY: Charles Scribner Sons 1929) 121.

Afterword MY FIRST IDEA for a new approach to epistemology surfaced in 1956, during a late-night debate among bookish graduate students at the University of California, Berkeley. The idea came to me as a kind of visual satori: a “cube of knowledge” whose three dimensions were behavioral, physical, and ideal. That geometric metaphor tumbled about in the back of my mind for the rest of my life. At Berkeley, I had learned classical philosophy from several mentors: epistemology and value theory from Stephen Pepper, logic and set theory from John Myhill, and the traditional philosophical schools from William Dennes and Bertram Jessup. After getting my degrees I joined the Kaiser Foundation Research Institute as assistant director of the Laboratory of Comparative Biology. My principal mentor there was Ellsworth Dougherty, who taught me the rudiments of organic physiology at the cellular level. Electronics had been my life-long hobby, so in 1962 I founded Berkeley Instruments, a small company that manufactured automatic weather stations. It was there that I became immersed in the technology of digitization. My principal mentor was Jack Hawley, an inventive engineer who later patented an implementation of Doug Englebart’s computer mouse—today the world’s most ubiquitous digitizer. Jack’s and my weather station designs spawned three more patents for digital technology. After several years of working with small companies I finally moved to Silicon Valley and joined the senior technical team at Apple Computer. That remarkable company became my working home for the next thirty-one years. My first book of epistemology, The Architecture of Knowledge, came out in 1980, before I started working for Apple. It had seemed to me that the “cube of knowledge,” tucked away in my memory, implied a story with details enough to fill a book. At the time, however, I had no clear idea what general principles might lie behind that raw idea. Apple in 1982 was making the Apple II and trying to figure out what desktop computing was about. I was bemused when they handed me the new Macintosh computer and explained its inner workings. It soon became apparent that the concept of a personal digital assistant—“wheels for the mind”—was revolutionary. The people inventing it were like the natural philosophers of earlier centuries, who envisioned completely new human helpers such as power engines and mechanical ways to handle language.

When I could find time away from Apple, I wrote three more books—Processes of Knowledge, The Reality of Knowledge, and Digital Reality—to document some of the ways I thought that personal computer designs implied principles of human understanding. In those books I acknowledged the help of several colleagues at Apple and elsewhere, who volunteered to nourish and challenge my philosophical speculations. In alphabetical order I want to thank Ellen Anders, David Cásseres, David Gatwood, Steven Gulie, Bill Harris, Michael Hinkson, Edward Jayne, Tanya Kelley, Thor Lewis, Tom Maremaa, Tom McArthur, Tim Monroe, Ted Nelson, Martin Schlissel, Darcy Skarada, Jim Stanfield, Bernard Tagholm, Shirley Walker, Allen Watson, Greg Williams, and Jeanne Woodward. Around the Apple campus I acquired the nickname “Gandalf.” The present book brings all this earlier work together into a compact narrative that I call Digital Reality Theory. The final pieces fell into place after I retired from Apple and had time to think through the concepts and evidence that had accumulated during fifty years. In Thinking Like a Computer, I have tried to present a compact and straightforward description of Digital Reality Theory. My thanks are owed to the staff at Austin Macauley Publishers, who guided the book through publication, and to everybody not mentioned here who helped me along the way. This story is far from complete, but at least the primitive cube of knowledge that I spied dimly in 1956 can now be understood as a metaphor for digital reality. George Towner [email protected]