Can Matter “Understand”?
Having discussed free will, let us now turn to the other faculty of human beings that is claimed to distinguish us from the lower animals and from machines, namely intellect or reason. The crucial question is whether the human intellect is something that can be explained in purely physical and mechanistic terms, or whether it points to the existence of a reality that goes beyond the physical. As we shall see in the next chapters, a revolutionary discovery made in the early part of the twentieth century casts considerable light on this question. This discovery, unlike those we have discussed up to this point, is in the field of mathematics rather than physics. It is a theorem proven in 1931 by the great mathematical logician Kurt Gödel. On the basis of this theorem it has been argued, notably by the philosopher John Lucas and the mathematician and physicist Roger Penrose, that all of the powers of the human intellect cannot be accounted for on the supposition that the human mind is merely a computer in action. Before getting into the details of the Lucas-Penrose argument and the technicalities of Gödel’s Theorem, it will be helpful to set the stage by approaching the question from a more philosophical point of view.
The first thing that we must do is be a little clearer about what we mean when we speak of the intellect. Without attempting to be comprehensive, we can certainly say that among the powers of the human intellect are conceptual understanding and rational judgment. The former is the ability to understand the meanings of abstract concepts and of the propositions that contain them, while the second is the ability to judge the adequacy of these concepts and the truth of these propositions. These are the human abilities with which we will be chiefly concerned in the chapters to come.
A point that is important to emphasize is that the powers of intellect and free will, even though we are discussing them separately, are intimately related. Both are aspects of human rationality. What makes will free, as we noted in the previous chapter, is that it is rational. We can have reasons for our decisions. We can deliberate about them. They can be based upon judgments about what is true and false or right and wrong. And, by the same token, our intellects if they are truly rational must be free rather than controlled by merely physical causes. Human freedom and human rationality stand or fall together.
When we say that the intellect has the power to understand abstract concepts, we have to explain what we mean by the word abstract. This is perhaps best done with a simple example. If you think about a particular woman, you are not thinking abstractly, but if you think about “womanliness,” you are. Abstract terms, such as womanliness, truth, impossibility, virtue, or circularity, are what philosophers call “universals.” The word circularity, for instance, is a universal because it does not refer to this or that circular object—this dinner plate or that wagon wheel, say—but to all real or possible circular objects. Indeed, it can be understood apart from any concrete example of circularity. When we think of a universal apart from any particular object, we are engaged in truly abstract thought.
Abstract thought has, therefore, in a sense, an unlimited reach. It transcends what is here and now and the particularities of specific objects to embrace a concept that is infinite in scope. For this reason, the philosopher Mortimer Adler,1 following a philosophical tradition going back at least to Aristotle,2 has argued that nothing which is merely material can engage in abstract thought. It is true that a material system can exemplify—or in philosophical jargon “instantiate”—a universal, the way a dinner plate exemplifies circularity. But it cannot contain the whole abstract meaning of a universal. For example, a dinner plate is composed of certain materials, exists at a certain time, and has a certain size, position, and orientation in the three-dimensional physical space of our world. The concept “circularity” has no such limitations. It applies to circles of any size, position, and orientation. Indeed, it applies even to circles in numbers of dimensions that cannot be “instantiated” in our physical world. (The human mind cannot visualize more than three dimensions, but the human intellect is nevertheless able to think abstractly about mathematical entities, like circles and spheres, in any number of dimensions. For example, any mathematics graduate student could easily prove that the volume of a four-dimensional sphere is π2/4 times its radius to the fourth power.)
The argument, then, is that because our brain is a finite material system, it cannot encompass within itself the whole meaning of an abstract concept. It may contain images that illustrate abstract concepts. It may even have words or symbols stored in it that “stand for” abstract concepts. But the full universal meaning of an abstract concept cannot be inscribed in it. There must, therefore, according to Adler, be some non-material component to our minds that enables us to think abstractly.
Adler also maintains that there is no scientific evidence that any animal other than human beings can understand universals. He admits that there are some facts that appear at first sight to contradict this. For example, even some species of fish can distinguish between a square object and a circular object. However, this is not an example of true abstract thinking, according to Adler, but rather of what he calls “perceptual abstraction.” These fish can only recognize a circle when presented with a circular object. In other words, the “abstraction” is tied closely to a perceptual act. In contrast, human beings can engage in what Adler calls “conceptual abstraction”; they can think about circularity in general, apart from any perceived object. They can relate the concepts circle and circularity to other concepts, or make them mathematically precise, or even prove theorems about them.
At this point, a materialist would be tempted to object that computers can prove theorems also, and therefore (since computers are material objects) it must be that Adler is wrong. But this objection raises the question of whether a computer really “understands” what it is doing. It manipulates symbols or numbers, and those symbols or numbers mean something to the human programmer. But do they mean anything to the computer itself? Does it know that the string of symbols it prints out refers to circles rather than to rocks or trees? Does it know that it is saying anything meaningful at all? The symbols or “bits” that it manipulates stand (in somebody’s mind—but not the computer’s) for concepts, but the symbols and bits are not, themselves, concepts.
Many materialists would reply that a computer can understand, because “understanding” information means no more than being able to put that information to appropriate use, to act in appropriate ways on the basis of that information. They would say that if a robot uses information from sensors to avoid bumping into a table, it “understands” that the table is there. However, it would seem that understanding is something more than this. There are many things that we understand that have no particular relevance to our behavior. We have insights that we cannot possibly put to practical use. If we understand something about spheres in six dimensions, what is the appropriate behavior that follows from that understanding? After all, we do not live in six dimensions. An aristocratic lady once asked the famous nineteenth-century Irish mathematician William Rowan Hamilton, who had invented “quaternions” (a kind of number), what quaternions were useful for. He replied, “Madam, quaternions are very useful for solving problems that involve quaternions.”
The idea that a computer “understands” if it can make use of information leads to rather bizarre conclusions. An ordinary door lock has “information” mechanically encoded within it that allows it to distinguish a key of the right shape from keys of other shapes. It uses that information to react in an appropriate way when the right key is inserted into it and turned: the lock mechanism pulls back the bolt and allows the door to open. Does the lock understand anything? Most sensible people would say not. The lock does not understand shapes any more than the fish understands shapes. Neither of them can understand a universal concept. However, many materialists do believe that even very simple non-living physical systems have mental attributes. For example, the man who invented the term artificial intelligence, John McCarthy, has written that “machines as simple as thermostats can be said to have beliefs.”3
In talking about thermostats, McCarthy was not choosing an example at random. A thermostat can be thought of as a very simple brain. For, just as an animal’s brain receives information about the world around it from sensory organs, a thermostat has a sensor which tells it what the temperature is in some particular locale, such as the living room of a house. And just as an animal’s brain controls a body, telling it how to react to its environment, a thermostat controls some apparatus, usually a heating or cooling device. We could say, then, that a thermostat “senses” one feature of its environment and responds to it. And we therefore could, in some very broad sense, attribute to thermostats “sensation,” “perception,” and even “cognition.” However, it remains the case that a thermostat cannot understand universal concepts or abstract ideas. To put it succinctly, a thermostat does not understand thermodynamics.
Any sane materialist would concede this, of course, but would explain it by saying that a thermostat is simply much too elementary a brain to have higher mental functions like abstract thought. What makes it possible for human beings to think abstractly, in the materialist’s view, is the enormous complexity of our brain’s neural circuitry.
While this view may sound plausible at first, it leads to conclusions that would be uncomfortable, I think, to most physical scientists, even, I dare say, to many scientists of a materialist bent. To see the difficulty, let us consider the difference between perceiving a physical object, like a tree, and thinking about an abstract idea like those encountered in mathematics, such as the number π.
The impression produced in our minds by a tree is made up of various sensations, both present and remembered, like the roughness of bark, or the rustling of leaves, or the filtering of light through the foliage, together with many associations to ideas related to trees, such as shade, forests, timber, wood, songbirds, and so on. We can easily imagine that all of these mental impressions are just a complicated set of responses of our nervous system and brain to the external physical reality we call a tree.
Thinking about an abstract idea like the number π is in some respects similar to this. Certainly, many images and associations crowd around the idea of the number π in the mind of the mathematician or physicist. However, in an obvious and important way there is a great difference between thinking about a tree and thinking about π. A tree is a physical object, and π isn’t. The tree presents no problem for the hard-nosed materialist who believes that only matter exists, for a tree is made of matter. But π is not made of matter. So what is the materialist to make of it, or indeed of any other number, or mathematical concept?
An obvious answer is that a number, while not itself a material thing, is an aspect of material things. While the number 4, for example, is not made of matter, a 4-sided table is, and 4 rocks are, and a 4-footed animal is. The number 4, then, might be considered a property of material objects. When we think about “4,” we are really thinking about things or groups of things in the real physical world that in some way have four-ness about them. While plausible for small “counting numbers” like 4, this point of view runs into serious difficulties when it comes to other kinds of numbers like π. One can have 4 cows, but one cannot have π cows; and one can have a 4-sided table, but not a π-sided table. Of course, in a sense, one can have a π-sided table: a circular table whose diameter is one meter will have a circumference of π meters. So can π perhaps really be thought of as simply a property of material objects, and specifically of objects that are circular?
This is a very common idea. In fact, if you ask “the man in the street” what π is, this is the answer you are likely to get. But it really will not do. One problem with it is simply that there are no exactly circular objects in the physical world; and for an object that is not exactly circular, the ratio of circumference to diameter is not π (except for very special shapes that are as unlikely to exist in nature as exact circles). It might be close to π, but close to π is not π, at least not the mathematician’s π. Nevertheless, π does at least have some connection to shapes that we see approximated in the physical world. However, most numbers do not have even this link to the world of matter. Consider, for example, the number 0.1011001110001111000011111 .… (The pattern is clear: one 1, one 0, two 1’s, two 0’s, three 1’s, three 0’s, etc.) This is a definite well-defined number, but it has no connection to any shape or figure that one will find in the physical world. Nor will most other numbers, such as the 17th root of 93.
We are left with a problem. If numbers and other mathematical concepts (unlike trees) are neither material things nor even aspects or properties of material things, then what are they? The most reasonable answer seems to be that they are mental things, things that exist in minds. Mathematics is a mental activity. Most schools of thought in the field called “the philosophy of mathematics” adopt some version of this view. But this raises the question of what a “mind” is and what “mental things” are.
To the non-materialist, minds and the ideas they contain can be real without being entirely reducible to matter or to the behavior of matter. To the materialist, however, there can be nothing to our minds besides the operations of our central nervous systems. In the memorable words of Sir Francis Crick, “you are nothing but a pack of neurons.”4 Consequently, to a materialist, it follows that “an explanation of the mind … must ultimately be an explanation in terms of the way neurons function,” to quote Sir John Maddox, the former editor of the scientific journal Nature.5 Now, if we say that abstract concepts, such as the number π, exist only in minds, and if we also say, with the materialist, that minds are only the functioning of neurons, then we are left in the strange position of saying that abstract concepts are in themselves nothing but patterns of neurons firing in brains. Not, mind you, merely that our neurons fire when we think about or understand these concepts, or that the firing of neurons plays an essential role in our thought processes, but that the abstract concepts about which we are thinking are in themselves certain patterns of neurons firing in the brain, and nothing but that. That is why one finds in a recent article the statement, “Numbers are … neurological creations, artifacts of the way the brain parses the world.”6 The author of that statement was summarizing the views of a “cognitive scientist” who had written a book subtitled “How the Mind Creates Mathematics” (by which he really meant, of course, “how the human central nervous system creates mathematics”).7
To the consistent materialist, then, the number π is a pattern of discharges of nerve cells. It has no more status, therefore, than a toothache or the taste of strawberries. This is a notion that many people who deal extensively with abstract mathematics would have a hard time accepting. The number π appears to the mathematician as something more than a sensation or a neurological artifact. It is not some private and incommunicable experience, like a toothache; it is a precise, definite, and hard-edged concept with logical relationships to other equally precise concepts. It is something that can be calculated with arbitrary precision; for example, to ten figures it is 3.141592653 (and it has been calculated to 50 billion figures). It has remarkable and surprising properties, which the mathematician feels that he is discovering rather than generating neurologically.
To take just a few examples, the sum of the infinite series of fractions 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + … is exactly π/4 ; the sum of the infinite series 1 + (1/2)2 + (1/3)2 + (1/4)2 + … is exactly π2/6 , and the natural logarithm of -1 is iπ, where i is the square root of -1. But what are all these precise and beautiful mathematical statements? According to the consistent materialist they too are “neurological creations.” Not only π itself, but the statement 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + … = π/4 is no more, ultimately, than a pattern of neurons firing in someone’s brain. The neurons firing in someone’s brain in a certain way may lead him to write certain figures on a piece of paper or blackboard (like the formula “1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + … = π/4”), and those shapes on paper or blackboard may in turn stimulate the neurons in someone else’s brain to fire in certain patterns. But whether it is patterns of ink on paper, of chalk on blackboard, or of neurons firing in brains, concepts, to the consistent materialist, are just patterns that exist in some material system.
Some materialists might be tempted to explain their position by saying that these physical patterns on paper or in the brain “mean” something. However, to say that a pattern means something is to say that it stands for some ideas: meanings are ideas that are understood by minds. And from the materialist’s point of view to say that a “meaning” is being “understood” by a “mind” is ultimately to say no more than that some pattern of electrical impulses is occurring in a brain. The materialist cannot go beyond patterns to the “meanings” of the patterns, because meanings themselves are ultimately nothing but patterns in brains.
I have used the number π as an example, but I could have used any other abstract concept, whether mathematical or not. However, since mathematics is the realm of the purest abstract thought, and is also the language of physical science, it is particularly relevant to our discussion. For if we reduce mathematical ideas to neurons firing we reduce all of scientific thought to neurons firing. What is the theory of relativity? What is quantum theory? What is the Schrödinger equation? What are Maxwell’s equations of electromagnetism? Just neurons firing? What are the statements theoretical physicists make, like “Observables are hermitian operators acting in a Hilbert space,” or “All Cauchy surfaces for a space-time are diffeomorphic,” or “Spontaneously broken gauge theories are renormalizable”? Nothing but patterns of nerve impulses? Squiggles on a page? It is the absurdity of this kind of conclusion that was the basis of the incisive critique of materialism made by Karl Popper, the eminent philosopher of science, especially in his later works.8
Most people do not like mathematics and physics much, and perhaps would be just as happy to think of them as some quirky phenomena occurring in the nervous systems of a small number of peculiar people. (Not surprisingly, most mathematicians and theoretical physicists do not share this view.) But it is not just the concepts of mathematics and physics that are at stake here; all concepts are at stake, including those of biology, neuroscience, and indeed the concepts of the cognitive scientist I just mentioned who thinks that numbers are “neurological creations.” Cognitive scientists talk about neurons, for example. But “neuron” itself is an abstract concept that arose from the researches of biologists. For the materialist, then, even this concept of “neuron” is nothing but a neurological creation; it also is a pattern of neurons firing in someone’s brain. If this sounds like a vicious circle, it is. We explain certain biological phenomena using the abstract concept “neuron,” and then we proceed to explain the abstract concept “neuron” as a biological phenomenon—indeed, a biological phenomenon produced by the activity of neurons. What we are observing here is the snake of theory eating its own tail, or rather its own head. The very theory which says that theories are neurons firing is itself naught but neurons firing.
This is an example of what G. K. Chesterton called “the suicide of thought.”9 All of human understanding, including all of scientific understanding, is reduced to the status of electrochemical processes in an organ of the body of a certain mammal. In the words of a recent Newsweek article, “Thoughts … are not mere will-o’-the-wisps, ephemera with no physicality. They are, instead, electrical signals.”10
Why should anyone believe the materialist, then? If ideas are just patterns of nerve impulses, then how can one say that any idea (including the idea of materialism itself) is superior to any other? One pattern of nerve impulses cannot be truer or less true than another pattern, any more than a toothache can be truer or less true than another toothache.
The intellect has not only the power of abstract understanding but also the power of judging the truth and falsehood of propositions. It has been argued by philosophers since antiquity that this power goes beyond the capacities of purely physical or mechanistic systems. There are many grounds for this conclusion; one of the most basic has to do with what might be called “openness to truth.” Human “openness to truth” was, in fact, referred to in one of the passages quoted in chapter 19 as a sign of our “spiritual” nature.
The question is this: If my thoughts follow a path set out for them by the laws that govern matter, how does “truth” enter into the picture? In the final analysis, my thoughts are not “reasonable” or “unreasonable,” they are just the thoughts that I must have given the way the molecular motions in my brain and the rest of the world have happened to play out. As the mathematician Hermann Weyl put it, “[there must be] freedom in the theoretical acts of affirmation and negation: When I reason that 2 + 2 = 4, this actual judgment is not forced upon me through blind natural causality (a view which would eliminate thinking as an act for which one can be held answerable) but something purely spiritual enters in.”11 He went on to explain that thought that is free, and therefore rational, must not be entirely determined by physical factors (in which case it would be “groundless” and “blind”) but must be “open” to meaning and truth.
Writing in the same year (1932), the biologist J. B. S. Haldane argued, “If materialism is true, it seems to me that we cannot know that it is true. If my opinions are the result of the chemical processes going on in my brain, they are determined by chemistry, not the laws of logic.”12 In Orthodoxy, his brilliant defense of Christianity written in 1908, G. K. Chesterton noted that the materialist skeptic must sooner or later ask, “Why should anything go right; even observation and deduction? Why should not good logic be as misleading as bad logic? They are both movements in the brain of a bewildered ape.”13 Recently, Stephen Hawking worried about the same issue in connection with the “theory of everything,” which many physicists are seeking.14 A theory of physics that explained everything would also have to explain why some people believed it and some people did not. Their belief (or disbelief) in the theory, then, would be the result of inevitable physical processes in their brains rather than being a result of the validity of the arguments made in behalf of the theory itself. This argument is very ancient. For example, Epicurus wrote, “He who says all things happen of necessity cannot criticize another who says that not all things happen of necessity. For he has to admit that the assertion also happens of necessity.”15 (Of course, the necessitarian could always make the clever comeback that not only can he criticize, but he must criticize—of necessity. However, Epicurus’s point is that the criticism would not then be a rational one.)
Haldane recanted his argument in 1954 because of the development of the computer.16 He was impressed by the fact that although a computer is made only of matter and obeys the laws of physics it is nonetheless capable of operating in accordance with truth. However, Haldane was wrong to recant, since the example of the computer does not really resolve the question that he originally raised. It is true that a calculating device can print out 2 + 2 = 4, or some equivalent formula, and in that sense can operate in accordance with the truth. However, its ability to do so is not to be sought simply in the laws of physics that it obeys. A calculating device (also obeying, of course, the laws of physics) could just as easily be produced that printed out 2 + 2 = 17. In fact, it would be even easier to build a device that printed out complete gibberish. The reason that most calculating devices do operate in a manner consistent with logic and mathematical truth is that they were programmed to do so. That is, they have built into them a precise set of instructions that tells them exactly what to do at every step. These programs are the products of human minds. More precisely, the acts of understanding that lie behind these programs took place in human intellects. Rather than illustrating, therefore, how an automatic device can give rise to intellect, artificial computers merely show that an intellect can give rise to a device. Not only do the design and programming of these devices occur as the result of human acts of understanding, but the meaning of their outputs can only be apprehended by human acts of understanding, not by the machines themselves. (These outputs can indeed be used by other machines, but only by machines designed to be able to do so by human intelligence.)
The mystique of the high-performance computer can obscure what is really going on. Let us therefore consider instead a humbler device, a vending machine. A vending machine contains a simple computing device that enables it to make change correctly. In spite of this, we do not attribute intelligence to vending machines. On the other hand, we might well attribute a great deal of intelligence to a child who was able to figure out for himself or herself how to make change correctly. What is the difference? The difference is that the child understands something and the vending machine does not. Of course, the vending machine could not do what it does without intelligence being involved somewhere along the line. At some point there was an understanding of numbers and the operations of arithmetic; there was an understanding of how to perform these operations in a routine fashion; and finally there was an understanding of how to build a machine to carry out these routine steps automatically. All of these acts of understanding took place in human intellects, not in vending machines. The point at which any task has become routinized so that it no longer requires acts of understanding is the point at which it can be done by a machine which lacks intellect.
The question raised by progress in developing computer hardware and computer programs is whether a sufficiently advanced computer could have genuine intellect, and whether in fact the human intellect can be explained as being the performance of an enormously powerful biological computer, the brain. When we “understand” abstract ideas, is no more going on than that our brains are following some canned instructions, some very complex but routine procedure? That is the issue that we will examine in the next chapters. For now let us suppose (contrary to what we shall argue later) that the human intellect can indeed be explained as the product of a sophisticated computer program.
A number of difficult questions then arise. For example, how is it that we know, as sometimes at least we do know, that our thought processes are reasonable and consistent? As we shall see later, no program (except a very trivial one) that operates consistently is able to prove that it does so. If we were merely machines, then, we could not be aware of our own consistency. Moreover, if our brains have in fact been programmed so that they can think consistently and reasonably, how did that happen?
If an electronic computer operates in a correct way, it is always because some human beings programmed it to do so. But who programmed those human beings to operate correctly so that they could impart that correctness to the computer? The only answer available to the materialist (and the one suggested by Hawking, among others) is natural selection. Natural selection programmed human beings to think in such a manner that our thoughts correspond in some way to reality. Obviously, an organism that could not think straight would be at a disadvantage in the struggle for survival. This answer is appealing at first sight, but it is really not adequate.
In the first place, the human mind is capable not only of dealing with the kinds of problems that our primitive forebears faced, like how to escape from a predator, or how to make a rude shelter; it is also capable of doing an enormous variety of things that have no conceivable application to survival in the wild, like playing chess, proving theorems in non-Euclidean geometry, doing research in nuclear physics, designing jet aircraft, or composing violin concertos. If the human mind is nothing but a computer program, this versatility is quite mysterious. A program that was designed to play chess would not, in general, be able to do other things, even to play other games, like Parcheesi. A computer program that could play many types of games would be much more complicated than a program that could only play one. Similarly, a program (assuming one were possible) that could replicate human abilities in every field of activity would be far more complicated in structure than a program that had only the ability to do the things that cavemen did in order to survive. How can one possibly explain that natural selection gave us a program than was vastly more sophisticated than was required for survival?
This kind of argument (among others) has led Roger Penrose to challenge the idea that the human intellect operates simply as a computer program. He argues that what enabled our forebears to survive was not simply a complicated program, but something which no mere program can have, namely the capacity to “understand”—that is, intellect.17 And since the capacity to understand is a very general kind of thing, it allows human beings to perform well in a wide variety of activities. I shall have more to say about Penrose’s ideas in the next chapter.
There is another problem with the idea that the human mind is merely a computer programmed by natural selection, and this has to do with two remarkable abilities which the human mind possesses: the ability to attain certainty about some truths, and the ability to recognize that some truths are true of necessity. (These may seem like the same thing, but they are not. I am certain that my first name is Stephen, but it could have been something else, so that is not a “necessary truth.” On the other hand, it is necessarily true that 147 × 163 = 23,961, in the sense that it could not have been anything else, and must be so in any possible universe, but someone who is not adept at multiplying large numbers might nevertheless be in a state of uncertainty about it.)
There are two aspects to this problem. In the first place, a creature’s “evolutionary success” (that is, its success in surviving to reproduce and in ensuring the survival of its offspring) does not require that it know things with absolute certainty or that it recognize truths as necessary ones. It is quite enough for it to have knowledge which is reliable for practical purposes and which is known to be generally true in the circumstances that it has to face. It does not have to know with absolute certainty that this branch will support its weight, that this fruit is not poisonous, or that fire will burn it, in order to survive. It is quite enough for it to be 99.99 percent sure or even 90 percent sure. Nor is it of any use to it to understand that the statement “2 + 2 = 4” is true of necessity, and therefore true in any possible world. It would be just as good to know that 2 + 2 can be trusted to come out 4 in its own situation.
The second aspect of the problem is that even if it were helpful to their survival for human beings to have absolute certainty in some matters, or to realize that some things are true of necessity, there seems to be no way that natural selection could possibly have programmed us to have that kind of knowledge. Natural selection is based ultimately on trial and error. Various designs are tried out, including various designs of brain hardware and brain software, and those that give the best results on average tend to lead to more numerous offspring. However, trial and error cannot produce certainty. Nor, obviously, can it lead to conclusions about what is necessarily true.
In a recent book about the philosophical implications of modern science the author asked, in all seriousness, “Is it so inconceivable that a reality could exist in which 317 is not a prime number?”18 The answer is, quite simply, yes. It is totally inconceivable, indeed absurd. I do not know of any scientist or mathematician who would admit any possibility of doubt about this. Not only is 317 prime here and now, it is indubitably prime in galaxies too remote to be seen with the most powerful telescopes. It was prime a billion years ago and will be prime a billion years hence. It would have to be prime in any other possible universe. However, there is no way that the processes of natural selection that operated upon our forebears could have had access to information about what will be true in a billion years, or in remote galaxies, or in other possible universes. How, then, can those physical processes have taught us these truths, or fashioned us so that we could recognize them?
It is important to be clear about what the issue is here. The issue is not how we came to be able to do arithmetic and figure out whether 317 is a prime number. Having the abilities that enable us to figure out the rules that will give correct answers to arithmetical problems may indeed have advantages for survival. One could even imagine that by evolutionary trial and error the right circuitry was “hard-wired” into our brains to do arithmetic correctly. And, therefore, assuming that “317 is a prime” happens actually to be a necessary truth, it is not surprising that evolution allows us to arrive at conclusions which in fact happen to be necessary truths. The question, however, is this: How do we recognize that necessity? How and why did natural selection equip us, not merely to say that 317 is prime, but to recognize of that truth that it is true of necessity?19
(An aside to philosophers: It may appear that I am implicitly endorsing the Kantian idea that mathematical statements are synthetic a priori truths. I am not. Even if one adopts the view, say, that mathematics is reducible to logic, precisely the same kind of question arises with respect to evolution: How was natural selection able to produce in us the knowledge that the laws of logic are universally valid and admit of no exceptions?)
The reaction of many materialists to such an argument, I dare say, would be to suggest that human beings are not actually capable of achieving certainty about anything, or knowledge of the necessity of truths. In their view, all we can ever claim to have is knowledge that has a high probability of being right. When we say we are “certain,” we are, according to this view, only expressing a strong feeling of confidence in what we are saying. And when we say something is “true by necessity,” we just mean that we have not been able to imagine a contrary situation. Absolute certainty, according to many materialists, is a chimera.
This skeptical position has some superficial plausibility. After all, we have all had the experience of claiming to be certain about something only to find out later that we were mistaken. In spite of its initial plausibility, however, this account of what we mean by “being certain” is, in my view, simplistic and untenable.
Consider the two statements, “The Sun will come up tomorrow” and “317 is a prime number.” I have great confidence in the truth of both. But they are radically different types of statement, in which I have radically different types of confidence. I admit that it is overwhelmingly probable that the Sun will come up tomorrow, but I do not believe that it is absolutely certain. It is quite conceivable that the Sun will not come up tomorrow, and in fact there are scenarios, not excluded by anything that we know about the laws of nature, in which the Sun would not come up tomorrow.
To take the most exotic such scenario, we could be in what particle physicists call a “false vacuum state.” That is, just as some radioactive nuclei with very long half-lives appear to be stable, but actually have a small chance of disintegrating suddenly and without prior warning, so the state of matter of our world may actually be unstable in the same way. It is possible that a large bubble of “true vacuum”—that is, a state of lower energy—will suddenly appear in our vicinity by a “quantum fluctuation.” If it does, it will expand at nearly the speed of light and destroy all in its path. We would never know what hit us. The Sun would not come up tomorrow, because the Sun would have ceased to exist. (No particle physicist loses even a moment’s sleep over this possibility, but none would claim that it is absolutely ruled out either.)
There are less exotic possibilities that also are not excluded by what is presently known about physics and that would prevent the next sunrise. And, aside from natural catastrophes, there is always the logical possibility of a miracle. Earth might miraculously stop rotating on its axis or the Sun might miraculously disappear. As the philosopher David Hume pointed out, one cannot rigorously deduce what will happen in the future from what has happened in the past. Therefore, the skeptical materialist is right about this case: when we say we are “certain” that the Sun will come up tomorrow, what we really mean is that we have an extremely high degree of confidence that it will.
However, it is far otherwise with “317 is a prime number.” No scientific phenomenon, however exotic, can make 317 not be a prime. The mediaeval theologians would have said that even the omnipotence of God could not do that.20 This is not just a question of something that is very highly probable, but of something truly certain. Someone might object that 317 is a fairly large number, and that the calculations which show it to be prime are too complicated to allow him to be certain about them. However, one can always take a case where this is not an issue, like 1 ≠ 0 (1 does not equal 0). I think most people would admit to knowing this with certainty, and not simply to having a lot of confidence in it.
Moreover, the idea that our certainty about such things as “317 is prime” is simply a sort of gambler’s confidence, a confidence born of practical experience, simply does not bear careful scrutiny. I have seen that the Sun came up about fifteen thousand times without fail; whereas in the last fifteen thousand arithmetical calculations I have done, I have not always gotten consistent answers. In fact, on many occasions I have not. If anything, then, my confidence that the Sun will come up tomorrow ought to be greater than my confidence in the consistency of arithmetic. I can make the argument even stronger. It is clear, from the absence of excited testimony to the contrary, that the Sun has come up every day without fail for the last several thousand years—about a million consecutive times.
However, our confidence in arithmetic is in fact stronger than our confidence in the Sun coming up. Why? Is it based on some logical analysis? But that just raises the question of how it is that we have the confidence we do in the consistency of logic.
Even if one were to concede that our certainty is never absolute (which I am not prepared to do), it would still remain the case that we have a certainty about some kinds of truths that far exceeds what we can derive from trial and error, and which it is very hard to explain as arising from natural selection.
It is still open to the materialist to retreat to an even more skeptical position. Yes, he might concede, we do actually have the conviction in some cases that we know something with absolute certainty, or know that something is true of necessity. But perhaps all such convictions are just illusions or feelings planted in us by nature. They are chemical moods, so to speak. For some reason our brains were fashioned by natural selection to have these feelings of certainty because they help us get through life. In Chesterton’s words, they are just movements in the brain of a bewildered ape. This is a possible position, but it means, ultimately, abandoning all belief in human reason. I would rather take my stand with the mathematician G. H. Hardy, who said, “317 is a prime number, not because we think it so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way.”21 And with Galileo, who said, “It is true that the divine intellect cognizes mathematical truths in infinitely greater plenitude than does our own (for it knows them all), but of the few that the human intellect may grasp, I believe that their cognition equals that of the divine intellect as regards objective certainty, since man attains the insight into their necessity, beyond which there can be no higher degree of certainty.”22
I have cited the capacity to understand universals, or abstract concepts, openness to truth, the ability to attain certainty, and the power to recognize that some truths are true “of necessity,” as being beyond the capacity of any merely material system, including the kind that the materialist conceives us to be—a computer programmed by natural selection. Another human intellectual ability involves all of these at once, namely the power to recognize that some truths hold in an infinite number of cases. Roger Penrose gives the following example.23 We all know that 3 × 5 = 5 × 3. As Penrose notes, mathematically speaking this is not the empty statement it appears to be. It really says that three groups of five objects and five groups of three objects contain the same number of objects. There is a simple pictorial argument that shows this: If we arrange fifteen objects in a three-by-five rectangular array, we see that it has five columns of three objects and three rows of five objects. Now, most of us, when we look at that array, will immediately “see” that the same thing works for a rectangular array of any size, so that a × b = b × a, in general, for any numbers a and b.
The ability to “see” this is a remarkable thing, as Penrose points out. We are seeing at once the truth of an infinite number of statements. Exactly the same observation was made by St. Augustine in his great philosophical work De Libero Arbitrio, completed around 395 A.D. In reference to a similar arithmetical statement he asked, “How do we discern that this fact, which holds for the whole number series, is unchangeable, fixed, and incorruptible? No one perceives all the numbers by any bodily sense, for they are innumerable. How do we know that this is true for all numbers? Through what fantasy or vision do we discern so confidently the firm truth of number throughout the whole innumerable series, unless by some inner light unknown to the bodily senses?”24
In this chapter I have used mostly mathematical examples. This is not essential to the arguments. Mortimer Adler’s Aristotelian argument, discussed earlier, applies to any universal concepts, not just mathematical ones. Similarly, there are things we know with certainty that are not mathematical in nature. Nevertheless, mathematics is a particularly pure example of abstract thought. In the next chapter we shall look more closely at what is involved in it.
IF NOT THE BRAIN, THEN WHAT AND HOW?
A lot of people who might admit that materialism has difficulties explaining the human intellect nevertheless embrace it because they simply see no alternative. Let us grant, they say, that something we call the intellect exists and that it has the ability to understand abstract ideas. Where, they ask, does this intellect reside, if not in the brain? Is it floating around somewhere in space? And how, they ask, does the intellect work? There must be some mechanism by which it operates. But if it is by some mechanism, then why deny that that mechanism takes place in the brain and is a physical mechanism?
These seem like reasonable questions, but it is not clear that they will ultimately turn out to be meaningful. There are many things that we explain in terms of more elementary constituents or concepts. For example, the properties of a gas are explained by saying that the gas is made up of molecules. The pressure that the gas exerts on its container is the result of countless molecules bouncing off the container’s sides. The temperature of the gas is a measure of how energetically its molecules are bouncing around. The sound that propagates in the gas is explained as waves of compression and rarefaction reducing or increasing the average spacing of the molecules. This kind of explanation of one thing in terms of something more basic is not always possible, however. It cannot be, since there must be some things that are the most basic of all and in terms of which everything else is explained. Until we have complete understanding, we will not know whether certain things we now treat as basic are reducible or not to something more basic still.
The first part of physics that was well understood was mechanics, which deals with the motion of material bodies. When electromagnetic phenomena began to be studied, therefore, it was natural to make “mechanical models” of electromagnetic fields. Only later was it realized that these “aether theories,” on which a great deal of ingenuity was expended, were largely a waste of time. Electromagnetic fields were recognized to be a reality in their own right, as basic as material bodies. They were something qualitatively new, and the effort to find a material aether that explained them was seen to be misguided.
Of course, electromagnetic fields are just as physical as the material bodies studied by Newtonian mechanics, and we now have an overarching, coherent physical theory in which both electromagnetic fields and material bodies find their proper place. The point, however, is that attempts to find a coherent picture of reality by shoehorning new and poorly understood phenomena into existing conceptual categories, as the aether theorists did, are sometimes intellectual dead ends.
It is conceivable that all mental realities, including “intellect,” “understanding,” abstract “concepts,” and so on, are ultimately explicable in terms of neurons firing, or some other basic physical events or entities, just as the phenomena of gases are explicable in terms of molecular motions. But it is just as conceivable that these phenomena involve something new, something that exists in its own right, and is not reducible to or explicable in terms of something more basic than itself. Science gives us examples of both possibilities.
But even if the intellect is not simply a material system, we would still want to know how it works. However, we must again be careful. It sometimes makes sense to ask “how” something works, in the sense of seeking a “mechanism” by which it happens, but sometimes it does not. For example, one can answer the question of how a phonograph produces sound, or how a television produces a picture. But it is not clear that it makes sense to ask “how” a mass produces a gravitational field, say. In Newton’s theory, “mass” and “gravitational field” are fundamental concepts. Newton’s law of gravitation posits the existence of a relationship between them and gives a quantitative account of that relationship. But it does not explain “how” the mass produces the field. As Newton himself said in the famous concluding words of his Principia: “I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses.… [It] is enough that gravity does really exist and act according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies, and of our sea.”25 Similarly, Einstein’s theory, while it gives a deeper understanding of what a gravitational field is, namely the curving of space-time, does not explain “how” a massive body causes that curvature, in the sense of a mechanism.
All we know about the human intellect is that it is capable of having insights, of understanding meanings. By what mechanism? “How” does the intellect act on the physical brain? What is the intellect made of? Far from requiring a materialist answer, these questions may not even turn out to make any sense. Sometimes we must have the patience to hold certain questions in abeyance until we have the conceptual equipment and level of understanding that allows us to distinguish the good questions from the bad ones. To repeat again the wise words of Hermann Weyl: “One of the great differences between the scientist and the impatient philosopher is that the scientist bides his time. We must await the further development of science, perhaps for centuries, perhaps for thousands of years, before we can design a true and detailed picture of the interwoven texture of Matter, Life, and Soul.”