PART III

 

A CASEBOOK

Chapter 7 - THEORIES IN MATHEMATICS

Dr. Roy A. Clouser

 

 7.1 INTRODUCTION

It is a very old idea in Western culture that the prime counter-example to the central claim of this book is mathematics. After all, the objection goes, doesn’t 1 + 1 = 2 for everyone regardless of their religious belief? Isn’t it therefore a neutral and universally accepted belief in exactly the sense you are denying?

This chapter will be devoted to answering that objection.

Let me start by saying that at a simple, commonsense level this objection has a prima facie — but ultimately deceptive — plausibility. Almost everyone has had the experience of finding simple arithmetic to be obvious. In the light of what has been said about abstraction, this would be because the things we experience exhibit quantity; there is a “how much” to them. These quantitative properties can be abstracted, allowing them to be represented by numerals, and relations among them can be noticed, symbolized, and formulated. In this way many mathematical truths and techniques can be discovered without the need to formulate any theories. And at this level there is, indeed, agreement.

Nevertheless, there are questions about the concepts involved in 1 + 1 = 2 that cannot be answered simply by abstracting and symbolizing quantities and noticing the most obvious laws that hold among them. These questions concern issues that are crucial to the understanding of precisely what that formula means. Once these questions are made explicit and answers are proffered for them, the answers can be seen to constitute entity and/or perspectival hypotheses. One of the most famous of these questions is what, exactly, do the symbols of the formula represent? In other words, what is a number? As soon as this issue is raised, we find there are serious disagreements among mathematicians as to how to answer it. Their disagreements are forced to light because this question requires a more extensive examination of the concept of number each thinker holds. In this respect, the agreements and differences concerning the concept of number are similar to the agreements and differences we noticed in the concept of a saltshaker discussed in chapter 4. As with the saltshaker, there is enough overlap between the various conceptions of number that 1 + 1 = 2 can be obvious and be agreed on. But, as with the saltshaker, once a concern arises that occasions a more detailed analysis of the concept of number, it turns out that examining its content more extensively shows that different thinkers hold very different notions of its nature. So as with the saltshaker, the wider concepts of number reveal that people include in them relations of the quantitative properties to properties of other kinds. These relations, when stated and defended, constitute a view about the nature of number. Such views may simply be assumed unconsciously, of course, in which case they aren’t theories. But if they are made conscious and defended, then they are, indeed, theories about the nature of math that reflects a general view of the nature of reality and thus of divinity. (If they are only assumed, then the control they exercise remains an unexamined faith that is, all the same, religious in character.) And the truth is that such differences in the concepts of number are so great that the major players in the history of mathematics have had radically conflicting ideas of what math is, how it is to be done, and what it can be relied on to  do! In fact, these disagreements are among the widest and sharpest in Western theory making.

So let’s now consider the question: what do the symbols 1 + 1 = 2 stand for? We can directly observe, of course, that one thing and another thing often make two things. That is not itself a theory, but neither is it what is meant by 1 + 1 = 2. That formula expresses a truth about abstract quantities, not about objects of pre-theoretical experience. Were the formula about ordinary objects, then it wouldn’t always be true. As Whitehead once observed, a spark is one thing and a pile of gunpowder is another, but together they make an explosion that is very unlike two things. Then he immediately added: “common sense  at once tells you what is meant.”¹ What Whitehead called common sense is the recognition that whatever the symbols 1 + 1 = 2 represent, it’s not simply objects of pre-theoretical experience but abstract numbers, and that’s what our question is asking for an explanation of. Thus the distinction between abstract number and ordinary objects to which we apply numbers is an important point. When little children first learn arithmetic, they often do suppose the numerals stand for things and events of their everyday experience. It is easy for them  to get that impression because the problems given in elementary textbooks usually have them practice arithmetic by making calculations about bales of hay, or pairs of shoes, or apples, and so on. But it soon becomes apparent that although numerals can be applied to the objects of ordinary experience, those objects are not what the numerals themselves stand for. If numerals stood for things, it would be impossible to subtract 8 from 5. But although we cannot take 8 things from a pile which contains only 5, we can subtract 8 from 5 and get -3.

So this brings us back to the question posed. If the symbols do not stand for the objects we experience, what do they stand for? Both mathematicians and philosophers have proposed very different theories to answer this question.

 

7.2            THE NUMBER-WORLD THEORY

One very famous hypothesis in answer to this last question is that the numerals and other markings of mathematics stand for real entities in another world or dimension of reality. These entities are never observed, nor are they locatable in space; we cannot look out the window and see anything which   is the number 2 in the backyard, even though we can see things to which that number concept can be applied. According to the theory, however, the world of mathematical entities is not only real, it is more real than the things which are observable and exist in space and time. It is more real for two reasons. One is that the entities populating it have independent existence, are eternal, and can never change or pass away. The other is that the mathematical laws governing this realm are not only independent and changeless, but all-governing. They determine what is possible and impossible for all reality, not just for the number world. Versions of this theory were held in ancient times by Pythagoras and Plato, and different versions of it are still popular among mathematicians today. The great mathematician G. W. Leibniz (who invented the calculus) also held a version of this theory, and had a very tidy way of putting it. When asked by a student how we can be sure of 1 + 1 = 2, Leibniz replied that it is, like  all other truths of mathematics, an eternal and necessary truth which would not be affected even if the whole [observable] world were destroyed and there were no one to count and no objects to be counted.²

Clearly, this is an entity hypothesis. It proposes that there is an infinitely large realm of mathematical entities in addition to the changeable, observable objects of our everyday experience. These entities include all the natural numbers, all the fractions, decimals, all perfect geometric figures, roots, etc. These are all entities distinct from, and independent of, the world of everyday experience. Nevertheless, the laws holding among these entities also rule the changeable everyday world as well as guarantee the truths expressed by mathematical formulas about the numbers. This, then, is how Pythagoras, Plato, and Leibniz answer the question as to what numerals and other mathematical symbols stand for.

Just as clearly, however, this entity hypothesis in turn presupposes a perspectival hypothesis. The perspectival hypothesis is about how the quantitative aspect of the things we experience relates to all the other aspects. This is because, for the number-world theory to be true, it would have to be the case that the quantitativeness of things relates to the other kinds of properties and laws true of them by being utterly independent of them all. Thus, the quantitative aspect is (at least part of) what things and their other kinds of properties depend on for existence. On this theory, then, the things we experience, along with their other aspects, are made possible (or possible and actual) by the entities and laws of the number world.³

Strictly from the standpoint of our direct experience prior to making theories, the quantitative aspect is but one among a multiplicity of aspects things display. But Plato, Leibniz, and others adopted the perspective that it is not merely one of the beads of the necklace but (at least part of) its string. It is basic to other aspects. Thus they had no trouble believing that mathematics does not merely deal with the quantitative properties and laws of the things we experience, but is the reflection in our experience and thought of a realm of unobservable, independent, and changeless entities on which all observable, changeable things depend.

 

7.3            THE THEORY OF J. S. MILL

Let’s now contrast this number-world theory of what numerals symbolize with the theory of John Stuart Mill. Mill’s theory was that numerals symbolize sensory perceptions. He thought it farfetched to say that any sort of knowledge could exceed the observations from which it arises. All we experience, said Mill, are our own sensations, so they are all we can know. Thus we can employ numbers in order to speak and calculate only about the feelings, sights, tastes, touches, smells, and sounds we experience.

Mill defended this view of math by arguing that not only the quantitative aspect, but all other aspects of our pre-theoretical experience are actually identical with its sensory aspect. That is, Mill’s theory was that the nature of all reality is sensory, period; all we can know is exclusively sensory in nature. Thus, on Mill’s theory, the fact that there seem to be many aspects which differ from the sensory is a mistake, and the commonsense experiences in which things appear this way are misleading. The things we experience, he held, are nothing more than bundles of sensations. This committed him to a theory in which all knowledge, including math, must be derived from purely sensory data.

Obviously, then, Mill disagreed with the theory about a realm of eternal numbers and laws which are not sensorily perceivable. He held instead that 1 + 1 = 2 and other mathematical formulas are nothing more than generalizations about our sensations, so that they represent only sights, tastes, touches, smells, sounds, or feelings. The formula about 1 + 1 is thus no more than a way of saying that we find by perception that whenever we have experienced one sensation and another sensation we are then experiencing two sensations.4 This view requires, as Mill admitted, that we do not know that 1 + 1 must equal 2, or that it always does. We are at best entitled to believe that 1 + 1 will probably make 2 in the future because it has so often made 2 in the past! It also means that what the numerals symbolize can exist only if there are objects to count and people to count them. And for the same reasons, it means we have no grounds for supposing that anything about math is either everlasting or changeless. Here we can see that, just as in the case of the number-world theory, a philosophical perspective on the general nature of reality is presupposed by Mill’s theory about what numerals symbolize.

 

7.4            THE THEORY OF RUSSELL

Yet another theory to answer the question of what the numerals symbolize is the one espoused by Bertrand Russell. Unlike Mill, Russell could not accept the proposal that the symbols of math refer to sensory perceptions since that would remove the necessity and exceptionless character of mathematical truths. But like Mill, he rejected the theory of an unseen eternal realm of math entities. He hastened to add, however, that by rejecting that theory he did not mean that 1 + 1 = 2 is false:

Clearly, then, Russell’s entity theory concerning what mathematical symbols represent also presupposes a view of the nature of reality, a view of how all the aspects interrelate. In his theory the logical aspect — at least so far as its laws are concerned — enjoys an independence from the others which the others do not have from the logical. The laws of logic hold for all reality actual or possible. Once again, we should notice that from the standpoint of our pre-theoretical experience, the logical is but one of many aspects. But having once abstracted it, Russell regards it as more than merely an aspect of our experience. Instead, as he said, it is the very “heart and immutable essence of all things.” So Russell’s philosophical perspective was that the ultimate nature of reality is (at least partly) logical, so that the non-ultimate aspects of things depend upon the logical.

 

7.5            THE THEORY OF DEWEY

Finally, in contrast to the theories above, John Dewey gives yet another answer to our question. In reply to the question as to what mathematical symbols stand for, Dewey says, “Nothing.” In keeping with this, he also affirms that the formula 1 + 1 = 2 is not true. Or, more precisely, he claims it is neither true nor false.

According to Dewey, humans are to be understood as essentially biological beings struggling to survive in a certain environment. All living beings do the same, of course, but humans cope with their environment by trying to alter it to suit themselves rather than by adjusting themselves to it. They manage to do this because they have been endowed by evolution with a superior intelligence, and the way they utilize this intelligence is to make tools or instruments. This idea of an instrument is much broader in Dewey’s thought than the way we normally think of tools. For Dewey, all human cultural products are instruments, even such things as values and institutions. Likewise, an idea, a language, a theory, or a concept is also a tool.

On his view, then, the very questions the other theories have been trying to answer were wrongly posed. The symbols of math do not stand for anything any more than a hammer or a shovel stands for anything. Like all other tools, the symbols of math merely do certain jobs. So just as it would be inappropriate to ask what a hammer stands for but appropriate to ask what it can do, so too for the apparatus of mathematics. Numbers and formulae don’t stand for something else but simply do certain jobs. The same point holds for the question of the truth of math. Just as it is inappropriate to ask whether a hammer is true or false, it is equally inappropriate to ask that of mathematical tools. 1 + 1 = 2 is thus neither true nor false, says Dewey, though it performs certain tasks well. It is the success of math in accomplishing certain tasks that we (misleadingly) refer to when we say that a mathematical formula is true. Dewey puts it this way:

And again, Mathematics is often cited as an example of purely normative thinking dependent upon [absolute rules] and [other-worldly] material. . . . The present-day mathematical logician may present the structure of mathematics as if it had sprung from the brain of a Zeus whose anatomy is that of pure logic. But . . . [math has] a history in which matter and methods have been constantly selected and worked over on the basis of [experiential] success  and failure.¹⁰

To summarize: on Dewey’s theory, math itself is neither true nor false in the traditional sense, but just works. Its symbols and formulae do not stand for unseen eternal realities, sensory perceptions, or logical classes, because they do not stand for anything at all. Their meaning is their use. They guide us in “reorganizing our environment.” Where they do that successfully we call them true, but that is a misleading way of saying no more than that we are successful when guided by them.

Applying this theory, often called instrumentalism, to 1 + 1 = 2 is another case of a view of math being guided and controlled by a view of the nature  of all reality. For clearly Dewey’s instrumentalism asserts a view about how all the aspects of experience relate. From the outset, his view of math and all other human conceptual activities is governed by a biological perspective. For him, humans are to be viewed essentially as living organisms struggling for survival. This perspective leads him to take an instrumentalist interpretation of truth and, consequently, an instrumentalist view of math. For him, the so-called truths of mathematics are, like all other “truths,” tools of biological survival. So if the truths of math are all tools of our own devising, there is then no reason to believe they show us the heart and essence of reality, or give us immutable truth. Rather, they are all the products of human invention, which depends, ultimately, on our evolution. This implies that had our brains evolved differently, we might now have a math so different that with our present brains we cannot even imagine it. Yet that math would appear as certain to us under those circumstances as our present math does to us with the brains we now possess.¹¹ In this way, the biological aspect of reality is given a status basic to all the other aspects of reality.

The perspectives cited above are not the only ones to have been adopted in the history of mathematics. Besides the number world of Pythagoras, Plato, and Leibniz, the logicism of Russell, the empiricism of Mill, and Dewey’s  instrumentalism, there are still other competing “schools of thought.” There are, for example, formalists such as David Hilbert, and intuitionists such as Henri Poincaré, Hermann Weyl, and Luitzen Brouwer.

 

7.6            WHAT DIFFERENCE DO SUCH THEORIES MAKE?

These differences among theories about the science of mathematics have created very important differences within it, resulting in wide disagreements over the practices and procedures for doing it. Take, for example, the resistance to the use of irrational numbers by the Pythagoreans. The Pythagoreans, like Plato and Leibniz after them, believed numerals represent a realm of invisible mathematical entities upon which the visible world depends. Since these mathematical entities are supposed to be the ultimate units or building blocks of the world, they were thought to be indivisible. Because of this conviction, the Pythagoreans had a horror of division, fractions, and irrational numbers. This is why they translated fractions into ratios or line segments and insisted there could not be genuinely irrational numbers. The discovery that there are in fact ratios which cannot be expressed as fractions — numbers which are unending decimals such as π — is said to have been made by Hippasus of Metapontum in the fifth century B.C. The story goes that at the time he thought of his discovery he was at sea with a boatload of Pythagoreans who were so incensed by it that they threw him overboard!¹²

Similar to the Pythagorean resistance to irrational numbers was the resistance of Leibniz to negative numbers. Though he allowed them into equations on the ground that their form was proper, he did so only with the proviso that they were to be regarded as purely imaginary quantities.¹³ In other words, he insisted that only positive numbers are real, while the negative numbers are fictions. He was forced to this (implausible) interpretation because he believed math to be a reflection in thought of the real, unseen, eternal realm of numbers. On this view, each numeral we use must stand for a real hypothetical entity, a collection of them, or the relations among them. This being the case, how could a number be negative? How could it stand for nothing? Hence this view has the implausible consequence that negative calculations can’t be true, for they would fail to assert anything that is in fact the case!

These instances of math being done differently because of a perspectival overview of reality may seem to be no more than historical curiosities, so let’s consider another which is a live issue in our own time — the differences in doing math caused by the perspective of the present-day intuitionist theory.

Intuitionists, like the advocates of the number-world theory, maintain an overview perspective which sees the mathematical aspect as utterly independent of all other aspects. But whereas Pythagoras, Plato, and Leibniz lumped logic in with math, intuitionists regard math as basic to logic in such a way  as to leave math partly independent of logical rules. They insist that the intuited mathematical truths are more basic and more reliable than those of any other aspect — including even logical axioms. Thus, for intuitionists, if logical paradoxes arise concerning a mathematical system, that is a problem for logic, but need not trouble the mathematician. As Morris Kline describes the position of the great intuitionist Luitzen Brouwer,

And the intuitionist mathematician Hermann Weyl put the point this way:

Classical logic was abstracted from the logic of finite sets and their subsets. . . . Forgetful of this limited origin, one afterwards mistook logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of set theory, for which it is justly punished by the antinomies.¹⁵

One of the practical consequences of this view is the rejection of the socalled “new math” introduced into the public school curriculum in the United States in the 1960s (and which is now discontinued). The new math was based on a view like that of Russell and proceeded first by teaching logical rules such as commutation, association, and distribution, and then by applying them to sets, which were taken to be what arithmetic is really about. In the quote above, Weyl warns that set theory runs into logical paradoxes and is therefore unsuitable as the basis for arithmetic.

Another important consequence of the intuitionist view is their rejection of any proof which rests upon the logical law of excluded middle. (This is the law which says a statement has to be either true or false and cannot be neither; there is no third or “middle” alternative to being true or false.) As a result, they reject any proof in the reductio ad absurdum form. They also disallow any proof which rests on the logical rule that if one of two options has to be true, and one of them can be shown false, then the other must be true. Both these consequences lead to sharp differences in what intuitionists will accept as proper proofs as over against what mathematicians holding other positions will accept.

Despite accepting a perspective similar to that of the number-world theory because of maintaining the complete independence of the mathematical aspect, intuitionists nevertheless differ with the hypothesis that numbers are real objects. Instead, they insist that math is exclusively mental, and for that reason everything which goes on in mathematics must completely correspond to what we can actually conceive. Thus many of them reject as meaningless both the complex numbers and the irrational numbers that had so upset the Pythagoreans even though these are accepted by Platonists, formalists, and logicists.

For the same reason, intuitionists also deny there is any actual infinity. As Henri Poincaré put it:

This denial of the existence of actual infinite sets forces intuitionists to  yet another denial. They reject an entire branch of mathematics, the theory  of transfinite numbers developed by Georg Cantor. Thus, despite the fact that most mathematicians regard Cantor’s work as the greatest advance in mathematics of the past hundred years, intuitionists insist it doesn’t so much as rise to the dignity of being false but is utterly meaningless!

These are only a few examples of how the differences in philosophical theories about the relations among aspects have affected the concept of what a number is, and thus the doing of mathematics. It is because of just such disagreements and their severe and important consequences that Kline says:

Of course, it is possible that people who work in math are never bothered by the sorts of issues we have been discussing. While many scientists can honestly report that they do their work without puzzling over which perspective is the right one, we should remember — once again — that a perspective need not be conscious to exert its influence. What is important is whether the procedures and techniques of mathematics are of such a character as to require that those using them presuppose some philosophical perspective, not whether everyone is consciously aware of doing so.

 

7.7            THE ROLE OF RELIGION IN THESE THEORIES

But even if a case has been made for the involvement of philosophical perspectives in mathematics, is it equally clear that they all in turn presuppose religious belief? By now it should be. The truth is that some per se divinity belief is at the heart of every one of these perspectives. Theories like Plato’s propose a separate, independent, realm of mathematical entities. On that view, our grasp of mathematical truths is the result of the dependency of our experienced world upon the entities of that independently existing mathematical realm. This is why the truths we obtain about that realm are supposed to be unaffected by the world we experience, as Leibniz affirmed. But anyone who regards this hypothetical realm as having independent existence has thereby accorded it divine status. Remember, it is not because the hypothetical realm is populated by entities supposed to be everlasting, changeless, or logically necessary, that they are divine. Those characteristics alone would not be sufficient. Even if everlasting, an entity could still be everlastingly dependent on something else. Likewise, mathematical truths could express necessary connections among quantities but still depend for their existence on something else. It is the independent existence of the hypothetical entities and laws from all other reality — the view that mathematical truths would be the same whether anything else existed or not — that is the same as regarding them as divine.¹⁸

Moreover, not only is such a view a per se divinity belief, but it is one which perfectly conforms to the pagan dependency arrangement. For although it is true that the realm of mathematical entities is supposed to be invisible, nonphysical, eternal, and changeless, it is still viewed as continuous with the rest of the world which we observe in two respects. The first is that even though it is taken to be more than just an aspect of the world of our everyday experience, it is still supposed to be true of this world. That is, the world we experience contains things that exhibit quantity. The second is that both the hypothetical realm and the observable world conform to mathematical laws. Indeed, the allegedly self-existent laws of the mathematical world are supposed to make the experienced world either possible or both possible and actual. For someone with two per se divinities, like Plato, the laws of the hypothetical realm make the cosmos possible by governing matter (which is also per se divine). For Pythagoras, on the other hand, the world is entirely made of numbers and relations between them. So for him numbers and their laws make the cosmos not only possible but actual.

To see its pagan character more clearly, consider a contrast between this theory’s idea of the relation between the observable world and the hypothetcal realm on the one hand, and the theistic idea of the relation between the universe and God on the other hand. On the Platonic view, the laws governing the hypothetical realm are the order of the observable world as well. In fact, this unseen (divine) realm is the very core of the being of the world which appears to us. As a result, this is clearly a pagan position since the divine is identified with an aspect of the world we experience. According to the biblical idea, God brings into existence (out of nothing) all that is true of the universe including all the kinds of orderliness which govern it. It is that order we attempt to capture in law-statements. So God brought it about that there are things with quantity, and that the relations among quantities conform to just the laws they conform to. (The fact that God is said to be “one” or “one in three” is the consequence of the way he has freely taken on created properties in order to make himself known to humans. This point will also be explained more fully in chapter 10.)

Having pointed out the pagan character of the number-world theory, I must at least mention here that there is a long-standing tradition of theistic thinkers who believe this theory can be adapted so as to be compatible with belief in God. This ploy was explained in chapter 5, where I dealt with the tradition I called “scholastic” and its proposal for de-paganizing theories that take one or another aspects of the world to be what all the rest of creation depends   on. I explained that this tradition agrees with me that regarding mathematical (or any other kind of) entities as self-existent would be unacceptable from a theistic point of view. But it adds that would only be so if the theory ended there. The pagan character of such a proposal can be neutralized, it says, while retaining its basic idea. This can be done by regarding the mathematical realm as, in turn, dependent on God. The most popular proposals for doing this are to say that mathematical truths are part of God himself or are ideas in God’s mind. In this way it is thought the theory can account for the necessity and eternality of the hypothetical realm without admitting it as something both self-existent and distinct from God. In chapter 5 I began an explanation of why this is objectionable on both religious and philosophical grounds. There we saw that simply tacking God onto a theory does not change the content of its explanatory power, which is therefore treated as religiously neutral. So I’ll not repeat all that here. I will, however, return to this objection and develop it in more detail in chapter 10.

Divinity belief plays a crucial role not only in the number-world theory, but in Russell’s theory as well. The chief difference is that for Russell the principles which govern all reality because they are divine are those of logic rather than math. The logical laws, he says, are not only those to which all reality — actual or possible — must conform, but they are “the heart and immutable essence” of all things. Once again, this position amounts to abstracting an aspect of our experience and giving it the priority of having divine status. Thus Russell’s theory also rests upon a pagan religious presupposition.

Some scholastic thinkers have also attempted to reconcile this logicist theory of number with belief in God, and have done so in roughly the same way others did for the number-world theory. They propose that logical laws, sets, etc., be regarded as part of the being of God or as ideas in God’s mind in order to preserve their eternality and necessity while still finding a sense in which they may be said to depend on God. In chapter 10 we will see why this view no more succeeds for logical laws and classes than it does for mathematical laws and entities.

The theory of Mill is, perhaps, even more obviously pagan. On Mill’s view, mathematical truths and laws are all generalizations about our sensations, and we have sensations because all objects are made up of purely sensory qualities; they are sense-bundles. To account for why we all observe the same sensebundles, Mill postulated for each of them the existence of a mysterious entity he called the “permanent possibility of sensation.” When Mill was asked why there are such permanent possibilities and what their causes are, he replied that we can never discover this. Thus, so far as we can ever know, they are just there, so that his theory leaves them with divine status by default (as was explained in chapter 2).¹⁹ Thus, Mill’s theory also presupposes a pagan religious belief. The same is true of the theory of Dewey, though he is much vaguer about the status of the physical-biological aspect(s) which he takes as the basic nature of reality. Dewey does not specifically say that those aspects have independent existence, so far as I know. But all through his theorizing, he regards all other aspects as dependent on the physical-biological and never regards them, in turn, as dependent on anything else. To further complicate this point, Dewey at times denies that anything is utterly independent (he called it “absolute”). But he is also adamant in denying that there is anything outside the universe for it to depend on. Thus we are entitled to say that, relative to our definition of divinity, he is simply inconsistent on this point: it cannot be that there is nothing but the universe and that the universe is not divine. For if there is nothing but the universe, then there is nothing for it to depend upon and it would, indeed, have “absolute” self-existence. Thus it seems fair to say that

Dewey’s theory is yet another case of a pagan religious belief controlling how a theory of the nature of mathematics is developed.

This same state of affairs — religious belief controlling a view of the nature of reality which in turn controls the theories of math — can be found in logic as well. Any number of thinkers in philosophy agree with Russell’s view, and regard the laws of logic as (at least part of) ultimate reality. Thus they see modern logic as a problem-solving method based on truths possessing absolute reality and religious neutrality. But the issue of what explains the connectedness between all the aspects cannot be shut out of logic any more than it can be shut out of math. So logical laws have also been understood not only as absolute, but as a product of the structure of language, as rules by which we cannot help but think because of the way our brains evolved, as the products of historical conditioning, and so on. Even some of the most highly formalized symbolic systems of logic are quite incompatible with one another because of these and similar differences.²⁰ In each case, these views either take logical truths to be self-existent or to be generated by some other aspect which is regarded as self-existent. They are therefore one and all under religious control. It may now be clearer how the varieties of pagan religious belief evince the restless and wandering character which I mentioned earlier. From the standpoint of biblical religion, paganism ransacks the dependent, relative universe for that which is self-existent and absolute. And each claim to have found the divine aspect(s) of creation evokes a counter-claim of divinity on behalf of other aspects which is just as plausible (and therefore just as implausible) as all the others.

But how should we spell out the difference for a theory of the nature of number were it to presuppose belief in God rather than a pagan belief in the divinity of some aspect of creation? To answer this adequately would require that we first elaborate and defend a theory of reality which presupposes belief in God, and then show the consequences of that overview of (created) reality for the concepts of the entities proposed in math and other scientific theories. A blueprint for such a theory of reality will be presented in chapters 11, 12, and 13. For now I can only ask you to recall the earlier, rough, indication of its shape given in chapter 4. There we saw why belief in a transcendent creator should lead us to the view that no aspect of creation is self-existent, nor does any generate any other since all are dependent on God alone. For that reason, we should regard all the aspects we experience as equally real, true of all created things, and irreducible to one another. This requires that the natures of the entities proposed in the sciences should never be limited to any one or two aspects; nothing has a nature that is either basically or exclusively one or two of its aspects in the ways pagan-based theories have always proposed. All these points will be explained more fully later on. For now, we can only briefly notice a few of the more obvious ways such a view of reality would guide a theory explaining the nature of what 1 + 1 = 2 stands for.