ON THE EXISTENCE OF NUMBERS: A CASE FOR FORMS, UNIVERSALS, AND ABSTRACT OBJECTS

Introduction

One of the premier debates of philosophy is on the problem of Universals, sometimes called the problem of Abstract Objects or Platonic Form. Numbers, if indeed they are real things, fall into the category of Abstract Objects. For the uninitiated, a discussion on whether or not numbers, and other so-called abstract objects, exist may seem very odd. I submit among the reasons why such a discussion may seems odd, that not least among them, is the fact that most people would say the answer is obvious. “Of course numbers exist,” they will say “I see them every day and use them every day.” The existence of numbers is seemingly apparent to all people. But, even though it may seem obvious there are those who have raised valid questions about whether such things as numbers have any real existence or whether they are merely conventions of speech for practical purposes and when we say “two” or “three” we do not really refer to anything at all.

It is the purpose of this paper to outline the debate about the problem of numbers as abstract objects, to weigh the arguments of both sides (i.e. the Realist versus the Nominalist positions, each of which have varied levels of adherence) and then to finally to demonstrate the real existence of numbers and, incidentally, other abstract objects as well. I will demonstrate that this question not only has major implications for mathematics and science but also that it has implications that reach beyond to questions of ontology, that is, reality itself and even theology and questions of the nature of God.

Defining Numbers

What kind of thing is a number?

Defining what a number actually is may be more difficult than many would suppose upon first considering the job. Consider any number whatsoever. For instance take the number two. What is the number two? What is the relationship between the number two and the number one, or three, or 10,000? In fact we might be justified in asking, “is the number ‘two’ the same thing as the number ‘2’?” Is the number two tied to any particular representation of that number in numeric or verbal form? If there sits before you a couple of objects, are they a manifestation of twoness? These questions may be easier to ask than to answer.

Bertrand Russell is one who has taken up the matter and attempted to give a definition of a number. In a short essay entitled Definition of Number he argues that many people approach the problem incorrectly. He states, “Many philosophers, when attempting to define number, are really setting to work to define plurality, which is quite a different thing. Number is what is characteristic of numbers, as man is what is characteristic of men. A plurality is not an instance of number, but of some particular number.” In other words, it seems, Russell is asking what the essence of number is. That thing which makes it what it is and without which it ceases to be what it is.

His reference to man or mankind is apropos because it is another philosophical question which has been wrestled with for several millennia, what makes man man? Mankind has been referred to as rational animals, image bearers, and a great many other things in trying to describe the distinction between ourselves and other living things. But there is a sense in which the question “What is man?” is actually very different from the question “What is number?” When it comes to Man we can refer to tangible qualities which at least separate man’s nature from that of other things, both living and inanimate. So even if the ultimate answer to what man is goes deeper than the physical attributes he carries it must be admitted that number does not even have this as a starting point in the discussion.

Let us continue a bit further with Russell’s attempt to define what a number is. Another interesting point he makes in regard to the definition of number is that “we cannot in any case, without a vicious circle, use counting to define numbers, because numbers are used in counting.” In other words we cannot say the number two is simply that which comes after one and before three when counting because this argument commits the fallacy of assuming the premise in our conclusion (i.e. circular reasoning). Furthering this point he writes, “In counting, it is necessary to take the objects counted in a certain order, as first, second, third, etc., but order is not of the essence of number: it is an irrelevant addition, an unnecessary complication from the logical point of view.” In other words a thing, or set of things, can be an instantiation of number without us counting to that number. In fact they could be an instantiation of a number that is beyond our ability to count to while, all the same, truly exemplifying that number.

So given these helpful preliminary remarks about what a number must not be, what does Russell take to be the proper definition of number? He tells us plainly that “A number is anything which is the number of some class.” While this definition may seem circular at first blush we find that it is, in fact, not. As he argues on, “We define ‘the number of a given class’ without using the notion of number in general; therefore we may define number in general in terms of ‘the number of a given class’ without committing any logical error.” In other words a number is that which corresponds to a class of things. The class of all apples sitting on a given table might might instantiate an example of the number seven. Likewise the number of pencils on the table may also instantiate an example of the number seven. In other words these are two different classes of things which instantiate examples of the same number, in this case, seven. Another way of stating this would be to say that both of those classes share the property of sevenness.

I believe this definition is a very good one. One point where I would differ from Russell is on a basic assumption about the possibility of actual infinities, or actual infinite sets of things. In this same essay he writes, “it is to be presumed, for example, that there are an infinite collection of trios in the world, for if this were not the case the total number of things in the world would be finite, which, though possible, seems unlikely.” It seems to me that Russell’s definition of number, based on classes which instantiate examples of that number, is based on his presumption here that there are an actual infinite number of things in the universe. This is where I would raise a contention up against Russell. I will explain my contrary contention momentarily but I want to state that although I think he is wrong on this point, and that his definition stems from his belief about this point, that his definition of number will still stand once I have constructed my counterpoint and placed his definition on a different foundation.

Now let me explain my contention. What is hopefully becoming clear is that numbers are immaterial, they do not extend into space. I think Russell would agree on this point. The way numbers are represented such as “two” or “2” are not in fact the same as the number itself. Numbers are concepts which correspond to a class of things which have that number. Where I disagree with Russell is on the following point, namely, that the number “two” is a discernable idea even apart from an instantiation of twoness such as a couple of spoons or apples before you on a table. In other words, classes of things in the world exemplify number, and numbers are that which correspond to a given class, but numbers are not, strictly speaking, the same as those classes. This is shown to be the case because more than one class can correspond to the same number. So then numbers are properties which classes exhibit but numbers, as real things, are not dependent on the existence of classes of things to exist. Rather numbers are only dependent on classes of things to exhibit their existence. It is entirely feasible that many numbers (perhaps an innumerable amount of numbers, we might say) exist which do not correspond to any actual class of things in the world.

Russell would, I believe, advocate a view called Nominalism. Given his status as an atheist and a philosophical naturalist I think this assumption is fair because Nominalism must be true if all that actually exists is material things. It is for that reason that Russell believes 1) that an infinite number of things exist in the universe and 2) that numbers correspond to actual existing sets of things from one to infinity. Were this not his position he would have to admit that there is a finite number of numbers which seems to be illogical (seeing as no matter how high of a number we name we can always add one more to it). And yet the idea of an actual infinite set (and by actual I mean a set which is exhibited in the natural world) is also logically impossible.

To demonstrate my last claim about the impossibility of an actual infinite set I turn to the work of William Lane Craig. In his book, Reasonable Faith, Craig discusses the Kalam Cosmological Argument for God’s existence and it is in this context that he demonstrates the logical impossibility of an actual infinite set of thing in the natural world. The Kalam Cosmological Argument is as follows:

1. Whatever begins to exist has a cause.
2. The universe began to exist.
3. Therefore, the universe has a cause.

In defense of the second premise of this argument craig demonstrates the logical impossibility of an actual infinite set of things in the world. By extension he argues that if an actual infinite set is impossible then it is impossible for their to be an infinite amount of days that have passed and, therefore, that the universe must have had a definite beginning a finite time ago in the past.

Here is one way by which Craig demonstrates the logical impossibility of an actual infinite set:

[I]magine a hotel with an infinite number of rooms and suppose...that all the rooms are occupied. There is not a single vacant room throughout the entire infinite hotel. Now suppose a new guest shows up, asking for a room. “But of course!” says the proprietor, and he immediately shifts the person in room #1 to room #2, the person in room #2 into room #3, the person in room #3 into room #4, and so on, out to infinity. As a result of these room changes, room #1 now becomes vacant and the new guest gratefully checks in. But remember, before he arrived, all the rooms were already occupied.

Craig continues by giving several other examples of contradictions that can be brought into play using “Hilbert’s Hotel.”

Another example that helps illustrate this point is that of an infinite line of falling dominoes. Imagine the idea that these dominoes have been falling forever without a beginning. If you can imagine that you are doing something rather impressive for it is an inconceivable notion that we might not be able to eventually track down the first domino that fell! Or try another case in which we imagine an infinite line of blue marbles. Now imagine that we paint every other marble red and separate the red marbles from the blue marbles forming two separate lines of marbles. How many blue marbles do we now have? The answer is necessarily a contradiction for this process certainly divided the line of marbles by two which means a reduction of the number of blue marbles by 50% and yet, if the line of marbles is an actual infinite, the line of blue marbles still has an infinite amount of marbles. We have, in this case, not reduced the number of marbles by division but we have doubled it because now we have two infinite lines of marbles. But then, again, there are actually the same amount of marbles as ever there were, an infinite amount.

So if the reasoning of Craig and other thinkers on this issue is correct that an actual infinite set is logically impossible how can it be the case that 1) it is logically necessary that the number of numbers are infinite and also 2) that actual infinite sets exhibiting those numbers are logically impossible? This is the problem that materialists (i.e. Nominalists) face but which is solved by the kind of Realism I am going to advocate, the view that what many philosophers have called “Universals” or “Abstract Objects” actually exist although in an immaterial way and which are not tied of necessity to things in the natural world but which may be exhibited my some things in it.

What is an Abstract Object?

Although I have just alluded to the concept of abstract objects above it will serve to give a more formal definition here. An Abstract Object is a thing which has an immaterial and necessary existence apart from things in the physical world but which may be exhibited or instantiated as a property of the things that exist in the physical world.  I want to draw attention to a few important words in this definition, namely, “immaterial” and also “necessary existence.”

By immaterial I mean a thing which has no tangibility. By way of example colors, laws of logic, thoughts you are thinking right now, and God are all immaterial. Numbers, as we have said, clearly fit in this group. It makes no sense to talk about touching red or holding the laws of logic in your hand, they are not material things. We may rightfully talk about holding a red object, but what we are really saying is that we are holding an object which has the quality of redness. No one has held pure redness in their hands. We might also speak poetically and talk about thoughts “bouncing around in his head.” But here again this is to speak, shall we say, materialmorphically. That is, it is to speak materialistically about the immaterial.

The other key phrase I used was “necessary existence.” Now this gets down to the heart of the matter because it is a claim that says not only do Abstract Objects like colors, numbers, laws of logic, etc., exist but that they do so of necessity. In other words, it could not be the case that they did not exist. Or, once again, there is no possible world in which these abstract objects are not real. The reason for this is because Abstract Objects, at least on a Realist account, belong to the realm of the eternal and are not tied to any specific world. They are more basic than the material world.

Now, admittedly, I have put the cart before the horse a bit. The definition I have given and fleshed out is a Realist definition (as opposed to a Nominalist definition). Various nominalists treat the idea of abstract objects in different ways but for the most part agree that when we speak of these properties that things have we speak of either something concrete belonging to the physical object or, if we speak of them as Universals, we are merely using these terms as useful fictions. In the next section I will consider these three competing views (Realism, and two different forms of Nominalism) and demonstrate the superiority of the Realist view. That being said, before I am done, I will argue for a uniquely Christian view of Realism that will differ in at least one significant way from the Platonist view which is the original Realism.

Are Abstract Objects Real?

The Platonist view

Mortimer J. Adler has said, “Anyone who classifies things or tries to make definitions may be led to wonder whether classifications are entirely verbal and definitions fictions of the mind, or whether things themselves belong together in some real community based upon an inherent sameness or similarity.” Indeed he is correct and this conversation goes as far back (and probably further) as the fifth century B.C. to a man named Socrates whose teaching has been preserved by his student, Plato.

Socrates view of Form is that which we have been calling “Realism.” More often this is referred to a Platonic Form since it is Plato who recorded the words of Socrates and who, it would seem, agrees and taught these same concepts to his own students. Here is one example of Socrates teaching about Form which directly relates to mathematics, and numbers:

You are aware that students of geometry, arithmetic, and the kindred sciences assume the odd and the even and the figures and three kinds of angles and the like in their several branches of science; these are their hypotheses, which they and everybody are supposed to know, and therefore they do not deign to give any account of them either to themselves or others; but they begin with them, and go on until they arrive at last, and in a consistent manner, at their conclusion? Yes, he said, I know. And do you not know also that although they make use of the visible forms and reason about them, they are thinking not of these, but of the ideals which they resemble; not of the figures which they draw, but of the absolute square and the absolute diameter, and so on—the forms which they draw or make, and which have shadows and reflections in water of their own, are converted by them into images, but they are really seeking to behold the things themselves, which can only be seen with the eye of the mind? That is true.

Socrates, in dialog with Glaucon, has demonstrated his view that mathematical representation are simply that, representative of immaterial concepts. Lines drawn on a piece of paper are not the same as the concepts they represent.

To that very point Socrates has the following conversation with a man named Cratylus:

Let us suppose the existence of two objects: one of them shall be Cratylus, and the other the image of Cratylus; and we will suppose, further, that some God makes not only a representation such as a painter would make of your outward form and colour, but also creates an inward organization like yours, having the same warmth and softness; and into this infuses motion, and soul, and mind, such as you have, and in a word copies all your qualities, and places them by you in another form; would you say that this was Cratylus and the image of Cratylus, or that there were two Cratyluses?

Cratylus answers that he would see it as there being two Cratyluses rather than one and an imitation of that one.

Just as it would be absurd to think that there could be two Cratyluses it would also be absurd to think that lines on a page were the same as the things which they were representing. A red brick is likewise not the same as redness, nor is any given brick the same as the Idea of what a brick is. Certainly, also, the idea of a brick preceded the actual making of a brick. Equally certain is that any given brick that is made is not the same as the idea of a brick. So then there is a separation, according to Socrates and Plato between the Form or Idea and the imitation of copy of that idea. Further the Form must be logically prior to the imitation.

One of the most famous passages in all of Plato’s writings is the allegory of the cave. It is also one of the most useful illustrations that Plato gives of his and Socrates’ view of Form. Consider the situation he sets up:

Behold! human beings living in an underground den, which has a mouth open towards the light and reaching all along the den; here they have been from their childhood, and have their legs and necks chained so that they cannot move, and can only see before them, being prevented by the chains from turning round their heads. Above and behind them a fire is blazing at a distance, and between the fire and the prisoners there is a raised way; and you will see, if you look, a low wall built along the way, like the screen which marionette players have in front of them, over which they show the puppets. I see. And do you see, I said, men passing along the wall carrying all sorts of vessels, and statues and figures of animals made of wood and stone and various materials, which appear over the wall? Some of them are talking, others silent. You have shown me a strange image, and they are strange prisoners. Like ourselves, I replied; and they see only their own shadows, or the shadows of one another, which the fire throws on the opposite wall of the cave? True, he said; how could they see anything but the shadows if they were never allowed to move their heads? And of the objects which are being carried in like manner they would only see the shadows? Yes, he said. And if they were able to converse with one another, would they not suppose that they were naming what was actually before them? Very true. And suppose further that the prison had an echo which came from the other side, would they not be sure to fancy when one of the passers-by spoke that the voice which they heard came from the passing shadow? No question, he replied. To them, I said, the truth would be literally nothing but the shadows of the images. That is certain.

This fascinating thought experiment captures the Platonic view of the world we live in. The things which we see with our eyes, feel with our hands, hear with our ears, etc., are all just shadows, imitations, copies of some truer Form of the things we experience in this present life.

For Plato, everything we experience in this world is but a shadow to true Form that exists unchangeably with the gods. Some things are easier to demonstrate this point with, namely, physical objects in the world. As with my above illustration of the brick, the tangible things are the simplest to conceive of the idea of having a perfect Form of which they are imitations. Less easy to grasp are certain properties like colors (e.g. Red that is exemplified by a certain brick). But the fact is that not only bricks but many other objects in the world exemplify Redness. They all participate in this property. But imagine for a moment that all of the red exhibiting things in the universe suddenly ceased to exist in the universe, at this point would red cease to be a thing? It seems obvious that the answer is no because we are still able to conceive of the color red apart from instances of it. But even more challenging in this discussion than Forms of physical objects or even Redness is concepts like truth, goodness and beauty.

We often attribute these qualities to people and to objects in the world, and we believe them to be really meaningful attributions, but what is the good? What is the beautiful? Or as Pilate once asked Jesus, “What is truth?” It must be the case, if these are to be meaningful kinds of statements to make, that there is an objectively existing goodness, and an objectively existing beauty and so on with truth. The apprehension of this truth is difficult, according to Plato, but is essential to understanding the way things really are. Picking up from where we left off in Plato’s Cave, Socrates continues:

And now look again, and see what will naturally follow if the prisoners are released and disabused of their error. At first, when any of them is liberated and compelled suddenly to stand up and turn his neck round and walk and look towards the light, he will suffer sharp pains; the glare will distress him, and he will be unable to see the realities of which in his former state he had seen the shadows; and then conceive some one saying to him, that what he saw before was an illusion, but that now, when he is approaching nearer to being and his eye is turned towards more real existence, he has a clearer vision—what will be his reply? And you may further imagine that his instructor is pointing to the objects as they pass and requiring him to name them—will he not be perplexed? Will he not fancy that the shadows which he formerly saw are truer than the objects which are now shown to him? Far truer. And if he is compelled to look straight at the light, will he not have a pain in his eyes which will make him turn away to take refuge in the objects of vision which he can see, and which he will conceive to be in reality clearer than the things which are now being shown to him? True, he said. And suppose once more, that he is reluctantly dragged up a steep and rugged ascent, and held fast until he is forced into the presence of the sun himself, is he not likely to be pained and irritated? When he approaches the light his eyes will be dazzled, and he will not be able to see anything at all of what are now called realities. Not all in a moment, he said. He will require to grow accustomed to the sight of the upper world. And first he will see the shadows best, next the reflections of men and other objects in the water, and then the objects themselves; then he will gaze upon the light of the moon and the stars and the spangled heaven; and he will see the sky and the stars by night better than the sun or the light of the sun by day? Certainly. Last of all he will be able to see the sun, and not mere reflections of him in the water, but he will see him in his own proper place, and not in another; and he will contemplate him as he is. Certainly. He will then proceed to argue that this is he who gives the season and the years, and is the guardian of all that is in the visible world, and in a certain way the cause of all things which he and his fellows have been accustomed to behold?

Here Socrates describes a truly painful process of coming to see things as they truly are. We are so accustomed to thinking that what we see is the real thing that it is difficult to accept that what we have experienced is, in fact, merely shadows of the real eternal world. But, Socrates tells us, this painful discovery is absolutely essential if we are ever to truly apprehend the good. If Plato’s forms do not exist then “goodness” is nothing more than social convention. However we all know that it is in fact more than that. Self sacrifice for your brother’s well being is good. Giving your life so that many others may live is praiseworthy. These statements are meaningful. Again Plato lets Socrates put this in own words:

This entire allegory, I said, you may now append, dear Glaucon, to the previous argument; the prison-house is the world of sight, the light of the fire is the sun, and you will not misapprehend me if you interpret the journey upwards to be the ascent of the soul into the intellectual world according to my poor belief, which, at your desire, I have expressed—whether rightly or wrongly God knows. But, whether true or false, my opinion is that in the world of knowledge the idea of good appears last of all, and is seen only with an effort; and, when seen, is also inferred to be the universal author of all things beautiful and right, parent of light and of the lord of light in this visible world, and the immediate source of reason and truth in the intellectual; and that this is the power upon which he who would act rationally either in public or private life must have his eye fixed.

Of course many have, since Plato, taken up the notion of Forms. Descartes is one example who, for instance, argues for the objective existence of triangles:

And what I here find to be most important is that I discover in myself an infinitude of ideas of certain things which cannot be esteemed as pure negations, although they may possibly have no existence outside of my thought, and which are not framed by me, although it is within my power either to think or not to think them, but which possess natures which are true and immutable. For example, when I imagine a triangle, although there may nowhere in the world be such a figure outside my thought, or ever have been, there is nevertheless in this figure a certain determinate nature, form, or essence, which is immutable and eternal, which I have not invented, and which in no wise depends on my mind, as appears from the fact that diverse properties of that triangle can be demonstrated, viz. that its three angles are equal to two right angles, that the greatest side is subtended by the greatest angle, and the like, which now, whether I wish it or do not wish it, I recognise very clearly as pertaining to it, although I never thought of the matter at all when I imagined a triangle for the first time, and which therefore cannot be said to have been invented by me.

Descartes demonstrates by his reasoning that the concept of a triangle is no mere convention or something which could be made up by himself, rather it is an objectively existing idea, a concept, which has meaning apart from it even existing in the world or even in his mind.

Much more could, and deserves to be, said about Platonic Form and the concept of Realism which has many representatives over the centuries. For now, however, we will rest with having laid out the concept as the existence of perfect, unchanging, immaterial objects which things in this world either imitate or participate in but are not, themselves, the thing they imitate or participate in. Realism makes things like references to numbers, moral values (i.e. justice, mercy, love), aesthetic judgment and even colors meaningful rather than mere useful fictions or brute material facts.

The Nominalist view

As with the Platonist/Realist view of abstract objects or Forms it should be noted that there are a slew of different views all of which fit under the umbrella of Nominalist theories. The fact that I am only looking at two of them may not do total justice to every nuance of nominalism but with the space permitted it must do for now. The two forms of nominalism I am going to critique are 1) that properties exist in individual concrete objects and have no universal qualities or 2) that properties are spoken of as universals but, in reality, these are useful fictions and they do not exist.

As to properties existing only individual, concrete objects it was not long at all after Plato that this push back against Socrates teachings were challenged. In fact it was Plato’s own student, Aristotle, who rejected the idea of universals with the exception as an abstraction of the human mind. That is to say that Aristotle believed Redness, for example, exists only in direct union with a physical substance. Now, in fairness to Aristotle, he claims that he is not departing from Socrates’ teaching but that he is agreeing with Socrates and it was those who came after him that, in his view, sort of put words in Socrates’ mouth. Aristotle states, “But when Socrates was occupying himself with the excellences of character, and in connexion with them became the first to raise the problem of universal definition… but Socrates did not make the universals or the definitions exist apart: they, however, gave them separate existence, and this was the kind of thing they called Ideas.” But, regardless of whether or not Aristotle is right here that Socrates (and presumably Plato) didn’t actually hold to the separate existence of the Forms/Ideas/Universals, and this is a very debatable point, the doctrine certainly preceded Aristotle and he is objecting to it.

Adler addresses Aristotle’s view of Form when he writes:

Aristotle’s denial of separate existence, or substantiality, to the Ideas or universals stands side by side with his affirmation of the place of forms in the being of substances and the role of universals in the order of knowledge. Furthermore, he limits his denial of the substantiality of Ideas to those Forms which seem to be the archetypes or models of sensible things. Particular physical things—familiar sensible substances, such as the stone, the tree, or the man—are not, in his opinion, imitations of or participations in universal models which exist apart from these things.

For Aristotle there is Form and matter which equal a composite, concrete thing in the world. One cannot have just matter without form, nor can form exist without matter except in imagination of people by abstraction (i.e. imagining red apart from an object it is attached to). There are  many representatives of this view in history including William of Ockham, Francis Bacon and a good deal of contemporary naturalists.

Now consider the second way of looking at universals under a nominalist account, namely, that they are mere conventions of the mind or useful fictions. George Berkeley represents this view as he speaks about numbers in his work Principles of Human Understanding.

That number is entirely the creature of the mind, even though the other qualities be allowed to exist without, will be evident to whoever considers that the same thing bears a different denomination of number as the mind views it with different respects. Thus, the same extension is one, or three, or thirty-six, according as the mind considers it with reference to a yard, a foot, or an inch. Number is so visibly relative, and dependent on men’s understanding, that it is strange to think how anyone should give it an absolute existence without the mind. We say one book, one page, one line, etc.; all these are equally units, though some contain several of the others. And in each instance, it is plain, the unit relates to some particular combination of ideas arbitrarily put together by the mind.

Clearly, in Berkeley’s mind, numbers are not real things. But what does he offer in support of this claim? His argument seems to be that the same thing can be called by more than one number and therefore numbers are arbitrary. But surely this does not follow! In the example given, the measurement known as a yard, he says it can be called “one, or three, or thirty-six” and because of this he concludes that numbers are arbitrary.

Berkeley’s argument might be more plausible if it were the case that the given length in question was one, three, and thirty-six in the exact same way and at the same time (thereby creating a contradiction) but that is not the case. It is perfectly valid to measure the same length by different units and to come up with different answers. How does this in any way suggest that those numbers don’t have an objective meaning? You could tell a man at the lumber yard that you need a 2x4 three feet long, or one yard, or 36 inches and, no matter what unit of measurement you use to communicate with, he will bring you the right length of board. It is just such a fact that shows these numbers are meaningful and that they can be used to communicate with clarity.

Berkeley’s position is, in my view, an extreme one on more than one front as he is of the opinion that no only do forms and universals not exist but that the external world, the world beyond our sense perception, cannot be known to exist either. He states, “In short, if there were external bodies, it is impossible we should ever come to know it; and if there were not, we might have the very same reasons to think there were that we have now.” This is, I believe, the logical extension of nominalism. The rabbit hole that nominalism leads us down takes us to a place where we can have very little conception of any reality.

Pros and Cons of Realism and Nominalism

The strength of Realism is that it is the common sense view or, at least, it is the view we all seem to assume by our speech. We all speak about things as though universals really do exist. When we say things like “Oh, that thing has the color red in it” we are making a very Universal kind of statement. It assumes Red is a thing in itself and that this object we are pointing out is somehow participating in that redness.

Further, If Realism is correct then we speak meaningfully when we call something true, good or beautiful. We all do, of course, think we are really saying something meaningful when we say “that woman is beautiful” or “that sunset is truly beautiful.” Likewise when we say that the person who beat and murdered his wife and children is “evil” we think that means something real and that it is more than just an expression of personal taste. Numbers, likewise, we speak of as real things which certain sets of things exemplify. The strength then of Realism then is that it corresponds with what we perceive to be the case everyday. If Realism is false it is not obviously so.

A strike against Realism might be that it is challenging to prove beyond the fact that we all seem to intuit it to be the case. Given that it refers to an immaterial ultimate reality we do not find it easy to investigate by typical scientific means and it must be a strictly philosophical inquiry. Another objection that has been raised by Christians is the problem of God’s Aseity, the doctrine that God alone is eternal. The Platonic doctrine of Forms suggests that these Forms exists eternally alongside the gods but this cannot hold true if Christianity is true. Therefore many Christians have rejected the concept of Platonic From out of a sense of piety.

The strength of Nominalism is that it accords with a materialist view of the universe. Whereas Realism can only be defended philosophically, Nominalism accords with the scientific evidence. All that we can see, hear, smell, touch, taste, etc., would lead us to believe that Nominalism is the correct view. There is no evidence, of this kind, for Realism and many people seem to think that this is the only evidence that counts when it comes to determining truth.

However, the weakness of this position are, obviously, the strengths of the other position. If the Nominalist position is correct than much of what we believe to be meaningful statements about reality, about things outside the self, are in fact not meaningful at all. Further, even laws of nature (gravity for instance) and laws of logic (law of noncontradiction) are not detectable by the scientific method. Even so, they are usually assumed by those who wield the Scientific method and those in the Nominalist camp in general.

Given the various strengths and weaknesses of each view does little, however, to tell us which view is ultimately the true one. At best it seems to suggest that Nominalism is very difficult to live with in a consistent fashion. It is my argument that the deciding factor between Realism and Nominalism is wrapped up in the question of God’s existence and, specifically, the God of the Bible. If the Christian God exists then a type of Realism must be true and no form of Nominalism can be true.

A Third, Uniquely Christian, View

The pros and cons for each position have been clearly laid out in what has been stated above and neither view is without merit or difficulties. If Realism is to be correct then there are two primary challenges for the Christian. First there is simply the challenge of demonstrating that Realism is true by use of purely philosophic means. Second, assuming the first challenge is met, there is the problem of God’s aseity, his sole eternal existence which is challenged by the Platonic conception of Form. I propose to offer a set of logical arguments that will answer both problems at the same time.

To begin with let us presuppose the existence of the Christian God. We might note that there are many good arguments for existence of God (one has been mentioned earlier in this paper) and that this presupposition is hardly a non-evidenced assumption but one that has been found cogent by many of the brightest minds of history. Since defending the existence of God goes well beyond the scope of this paper, let’s just assume the truth of the proposition “The God of the Bible exists” for the time being. I mean to show you now that if this proposition is true then a type of Realism is necessarily true also and, therefore, Nominalism is false.

I offer the following three logical deductions based on our assumption of the existence of the Christian God:

Deduction one: Demonstrating that what is created exists first in the mind of the creator.

1. If God creates something it must have existence in his mind logically prior to creation.
2. God has created things.
3. Therefore, things exist in the mind of God logically prior to their creation.

Deduction two: Demonstrating the properties exist apart from things that exemplify them.

1. If a Universal property (such as redness) exists in any instance of a thing it must have existed logically prior in the mind of God which is apart from an instance of a thing.
2. Properties do exist in instances of things.
3. Therefore Universal properties exist first in the mind of God apart from any instance of a thing.

Deduction three: Demonstrating that all things that exist apart from God have always existed in the mind of God.

1. Whatever exists, other than God himself, is a creation of God.
2. All creations of God exist logically prior in the mind of God.
3. God does not change.
4. Therefore, all things that exist have always existed in the mind of God.

I conclude, therefore, from what was demonstrated in the three above deductions that all things that exist apart from God have always existed in the mind of God. Further that properties such as color, or abstract objects such as number, or values such good, true and beautiful, exist eternally in the mind of God apart from things which exemplify those qualities. All of these things, existing in the mind of God from eternity, are what we may call Forms, Universals and Abstract objects, etc.

I believe this argument to be not only logically valid, which is undeniable, but also sound. I am not alone on this point as I have two giants of the faith to appeal to in regard to my given arguments. Aquinas and Augustine both are representative of the view I have just expounded. Consider the words of Thomas Aquinas:

It is necessary to place ideas in the divine mind. For the Greek word Ἰδέαis in Latin Forma. Hence by ideas are understood the forms of things, existing apart from the things themselves. Now the form of anything existing apart from the thing itself can be for one of two ends: either to be the type of that of which it is called the form, or to be the principle of the knowledge of that thing, according as the forms of things knowable are said to be in the knower. In either case we must suppose ideas, as is clear for the following reason. In all things not generated by chance, the form must be the end of any generation whatsoever. But an agent does not act on account of the form except in so far as the likeness of the form is in the agent, as may happen in two ways. For in some agents the form of the thing to be made pre-exists according to its natural being, as in those that act by their nature; as a man generates a man, or fire generates fire. But in other agents (the form of the thing to be made pre-exists) according to intelligible being, as in those that act by the intellect; and thus the likeness of a house pre-exists in the mind of the builder. And this may be called the idea of the house, since the builder intends to build his house like to the form conceived in his mind. As then the world was not made by chance, but by God acting by His intellect, as will appear later, there must exist in the divine mind a form to the likeness of which the world was made. And in this the notion of an idea consists.

Aquinas’ example of the builder of a house necessarily conceiving of the house before he builds it is excellent.

It is important to not here, however, that God is no normal builder. For men come to have ideas that they did not always have but God who is eternal and omniscient does not learn, nor think in a linear process and does not come to have new thoughts or ideas but is in a perfect state of knowledge. This is to say, as I argued in my third deduction, that whatever exists in the mind of God has always existed in the mind of God. The Forms are therefore eternal. But note also that locating Forms in the mind of God does away with the problem of God’s aseity. He alone is eternal, and the Forms are also eternal and there is no contradiction or difficulty in both of these things being true. In this way the Christian conception of God is superior to the Greek pantheon which could not have located the forms in the gods themselves who are subject to change and who are themselves created.

To this point Aquinas also quote Augustine who speaks to the eternality of Forms in the mind of God:

Augustine says, “Ideas are certain principal forms, or permanent and immutable types of things, they themselves not being formed. Thus they are eternal, and existing always in the same manner, as being contained in the divine intelligence. Whilst, however, they themselves neither come into being nor decay, yet we say that in accordance with them everything is formed that can arise or decay, and all that actually does so.”

It would seem then that this is a truly Christian solution to the problem of Form. In fact, given an orthodox view of God’s eternal nature and omniscience it would seem that this position is the undeniably Christian view. It does justice to who God is and yet is provides a sufficient grounding for truth, goodness and beauty. It makes our words meaningful because they refer to concepts that God knows from eternity. Numbers are real objects that are attached to concept of classes of things. God knows the exact amount of things in each class that could be formed and these are what numbers are. Further, back to the dilemma I raised in the first section of this paper, about how can it be both true that numbers are essentially infinite and that an actual infinite set of things cannot exist, is resolved by God as well. God knows not only the actual created things, which are finite, but God knows things which do not actually exist. The idea of an infinity is not incoherent in the mind, we all can conceive of an endless set of things, but it is impossible in the physical world. God knows both what is created and that which is conceptually knowable even if not possible in a world such as our own.

Conclusion

I began this paper by defining numbers with the help of Bertrand Russell who helped us see that number are what correspond to a class of things. We found, however, that Russell had some troubling ideas about the possibility of an infinite set of things in the material world, a supposition that leads to contradictory notions. This view held by Russell seems to stem from his commitment to materialism and the necessity that numbers are infinite, therefore necessitating the need for actual infinite sets. Proving that this cannot be the case we set off to find a remedy to the problem of numbers being infinite and yet there being no actually infinite set of things in the material world. I then brought the discussion to concept of Forms, first introduced by Plato and the ensuing conversation about it between Realists and Nominalist. Both views had troubles of their own but Realism was greatly preferable if one could reconcile the view with Christianity and demonstrate that it was actually true. I then demonstrated that if God exists then a modified type of Realism (which locates the Forms in the eternal changeless mind of God) is necessarily true and Nominalism necessarily false.

Certainly some would charge this inquiry into the existence of Forms as little more than an esoteric pursuit which has little to do with real life. Whereas the discussion may be a little esoteric, in the sense that it is technical, it could not have more practical import than it does. The truth is the discussion about the existence of number is part of a much larger conversation as we have seen. The existence of Forms is really about meaning in the universe. Is everything just a brute fact of natural process without meaning or, alternatively, is everything a mere social convention which is ultimately arbitrary? I argue that if there are no Universals, no Forms, then that is all we are left with, a dilemma between two equally disagreeable options.

Thankfully, as we have seen above, we have not only a third option but we have a truth, tied to the existence of God, that Forms do exist and life is full of meaning. Numbers are real things that God knows. Language has objective meaning because words, no matter what language, refer to concepts that God knows. Beauty and moral values also exist as things God knows. The world makes sense because God exists and is the source of Universals and therefore knowable, communicable truth is possible. If God exists then Forms exist and, consequently, Nominalism is false and should be utterly rejected by all Christians.