ON THE OBJECTIVE REALITY OF MATHEMATICAL OBJECTS
William A. Wisdom
May 2006

This essay is occasioned by some claims made well over a year ago by Al Erpel, my friend and fellow member of the Philadelphia Association for Critical Thinking. He has written: "Everything we know to be real is 'physical' by definition....This is not my opinion, this is scientific observation." Of course the definition of "real", or of any word, could not possibly be a matter of scientific observation. This is Al's opinion, or rather his stipulation. "Only [and he clearly intends 'all' as well] things which exist can be defined in terms of matter, energy, or space." "Everything...we know to be real in the universe [can be defined] in terms of mass, space, energy or time." Al may not be the only person who holds this view; but he's the only person I know who has articulated it so clearly and succinctly.

I will explain and defend "mathematical realism" or "mathematical Platonism" in this sense: the things studied by mathematics (numbers, sets, points, spaces, properties, etc.) are objectively real--i.e., they exist independently of any minds, and statements about them are true or false independently of our knowledge of them or our ability to prove them--or even the fact that we think about them.

They are abstract, timeless, and universal: they do not exist in physical space, and do not causally interact with the physical world. While they are not themselves sensible, or definable "in terms of mass, space, energy or time", our reasons for believing that they are objectively real are exactly the same as our reasons for believing that the physical objects of sense and the theoretical objects of physics are objectively real--that they actually exist. Numbers, for example, are not just creations of the human imagination. When we think about them, we are not just thinking about words or ideas.

A first refinement of this thesis is necessary. Neither science nor observation commits us to the existence of numbers (or of physical objects or quarks, for that matter). I address this question: to the existence of what things are we committed by the sentences--particularly the sentences in the sciences--that we deem true? I will argue that it is impossible to avoid commitment to the existence of abstract entities (such as sets or numbers or whatnot) by our scientific talk. I don't just mean that we have to use numerals. I mean that scientific discourse, whether or not it requires the use of numerals, commits us to the objective (mind-independent) existence of numbers--or of the sets in terms of which numbers can be defined. I shall talk in terms of numbers here, to simplify matters.

Most mathematicians and distinguished thinkers about mathematics--including such giants of the last century as Frege, Gödel, Hardy, Putnam, and Quine--have been mathematical realists. Noting this fact is not an argument for realism; but it does mean that realism should be taken seriously. It can't be dismissed casually. In particular, it can't be rejected as absurd or obviously false.

THE LANGUAGE L

First, some preliminaries must be taken care of. Some of what follows will be rather intricate, but that is unavoidable. On first reading, try to absorb as much as possible; but don't give up. After you've finished reading the whole article, you may want to come back to the more complicated parts to get a better mastery of them. If you don't know what "infinite", "denumerable", "uncountable", and the rest of the elementary terms of the theory of sets (or classes) mean, see my article on the concept of infinitude.

I want to identify a "canonical language" for science. We'll call it "L". This will be a single language adequate for the systematic expression of all of science. Not only will this mean that we won't have to use one language for physics and a different language for economics and a third for mathematics. It will also make possible a single uniform and perfectly unambiguous definition of truth for statements in what we can now justly call the language of science. This means, among other things, that we will not be burdened with one notion of physical truth and a different notion of mathematical truth and a third notion of sociological truth and so on.

Now some other language might serve as a canonical language for science. But I know of no other candidate anywhere close to being as clear and useful as the one I will lay out. I will first present the vocabulary for L. And then I will define truth for sentences of L. (I will occasionally use the variable letters "A", "B", "C", ... to refer to expressions of L. This usage should be clear from the context.)

The predicates of L will be represented by lower case letters (except the last three, which have another role in L): often for examples we shall use "f", "g", "h", etc., with or without prime marks, as predicates of L. A 1-place predicate will correspond to a property of something: e.g., "...is prime" or "...is red". L will contain as many 1-place predicates as we want, up to denumerably many--i.e., if we want, we can have as many 1-place predicates as there are positive integers 1, 2, 3, 4, ... . A 2-place predicate will correspond to a relation between two things: e.g., "...loves...", or "...is larger than...". We can have up to denumerably many 2-place predicates. A 3-place predicate corresponds to a relation between three things: e.g. "...equals...plus..." or "...is sitting between...and...". We can have up to denumerably many 3-place predicates. We can have denumerably many 4-place predicates: e.g., "...is as far from...as...is from...". And for each positive integer n, we can have as many as denumerably many n-place predicates.

For what are, or will become, obvious reasons, we will include in the vocabulary of L a special 2-place predicate "=". "...=..." will always means "...is identical to...".

L will contain denumerably many individual variables "x", "y", "z", "x´", "y´", "x´´", and so on: "x", "y", and "z", each followed by zero or more prime marks. If, when meanings are assigned to the predicates, the 1-place predicate "f" means "...is prime" and the 2-place predicate "g" means "...is larger than..." , then "fx", for example, is to be read "x is prime", and "gzy", for example, is to be read "z is larger than y". Variables, then, stand in the places where designating or referring expressions might stand.

Although we could get along with just one quantifier, defining the other one in terms of that one, we shall simplify things by adopting two: the universal quantifier ∀, and the existential quantifier ∃. An expression like "(∀y)fy" means "For each thing y, y is prime"--or, more compactly, "Everything is prime". An expression like "(∃z)fz" means "There is at least one thing z such that z is prime", or, more compactly, "Something is prime" or "There are primes" (in the sense that there is at least one prime).

Combining the quantifiers gives us expressions like "(∃x)(∀z)gxz", which means "There is at least one thing x such that for each thing z, x is larger than z" ("There is something that's larger than everything"). "(∃x)(∀z)gzx" means "There is at least one thing x such that for each thing z, z is larger than x" ("There is something than which everything is larger"). To continue the example, "(∀z)(∃x)gxz" means "For each thing z there is at least one thing x such that x is larger than z" ("For everything there's something larger"). And "(∀z)(∃x)gzx" means "For each thing z there is at least one thing x such that z is larger than x" ("Each thing is larger than at least one thing.") No one of these last four expressions has the same meaning as any other, as some reflection will reveal.

You've noticed that parentheses are also part of the language L. So far they've not been necessary. But they become necessary when we add truth-functional connectives to L. We'll use the singulary "connective" ~, which precedes a single expression to form its negation. For example, whatever the expression A is, "~A" means "It is not the case that A". "&", as you might expect, is a binary connective. Insert it between any two expressions and you form their conjunction. Whatever expressions C and A are, "C&A" means "Both C and A".

While there are (infinitely) many other truth-functional connectives, we need not bother with them, since they are all demonstrably definable in terms of, and hence eliminable in favor of, combinations of the two we've adopted. For example, let us introduce the binary connective "⊃" by definition in terms of "~" and"&". For any two expressions A and B, their conditional "A⊃B" will be short for "~(A&~B)", and can be read "If A then B" ("It's not the case that A but not B"). And let us introduce the binary connective "≡" by definition in terms of "&" and "⊃". For any two expressions A and B, their biconditional "A≡B" will be short for "(A⊃B)&(B⊃A)" and can be read "A if and only if B" or "A just in case B".

[In fact, all truth-functional connectives are definable in terms of just one binary connective "|", "A|B" being read "not both A and B"--i.e., "either not A or not B". But it becomes difficult if not impossible to understand even moderately long expressions in which the only connective is "|", so we won't use it. Our interest in economy of symbols is not that great.]

We said a little earlier that we could get along with just one quantifier, defining the other one in terms of that one. Here is how that would work. Again, let "f" mean "...is prime". We could regard "(∃x)fx" as an abbreviation for "~(∀x)~fx" ("It is not the case that for every thing x, x fails to be prime", or, less formally, "it's not the case that nothing is prime"), thus eliminating "∃" from the basic vocabulary of L. Or we could regard "(∀x)fx" as an abbreviation for "~(∃x)~fx" ("It is not the case that for at least one thing x, x fails to be prime", or less formally, "it's not the case that there's something that's not prime") , thus eliminating "∀" from the basic vocabulary of L. But instead, we'll consider both "∀" and "∃" as basic elements of the vocabulary of L.

Parentheses, as you may have noticed three paragraphs back, are now essential because we have to be able to distinguish between, say, a sentence of the form ~(A&~B) and one of the form (~A&~B). The first is the negation of a conjunction, and the second is the conjunction of two negations. They do not mean the same thing. (Incidentally, the expressions of L can be of any finite length.)

You may have noticed that we have no symbols in the vocabulary of L which would serve as the names of things--no symbols to name Al Erpel, and no numerals to name the number seven. Of course we need to be able to refer to or designate such things somehow. But we don't need proper names in order to do that. We can use definite descriptions, or what we'll simply call descriptions, to serve this function. I'm not going to pause here to explain how this is done--I'll treat descriptions, which serve the function of designators, shortly. Suffice it to say that combinations of quantifiers, variables, truth-functional connectives, and parentheses can do precisely the job that we would want proper names to do--like "the Empire State Building" or "God")--or phrases meant to be uniquely descriptive of something (like "the place of eternal torment to which those who reject Christ go after death" or "the fortieth President of the United States"). How this is accomplished is explained in a wide variety of intermediate level textbooks in logic and in the philosophy of language. (See, e.g., Quine's Word and Object, §§ 37-39.) I will discuss it briefly below.

Now for the truth or falsehood--the "truth-value"--of statements in L. The first thing you have to do is identify some non-empty set of things as your domain or universe of discourse D. These are the things about which you will talk in L. The non-empty set D can have as many members as you want. And they can be of any sort that you want--rocks, rainbows, sets, regions of space, dreams, quarks, tables, numbers, the Equator...anything you want. (We may want to trim down your original selection later.)

Now to each 1-place predicate P, assign as the meaning (the "extension") of P some (possibly empty) subset of the domain D. These will be the things of which you consider P true. (The set assigned to some predicate P will be empty just in case P holds of nothing.) To each 2-place predicate P, assign some (possibly empty) set of ordered pairs of members of D. These will be the pairs of things of which you take the relation expressed by P to hold (in the order in which they appear in the pair). To each 3-place predicate P, assign some (possibly empty) set of (ordered) triples of members of D. These will be the triples of things of which you take the relation expressed by P to hold (in the order given). And so on for every predicate in L. Of course, to the predicate "=" will be assigned the pair <d,d> for every member d of the domain D.

We need another notational convention in order for us to define the truth of quantifications. Just as we used the general letters "A", "B", etc. to refer to expressions of L, and "P" to refer to predicates of L, we'll use "X" and "Y" to refer generally to the variables "x", "y", "z", ... of L. Note that "A", ... , "P", "X", and "Y" are not parts of the language L, but are used to facilitate general talk about the elements of L. Incidentally, we'll occasionally abbreviate an expression like "~X=Y" as "X≠Y".

Let A be an expression of L in which the variable Y does not occur, and let A(Y/X) be an expression exactly like A except for showing occurrences of Y at all the places where A shows occurrences of X.

A universal quantification (∀X)A will count true just in case A(Y/X) is true regardless of what member of the domain D you might associate with Y. Put technically, (∀X)A is true just in case A(Y/X) is true for every value of the variable Y.

And an existential quantification (∃X)A will count true just in case A(Y/X) is true for at least one value of Y--true so long as the association of at least one member of D with Y makes A(Y/X) true.

We'll look at some examples. "(∀y)fy" counts true just in case "fz" is true for every value of "z"--i.e., just in case each member of D is a member of the set associated with "f". After all, we want "(∀y)fy" to mean "Every member of D is prime" or "Everything is prime", which it does.

"(∃z)fz" counts true just in case "fx" is true for at least one value of "x"--i.e., just in case at least one member of D belongs to the set of members of D associated with "f". After all, we want "(∃z)fz" to mean "Some things are prime" or "There are primes" or "Primes exist" (in the sense that "At least one thing is prime"). And it does.

"(∀x)x=x" ("Everything is self-identical") will count true so long as "y=y" is true for every value of "y", which it is--because <d,d> has been assigned to "=" for every member d of D.

"(∃z)(∀y)y=z" is true just in case there exists exactly one thing: there is at least one thing z such that everything that exists is z.

Now for a more complex example. "(∀z)(∃x)gxz" counts true just in case "(∃x)gxy" is true for every value of "y" (i.e., is true of every member of D). And "(∃x)gxy" is true for any particular value of "y" just in case "gxx´" is true for some value of "x´": i.e., each member of D has got some member of D that's larger than it (the former). We said above that we want such an expression to mean "For everything there's something larger", which it does. Follow it through: "(∀z)(∃x)gxz" will count true just in case, for every member d of the domain D, there is some member d´ of D such that the pair <d´,d> is a member of the set of pairs assigned to the two-place predicate "g" as its meaning.

Finally, we have to define truth for truth-functional compounds: negations and conjunctions. This is easy and perhaps obvious. The negation ~A of a statement A of L is true just in case A is not true. The conjunction A&B of the statements A and B of L is true just in case both A is true and B is true. (So a "conditional" A⊃B is true just in case it's not the case that both A is true and B is not--in short, either A is not true or B is true, or both. And a "biconditional" A≡B) is true just in case either A and B are both true or they are both false--i.e., just in case A and B have the same truth-value.)

Finally, any statement of L is false if and only if it is not true.

DEFINITE DESCRIPTIONS

We noted earlier that our language L would do without proper names like "2" or "Pegasus"--words or phrases whose typical function is to designate things--whether or not they actually do designate. Their job would be done in L by descriptions. In English, these are expressions like "the winged horse of Bellerophon", "the smallest prime number", "the author of Huckleberry Finn", and "the oldest man alive"--each a descriptive phrase purporting to refer to exactly one thing. The first would refer to Pegasus, the second to the number two, the third to Mark Twain, and the fourth to someone I know not whom. That's the sense in which descriptions function like names.

In symbols, the description "the one and only thing that f's" is represented by "(⌉x)(fx)"--the thing x such that fx, or sometimes just "the f-er". Such an expression is not defined, but is understood in context. In the simplest case, "(⌉x)(fx) exists" is expanded to "(∃y)(∀x)(fx≡x=y)"--"There is at least one thing y such that any thing x f's if and only if it (the latter) is that (former) thing." A bit of reflection on the truth-conditions for quantifications and biconditionals will reveal that "(∃y)(∀x)(fx≡x=y)" will be true in L just in case there exists exactly one f-er, and will be false if there are less than one or more than one (which of course is exactly what we would want in this case).

In a slightly more complicated case, "g(⌉x)(fx)"--i.e., "The one f-er g's" becomes "(∃y)(∀x)((fx≡x=y)&gy)". Again, as wanted, this will be true in L just in case there is exactly one f-er, and that thing g's.

"The f-er does not g" poses problems with the "scope" of the negation. It might mean "It is not the case that the f-er g's; or it might mean "Exactly one thing f's, and that thing does not g." The first is symbolized "~(∃y)(∀x)((fx≡x=y)&gy)", and is true if and only "(∃y)(∀x)((fx≡x=y)&gy)" is not true. The second is symbolized "(∃y)(∀x)(fx≡x=y)&~gy)", and is true if and only if there is exactly one f-er, which thing does not g.

More complex treatments of descriptions yield to similar contextual "definitions". This should at least suggest how we can dispense with individual names in L with no loss of expressive power.

Let's look at a simple example: "The number two is even." Let's use "s" for the predicate "...is the smallest prime" and "e" for "...is even. In L, this becomes "∃y)(∀x)(sx≡x=y)&ey)". You should satisfy yourself that this is true if and only if the number two is even.

How about "Pegasus does not exist"? Let's use "h" for "...is the winged horse of Bellerophon". Then we have "~(∃y)(∀x)(hx≡x=y)", which is true just in case there is no unique thing that's the winged horse of Bellerophon--i.e., just in case Pegasus doesn't exist.

Given these definitions of truth and falsehood, it is possible to define further key logical notions: the logical ("necessary") truth of a sentence, the logical inconsistency of a collection of sentences (the impossibility of their simultaneous truth), the logical entailment or implication of some sentence by a set of sentences (the impossibility of the falsehood of the former given the truth of all the members of the set), and others.

THE WEB OF BELIEF

[Much of what follows is due to W. V. O. Quine, as well as to P. Duhem, H. Putnam, P. Benacerraf, and others.]

The diagram below is a picture of Al Erpel's mental life.

This is not the chaotic mess that you might suppose. In fact, it is a well-organized mental life. The dots represent (only a few of) his beliefs. The lines between them represent connections of dependency: some beliefs support others, and some are supported by others. At the periphery of the "web of belief" is experience--sensory stimulation. The distance from the perimeter represents both how many other beliefs are connected to any particular belief--how deeply it is imbedded in the web--and how reluctant Al (or anyone) would be--or should be--to change a belief in the face of recalcitrant experience. This is because the more other beliefs with which any one belief is connected, the more "costly" it would be to change that one: the more other beliefs would also have to be reconsidered, and perhaps modified or abandoned altogether.

Beliefs at the periphery of the web are those based on only one or very few experiences, and on few or no other beliefs. The sound behind me leads me to believe that my cat is in my office. But my confidence in this is very low--which means that further experiences could easily induce me to change my mind. I turn around and see her in the room across the hall. So I was wrong, and I change my mind with no cost to the rest of my belief system.

Repeated experience has led me to believe that there are no bars on Diamond Street. My evidence is better here, and the belief is a bit more deeply imbedded in the web of my belief, being evidentially linked to other beliefs about what a bar looks like, the reliability of my senses and my memory, and my guess that no bar has opened on Diamond Street since my last visit there. It would take rather little to get me to abandon or alter this belief if I see a bar on Diamond Street. I might have to rethink what I take to be a bar. And so on. But I'd be ready enough to change my mind with respect to bars on Diamond Street.

Deeper still is my belief that all sodium salts burn yellow. Being more deeply imbedded in my web, it depends on many other beliefs, and many other beliefs depend on it. So I am relatively reluctant to abandon or alter this belief. But I would be willing to if my experience were rather different over the range of experiences on which this belief ultimately depends.

We notice that the more deeply we burrow into the web of our belief, the wider the range of experience with which any particular belief is evidentially connected, and hence the more scientific disciplines are implicated in that belief. Take for example the current version of the theory of evolution. It gets its support from biology, biochemistry, genetics, geology, paleontology, anthropology, etc.--and ultimately from the experiences on which these sciences depend--and in turn lends its accumulated credibility to those and other sciences. So one's belief in the theory of evolution would be a very costly and hence intellectually difficult one to abandon, though developing experience in a number of areas of one's experience could cause us to adjust the theory accordingly.

At the center of one's web of belief are those beliefs that one could not imagine changing, because that person could not imagine how new experiences could make a relevant difference to their truth. Some simple examples are: "All red fire engines are red" and "5+7=12"--what are sometimes called truths of logic and mathematics, respectively. A "truth of logic" less obviously true is "If it's not one thing, it's another"--in symbols: "(∀x)(∀y)(x≠y⊃(∃ z)(z≠y&x=z))". Consider any two things at all: if the first is not identical to the second, then there is something other than that second to which the first is identical.

We may be extremely reluctant to alter such beliefs. But there is no reason absolutely to refuse to do so. They are supported just like any of the claims of physics or chemistry by the (enormous) number of other beliefs on which they depend and which depend on them. Our failure to imagine falsifying experiences is (merely?) a failure of imagination--of course a thoroughly understandable failure.

CONFIRMATIONAL AND SEMANTIC HOLISM

Willard Quine is famously known (among other things) for his claim that "our statements...face the tribunal of sense experience not individually but only as a corporate body" ["Two Dogmas of Empiricism"; Duhem argues for much the same point in La Théorie physique]. Single statements are not confirmed by experience one at a time: only whole collections of statements (e.g., theories) are, this because given any "recalcitrant" experiences--experiences that turn out differently than our prior beliefs would lead us to expect--we can consistently accommodate any one statement to the new collection of experiences in indefinitely many ways.

Ultimately, all of our beliefs are jointly tested and confirmed by experience, since adjustments to accommodate unexpected experiences can normally be made at any number of different places in the web, in any number of different ways. And this is because our beliefs are all interdependent upon each other in so many intricate ways. As an extreme example, we could consistently believe that we are living on the inside rather than on the outside of the big ball we call the Earth, if we were willing to make enough changes in our beliefs about gravitation, light, meteorological and astronomical phenomena, and so on.

The elaborate interconnection of all of our beliefs is a major reason why we want to have a single language of science, a single universe of discourse comprising at once all the things about which we might want to "do" science, and a single definition of truth for all the statements of science. If we had separate domains and separate languages and separate notions of truth for different subject-matters, we could not recognize relations of evidential connection across the barriers between them.

Incidentally, some people do seem to operate as if they lived in two or more distinct worlds or domains, with distinct languages and concepts of truth therefor. Often we call them fanatically religious--sometimes just plain crazy.

THE ENTITIES POSTULATED BY SCIENCE

Of course all sorts of "things" are employed or postulated or posited in scientific and everyday discourse: physical objects like rocks and tables; as well as forces, classes, numbers, gods, and a wide variety of other hypothetical or otherwise unobservable things--frictionless surfaces; physical systems unaffected by external forces; geometrical points, lines, surfaces, and figures; and so on and on. None of these are the experiences, or sensory stimulations, on which our science ultimately rests. They are offered to make sense of those experiences. Roughly, our conceptual schemes, our scientific beliefs and theories, the apparatus of science are used to lead us from experiences to other experiences: given such and so experiences, our science leads us to expect certain other experiences, in the absence of which, adjustments must be made in the machinery that led us to expect the latter experiences. Even our commitment to the existence of the chair on which we sit is an hypothesis to account for certain regularities in our past and present experiences. This is even more clearly so of electrons.

The question immediately presents itself to us: by what criteria do we evaluate the plausibility of the body of scientific claims to which we are presently, and in the future will be, committed? They are the familiar standards of simplicity, consistency with other well-established beliefs, conservatism, familiarity, scope, etc. In short, we want our science as a whole to provide a coherent systematization of the whole of our experience.

An incidental question concerns the status of the truths of logic and mathematics: "If Socrates is a human being and all human beings are mortal, then Socrates is mortal", or "5+7=12". These, like all our other beliefs, are parts of the web of organized beliefs supported ultimately by experience. It may be difficult to imagine how experiences bear on their truth. But this is difficult because the truths of logic and mathematics, at least for minimally rational people, lie at or near the center of their webs of belief, intimately connected with virtually all of their other beliefs. So their beliefs would have to be radically different than earlier for the criteria of justification to warrant changes in the beliefs about what they had previously considered truths of logic or mathematics. But the fact is that both logicians and mathematicians have proposed changes in their fields in order to accommodate exotic experiences (like claims in fiction, or the claims of quantum mechanics).

ONTOLOGICAL COMMITMENT

Remember that the members of your domain D are all the things--probably infinitely many; even uncountably many--that you might ever have occasion to talk about or make reference to in your scientific discourse. Put otherwise, they are the things to whose existence you are prepared to commit yourself to make the beliefs you express in L true. They are the things whose existence is indispensable to the practice of your science.

Putting the matter in this latter way recognizes that we might find ourselves committed to the existence of numbers but not of sets (which can be defined in L in terms of numbers), or we might be committed to the existence of sets but not of numbers (which can be defined in terms of sets). That is, numbers might be dispensable, or sets might be dispensable. But it is inconceivable that we could practice science without acknowledging either numbers or sets or some other insensible abstract entities--perhaps functions--in terms of which the others can be defined. (If you have scruples about this, those scruples will have to extend to some other abstract, timeless, immaterial, insensible universals equally objectionable.) So for simplicity's sake, we might as well acknowledge the existence of numbers. This means that there will be numbers in D, and that our variables will range over those numbers and everything else in D. "To be is to be is to be the value of a variable." Put otherwise, we are committed to the existence of the things included in our domain D--i.e., to the things that would have to exist in order for our statements to come out true in the language L of science.

Putnam says, in agreement, that "Quine...has for years stressed both the indispensability of quantification over mathematical entities and the intellectual dishonesty of denying the existence of what one daily presupposes" (Philosophy of Logic, 1971, p. 57). This summarizes in a particularly dramatic way what we've already said: that (a) science requires the inclusion of mathematical entities--in our case, particularly numbers--in the domain of its canonical language, and that (b) we cannot conscientiously deny the existence of what we are daily committed to.

What that boils down to is that we commit ourselves to the existence of numbers by the practice of science--we could not commit ourselves to the truth of the claims of science and everyday life without acknowledging the existence of numbers, for which the whole of our experience is a massive body of empirical evidence.

Copyright © 2006, William A. Wisdom

SELECTED QUOTES ON THE OBJECTIVITY
(MIND-INDEPENDENT REALITY) OF MATHEMATICAL OBJECTS


G. W. Hardy, A Mathematician's Apology (1940), pp. 122-30

In the first place, I shall speak of "physical reality", and here...I shall be using the word in the ordinary sense. By physical reality I mean the material world, the world of day and night, earthquakes and eclipses, the world which physical science tries to describe.

...For me, and I suppose for most mathematicians, there is another reality, which I will call "mathematical reality"....

...I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations", are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards.... (122-24)

There is another remark which suggests itself here and which physicists may find paradoxical, though the paradox will probably seem a good deal less than it did eighteen years ago. I will express it in much the same words which I used...in an address to Section A of the British Association. My audience then was composed almost entirely of physicists, and I may have spoken a little provocatively on that account; but I would still stand by the substance of what I said.

I began by saying that there is probably less difference between the positions of a mathematician and of a physicist than is generally supposed, and that the most important seems to me to be this, that the mathematician is in much more direct contact with reality. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real'; but a very little reflection is enough to show that the physicist's reality, whatever it may be, has few or none of the attributes which common sense ascribes instinctively to reality. A chair may be a collection of whirling electrons, or an idea in the mind of God: each of these accounts of it may have its merits, but neither conforms at all closely to the suggestions of common sense.

I went on to say that neither physicists nor philosophers have ever given any convincing account of what 'physical reality' is, or of how the physicist passes, from the confused mass of fact or sensation with which he starts, to the construction of the objects which he calls 'real'. Thus we cannot be said to know what the subject-matter of physics is; but this need not prevent us from understanding roughly what a physicist is trying to do. It is plain that he is trying to correlate the incoherent body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme which he can borrow only from mathematics.

A mathematician, on the other hand, is working with his own mathematical reality. Of this reality,...I take a 'realistic' and not an 'idealistic' view. At any rate (and this was my main point) this realistic view is much more plausible of mathematical than of physical reality, because mathematical objects are so much more what they seem. A chair or a star is not in the least like what it seems to be; the more we think of it, the fuzzier its outlines become in the haze of sensation which surrounds it; but '2' or '317' has nothing to do with sensation, and its properties stand out the more clearly the more closely we scrutinize it. It may be that modern physics fits best into some framework of idealistic philosophy--I do not believe it, but there are eminent physicists who say so. Pure mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is a prime, not because we think 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. (128-30)

Antony Flew A Dictionary of Philosophy (2nd ed. rev., 1984), p. 223

Many mathematicians...take a realist view about mathematical truth and the existence of mathematical objects. They hold that the latter exist independently of our thought and hence that mathematical statements are true (or false) independently of our knowledge of them or our ability to prove them. This view is known as Platonism since it derives from and often includes Plato's view that the subjects of mathematical statements--numbers--are abstract entities and that, if true, these statements describe relations holding between the entities. Abstract entities are timeless, do not exist in physical space, and do not causally interact with the physical world.

S. Blackburn (ed.), The Oxford Dictionary of Philosophy (1994), p. 289

Platonism: The view...that abstract objects, such as those of mathematics, or concepts such as the concept of number or justice, are real, independent, timeless, and objective entities.

>Word IQ Encyclopedia (on line)

Mathematical Realism, or Platonism

Mathematical realism holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. The term Platonism is used because such a view is seen to parallel Plato's belief in a "heaven of ideas", an unchanging ultimate reality that the everyday world can only imperfectly approximate. Plato's view probably derives from Pythagoras, and his followers the Pythagoreans, who believed that the world was, quite literally, built up by the numbers. This idea may have even older origins that are unknown to us.

Many working mathematicians are mathematical realists; they see themselves as discoverers. Examples are Paul Erdös and Kurt Gödel.... Gödel believed in an objective mathematical reality that could be perceived in a manner analogous to sense perception. Certain principles (e.g., for any two mathematical objects, there is a collection of objects consisting of precisely those two objects) could be directly seen to be true....

The major problem of mathematical realism is this: precisely where and how do the mathematical entities exist? Is there a world, completely separate from our physical one, which is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities?...

An important argument for mathematical realism, formulated by Quine and Putnam, is the Indispensability Argument: mathematics is indispensable to all empirical sciences, and if we want to believe in the reality of the phenomena described by the sciences, we ought also believe in the reality of those entities required for this description. In keeping with Quine and Putnam's overall philosophies, this is a naturalistic argument. It argues for the existence of mathematical entities as the best explanation for experience.

Stanford Encyclopedia of Philosophy (on line)

Indispensability Arguments in the Philosophy of Mathematics
Mark Colyvan

One of the most intriguing features of mathematics is its applicability to empirical science. Every branch of science draws upon large and often diverse portions of mathematics, from the use of Hilbert spaces in quantum mechanics to the use of differential geometry in general relativity. It's not just the physical sciences that avail themselves of the services of mathematics either. Biology, for instance, makes extensive use of difference equations and statistics. The roles mathematics plays in these theories is also varied. Not only does mathematics help with empirical predictions, it allows elegant and economical statement of many theories. Indeed, so important is the language of mathematics to science, that it is hard to imagine how theories such as quantum mechanics and general relativity could even be stated without employing a substantial amount of mathematics.

From the rather remarkable but seemingly uncontroversial fact that mathematics is indispensable to science, some philosophers have drawn serious metaphysical conclusions. In particular, Quine and Putnam have argued that the indispensability of mathematics to empirical science gives us good reason to believe in the existence of mathematical entities. According to this line of argument, reference to (or quantification over) mathematical entities such as sets, numbers, functions and such is indispensable to our best scientific theories, and so we ought to be committed to the existence of these mathematical entities. To do otherwise is to be guilty of what Putnam has called "intellectual dishonesty". Moreover, mathematical entities are seen to be on an epistemic par with the other theoretical entities of science, since belief in the existence of the former is justified by the same evidence that confirms the theory as a whole (and hence belief in the latter). This argument is known as the Quine-Putnam indispensability argument for mathematical realism. There are other indispensability arguments, but this one is by far the most influential, and so in what follows I'll concentrate on it....

1. Spelling Out the Quine-Putnam Indispensability Argument

The Quine-Putnam indispensability argument has attracted a great deal of attention, in part because many see it as the best argument for mathematical realism (or platonism)....Many platonists...rely very heavily on this argument to justify their belief in mathematical entities. The argument places nominalists who wish to be realist about other theoretical entities of science (quarks, electrons, black holes and such) in a particularly difficult position. For typically they accept something quite like the Quine-Putnam argument) as justification for realism about quarks and black holes. (This is what Quine calls holding a "double standard" with regard to ontology.)

For future reference I'll state the Quine-Putnam indispensability argument in the following explicit form:

(P1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.

(P2) Mathematical entities are indispensable to our best scientific theories.

(C) We ought to have ontological commitment to mathematical entities.

Thus formulated, the argument is valid. This forces the focus onto the two premises. In particular, a couple of important questions naturally arise. The first concerns how we are to understand the claim that mathematics is indispensable. I address this in the next section. The second question concerns the first premise. It is nowhere near as self-evident as the second and it clearly needs some defense. I'll discuss its defense in the following section. I'll then present some of the more important objections to the argument, before considering the Quine-Putnam argument's role in the larger scheme of things - where it stands in relation to other influential arguments for and against mathematical realism.

2. What is it to be Indispensable?

The question of how we should understand ‘indispensability’ in the present context is crucial to the Quine-Putnam argument, and yet it has received surprisingly little attention. Quine actually speaks in terms of the entities quantified over in the canonical form of our best scientific theories rather than indispensability. Still, the debate continues in terms of indispensability, so we would be well served to clarify this term.

The first thing to note is that ‘dispensability’ is not the same as ‘eliminability’. If this were not so, every entity would be dispensable (due to a theorem of Craig). What we require for an entity to be ‘dispensable’ is for it to be eliminable and that the theory resulting from the entity's elimination be an attractive theory. (Perhaps, even stronger, we require that the resulting theory be more attractive than the original.) We will need to spell out what counts as an attractive theory but for this we can appeal to the standard desiderata for good scientific theories: empirical success; unificatory power; simplicity; explanatory power; fertility and so on. Of course there will be debate over what desiderata are appropriate and over their relative weightings, but such issues need to be addressed and resolved independently of issues of indispensability.

These issues naturally prompt the question of how much mathematics is indispensable (and hence how much mathematics carries ontological commitment). It seems that the indispensability argument only justifies belief in enough mathematics to serve the needs of science. Thus we find Putnam speaking of "the set theoretic ‘needs’ of physics" and Quine claiming that the higher reaches of set theory are "mathematical recreation...without ontological rights" since they do not find physical applications. One could take a less restrictive line and claim that the higher reaches of set theory, although without physical applications, do carry ontological commitment by virtue of the fact that they have applications in other parts of mathematics. So long as the chain of applications eventually "bottoms out" in physical science, we could rightfully claim that the whole chain carries ontological commitment....

3. Naturalism and Holism

Although both premises of the Quine-Putnam indispensability argument have been questioned, it's the first premise that is most obviously in need of support. This support comes from the doctrines of naturalism and holism.

Following Quine, naturalism is usually taken to be the philosophical doctrine that there is no first philosophy and that the philosophical enterprise is continuous with the scientific enterprise (Quine 1981b). By this Quine means that philosophy is neither prior to nor privileged over science. What is more, science, thus construed (i.e. with philosophy as a continuous part) is taken to be the complete story of the world. This doctrine arises out of a deep respect for scientific methodology and an acknowledgment of the undeniable success of this methodology as a way of answering fundamental questions about all nature of things. As Quine suggests, its source lies in "unregenerate realism, the robust state of mind of the natural scientist who has never felt any qualms beyond the negotiable uncertainties internal to science". For the metaphysician this means looking to our best scientific theories to determine what exists, or, perhaps more accurately, what we ought to believe to exist. In short, naturalism rules out unscientific ways of determining what exists. For example, naturalism rules out believing in the transmigration of souls for mystical reasons. Naturalism would not, however, rule out the transmigration of souls if our best scientific theories were to require the truth of this doctrine.

Naturalism, then, gives us a reason for believing in the entities in our best scientific theories and no other entities. Depending on exactly how you conceive of naturalism, it may or may not tell you whether to believe in all the entities of your best scientific theories. I take it that naturalism does give us some reason to believe in all such entities, but that this is defeasible. This is where holism comes to the fore: in particular, confirmational holism.

Confirmational holism is the view that theories are confirmed or disconfirmed as wholes. So, if a theory is confirmed by empirical findings, the whole theory is confirmed. In particular, whatever mathematics is made use of in the theory is also confirmed. Furthermore, as Putnam has stressed, it is the same evidence that is appealed to in justifying belief in the mathematical components of the theory that is appealed to in justifying the empirical portion of the theory (if indeed the empirical can be separated from the mathematical at all). Naturalism and holism taken together then justify P1. Roughly, naturalism gives us the "only" and holism gives us the "all" in P1.

It is worth noting that in Quine's writings there are at least two holist themes. The first is the confirmational holism discussed above (often called the Quine-Duhem thesis). The other is semantic holism which is the view that the unit of meaning is not the single sentence, but systems of sentences (and in some extreme cases the whole of language). This latter holism is closely related to Quine's well-known denial of the analytic-synthetic distinction and his equally famous indeterminacy of translation thesis. Although for Quine, semantic holism and confirmational holism are closely related, there is good reason to distinguish them, since the former is generally thought to be highly controversial while the latter is considered relatively uncontroversial.

Why this is important to the present debate is that Quine explicitly invokes the controversial semantic holism in support of the indispensability argument. Most commentators, however, are of the view that only confirmational holism is required to make the indispensability argument fly, and my presentation here follows that accepted wisdom. It should be kept in mind, however, that while the argument, thus construed, is Quinean in flavor it is not, strictly speaking, Quine's argument.

Apart from the indispensability argument, the other major argument for mathematical realism is that it is desirable to provide a uniform semantics for all discourse: mathematical and non-mathematical alike. Mathematical realism, of course, meets this challenge easily, since it explains the truth of mathematical statements in exactly the same way as in other domains. It is not so clear, however, how nominalism can provide a uniform semantics.

Ted Honderich (ed.), The Oxford Companion to Philosophy, p. 536-37

If there is a received view concerning ontology [in mathematics], it is realism, the view that the subject-matter of mathematics is a realm of objects that exist independently of the mind, conventions, and language of the mathematician. Most realists hold that mathematical objects--numbers, functions, points, sets, etc.--are abstract, eternal, and do not enter into causal relationships with material objects.
...
W. V. O. Quine and Hilary Putnam, among others, have proposed a hypothetico-deductive account of mathematical epistemology. The view begins with the observation that virtually all of science is formulated in mathematical terms. Moreover, as far as we know, this is the only way to formulate scientific theories. So mathematics is ‘confirmed’ to the extent that scientific theories are. The argument is that because mathematics is indispensable for science, and science is well-confirmed and (approximately) true, mathematics is well-confirmed and true as well. On this view, mathematical objects, like numbers and functions, are theoretical posits. They are the same kind of thing as electrons, and we know about them the same way we know about electrons--via their role in mature, well-confirmed scientific theories. Articulations of this view should (but usually do not) provide a careful analysis of the role of mathematics in science, rather than just noting the existence of this role. Such an account would shed some light on the ‘abstract’ nature of mathematical objects and the relationships between mathematical objects and scientific or ordinary material objects. Typically, an advocate of the Quine–Putnam indispensability argument denies the necessity and apriority of mathematics. Mathematics is only known through its role in science, which is clearly a contingent, a posteriori affair. Because mathematics plays a central role in virtually every science, its disconfirmation is unlikely, but still possible in principle.

Michael D. Resnick, "Quine and the Web of Belief:, pp. 429-32
(Sec. 4: "The Indispensability Argument for Mathematical Realism")
The Oxford Handbook of Philosophy of Mathematics and Logic (Ch. 12)
(ed. Stewart Shapiro)

Everyone grants that mathematics is very useful to the pursuit of science. It gives science the wherewithal for representing empirical findings through statistical and other numerical means and for explaining these findings using such concepts as those of acceleration, state vector, random mating, allelic frequency, expected utility, and welfare function. Moreover, mathematical laws permit scientists to deduce nonmathematical conclusions from assumptions, such as Newton's laws of motion, that are formulated in a mix of scientific and mathematical vocabulary. Eliminating mathematics would thus drastically alter the practice of working science.

But what if the theoretical purposes of mathematics could be accomplished using a more parsimonious ontology without any reduction in the overall simplicity and economy of the resulting scientific theory? Quine would heartily approve, but he would not ask scientists to stop using mathematics. He would merely claim that since mathematics could be excised from the canonical formulation of science, science (and thus we) should no longer acknowledge its truth or ontological commitments. "...[N]ot that the idioms thus renounced are supposed to be unneeded in the market place or in the laboratory....The doctrine is that all traits of reality worthy of the name can be set down in the idiom of this austere form if in any idiom." (Quine [Word and Object] 1960, 228) Although Quine attempted to eliminate mathematics from science and applauded efforts aimed at showing that the mathematical needs of science can be reduced, he came to believe that most classical mathematics is indispensable to science (Quine 1960, 270).

Since there is, so far, no way of eliminating mathematics from the "austere idiom" of the canonical formulation of science, we are bound to admit the existence of those mathematical objects that science posits. This argument, which is rooted in Quine's writings and was propounded explicitly by Hilary Putnam, has become known as the Indispensability Argument for Mathematical Objects.

[Note: Cf. Putnam: "So far I have been developing an argument for realism along roughly the following lines: quantification over mathematical entities is indispensable for science, both formal and physical; therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question. This type of argument stems, of course, from Quine, who has for years stressed both the indispensability of quantification over mathematical entities and the intellectual dishonesty of denying the existence of what one daily presupposes." (Philosophy of Logic, 1971, p. 57)]

We can formulate a more explicit version of an indispensability argument as follows: First, mathematics is an indispensable component of natural science. Second, thus, by holism, whatever evidence we have for science is just as much evidence for the mathematical objects and the mathematical principles it presupposes as it is for the rest of its theoretical apparatus. Third..., by naturalism, this mathematics is true, and the existence of mathematical objects is as well grounded as that of the other entities posited by science. I call this the Holism-Naturalism (H-N) Indispensability Argument It is clearly based upon the principles that Quine accepts....

Now lots of philosophical energy and talent--including some of Quine's--has been spent trying to undermine the first premise of this argument by showing that mathematics is dispensable from science....
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...One can set aside [at least some of these worries by moving to another version of the indispensability argument. For whatever attitude scientists take toward their own theories, they cannot consistently regard the mathematics they use as merely of instrumental value. Take Newton's account of the orbits of the planets as an example. He calculated the shape of the orbit of a single planet, subject to no other gravitational forces, traveling about a fixed star. He knew that no such planet exists, but he also believed that there are mathematical facts concerning its orbit. In deducing the shape of such orbits, he presumably took for granted the mathematical principles he used. For the soundness of his deduction depended upon their truth. Furthermore, in using his (mathematical) model to explain the orbits of actual planets, he presumably took its mathematics to be true. For he explained the orbits of planets in our solar system by saying that they approximate the behavior of an isolated system consisting of a single planet orbiting a single star. For this explanation to work, it must be true that the type of isolated system (Newtonian model) has the mathematical properties Newton attributed to it. This illustrates that even when applying mathematics to idealizations or theories they know are wrong, scientists use it in a way that commits them to its truth and ontology.

Reflecting on this leads one to the Pragmatic Indispensability Argument, which runs as follows:
1. In stating its laws and conducting its derivations, science assumes the existence of many mathematical objects and the truth of much mathematics.
2. These assumptions are indispensable to the pursuit of science; moreover, many of the important conclusion drawn from and within science could not be drawn without taking mathematical claims to be true.
3. So we are justified in drawing conclusions from and within science only if we are justified in taking the mathematics used in science to be true.

Notice that...this [version of the Indispensability Argument] does not presuppose that our best scientific theories are true or even that they are well supported. It applies wherever science presupposes the truth of some mathematics. Thus...it applies even to the mathematics contained in those refuted scientific theories that scientists still use and to the mathematics of idealized scientific models....We can extend this argument to infer that we should acknowledge the truth of mathematics on pragmatic grounds. For given that we are justified in doing science, we are justified in using (and thus assuming the truth of) the mathematics in science, because we know of no other way of obtaining the explanatory, predictive, and technological fruits of science.

Since much standard mathematics is used in science, the indispensability arguments support realism about many parts of mathematics. Yet, as Quine was aware,...indispensability arguments fail to cover the more theoretical and speculative branches of mathematics. Currently science neither needs nor employs this mathematics, and it does not even help in simplifying and systematizing the mathematics that science does apply. Thus it is not part of the Web of Belief, and not connected even indirectly to experience.

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See Putnam's little Philosophy of Logic (Harper Torchbooks, 1971) for a fuller argument for mathematical realism, and a refutation of some of the major attacks on it.