Why do numbers exist

It comes indirectly from my work on languages in the Amazon. In the book, you talk at length about how our fascination with our hands—and five fingers on each—probably helped us invent numbers and from there we could use numbers to make other discoveries.

So what came first—the numbers or the math? There are obviously patterns in nature. There are lots of patterns in nature, like pi, that are actually there. These things are there regardless of whether or not we can consistently discriminate them.

When we have numbers we can consistently discriminate them, and that allows us to find fascinating and useful patterns of nature that we would never be able to pick up on otherwise, without precision. Numbers are this really simple invention. These words that reify concepts are a cognitive tool. Without them we seem to struggle differentiating seven from eight consistently; with them we can send someone to the moon.

A lot of people think because math is so elaborate, and there are numbers that exist, they think these things are something you come to recognize. Another interesting parallel is the connection between numbers and agriculture and trade. What came first there? I think the most likely scenario is one of coevolution.

You develop numbers that allow you to trade in more precise ways. As that facilitates things like trade and agriculture, that puts pressure to invent more numbers. In turn those refined number systems are going to enable new kinds of trade and more precise maps, so it all feeds back on each other. This counterfactual independence as we may call it is accepted by most analytic philosophers.

To see why, consider the role that mathematics plays in our reasoning. Were we to build a bridge across this canyon, say, how strong would it have to be to withstand the powerful gusts of wind? Sadly, the previous bridge collapsed. Would it have done so had the steel girders been twice as thick? This form of reasoning about counterfactual scenarios is indispensable both to our everyday deliberations and to science.

The permissibility of such reasoning has an important consequence. Since the truths of pure mathematics can freely be appealed to throughout our counterfactual reasoning, it follows that these truths are counterfactually independent of us humans, and all other intelligent life for that matter. That is, had been there been no intelligent life, these truths would still have remained the same.

Pure mathematics is in this respect very different from ordinary empirical truths. Had intelligent life never existed, this article would not have been written. More interestingly, pure mathematics also contrasts with various social conventions and constructions, with which it is sometimes compared Cole , Feferman , Hersh Had intelligent life never existed, there would have been no laws, contracts, or marriages—yet the mathematical truths would have remained the same.

Thus, if Independence is understood merely as counterfactual independence, then anyone who accepts object realism should also accept platonism. It is doubtful that this understanding of Independence is sufficient, however. For Independence is meant to substantiate an analogy between mathematical objects and ordinary physical objects.

Let us now consider some views that reject this stronger understanding of Independence in terms of the mentioned analogy. These views are thus lightweight forms of object realism, which stop short of full-blown platonism. This view is characterized by a plenitude principle to the effect that any mathematical objects that could exist actually do exist.

For instance, since the Continuum Hypothesis is independent of the standard axiomatization of set theory, there is a universe of sets in which the hypothesis is true and another in which it is false. And neither universe is metaphysically privileged.

By contrast, traditional platonism asserts that there is a unique universe of sets in which the Continuum Hypothesis is either determinately true or determinately false. One alleged benefit of this plenitudinous view is in the epistemology of mathematics. If every consistent mathematical theory is true of some universe of mathematical objects, then mathematical knowledge will, in some sense, be easy to obtain: provided that our mathematical theories are consistent, they are guaranteed to be true of some universe of mathematical objects.

Colyvan and Zalta criticize it for undermining the possibility of reference to mathematical objects, and Restall , for lacking a precise and coherent formulation of the plenitude principle on which the view is based. Martin proposes that different universes of sets be amalgamated to yield a single maximal universe, which will be privileged by fitting our conception of set better than any other universe of sets.

In object theory, moreover, two abstract objects are identical just in case they encode precisely the same properties. Assume that object realism is true. For convenience, assume also Classical Semantics. These assumptions ensure that the singular terms and quantifiers of mathematical language refer to and range over abstract objects.

Given these assumptions, should one also be a mathematical platonist? In other words, do the objects that mathematical sentences refer to and quantify over satisfy Independence or some similar condition? It will be useful to restate our assumptions in more neutral terms.

We can do this by invoking the notion of a semantic value , which plays an important role in semantics and the philosophy of language.

In these fields it is widely assumed that each expression makes some definite contribution to the truth-value of sentences in which the expression occurs. This contribution is known as the semantic value of the expression.

It is widely assumed that at least in extensional contexts the semantic value of a singular term is just its referent. Our assumptions can now be stated neutrally as the claim that mathematical singular terms have abstract semantic values and that its quantifiers range over the kinds of item that serve as semantic values. What is the philosophical significance of this claim? In particular, does it support some version of Independence? The answer will depend on what is required for a mathematical singular term to have a semantic value.

It suffices for the term t to make some definite contribution to the truth-values of sentences in which it occurs. The whole purpose of the notion of a semantic value was to represent such contributions. It therefore suffices for a singular term to possess a semantic value that it makes some such suitable contribution. This may even open the way for a form of non-eliminative reductionism about mathematical objects Dummett a, Linnebo Although it is perfectly true that the mathematical singular term t has an abstract object as its semantic value, this truth may obtain in virtue of more basic facts which do not mention or involve the relevant abstract object.

Compare for instance the relation of ownership that obtains between a person and her bank account. Although it is perfectly true that the person owns the bank account, this truth may obtain in virtue of more basic sociological or psychological facts which do not mention or involve the bank account. If some lightweight account of semantic values is defensible, we can accept the assumptions of object realism and Classical Semantics without committing ourselves to any traditional or robust form of platonism.

We conclude by describing two further examples of lightweight forms of object realism that reject the platonistic analogy between mathematical objects and ordinary physical objects.

First, perhaps mathematical objects exist only in a potential manner, which contrasts with the actual mode of existence of ordinary physical objects. According to Aristotle, the natural numbers are potentially infinite in the sense that, however large a number we have produced by instantiating it in the physical world , it is possible to produce an even larger number.

But Aristotle denies that the natural numbers are actually infinite: this would require the physical world to be infinite, which he argues is impossible. Following Cantor, most mathematicians and philosophers now defend the actual infinity of the natural numbers.

This is made possible in part by denying the Aristotelian requirement that every number needs to be instantiated in the physical world. When this is denied, the actual infinity of the natural numbers no longer entails the actual infinity of the physical world.

No matter how many sets have been formed, it is possible to form even more. If true, this would mean that sets have a potential form of existence which distinguishes them sharply from ordinary physical objects. Second, perhaps mathematical objects are ontologically dependent or derivative in a way that distinguishes them from independently existing physical objects Rosen , Donaldson For example, on the Aristotelian view just mentioned, a natural number depends for its existence on some instantiation or other in the physical world.

There are other versions of the view as well. For example, Kit Fine and others argue that a set is ontologically dependent on its elements. This view is also closely related to the set-theoretic potentialism mentioned above. I gratefully acknowledge their support. What is Mathematical Platonism? The Fregean Argument for Existence 2. Objections to Mathematical Platonism 3. Between object realism and mathematical platonism 4. Mathematical platonism can be defined as the conjunction of the following three theses: Existence.

There are mathematical objects. Mathematical objects are abstract. Morton and S. Stich, eds. Benacerraf, Paul and Putnam, Hilary eds. Second edition. Burgess, John P. Cole, Julian C. Pataut ed. Colyvan, Mark and Zalta, Edward N. Feferman et al, ed.

III, — Benacerraf and H. Putnam, eds. Reprinted in Hale and Wright Hersh, Reuben, , What is Mathematics, Really?

Linsky, Bernard and Zalta, Edward N. Martin, Donald A. Butts and J. Hintikka eds. Quine, W. Rees, D. Polkinghorne ed. Academic Tools How to cite this entry. Enhanced bibliography for this entry at PhilPapers , with links to its database. Other Internet Resources [Please contact the author with suggestions. Related Entries abstract objects mathematics, philosophy of: indispensability arguments in the mathematics, philosophy of: naturalism physicalism Plato: middle period metaphysics and epistemology.

Open access to the SEP is made possible by a world-wide funding initiative. Mirror Sites View this site from another server:. Mathematicians are reliable, in the sense that for almost every mathematical sentence S , if mathematicians accept S , then S is true. Every object you could possibly interact with is three-dimensional — no matter how thin a piece of, say, plastic you create, it always has a height and a thickness, giving it three dimensions.

Nothing, therefore, in the concrete world, is a real geometric line segment. We have things that approximate line segments — very straight, very thin objects.

But none of those things will ever be perfectly straight and with zero thickness. One of the most damning aspects of platonism is its failure to come to terms with how we learn things about abstract objects.

The general picture of how we acquire knowledge goes something like this: We perceive an object in the physical world, via physical means e. It is non-physical, and so, e. So our usual causal theory of knowledge acquisition fails for things like numbers. Well, then, how is it that we come across any knowledge of abstract objects, if they indeed exist?

As he wrote:. But, despite their remoteness from sense experience, we do have a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. But this is clearly an unacceptable answer to the problem of knowledge of abstract objects. How exactly do the axioms of set theory force themselves upon us? How does some feature of a non-spatiotemporal object force itself upon our spatiotemporal brains?

The only way would be somewhat magical, and you could look to Descartes to see the folly of such a plan. Descartes posited that minds are distinct substances from brains, and indeed were non-spatiotemporally located. Of course, this leaves the problem of how the mind somehow slips into the brain and affects it.

But this is no answer; it merely delays the answer for a moment. How does the non-spatiotemporal mind creep in through the pineal gland, and then into the brain? Descartes had no answer for this, of course, because the whole thing would be terribly mysterious, explaining how the non-physical interacts with the physical. Hardcore nominalists are often quite scientifically-minded, scientifically-motivated philosophers.

And it is this love of science that gets them into trouble with denying the existence of numbers. The argument runs, in broad strokes, like this:. But, as a nominalist you claim that there are no abstract objects! And you are caught in an intractable dilemma. Quantification over mathematical entities is indispensable for science…; 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. Science also uses frictionless planes, for example, and yet no scientist feels committed to the existence of those. Perhaps there is a way out of our commitment to numbers in the same way.

Or perhaps, one might argue, frictionless planes actually do exist as platonic, abstract objects. First, you could argue that numbers exist, and are actually physical objects. Penelope Maddy argues something close to this in her early work, Realism in Mathematics. She actually is here arguing for a version of naturalized platonism — the idea being that what is usually thought of as abstract objects are actually somehow existent in the physical world. But, platonist labels aside, the gain for nominalism on this take would be obvious: numbers, if they are physical objects, would be just another part of the down-to-earth nominalist physical world, like cats, trees, and quarks.

This brave strategy, however, ultimately fails. It would take us into some metaphysical thickets to explain why, so I have relegated this to a paragraph at the very end of this post. Hartry Field famously tried this strategy, claiming that science in fact only seems to rely on mathematics. In order to prove this Field attempted to nominalize a chunk of physics, by removing all reference to numbers within it.

This complicated, counterintuitive project has met with equal parts awe and criticism. The consensus is that his project is untenable in the long term.

And further you would claim that they possess a sort of existence that is abstract — different from the sort of existence that stones, trees, and quarks enjoy.