# How do mathematicians think about the existence of numbers?

Question: How do mathematicians think about the existence of numbers? And how did Newton, Euler, and other famous mathematicians thought about this concept?

I know that existence of numbers is a big ongoing debate in the philosophy of mathematics. I've searched online about this and found a lot of information (e.g. Aristotelianism, platonism, etc) , but nothing about the famous mathematicians.

Thank you

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Relatively few mathematicians have given detailed expositions of their views. The problem belongs to a branch of knowledge that is equally serious, but different. – André Nicolas Dec 10 '12 at 23:25
I heard that Gödel believed existence of real numbers. I would like someone to confirm this. – Makoto Kato Dec 11 '12 at 0:17
What do you mean by "the existence of numbers"? (I am not sure what you mean either by "existence" or by "numbers.") – Qiaochu Yuan Dec 11 '12 at 0:57
It has been said (I forget the source) that in their work, all mathematicians are naive Platonists, but in discussing the matter, many take a Formalists position. I am not comfortable with the use of the universal quantifier, particularly since as far as I know the assertion has not been experimentally tested. – André Nicolas Dec 11 '12 at 1:45

The famous British mathematical physicist Roger Penrose wrote an entire book on this subject: The Road to Reality: A Complete Guide to the Laws of the Universe (Knopf, 2005). In fact you can get a very good idea of his version of the Platonic theory just from Chapter 1, pages 7-24. He sees a tripartite world, divided into physical, mental, and mathematical domains. It's an interesting approach, and perhaps deserves special attention due to his prominence within both mathematics and physics. The book itself is just over a thousand pages long, and requires some mathematical maturity to comprehend. If you have what it takes, then it is well worth the effort.

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Number is a property of collections.

Imagine a very long train passes before you, with each car containing two objects and the contents of each car open to plain view: [two goats]--[two light bulbs]--[two shoes]--[one dolphin, one boat]--[one rock, one picture]--[one book, one dish]--, so on so forth. Eventually, it will strike you that the contents of these cars are all couples. That is to say, two is one of the common properties of the contents of all these cars. This is the psychological foundation why humans can sense numbers. It is the same as why humans can understand words like red, yellow, blue. No one ever saw colour independent of other properties such as shape, area, etc. When you see a red apple, red towel, red roof and a red shoe, you will notice that red is what these things have is common, although red has never been seen alone.

The longer the train, the fewer properties the contents of those cars have in common. As the train grows longer, eventually the contents of those cars will have only their number in common. Thus, one is what ALL singles have in common; two is what ALL couples have in common; three is what ALL triples have in common; so on so forth.

Technically, a number is a class whose members are also classes that are similar to each other but not with any classes outside of the parent class.* By "similar" we mean one-one relation. Notice that we can't say "all classes of the same size," because size is a number and number is what we are trying to define at this point. This definition of number is called ostensive definition, as opposed to dictionary definition.

For example: two is the class of all couples: { {foo, bar}, {a, b}, {c, d}, {Kramer, Seinfeld}, {Elaine, George}, {a goat, a truck}, ... }

For precise definition, see Introduction to Mathematical Philosophy, "Definition of Number", by Bertrand Russell.

*This limitation only applies to one particular type. Of course, a member class can always have similar classes from a different type, but it is meaningless to group these similar classes of different type within one parent class.

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I stalked you because of your other question and now I found a minor mistake in this answer: It’s Seinfeld, not Seinfield. ; – ) – k.stm Mar 27 '14 at 21:56
Corrected. Thanks, @k.stm – George Chen Mar 28 '14 at 2:16
"A number is a class" could be foundationally difficult to justify since this class appears to be proper. So the concept of a set of numbers would not be tenable. – Rachmaninoff Mar 30 '14 at 6:47
This definition pre-dates Von Neumann–Bernays–Gödel set theory. Class is used to sand for collections without assuming there are such things as classes. – George Chen Mar 30 '14 at 7:28

The case of the work of Godfrey Harold "G. H." Hardy on real analysis is an interesting case that shed light on the issue. Hardy wrote an analysis text around the turn of the century where he championed the case of the construction of the real numbers (via Cauchy sequences or Dedekidn cuts) and argued that these should be the basis of analysis. What is interesting is to compare the tone of the first edition of his book with the tone of the second edition. In the second edition that came out several decades later, Hardy seems a bit embarrassed about the "propaganda" tone of the first edition, and is more matter-of-fact about the real field. What happened in the meantime is that a transformation took place in the mathematical community and the real numbers entered the pantheon of mathematical concepts with impeccable ontology; i.e. in the meantime mathematicians embraced the (then-new) intuition that the so-called "real numbers" are just that, oh so real. What this arguably illustrates is that the perceived reality of mathematical objects is a function of time. The same was arguably the case for the rationals, as well; once upon the time only the natural numbers were thought of as "really existing". And certainly for the negative numbers.

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Interesting information. – goblin Jan 10 '14 at 18:32

The word "number" itself has no accepted meaning. However, given an algebraic structure $X$ with underlying set $U$, it is sometimes useful to call the elements of $U$ "numbers", to create an analogy with the algebraic structures $\mathbb{N}, \mathbb{Z}, \mathbb{Q}$ etc.

Therefore, rather than thinking about the existence of "numbers", mathematicians tend to think about the existence of algebraic structures. Furthermore, to answer the question: "Which algebraic structures exist?" we mainly use ideas from set theory and model theory, and occasionally, type theory.

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Arguably numbers are more basic than sets. Mathematicians and logicians studying Peano Arithmetic, for example, aren't necessarily concerned with set-theoretic structures. Besides, reducing "numbers" to "sets" merely pushes off the OP's question another level. His question can equally well be formulated for sets. – user72694 Mar 30 '14 at 17:53
@user72694 I'm not advocating reducing numbers to sets; rather, I'm advocating thinking about the existence of number systems, rather than the existence of numbers. A major tool for thinking about whether or not a number system exists satisfying some given specifications is set theory. – goblin Mar 30 '14 at 17:56
OK, but arguably number systems are more basic than sets. Mathematicians and logicians studying Peano Arithmetic, for example, aren't necessarily concerned with set-theoretic structures. Besides, reducing "number systems" to "sets" merely pushes off the OP's question another level. His question can equally well be formulated for sets. – user72694 Mar 31 '14 at 15:41
@user72694, I disagree. Collections of things are utterly fundamental; if we're thinking about the natural numbers, we're thinking about the collection of all natural numbers, or at the very least some initial portion. You're partially correct, though, in the sense that: its true that when set-theorists say "set," they don't just mean "collection," they mean "idealized collection having relative complements, a powerset, etc." which isn't realistic if by "collection" you mean something that can be computably realized. So bringing "set theory" into it (as opposed to "collection theory")... – goblin Mar 31 '14 at 15:49
... may have been slightly intellectually dishonest. However, the idea that "numbers are more fundamental than collections" is simply bonkers. Numbers form a collection, but collections do not form a number. More to the point: without collections, you cannot think. – goblin Mar 31 '14 at 15:50