# 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 the common property 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, Seinfield}, {Elaine, George}, {a goat, a truck}, ... }

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

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The word "number" itself has no intrinsic meaning. Instead, I would say that given an algebraic structure $X$ with underlying set $U$, it is sometimes useful to refer to the elements of $U$ as "numbers", mainly to draw 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, model theory and universal algebra.

So you could say that mathematicians think about the existence of number systems using the methods of set theory, model theory and universal algebra.

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