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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.

Are there any books/articles about this concept?

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
There was a interesting video about this on Numberphile (youtu.be/1EGDCh75SpQ) about this. It is worth a watch ;) –  CBenni Dec 11 '12 at 19:16
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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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