TITLE: How do mathematicians think about the existence of numbers?
QUESTION [14 upvotes]: 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

REPLY [3 votes]: 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.