"It’s quite astonishing and I still don’t understand it, having been a mathematician all my life. How can things be there without actually being there? There’s no doubt that 2 is there or 3 or the square root of omega. They’re very real things. I still don’t know the sense in which mathematical objects exist, but they do. Of course, it’s hard to say in what sense a cat is there, too, but we know it is, very definitely. Cats have a stubborn reality but maybe numbers are stubborner still. You can’t push a cat in a direction it doesn’t want to go. You can’t do it with a number either. I’m only using the word number because you’ll have a vague idea in your head as to what I mean. The objects that a mathematician studies are more abstract than numbers but very real.
"I often think of cats. I think of trees. I think of dogs occasionally but I don’t think of them all that much because dogs are agreeable. They do what you want them to do to some extent. Some people believe that mathematics is what we think it is and it’s created by our thoughts. I don’t. I’m a Platonist at heart, although I know there are very great difficulties in that view." -- John Conway
Not to take anything away from our Veterans, but this is a math blog, and I'll use the opportunity of Memorial Day to once again remember Martin Gardner, whose death just over 3 years ago inspired me to start this endeavor (with no idea it would still be up-and-running 3 years later!!).
The above quote from John Conway, one of the most creative, productive mathematicians around, is taken from a book review Gardner wrote for a 2009 volume by Mariana Cook, covering 92 mathematicians, entitled "Mathematicians: An Outer View of the Inner World."
It's not a book I've personally seen, but read Gardner's review here:
Gardner (a vocal math Platonist) uses the review to go off on the topic of Platonist vs. non-Platonist viewpoints, writing at one point, "I suspect that almost every mathematician in the book is a Platonic realist, one who believes that mathematical theorems are forever true in all possible worlds and are independent of human culture," before continuing on to offer the above quote from Conway. If anything, the Platonism divide has only deepened since 2009, with brilliant adherents on both sides -- I can't help but think some of it is little more than muddy semantics, while also recognizing that there does exist a core of real (and perhaps non-resolvable) disagreement.
On a side note, I see that Martin Gardner's forthcoming autobiography "Undiluted Hocus-Pocus: The Autobiography of Martin Gardner" is already listed on Amazon, and I suspect will include more of his Platonist evangelism (available in September):
….I almost wish I didn't know it was on the way... because the 4-month wait will now be excruciating!! :-/
Lastly, if in the mood for some more memories of Martin see here:
also, this great 2005 AMS interview with Martin: