Yuri Gurevich shares an anecdote about beginnings which he refers to as “Research-Life Prehistory”
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Mathematics is the last refuge of platonism. In what sense, mathematical objects – from real numbers to Banach spaces – exist? Where are they to be found, detected by experimental means? Mathematical platonism comes very naturally to working mathematicians. When you dig into mathematical “reality”, you realize that it is objective. You work as an archeologist excavating stuff, rather than a gardener creating beautiful things. You have a strong feeling that mathematical world is out there for you to discover rather than to invent.
But mathematical platonism does not necessarily comes so naturally to uninitiated. I remember my own struggle with it. My middle school teacher was proving the congruence of two triangles. “Let’s take a third triangle,” she said. And I wondered where she would take it from. The life in Russia at the time was hard: shortages of meat, vegetables, even bread. Would there be enough triangles there? What if they ran out of triangles? So I raised my hand and asked her: “Where do we find that third triangle?” She stumbled, and the whole class looked at her. The students waited for her to reply even though most of them paid little attention to her up to that point. There was silence. Nobody moved. And then the teacher told me: “Shut up.”
Windows On Theory 
My first encounter with the question of the platonic existence of mathematical objects occurred even later — the question was asked to me by my math teacher in first-year college, right after I had given a small presentation related to knot theory and the Jones polynomial. I remember very well being caught completely off-guard; I had expected technical questions and I was suddenly asked about my opinion on the existence of mathematical objects!
I gave what I thought was the obvious answer: axioms are invented, objects and proofs are discovered (both pre-exist as long as they satisfy the axioms). It took me some time to realize there was much more to the question than it seemed: even though axioms can in principle be chosen arbitrarily, in practice this is not so true. Beyond consistency, a subtle requirement by itself, I think there is a strong and surprisingly shared sense of what are “correct” or “incorrect” axioms, just like there is often a “right” proof or definition.
Does the “correct” proof have a platonic existence?
Great post!
Gil Kalai tells us of a somewhat similar experience, from the teacher’s point of view, http://gilkalai.wordpress.com/2009/07/22/a-proof-by-induction-with-a-difficulty/