A discussion in the n-category cafe about Manin's 'emotional Platonism' made me wonder how such a perception of mathematics is distributed among mathematicians and how that influences attitudes towards specific concepts and themes. How is it with you? Is 'Platonism' something you can connect with, something (ir)relevant, or even contrary your attitude towards math for you?
closed as not a real question by Anton Geraschenko Nov 9 '09 at 20:41
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I don't know if it makes much sense at all to talk of such matters since all the words like "invention", "discovery", "exist", "create" are not something that is well-defined with sharp boundaries and the same clear meaning to everyone. Still, risking to bring the wrath of our respected administrator, who has already made it clear that this forum is ill-suited for such discssions, on my poor head, I'll try to answer.
I am quite strongly on the Platonic side myself, which, in layman terms, means that I believe that in the phrase "five trees" or "five dinosaurs" the object (or, if you want, relation but the distinction between "objects" and "relations" is also quite mirky) described by the word "five" exists in much stronger sense than the objects described by the words "tree" and "dinosaur" and that, while the language we use to describe mathematics and the order in which we look at things are human-dependent (and, thus, "created"), the mathematics itself exists out there independently of the mankind (and, therefore, "discovered").
The simplest example is the notion of a number. The words "one, two, three" are arbitrary and created by humans but the fact that sometimes you can give each person in the tribe one egg (or whatever they used to eat in the prehistoric times) and everybody will get his share and sometimes you cannot and that this result does not depend on whether you start distributing eggs starting from the chief or from the youngest and least distinguished member of the tribe is not. But this fact is just one (and rather trivial) manifestation of a relation called "number" in English. This manifestation certainly was observed (="discovered") and not created. This seems hard to argue with, but if you accept that the elementary arithmetic "exists" independently of the observer and (which is pretty much the same) its properties do not depend on who is watching its manifestations, you'll have to agree that the entire mathematics as we know it is out there. The question for me is rather whether mathematics is everything that is out there or there is also something else that escapes the nets of logic and counting.
Of course, this all is just my personal view on things and some of you may have radically different systems of beliefs. It all doesn't really matter. One can believe whatever he wants about mathematics as long as he sticks to the rules of the game when doing it and considers mathematical research a decent human activity not less (though, probably, not more either) important than, say, growing strange green things, fighting small invisible creatures, programming mischievious electrochemical machines, and designing crazy contraptions.
I must admit that I don't understand when mathematicians debate whether doing mathematics is a process of creation or discovery. I see it as a cycle of creation and discovery: I discover an interesting pattern within a logical structure which is already in use, explore whether the pattern is regular within that structure, or whether I can create a new, related structure where that pattern holds true.
There is a similar process within study of human languages; a conjecture whose name for a moment escapes me claims, roughly speaking, that a person can't have a clear concept for unless she has a word for it. This conjecture is false, but it does point to the process that a person goes through: to clarify a vague concept, she explores it with the language tools at her disposal, and this exploration changes the concept.
So, do we create concepts, or do they exist prior to the exploration?