What Would Plato Think of a Metric Space?

On one side, Platonism concerns the existence of an abstract world of mathematical objects which is independent of our concrete world. On the other side, some think that mathematics is grounded in the human world and is influenced by culture, time, place, and some fortuitous circumstances. Are those two views contradictory in the sense that accepting one necessarily forces rejecting the other? Or do they complement each other in the sense that one agrees and possibly clarifies the other?

A possible argument for the antagonistic nature of those two views is that, if there really is such an abstract world out of touch with the concrete world of humans, those objects should not be subject of any human influence; they should transcend cultures, time, and places. With such a view, mathematics seems to become this universal activity, where to talk of a European mathematics or Asian mathematics becomes meaningless, but mathematics tout court. Nevertheless, could this approach be seen as rather biased in that Platonism can be argued to be a product of Western thought thus could not and should not present itself as an encompassing interpretation for all other ways of thinking?

An argument for the rejection of Platonism and for an adoption of a mathematics more engrained with the human world might be that to talk of such an abstract world is epistemologically irrelevant in that we only know that humans have been able to devise new mathematical theories and that to assert that such mathematics comes from some abstract world is pure speculation. Another possible objection might be that, if one knows that all questions about mathematics already have an available answer and that all one needs to do is to look for such an answer, should one then see mathematical research not as this quest with a possibly unknown outcome but as a pure game in which one pretends to not even know if an answer exists while in reality knows it is just a matter of looking for something that already exists but is hidden from view? Furthermore, it seems to accept the Platonist view inevitably leads one to act on faith, but I wonder if this attitude contradicts the perpetual skepticism that seems to be essential to practice mathematics (I have the impression that to learn mathematics, at least the way it currently is done, one needs to be obsessed with the question “why?”.).

Could the rejection of the Platonist world be understood as an opposition to all mathematics inspired by Platonism? I suspect that the contention usually centers around the existence of such a world; past this point, it seems two mathematicians, upon working on a problem, might proceed in a similar manner although they may have different interpretations of their answers: with one thinking that the answer is just discovered and the other that the answer is created. A more reconciliatory approach might be to consider these interpretations as irrelevant, but I suspect that some mathematicians might want to consider their answers as their own creations instead of mere discoveries. As a result, such a conflict between those two interpretations may make any agreement between those views difficult.

So, do you consider yourself as a Platonist or non-Platonist? Or do you consider such distinction to be irrelevant?

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