When calling themselves "Platonists" mathematicians usually mean that they feel they discover ideal facts that eternally exist in some way. My question is if this sentiment is consistent with Plato's particular form of idealism.
Already in Plato's time it came in conflict with existing mathematical practice. Plato mocked in Republic the appearence of "becoming" in the way geometers operated "the science itself is utterly opposed to the proofs as stated in it by its practitioners... They speak, I suppose, very laughably and perforce, for they mention squaring, applying and adding, and state all their claims as if they are engaged in action and fashioning all their proofs for the sake of action." According to Plutarch he also came out very strongly against the use of motion, mechanically generated curves in particular, by Archytas and Eudoxus, "for thus is destroyed and corrupted the good of geometry which recurs again to sensibles and neither ascends nor lays hold of eternal and incorporeal images". Motion was acceptable to "save the phenomena" only, not in pure mathematics as such.
With the notable exception of Euclid Greek geometers largely ignored Plato's proscriptions on motion, mechanical curves proliferated (Archimedean spiral, etc.). Plato's idea that the language of becoming was "perforce" was disputed by Menaechmus, the founder of conic sections theory, who in a debate with Plato's successor Speusippus implied that geometers meant what they said. Indeed, ideal realm is at odds with hierarchical organization of mathematics by constructing complex objects from simpler ones that clearly appears even in Euclid. What possible status would mathematical constructions have for eternally co-existent unchanging objects? Since then calculus "brought motion into mathematics", and hierarchical constructions are essential in shaping the structure even of pure mathematics, and part of its appeal, rather than "perforce" figures of speech.
QUESTION: So was Plato misapplying his philosophy, and in itself it does not lead to his conclusions about the mathematical practice? Or is Platonist self-identification a misnomer, and mathematicians are really sympathetic to some other kind of idealism? Were the motion and mathematical constructions discussed in the context of Platonism post-Renaissance? Can consistent Platonism be reconciled with modern mathematics at all?
EDIT: I'd like to clarify that mathematical construction is not meant in the narrow sense of constructivism, although those certainly qualify. But modern mathematics is also full of highly non-constructivist constructions, like Cantor's generation of ordinals and cardinals, or maximal ideals, algebraic closures, and everything else involving the axiom of choice. The whole architecture of pure mathematics is based on this, relations and functions are constructed from sets, groups, posets, etc. are sets with relations and functions, then there are spaces of functions on them, operators and functionals on those, and on and on.
Agreeing with Plato seems to diminish the import of these hierarchic constructions, if not dismiss them as chatter. After all, proofs are about reducing facts about complex structures to simpler pieces, not "ascending and laying hold" of their "eternal and incorporeal images" in their unmoving finality.