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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.

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Mathematical platonism -- or platonism more generally (with the lower case 'p') -- holds the following three theses about mathematical objects: they (i) exist, (ii) are abstract, and (iii) are independent of intelligent agents. This is typically all that mathematicians mean when they say they are platonists, and the differ in their further commitments. Plato would hold these three premises as well (in that way Platonism is compatible with platonism) but his metaphysics is traditionally taken to extend further, committing him to stances that most mathematicians do not hold. Thus, I would say that mathematical platonism is not compatible with Platonism.

For a first example, let us consider ethics and the existence of moral truths. Obviously, for Plato, forms like the Good and Justice have the same sort of existence as Triangle. For mathematicians, there is a lot of variation. Some, like Godel, will commit themselves to a thorough metaphysical platonism that extends to ethics, hence his ontological argument for the existance of God, for instance (although note an important difference, Plato would not go to the same extent as Godel to see if some Form exists or not). Others, like Russell, do not think there was an objective Good; to see this read over his very pithy "Is there an Absolute Good?". Note that with Russell it is a little difficult to see if he held mathematical platonism and his moral positions at the same time, given that he went through at least three distinct stages and mathematics is mostly early Russell while ethics is middle and late Russell. However, from my personal experience, many mathematicians would not be platonists with respect to ethics.

For a second example, let us consider epistemology. For Plato, we simply 'remembered' the Forms, we do not discover them. A lot of mathematicians might object to this perspective. Although again, there is some heterogeneity here, for some we simply 'see with our intuition' (not that different from Plato's remembering) a mathematical truth and then construct a proof to guide others to what we saw, much like Socrates might guide the slave boy.

For a third example, let us consider particulars. For Plato -- most famously in the allegory of the cave -- particulars are of the same 'type' as the Forms but are degraded by being corporeal. The Forms are, in some way, causes of the particulars. I don't think that many mathematicians would subscribe to this view. I think that many of them would implicitly subscribe to a form of dualism similar to Frege's -- another notable mathematical platonist -- and think of separate worlds of the abstract and of the empirical with the effectiveness of mathematics in the sciences being either a selection-bias or unreasonable.

For a fourth example, let us consider structure. As you noted, the mathematicians' forms are not solitude and the universe mathematicians imagine is structured in some ways and they use that structure to navigate that universe. It is not clear that Plato's Forms have this structure, under most readings the various forms are rather isolated from each other.

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    A very thoughtful answer, thank you. Is there a philosophical system that better approximates mathematicians' intuitions, especially with respect to epistemology and structure? – Conifold Jan 11 '15 at 22:35
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    @Conifold I am not sure if mathematicians' views on this are sufficiently homogeneous (although that is a sociological question on which I'd be interested to see a study) to fit nicely into a big fully-fleshed out system. I think that is why mathematical platonism is so loose and remains agnostic on so many issues. – Artem Kaznatcheev Jan 11 '15 at 22:47
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There is a saying that mathematicians are formalists on workdays and Platoists on weekends. This suggests that most "real" work needs to be formal, but many mathematicians believe the underlying objects to be "real". This is not contradictory, in my view. It is simply a useful approximation.

Formalism requires no reality and gives us all the machinery to do our work. But that does not mean that we don't think that the formalism points to some possibility that certain mathematical structures have to actually exist. I think there is strong empirical and heuristic evidence that natural numbers exist, for example. And if this is so, then many of the constructions from natural numbers (integers, rationals, reals) should also exist.

So we have rules which define infinity. But they seem consistent with finite reality.

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    "So we have rules which define infinity. But they seem consistent with finite reality." Well .... my understanding is that, in physics at least, infinities tend to be an indication that Something Has Gone Wrong (i.e. in our understanding, and mathematical modelling of, the observable material universe). – Alexis Jan 8 '15 at 9:29
  • @Alexis Amen! Go Construcitvists (en.wikipedia.org/wiki/Constructivism_%28mathematics%29)! – R. Barzell Jan 8 '15 at 14:52
  • @Alexi: in physics, I'd agree, but not all math is physics, or programming, or whatever. And I think constructivism is interesting and insightful, but you can do it inside of "classical" math. I don't have a problem with people not wanting to do nonconstructive math, but I'm generally not in favor of limiting inquiry (i.e. wanting to actively exclude nonconstructive math). – user2170 Jan 8 '15 at 19:53
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    I think this misses the point of the question. As far as I can tell OP isn't asking the much simpler question of "are a lot of mathematicians idealists" nor is he asking the imprecise question of "is this vague form of idealism an appropriate belief to have". He is asking the precise question of "is mathematical platonism compatible with Platonism" where the first term refers to the 'mathematical objects are non-physical but real' stance of some mathematicians, and the second refers to the actual philosophy of Plato. – Artem Kaznatcheev Jan 8 '15 at 23:42
  • @Artem Kaznatcheev: You know, you're correct. I would strongly advise the OP to change the title of the question, because I'm not exactly certain what to change it to. And then the answer is probably "no" in the sense of the OP. – user2170 Jan 8 '15 at 23:58

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