I see the interesting question and the reference to the visualization of lines to be independent. You don't need to be a platonist to acknowledge the powerful effect of drawn lines or indeed linear objects on our imagination - the link between the sign (drawn line) and the signifier (in Saussaurian terms) the visual concept image - does not need to be mediated by a platonic ideal line. Indeed Occam's razor suggests this third item (Peirce's semiotics notwithstanding) is unnecessary, and indeed otiose/superfluous. In my view platonism results from not acknowledging the difference between personal knowledge and belief and social, cultural knowledge. The latter is very solid and convincing without having to imagine a quasi spiritual extra-material realm where platonic objects exist. In my view Plato was a heroic victim . of nominalization - to a name there must correspond a real object somewhere. But then I am an unashamed social constructivist!.

Paul,

Thank you for your answer. I agree that platonism is not necessary, yet I feel that the concept is so ingrained in the field when talking to fellow mathematicians, physicist, chemist and computer scientists.

Thank you Arturo for your welcoming answer. I agree that the question of platonism is interesting - how could such an absurd and implausible philosophy be so effective, convincing and useful? The answer for me is that cultural objects partake of objectivity - they are independent of individuals and represent the convergence of many lines of thought so that they seem solid, real, eternal and superhuman - and seem to exist independently of us.

Not everything that exists exists independently. A fairly obvious example: Beethoven's fifth symphony or, for that matter, any cultural object. Objects invariably present themselves to consciousness immersed in a "cloud" of meaning, and meaning invariably irradiates from the perceiving consciousness. This is true even for objects of ordinary sense perception. Mathematical objects are no exception. Platonism is simply a philosophical naïveté.

While Platonism in the narrow sense (that getting ideas is merely recollecting them) is, I believe, quite uncommon, Platonism in a broader sense (that mathematical ideas exist in some way independent of human mind and are discovered rather than invented) is shared by quite a number of prominent mathematicians, e.g. Yuri Manin (see e.g. this interview: http://www.ams.org/notices/200910/rtx091001268p.pdf ) and Vladimir Arnold (see e.g. here: http://www.ams.org/notices/201203/rtx120300378p.pdf ).