Even with the advances in analysis, there are still notions such as vector spaces which we intuitively visualize in terms of three dimensions in order to grasps the implications of our analysis. While one may argue that the analysis of mathematical objects is based on pure abstraction, there is still the part of the intuition of the concept which can (not a necessary condition but might) influence the abstract analysis of mathematical objects.
While the intuition does not necessarily imply that there are pure lines some might argue based on Platonism that it is our "recollection" of those pure objects. How do other movements counter such arguments?
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.
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