Timeline for What is the ontological difference between platonism and non platonism?
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10 events
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| Nov 28 at 8:42 | comment | added | Double Knot | Indeed the persistence of mental models is like the right adjoint forgetul functor from the category of free group to the category of sets in modern category theory, persisting simplified glued-together landscape-fixing objects such as colimits, while the mentally constructed variation of model images is like its left adjoint free functor, freely forming idiosyncratic models up to various nonstandard models via either ultrpowers or forcings as their limits... | |
| Nov 27 at 11:02 | comment | added | Mikhail Katz | I'm not sure what you're saying but the issue concerning mental models of a given mathematical thing is an interesting one. Does the persistence of mental models furnish evidence in favor of Platonism? Does variation of mental models furnish evidence in favor of Formalism? As far as automated theorem provers (Coq) are concerned, presumably the operations they are capable of performing are all finitist. With regard to finitist mathematics, Robinson would agree with a Platonist (and disagree with a nominalist or logical positivist) that mathematicians are capable of grasping it. @DoubleKnot | |
| Nov 27 at 8:35 | comment | added | Double Knot | I don't necessarily disagree. Like for formal syntactical axioms both competent platonists and formalists can share exactly and agree like an automated theorem prover Coq machine, so this implies the really common landscape-fixing objectivity lies in something ontic which precedes our conceptual models built from Baudrillard's simulucra and simulation... | |
| Nov 27 at 8:23 | comment | added | Mikhail Katz | @DoubleKnot, at bottom such debates revolve around the philosophical belief in the existence of a so-called "standard model" (a.k.a. intended interpretation). I would agree with you that modulo such a belief, one would have difficulty denying that mathematicians likely have similar mental models of a given mathematical entity. However, as I pointed out earlier, this is a philosophical assumption rather than a mathematical fact. In fact, it may be interpreted as the basis of the debate between Platonists and Formalists. | |
| Nov 27 at 8:18 | comment | added | Double Knot | If they share nonstandard models exactly then there won't be IST vs SPOT, there'll be a uni-math foundation for all competant mathematicians, at least in a same era. Everyone has their own subtle preference or emphasis regarding the shape or constitution of their intended model(s)... | |
| Nov 27 at 8:17 | comment | added | Mikhail Katz | @DoubleKnot, I am not sure I can follow either your comment or the linked post. At any rate, what I tried to point out is that your assumption (that different mathematicians necessarily share exactly the same mental concept of a given mathematical entity) can be challenged. | |
| Nov 27 at 7:55 | comment | added | Mikhail Katz | Meanwhile, R_Hrbacek (or R_Nelson) includes infinitesimals. @DoubleKnot | |
| Nov 27 at 7:55 | comment | added | Mikhail Katz | @DoubleKnot, mathematicians in fact do not "exactly the same mental concept" as you put it. For example, Leibniz had very different intuitions about infinity than we do in the post-Cantor-Weierstrass era. Leibniz rejected infinite wholes as contradictory; modern mathematicians think they are indispensable. Another example would be the mental concept of an entity like R. Mathematicians traned in the Weierstrassian paradigm have a very different mental concept of R than do those working in axiomatic nonstandard analysis. For example, R_Weierstrass is believed to be infinitesimal-free. ... | |
| Nov 25 at 20:19 | comment | added | Double Knot | Over time Bernays came to recognize the limitations of a purely formalist philosophy especially in light of Godel's incompleteness theorems undermining Hilbert's formalism to become sui generis as the nature of mathy truth could be independent from formal systems. Therefore his objectivity sui generis reflects this middle ground in terms of structure not necessarily Platonic realm. Or viewed alternatively, if without objectivity, how different mathematicians can share the exactly same mental concept of certain mathematical propositions in their inherently idiosyncratic subjective way?... | |
| Nov 25 at 12:17 | history | answered | Mikhail Katz | CC BY-SA 4.0 |