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Descartes considered that he needed Euclid's parallel postulate: Descartes identified space and the extension of matter, so geometry was, for him, about real physical space. But *geometric space, for Descartes, had to be Euclidean. This is because the theory of parallel lines is crucial for Descartes' analytic geometry - not for Cartesian ...


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SEP itself refers to Platonism and Mathematical Intuition in Kurt Gödel's Thought by Parsons and On the Philosophical Development of Kurt Gödel by van Atten and Kennedy as the sources for this interpretation. Further discussion can be found in van Atten's book Essays on Gödel’s Reception of Leibniz, Husserl, and Brouwer and Conversations with Gödel chapter ...


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No it is not stupid. So far it goes, it is right about Kant. As to Platonism in philosophy of mathematics, people give widely different definitions, often explicit that they do not mean to describe Plato's own view. So anything you say about that is likely to be right according to someone's understanding of the term.


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In addition to @AdamSharpe's analysis of "break it down into [six] cases", you can also analyze the situation your intuitive way, i.e., first choose a box, and then choose a coin from that box. Start with all three of your boxes: let's call them GG, SS and GS, with the obvious meanings. Then, first of all, if you happen to choose the SS box, then your ...


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In his book Inconsistent mathematics, Chris Mortensen introduces complement-topoi and closed set sheaves - which I think do what you want. Unfortunately, there's some sloppiness in his presentation. Here's how I would choose to describe the situation: We define "complement-topoi" and "closed set sheaves" and show (i) how complement-topoi can be built from "...


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I recommend the paper: B. Buldt, The Scope of Godel’s First Incompleteness Theorem, Log. Univers. 8 (2014), 499–552 , especially pages 530 - 531.


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