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1. • CommentRowNumber1.
   • CommentAuthorColin Tan
   • CommentTimeJan 21st 2014
   • (edited Jan 21st 2014)
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   Author: Colin Tan
   Format: TextIn contrasting synthetic versus analytic differential geometry, is the analytic-synthetic distinction as per that of Kant? From reading Kock's book "Synthetic differential geometry" and seeing how he introduces axiom after another and seeing the care he takes to inform the reader which axiom he is using in which section, it would seem to me that synthetic differential geometry is a syntactic way of doing geometry. In a later part his book, Kock them introduces models of this geometry. As such, analytic differential geometry appears to be our original semantics of his theory. The theory of synthetic differential geometry appears to be a Rusellian syntactical crystallization from philosophers of real mathematics, made after careful observation of the manner in which differential geometers create arguments about their original model.

   In contrasting synthetic versus analytic differential geometry, is the analytic-synthetic distinction as per that of Kant?
   
   From reading Kock's book "Synthetic differential geometry" and seeing how he introduces axiom after another and seeing the care he takes to inform the reader which axiom he is using in which section, it would seem to me that synthetic differential geometry is a syntactic way of doing geometry. In a later part his book, Kock them introduces models of this geometry. As such, analytic differential geometry appears to be our original semantics of his theory.
   
   The theory of synthetic differential geometry appears to be a Rusellian syntactical crystallization from philosophers of real mathematics, made after careful observation of the manner in which differential geometers create arguments about their original model.
2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 21st 2014
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   Author: Urs
   Format: MarkdownItexThat sounds exactly right to me as far as the maths is concerned. (I am however not well informed about the philosophical history of the terminology.)

   That sounds exactly right to me as far as the maths is concerned. (I am however not well informed about the philosophical history of the terminology.)

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 21st 2014
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   Author: David_Corfield
   Format: MarkdownItexWe discussed a similar point at the [Cafe](https://golem.ph.utexas.edu/category/2013/07/synthetic_groupoid_theory.html#c044296). I think an important source of the terminology is the nineteenth century distinction between synthetic geometry (done with figures) and analytic geometry (done with coordinates) (see, e.g., [here](http://books.google.co.uk/books?id=6Jya7zz9AkIC&pg=PA125)). Someone like Sophus Lie is using geometric intuition to do his work, and this includes intuition of infinitesimals. For Kant, all of mathematics concerns the synthetic a priori, arithmetic relies on the inner sensual intuition and geometry the outer.

   We discussed a similar point at the Cafe. I think an important source of the terminology is the nineteenth century distinction between synthetic geometry (done with figures) and analytic geometry (done with coordinates) (see, e.g., here). Someone like Sophus Lie is using geometric intuition to do his work, and this includes intuition of infinitesimals.

   For Kant, all of mathematics concerns the synthetic a priori, arithmetic relies on the inner sensual intuition and geometry the outer.

4. • CommentRowNumber4.
   • CommentAuthorColin Tan
   • CommentTimeFeb 2nd 2014
   • (edited Feb 2nd 2014)
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   Author: Colin Tan
   Format: TextRegarding this distinction, should I post here or at the Cafe? Is a synthetic theory necessarily elementary (that is, first-order)? As an example, synthetic differential geometry is elementary while analytic differential geometry is maybe third-order inside a set theory. As another example, HoTT is (to my knowledge) elementary while oo-topos theory is maybe third-order inside a set theory. I do not know how to account for Euclid's elements.

   Regarding this distinction, should I post here or at the Cafe?
   
   Is a synthetic theory necessarily elementary (that is, first-order)? As an example, synthetic differential geometry is elementary while analytic differential geometry is maybe third-order inside a set theory. As another example, HoTT is (to my knowledge) elementary while oo-topos theory is maybe third-order inside a set theory. I do not know how to account for Euclid's elements.
5. • CommentRowNumber5.
   • CommentAuthorZhen Lin
   • CommentTimeFeb 2nd 2014
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   Author: Zhen Lin
   Format: MarkdownItexSDG is not elementary (inside a topos). The Kock–Lawvere axiom involves an exponential object, which is not first-order.

   SDG is not elementary (inside a topos). The Kock–Lawvere axiom involves an exponential object, which is not first-order.