Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeFeb 14th 2011
   • (edited Feb 14th 2011)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexAt [effects of foundations on "real" mathematics](http://ncatlab.org/nlab/show/effects+of+foundations+on+%22real%22+mathematics) I've put in the example of Fermat's last theorem as being potentially derivable from PA, and pointed to two articles by McLarty on this topic. (Edit: the naive wikilink to the given page breaks, due to the " " pair)

   At effects of foundations on “real” mathematics I’ve put in the example of Fermat’s last theorem as being potentially derivable from PA, and pointed to two articles by McLarty on this topic.

   (Edit: the naive wikilink to the given page breaks, due to the ” ” pair)

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeFeb 14th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexThe problem with minimalistic approaches to foundations is always that they require much harder paths for proving anything more involved. Of course, it can be fun for understanding such points for easy/simple things, but when something is really complex, than the subject usually grows more interesting and one considers it a content beyond any particular choice of foundations. It is hard to expect that redoing something as complex as Fermat would be really proportionally insightful for a person in foundations.

   The problem with minimalistic approaches to foundations is always that they require much harder paths for proving anything more involved. Of course, it can be fun for understanding such points for easy/simple things, but when something is really complex, than the subject usually grows more interesting and one considers it a content beyond any particular choice of foundations. It is hard to expect that redoing something as complex as Fermat would be really proportionally insightful for a person in foundations.

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 15th 2011
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexI don't think that the particular example of FLT is useful except as an exercise before the general ideas are understood, or as an example for sceptics unwilling to learn the general ideas. The point should be (if it is in fact true) that the methods used to prove FLT are *obviously* valid in these weak foundations.

   I don’t think that the particular example of FLT is useful except as an exercise before the general ideas are understood, or as an example for sceptics unwilling to learn the general ideas. The point should be (if it is in fact true) that the methods used to prove FLT are obviously valid in these weak foundations.

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeFeb 16th 2011
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI guess my motivation was to show that as far as foundations go, you can do a lot with little. Like Friedman (wow, I didn't think I'd ever say that), me picking on a particular theorem in a particular paper in a particular journal just pins down the conjecture. The fact it uses a lot of seemingly complicated machinery helps to show that the _logical_ strength of that machinery is far less than what the set theorists like to promote as standard. Perhaps the example should have been called something like 'the non-effects of foundations'. At least, I would like to keep McLarty's (meta)result that all SGA works in MacLane set theory with one universe, if not the application of this program of paring down foundations to find what is necessary for FLT.

   I guess my motivation was to show that as far as foundations go, you can do a lot with little. Like Friedman (wow, I didn’t think I’d ever say that), me picking on a particular theorem in a particular paper in a particular journal just pins down the conjecture. The fact it uses a lot of seemingly complicated machinery helps to show that the logical strength of that machinery is far less than what the set theorists like to promote as standard. Perhaps the example should have been called something like ’the non-effects of foundations’. At least, I would like to keep McLarty’s (meta)result that all SGA works in MacLane set theory with one universe, if not the application of this program of paring down foundations to find what is necessary for FLT.