[FOM] Alain Connes' approach to Analysis
lp15 at cam.ac.uk
Mon Sep 3 07:59:01 EDT 2018
Non-Standard Analysis can be developed from basic set theoretical principles and no additional axioms, via an ultrafilter construction. This approach was formalised in Isabelle nearly 20 years ago by one of my PhD students, Jacques Fleuriot.
Moreover, results derived using non-standard methods can be transferred (again making no additional assumptions) to the standard real numbers. Surely this leaves no room for philosophical objections, provided one accepts higher order logic, which is a weak fragment of set theory.
> On 30 Aug 2018, at 08:06, Sam Sanders <sasander at me.com> wrote:
> Some people, including Alain Connes, view the aforementioned power of Nonstandard Analysis as something unholy. To reject Nonstandard
> Analysis based on this or similar intuitions, a lot of smart people have come up with faulty arguments that revolve around the idea that the afore-
> mentioned power must somehow make Nonstandard Analysis fundamentally non-constructive, ineffective, or not-applicable in physics
> (millennia of intuitive infinitesimal calculus in math and physics notwithstanding).
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