Analysis of Connes' criticism of Robinson's framework
| Critic | Venue where rebuttal appeared | Link to article/venue containing rebuttal |
| Alain Connes | Annals of Pure and Applied Logic | 21e, Section 8.6 |
| Bishop-Connes | Synthese | 17i |
| Alain Connes | Foundations of Science | 13c |
| Alain Connes | American Mathematical Monthly | 13d |
| Alain Connes | Math Overflow | Q&A thread |
Some leading mathematicians have been enthusiastic about Robinson's
framework not merely in word but in deed. Thus, Terry Tao has
published widely using ultraproduct-related techniques. Alain Connes
(1947- ), for whatever reason, took a dim view of Robinson's
framework. One could mention the following points.
(1) Connes' critique of the hyperreals on the grounds that they "lead canonically to a nonmesurable set" seem to have some kind of prurient appeal and refuses to fade away and therefore some clarifications seem appropriate.
(2) A nonstandard integer H in ℕ* leads "canonically" to a free ultrafilter {A: H ∈ A*} on ℕ. The fact that every set A has a natural extension to A* is part of the package that comes along with Robinson's Transfer Principle.
(3) A free ultrafilter naturally leads to a nonmesurable set.
(4) Therefore the criticism of "canonically leading to nonmeasurable sets" actually targets Tarski (rather than Robinson), who invented ultrafilters in 1930 (or Bourbaki, who invented ultrafilters in 1935 as believed by many in France including Connes).
(5) Connes routinely uses ultrafilters in many of his own articles and books, without mentioning anything about their leading to nonmeasurable sets.
(6) This occurs also in Connes' papers and books that voice the "nonmeasurable set" criticism against Robinson.
(7) Skolem's nonstandard integers embed in ℕ*. Hence by Connes' logic, a Skolem nonstandard integer should also lead to a nonmeasurable set.
(8) Yet Skolem's construction takes place in ZF (without choice) - or even weaker logical systems - which is not strong enough to prove the existence of a nonmeasurable set.
(9) What this illustrates is the power of Robinson's transfer principle that stands behind item (2) above, rather than any weakness of his framework.
(10) Connes' claim to the contrary amounts to an attempt to dress down a feature to look like a bug, to reverse a familiar quip from software developers.
(11) Analysis with infinitesimals can be done in an axiomatic theory SPOT conservative over ZF. This exposes as factually incorrect a claim by Alain Connes to the effect that "as soon as you have a non-standard number, you get a non-measurable set" (Connes 2007, 26). See Infinitesimal analysis without the axiom of choice.