[FOM] Alain Connes' approach to Analysis

[Mikhail Katz]

I hesitated answering this particular post on Connes' criticism of
Robinson and the hyperreals, since I was hoping somebody from the
reverse math community would respond.  Since the thread seems to be
still officially open, perhaps a reply is in order.  I would like to
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 a
prurient appeal and refuses to fade away and therefore some
clarifications seem appropriate.

(2) The fact is that a nonstandard integer H in *N leads "canonically"
to an ultrafilter {A: H\in *A} on N.

(3) An 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 also leads to a nonmeasurable set.

(8) Yet Skolem's construction takes place in ZF (without choice).

(9) What this illustrates is the POWER of Robinson's transfer
principle that stands behind item (1) 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.


On 3 september 2018, Jose Caballero wrote:

"Connes criticizes Leibnitz's approach as follows (free translation
from French): Every nonstandard real number determines, in a canonical
way, a subset of [0,1] which is not Lebesgue measurable, hence such
numbers do not exist. Reference (page 6) in Kanovei-Katz-Mormann's
paper: https://arxiv.org/pdf/1211.0244.pdf"