[FOM] Alain Connes' approach to Analysis

[José Manuel Rodriguez Caballero]

>
> MK wrote:
> (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.


Maybe Connes rejects non-measurable sets because he associates the
mathematical notion of measurability with the physical notion of
measurability. In the same way that a physicist rejects notions like
metaphysical time because it is non-measurable, Connes's may reject
non-measurable sets in mathematics. If this is the case, then Connes'
criticism of hyperreal numbers is a self-criticism as it was pointed out by
MK. This suggests a program of foundations of mathematics in which
noncommutative geometry should be developed omitting the possibility of
non-measurable sets.

Reference about quantum measurability:
https://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics

Kind Regards,
Jose M.
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