[FOM] Alain Connes' approach to Analysis

[katzmik at macs.biu.ac.il]

There was a clear connection in Connes' mind between his critique of NSA and
his proposal of his own theory of infinitesimals.

Connes' idea was, thinking of an infinitesimal as a sequence tending to zero,
to place the entries on the diagonal of an infinite matrix, thereby producing
a compact operator.  He then removed the "diagonal" assumption, working
instead with the sequence of eigenvalues.  This naturally creates a context
where "infinitesimals" don't necessarily commute.  He imposed a certain decay
condition on the eigenvalues, so as to be able to apply the Dixmier trace to
the operators.  This is the basic framework of his non-commutative analysis. 
For details one can consult the longer source

Kanovei, V.; Katz, M.; Mormann, T. "Tools, Objects, and Chimeras: Connes on
the Role of Hyperreals in Mathematics." Foundations of Science 18 (2013), no.
2, 259-296. See  http://dx.doi.org/10.1007/s10699-012-9316-5

MK

On Tue, August 28, 2018 06:47, Noah Schweber wrote:
> Hello,
>
> Are you sure that's the correct source? As far as I can tell, Connes is
> only referenced there in the context of (arguing against) his critique of
> NSA, not "his own approach to Analysis, using ideas from non-commutative
> geometry."
>
>  - Noah
>
> On Mon, Aug 27, 2018 at 3:21 AM, <katzmik at macs.biu.ac.il> wrote:
>
>> The by now STANDARD reference on this is the article
>>
>> Sanders, S. "To be or not to be constructive, that is not the question."
>> Indag. Math. (N.S.) 29 (2018), no. 1, 313-381.
>> https://doi.org/10.1016/j.indag.2017.05.005
>>
>> MK
>>
>> On Mon, August 27, 2018 03:07, José Manuel Rodriguez Caballero wrote:
>> > Hello, I read that Alain Connes' developed his own approach to Analysis,
>> > using ideas from non-commutative geometry. It would be nice to read
>> several
>> > points of view concerning Alain Connes' approach from the point of view
>> of
>> > the foundations of mathematics.
>> >
>> > Wiki-Reference:
>> > https://en.wikipedia.org/wiki/Criticism_of_non-standard_analysis
>> >
>> > Kind Regards,
>> > Jose M.
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>> >
>>
>>
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>