[FOM] 31st Novembertagung 2020
Sam Sanders
sasander at me.com
Sun Apr 26 15:24:03 EDT 2020
Dear Tim,
take a look at Errett Bishop’s writings around 1967: he seems to express the idea you have in mind, namely
that adopting a formal system for constructive mathematics would hamper its progress.
Note that he changed his mind later (see “mathematics as a numerical language”).
Best,
Sam
> On 26 Apr 2020, at 15:52, Timothy Y. Chow <tchow at math.princeton.edu> wrote:
>
> On Sun, 26 Apr 2020, Louis H Kauffman wrote:
>> The distinction between what can be accomplished with given axioms and what may be done without them cannot be made without a clear sight of those axioms. A good example is the remarkable success of methods of functional integration in physics and related topology that is still up in the air. Here is mathematics proceeding without acceptable axioms. The research problem to find those axioms or to ground these procedures in a known system (such as ZFC) is a well-defined problem even if the methods of functional integration appear to be ill-defined. That we have axioms allows us to see our own viewpoints in formal terms.
>
> I don't really disagree with what Louis Kauffman says here, but it makes me wonder whether *any* approach can be "harmful to mathematics." Toward any approach whatsoever, we can adopt a take-it-or-leave-it attitude. If it helps, adopt it; if it doesn't help, ignore it. This way, no harm can be done. Even sociological influences can be dismissed; suppose that Cartan's demand for axioms were actually inappropriate at that particular juncture, and that an intuitive approach without regard to axioms were the right way to make progress at that moment. The topologist could just tell Cartan to shut up and wait his turn, and no harm would be done.
>
> If this is all that is meant by "the axiomatic method is not harmful to mathematics" then it verges on being vacuously true, and hence not a very interesting claim.
>
> Tim
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