[FOM] 31st Novembertagung 2020

Timothy Y. Chow tchow at math.princeton.edu
Sun Apr 26 09:52:13 EDT 2020

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.


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