[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.
Tim
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