[FOM] Simpson on Tymoczkoism

[Neil Tennant]

Thanks to Ron Rood for an enlightening message.

There is another philosopher's distinction that might be relevant to
Steve Simpson's exchange with David Corfield. This is Reichenbach's
distinction between the context of discovery and the context of
justification. (Apologies to anyone who might have made this point
already.)

In the 23-page sample of his book which I downloaded---David,
why can't CUP cough up the whole of the first chapter for us?!---Corfield
emphasizes the relevance of real/core mathematical practice to the
philosophy of mathematics, and complains that philosophers pay too little
attention to it. Perhaps his emphasis here is really on the context of
discovery---involving the Lakatosian process of conjecture, failed
attempts at proof, subsequent conceptual tinkering, and eventual success
with proof---rather than the context of justification.

In response to this, the philosopher of mathematics who emphasizes instead
the context of justification will be waiting on the sidelines, as it were,
for the mathematician (or mathematical community) to "clean up its act",
and present us with the finished product. Then, and only then, does the
mathematical foundationalist of the Simpsonian kind roll up his or her
sleeves and get down to the business of anatomizing the conceptual
structures and deductive resources underlying the results proved.

"Proving" is (ideally) a success verb. The Simpsonian likes to look at
mathematics as a body of past and completed successes, ready for, as it
were, the 'static' analysis of its conceptual and deductive structure. The
Corfieldian likes to look at mathematics instead as an ongoing practice, a
striving, a process of refining and sharpening, demanding, as it were, a
more 'dynamic' analysis of its evolution. This would involve looking at
creative sources of conceptual innovation and proof-insights, and the
selective filters of aesthetic tastes, the interests and preferences of
influential contemporaries and precursors, economic needs, scientific
applicability, etc.

Surely there is no essential conflict here? Isn't it just like the
situation in biology, where one has both (comparative) anatomy [= the
'static' view] and evolutionary genetics [= the 'dynamic' view] ?

Just as rational mechanics (of the Suppes kind) need have no regard for
the possible fact that Newtonian mechanics might have arisen in large part
in response to the needs of European mercantilism, so too FOM need have no
regard for the 'whorl of organism' or creative foment in any department of
core mathematics. 

As Corfield explains in his book, the phrase "real mathematics" was
modelled after the British phrase "campaign for real ale", whose
organizational acronym was CAMRA. CAMRA was founded in response to 
the watering-down of the beloved amber liquid by large industrial
breweries. Corfield and his friends thought up their phrase "real
mathematics" in response to the dumbing-down of the British A-level
syllabus and exam in mathematics. So "real" is best translated not as
"core" but as "hard core". To make it easier for the mathematically
minded, let us call this XXX-mathematics.

The question I would like to ask Corfield is: What is it about the process
of *discovery* within XXX-mathematics (Hopf spaces, category theory, you
get to name it ...) that makes you think that its eventually *justified*
product (a body of axioms, definitions, and proofs of theorems) will
exhibit anatomical features calling for distinctive innovations from
*philosophers* of mathematics in response to them?

Put more succinctly: Why is the XXX-mathematics of today philosophically
sexier than the XXX-mathematics of yesteryear?

Neil Tennant

___________________________________________________________________
Neil W. Tennant
Professor of Philosophy and Adjunct Professor of Cognitive Science

http://www.cohums.ohio-state.edu/philo/people/tennant.html

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