[FOM] Simpson on Tymoczkoism

Stephen G Simpson simpson at math.psu.edu
Sat Oct 4 15:34:07 EDT 2003

Responding to Hazen's characterization of foundationalism, I said:

 > >Under this notion of "foundationalism", I am certainly a
 > >foundationalist.  Indeed, I firmly insist that the best method of
 > >organizing mathematics -- the "gold standard" -- is the orthodox,
 > >rigorous method, involving axioms, definitions, lemmas, theorems, and
 > >proofs.  [...]  In this sense, virtually the entire mathematics
 > >community -- core and non-core -- is foundationalist.

P.T.M. Rood, Fri, 26 Sep 2003 12:36:14 +0200 replied:

 > This seems to suggest that foundationalism is primarily about the
 > *organization* of mathematics, not about the justifcation of our
 > mathematical beliefs or our mathematical knowledge. [...]

I intended no such suggestion.

Because Hazen and Corfield had emphasized the *organization* of
mathematical ideas, I did too.  However, a look at the mathematical
literature reveals that the mathematics community's foundationalism
embraces not only organization but also *justification*.  The
definition-theorem-proof methodology plays a dominant role, not only
in organizing mathematical concepts and theorems, but also in
justifying them.

 > How the notion of direct justification needs to be analyzed is
 > considered to be a difficult question. 

Whether difficult or not, "direct" (i.e., non-deductive) justification
of axioms is an essential feature of the foundationalist landscape.

 > epistemologists understand knowledge as propositional knowledge,
 > i.e., to know is to know a proposition. 

The emphasis on *propositional* knowledge to the exclusion of
everything else has always struck me as a somewhat strange and
peculiar feature of contemporary epistemology.  Regular people
understand that knowledge of *concepts* is possible, so why can't
contemporary epistemologists deal with this?

 > I do not think that one can unconditionally say that mathematical
 > propositions are true. For example, if the axiom of parallels is
 > true, then (let us assume) its negation is false.  Therefore, if I
 > know the axioms of Euclidean geometry, then I do not know (indeed,
 > cannot know) those of non-Euclidean geometry. But that seems
 > absurd.

This paradox serves only to point up one of the difficulties of
focusing solely on propositional knowledge, to the exclusion of every
other kind of knowledge.  The truth is that Euclidean and
non-Euclidean geometry both fit perfectly well into a common
mathematical framework.  The currently accepted framework is
set-theoretical f.o.m.


Stephen G. Simpson
Professor of Mathematics
Penn State University

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