[FOM] Simpson on Tymoczkoism

[P.T.M.Rood@ph.vu.nl]

On Thu Sep 25 20:19:03 2003,
Stephen G Simpson <simpson at math.psu.edu> quotes from 
A.P. Hazen (Wed, 24 Sep 2003 18:01:15 +1000):
>
> > "Foundationalism" is term of art among philosophers, contrasting
> > with "coherentism".  It denotes a certain (family of)
> > epistemological theory (-ies) according to which the propositions
> > that can, rationally, be believed are organized into a well-founded
> > hierarchy, [...]

To begin with, Hazen's characterization of foundationalism seems not quite
exact and may easily lead to confusion.

Although I am not an expert in epistemology, I think that the following comes
down to a better characterization than Hazen's. Foundationalism in 
epistemology
is, roughly, a theory about the structure of justification or knowledge. (And 
not,
as Hazen says, *propositions* that can [sic] be rationally believed.)

There seem to be different variants of  foundationalism that all circle around
the following idea. Beliefs can be  partitioned into two classes. In the first 
class,
there are beliefs that are inferentially based on other beliefs one has. In 
the
second class, there are beliefs that are not inferentially based on other 
beliefs
one has. Ultimately, every belief in the first class is based on one or more
beliefs in the second class.

(I've looked this up in L. Bonjour "Foundationalism and coherentism", in: The
Blackwell Guide to Epistemology. Greco, Sosa (eds.), Oxford: Blackwell (1999).
See also the lemma "Foundationalist theories of epistemic justification" in 
the
Stanford Encyclopedia of Phillosophy.)

Beliefs in the former class may be called "indirectly justified"; those in the 
latter
"directly justified".

How the notion of direct justification needs to be analyzed is considered to 
be
a difficult question. Traditionally, the notion seems to have been associated 
with
such things as self-evidence etc.. Nowadays, the notion of self-evidence is 
taken
to be highly problematic.

To this I wish to add that epistemologists (at least those of a 
foundationalist variety)
seem to take the notion of indirect justifcation as fairly unproblematic. 
Presumably,
they do so because they think that the central underlying notion of inference 
is
unproblematic: inference is simply the application of a logical rule of 
deduction.
However, it is not clear whether inference in mathematics should be equated
with logical deduction.

I might also add the following. First, epistemologists understand knowledge 
as propositional knowledge, i.e., to know is to know a proposition. Second,
that one knows a proposition implies (1) that one believes it, (2) that one is
justified in believing it, and (3) that the proposition one believes is true. 
Whether
those three conditions are also sufficient for knowledge is a controversial 
issue.

Taking up Hazen's characterization of foundationalism, Simpson says:

>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.

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. However in order to be able to evaluate what Simpson
says we need to understand what he understands by "mathematics." Does he mean
a certain set of propositions (e.g. a theory in the logician's sense)? Or does 
he mean
a certain body of known propositions? Or perhaps yet something else? 

The question is important since, for example, the structure of known 
propositions
needs not be the same as the structure of propositions per se. For example, I 
take it that
I know directly that 1 + 1 = 2. (Let us assume that I do know this 
proposition, or that it
makes sense to say that I know it.) However, in standard formalizations if 
arithmetic
(e.g. Peano arithmetic) the proposition 1 + 1 = 2 is logically deducible from 
other
propositions. Thus, to put the point metaphorically, within the (?) structure 
of knowledge
1 + 1 = 2 lies "at the bottom" but withing the (?) structure of propositions 
it doesn't.

>Furthermore, a look at the contemporary mathematical literature
>reveals that the definition-theorem-proof methodology continues to
>pervade contemporary mathematical practice.  This methodology is a key
>component of the high standard of rigor which continues to prevail in
>mathematics textbooks, mathematics research articles, mathematics
>Ph. D. theses, etc.  In this sense, virtually the entire mathematics
>community -- core and non-core -- is foundationalist.

I am not so sure of this last claim. First, I do not think that, upon close
inspection,  the "entire mathematics community" is (or indeed, can be)
foundationalist in the way epistemologists understand it. For the notion of
direct justification is too controversial and problematic. Moreover, it is
questionable whether there generally is mathematical knowledge *in the sense 
of
propositional knowledge*. For to know a proposition implies that it is true
(see above). However, 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.

Second, I am not sure whether "mathematical practice" (whatever that is)
has, as Simpson seems to suggest, the organization of mathematics as its sole
target. Such concerns come only into play when a branch of mathematics
has reached a state were the most open problems are solved. Mathematicians,
I would say, are first of all interested in proving theorems, or, more 
generally,
solving problems. And proving theorems is not quite the same as organizing
mathematics, although the two may of course be related.

Furthermore, it is not clear whether proving a theorem from the point of view
of mathematical practice is the same thing as logically deducing it within an
organized body of mathematics. True, Simpson does not say this, but I suspect
that many are easily inclined to make such a slip. The methodology of 
mathematical
practice need not match the structure of an organized body of mathematics. I 
fact,
they may very well diverge to a considerable degree.

Ron Rood
-- 

*****************************
Ron Rood
Department of Philosophy
Vrije Universiteit Amsterdam
De Boelelaan 1105
1081 HV Amsterdam
The Netherlands
e-mail: p.t.m.rood at ph.vu.nl
FAX: +31-20-4446620
phone: +31-20-4446614
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