[FOM] Coherentism, pt. 2
A.P. Hazen
a.hazen at philosophy.unimelb.edu.au
Sun Sep 28 01:31:00 EDT 2003
Having gone seriously off topic in part 1, it's time for me to say
how these philosophical notions (F-ism, C-ism) are relevant to FoM. (At the
end I will add a note of thanks to two people who have tried to make my
characterizations of them less caricaturish.)
In my "Simpson on Tymoczkoism" post I claimed that people
interested in -- and convinced of the PHILOSOPHICAL importance of -- FoM
didn't have to be FOUNDATIONALISTS (in the sense in which that word is used
by philosophers). Simpson in reply pointed to the absolutely central role
of PROOF in mathematics: mathematicians establish theorems by deducing them
from (ultimately) axioms. (SOME philosophical "quasi-empiricists," and
some people who enjoy playing with computers, have seemed to deny the
centrality of proof to mathematics: Tymoczko, at times, sounded like that.
At least in his postings to this forum, f.w.i.w., Corfield doesn't seem to
be denying the centrality of proof so much as urging that attention be paid
to other features of mathematics as well.)
The role of proof in mathematics LOOKS like the kind of thing
philosophical F-ists describe: the justification for believing a theorem
(the "evidence" for it) lies in the previously established theorems (&
axioms) from which it is derived. Because of this, mathematics has seemed
like the area of knowledge where epistemological F-ism is most plausible,
and "quasi-empiricist" philosophers tend to interpet an interest in proof
and axiomatics as evidence of F-ist commitments. I think it is probably
fair to say that the structure of evidential relations within mathematics
is at least LOCALLY foundationalist: it is not an accident that proof
theorists represent proofs as WELL-FOUNDED trees of formulae! I think,
however, that this is consistent with a denial of GLOBAL F-ism, and that
much FoM work seems, if anything, MORE interesting in a more C-ist
framework.
The problem is with the AXIOMS. What makes it rational to accept
an axiom? Someone could say that mathematics rests, ultimately, on axioms
that are just self-evident, given to us by a kind of "intuition". (I tend
to think of "intuition," in epistemological contexts, as a negative term:
when you say someone has intuited something, you're saying they know it but
admitting that you can't say anything useful about HOW they know it!) I
think a consistent epistemological F-ist would have to say this. I don't
think it is very plausible, for at least two reasons. FIRST: Many F0M-ers
are interested in the possibility of NEW axioms, axioms going beyond
standard ZFC. Just how one is to find new axioms, and what makes it
rational to accept one, is a ... difficult ... question, but it's not just
a matter of waiting for the intuitive light to shine: it's something to
which argument and the search for evidence is relevant. SECOND: In the
decades following the discovery of Russell's paradox (& the other p's),
even the standard ZFC axioms didn't seem to be "given" by an indubitable
and unanalyzable intuition. (They still don't, to some people.)
I think some of Bertrand Russell's comments about axioms are very
plausible. (E.g. in his 1907 essay "The regressivemethod of discovering
the premisses of mathematics," pp. 272-283 in Russell (D. Lackey, ed.),
"Essays in Analysis.") It was nice if the axioms had some intrinsic
plausibility, or even obviousness, but that wasn't the whole story of what
justified their acceptance. Often some of the theorems derived from the
axioms were more obviously true than the axioms themselves! What
justified the adoption of the axioms was that they gave the best
systematization of mathematics; part of the evidence for them, therefore,
was that obvious theorems could be derived from them. On the face of it,
this is a lot closer to C-ism than to F-ism!
One of the interesting developments in FoM in recent decades (to
which Simpson has made important contributions) is the program of "Reverse
Mathematics": the detailed analysis of just which axioms are presupposed by
various theorems. A quite striking pattern of dependencies has been
revealed. It seems to me, however, that if all "axioms" were simply and
intuitively self-evident, the sort of investigation represented by Reverse
Mathematics would be of merely specialist interest. In the context of a
C-ist mathematical epistemology, where even axioms are supported
evidentially by their logical relations to other propositions, however, it
becomes central and vital.
------
My attempts to characterize Foundationalism and Coherentism have
been very rough. A couple of people have posted to FoM to improve on
them: Thank you!
FIRST (at least in my e-mail log), Panu (praatika at mappi....) has pointed
out a distinction between "Strong" and "Modest" F-ism. This is useful: I
should not have left the impression that there were just TWO positions in
epistemology, and most of what I have said about F-ism applies to the
extreme, Strong, version. (In my defense, maybe some of the
"quasi-empiricist" attacks on F-ism have also assumed this narrow version
as their target. But that's just me making excuses.)
SECOND, Ron Rood ("P.T.M. Rood") has added several details. (He has also
suggested a couple of useful resources for people who want to know more:
the (on-line, at http://plato.stanford.edu ) "Stanford Encyclopedia of
Philosophy" and the (print) Blackwell "Companion to Epistemology," edited
by J. Dancy and E. Sosa.) I had (bizarrely!) tried to write about
epistemology without using the words "knowledge" or "justification." I
talked instead about what it was "rational" to believe or accept-- I hope
the connection is at least roughly clear (but getting the connection
PRECISELY clear is almost incredibly difficult!). Rood rightly points out
that much epistemological theorizing has been in terms of the distinction
between "directly" and "indirectly" justified beliefs: the directly
justified correspond to the ground level in my description, and the
indirectly to all the rest.
---
Allen Hazen
Philosophy Department
University of Melbourne
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