FOM: Maddy's views
James Robert Brown
jrbrown at chass.utoronto.ca
Tue Feb 15 12:45:51 EST 2000
I'm very glad to see the discussion of Maddy's views in recent fom postings,
since I'm working on math naturalism in general and Maddy's version in
particular. As mentioned by others, one of her chief claims (in fact it's how
she defines her naturalism) is that mathematics is driven by its own internal
considerations (including methodological considerations), and not by external
factors such as science or philosophy. By philosophy she means ontology (what
are mathematical entities?) and epistemology (how do we come to know
mathematical truths?). Instead she claims mathematicians have adopted (if only
implicitly) principles such as MAXIMIZE ("the goal of founding mathematics
without encombering it" P 211). An axiom such as V=L violates this principle
while MC and other large cardinal axioms tend to be in accord. The
"encombering" refers to the apparent restrictions put on the idea of "arbitrary
set" by V=L. She cites the history of the concept of function as exemplifying
MAXIMIZE, ie, progressively more inclusive notions of function until the
currently accepted arbitrary correspondence.
Maddy's argument that philosophy (eg, Platonism) has played no role is very
simple: The methodological debate has been settled, but the philosophical
issues have not; so the philosophical issues have played no role.
I think there are at least a couple of problems with this argument, but there
is only one I want to pursue here. It MIGHT be the case that the philosophical
issue remains open in the larger intellectual community, but not in the set
theory community. Only a small number of people care about the V=L vs large
cardinals debate. Perhaps for this subset the issue of Platonism is also
settled and played a determining role in the outcome of debates about axioms.
This is a sociological question, so a quick survey is called for.
First, the issue. I'll characterize Platonism as having the following
ingredients (which, I think, Godel would endorse): 1 (Ontology) Mathematical
objects and mathematical facts exist independently of us, outside of space and
time. We do not create mathematical truths, we discover them. And those
truths do not depend on how we prove them, nor on our mathematical language,
and so on. 2. (Epistemology) Some of our mathematical knowledge stems from a
"grasp" of mathematical objects and truths, a kind of "seing with the mind's
eye". 3. (MM) Almost every Platonist takes the Platonic realm to be as full
and as rich as it could possibly be. Call this principle MM for "The more the
merrier".
MM clearly suggests that the set theory heirarchy is as rich as logic allows.
Since V=L tends to restrict this, Platonism is at odds with this axiom. What
is the relation to Maddy's MAXIMIZE ? Simply this: Platonism is a sufficient
condition for MAXIMIZE, but not a necessary one. So, it is possible that Maddy
is right, set theorists have adopted MAXIMIZE independently of Platonism. And
her argument for this independence, to repeat, is that Platonism is still an
open question while MAXIMIZE is settled.
Here is a rival hypothesis: Thinking about the issues surrounding large
cardinals has made people think about Platonism in general and MM in particular
(at least implicitly). They felt compelled to adopted Platonism, and so in
consequence adopted MAXIMIZE. Thus, issues of ontology have played a role in
math, contrary to Maddy's claim. On the other hand, the larger philosophical
and mathematical community which does not think about large cardinals has not
been faced with ontological issues in the same compelling way, so the question
of Platonism has remained open for them.
Since these (Maddy's view and my rival hypothesis) are at least in part
sociological claims, I'd like to survey fom readers for their views. I'd be
very grateful for answers to the following questions. Send me your answers off
the list. I'll report back, if the results are at all interesting.
Please reply, even if you don't care or know much about large cardinal issues,
since this too is valuable information in determining the correlation of
views. The questions are crude, but they are probably adequate to establish a
trend if one exists.
****************************************************************
A. How much do you know about V=L, MC, etc.?
1 little or nothing at all
2 read some literature, perhaps teach course in which these
issues
arise
3 do research and publish in the area
B. What are your philosophical views?
1 Platonist (including MM)
2 anti-Platonist
3 other (explain in a sentence or two)
C Which axiom do you accept?
1 V=L
2 MC or other large cardinal axioms
3 other or no view on the matter
Comments?
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Thanks,
Jim Brown
*********************************************************
James Robert Brown
Department of Philosophy
University of Toronto
Toronto M5S 1A1
Canada
Phone: office (416) 978-1727, home (519) 439-2889
Email: jrbrown at chass.utoronto.ca
Home page: http://www.chass.utoronto.ca/~jrbrown/index.htm
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