FOM: What Label to apply to this philosophy?
halpcom at worldnet.att.net
Tue Jan 20 16:20:13 EST 1998
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I enjoyed your attempt (posting of Jan 5: "What label to apply to this
philosophy?" fom-digest 33) to give a philosophy of mathematics that
barbers (or as in my case - janitors) can understand. But maybe you've
left out a crucial ingredient that, when added, brings you back to
Hersh's humanism. As I see it you're describing a formal theory and
talking about "operational correspondences" between its elements and
the real world. Isn't there something that precedes the formal theory,
namely the concept which the formal theory is trying to capture? For
the positive integers, Feferman describes the concept in his posting
of Jan. 3 as follows: "The positive integers are conceived within the
structure of objects obtained from an initial object by unlimited
iteration of its adjunction, e.g. 1, 11, 111, 1111, .... , under the
operation of successor." For the set theoretic universe Feferman's
description is: "Set theory is supposed to be about the cumulative
hierarchy, conceived as the transfinite iteration of the power set
operation. At base that depends on the conception of the totality
of arbitrary subsets of any given set under the membership relation."
Any formal theory of natural numbers or of sets is an attempt to capture the
respective concept. We know from Godel that such attempts fall short
in some sense.
>From this perspective, perhaps the interesting foundational question for
mathematical theories is, where does the concept come from and how does it
evolve? In the case of positive integers, the concept is, I guess, an
abstraction that came from human attempts to count and measure. The
evolution of the concept, whether it be positive integers or the set
theoretic hierarchy, is what Hersh talks about when he uses phrases like
"consensus" and "socially constructed". I may have this wrong because
it seems like some of the criticism of Hersh has been about consensus
as regards simple manipulations in the formal system.
Of course there are some theories, e.g. group theory, where the concept is
completely captured by the formal theory. I think these are abstractions
from more basic theories like number theory. For me, questions of their
truth (or relevance) has no additional problems beyond the relevance of
their parent theories. The relevance of the parent theories are hardly
surprising or problematic if the scenario suggested above is correct.
On a related point, I've been following the discussion between Hersh
and Davis and have thought about Davis's Jan. 5 critique of Hersh's
distinction between mathematics and theology. My guess is that
some aspects of theology are very much like mathematics as Davis
suggests. I wonder if what distinguishes most mathematics from
theology and most other such social constructs is:
1. Its intent to be a tool of science and its ultimate origin in science,
(By science I mean the attempt to understand the physical/material world)
This gives it universality and tests of fire that most theology avoids.
2. Its levels of iterated abstraction, and
3. The length and complexity of its deductions from sparse sets of basic
assumptions. (Perhaps somewhat a consequence of 2)
Dean, I'm cc'ing FOM in the hope that some of the professionals will
comment. Despite what the owners claim, they're probably only
interested in talking to others of their caliber. I'm also bcc'ing
some of the professionals in case the posting of this on FOM is delayed.
The following is added to bring me into compliance with FOM NRP:
Since I'm not currently a professional, I come to this list in supplication
(as has been suggested by the owners), and would like the above assertions
to be viewed as humble questions to the cognoscente.
Research Interest: The use of coercion in achieving consensus
in science (used in the broadest sense here) and the role of humor as
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