FOM: What Label to apply to this philosophy?

Dan Halpern halpcom at
Tue Jan 20 16:20:13 EST 1998

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Dear Dean,

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.

Dan Halpern
Facilities Engineer 
Belmont, Calif

Research Interest: The use of coercion in achieving consensus 
in science (used in the broadest sense here) and the role of humor as
a counterbalance.

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