FOM: human well-being; constructivism; anti-foundation

[Jesse F. Hughes]

Stephen G Simpson <simpson at math.psu.edu> writes:

> 1. HUMAN WELL-BEING
[snip]
> 
> Philosophy profoundly influences history.  There are some historical
> periods (e.g., the Dark Ages) when the human mind is philosophically
> shackled or subjected to an unwholesome preoccupation with otherwordly
> concerns.  And there are other historical periods (e.g., the
> Enlightenment) when the human mind looks boldly outward, seeking to
> understand nature and harness it for human purposes, to the enormous
> benefit of the human race.  And mathematics is one of the stages upon
> which this great drama plays out.  Thus the philosophy of mathematics
> and f.o.m. research may have the capability of contributing to human
> well-being over the long run, by discovering and validating a
> rational, scientific foundation for a kind of mathematics that is
> real-world oriented and applications oriented.
> 
> In my view, it is appropriate to examine the existing philosophies of
> mathematics (formalism, intuitionism, constructivism, Platonism, etc)
> and the existing research directions in mathematical logic (set
> theory, proof theory, recursion theory, model theory, etc), in terms
> of the historical perspective of the preceding paragraph.  Such issues
> may be seldom discussed, but they are obviously very important and
> worthy of discussion here on FOM.
> 
I am concerned with this criteria for judging philosophies of
mathematics and foundational research for a number of reasons.  It
seems a crude and indirect method of judging the utility of a 
branch of research, and one fraught with difficulty.  I should think
that the readers of this list are more or less well-trained to answer
questions like: Is this branch fruitful for stimulating work in
related areas? I think that we are less well-trained for answering
questions like: Is this theory likely to positively influence the
course of human development in a broad sense? This sort of question
seems to require some sort of understanding of mass psychology,
history, and many other courses that I personally avoided by studying
mathematics.  These problems persist even if we ignore the
controversial questions contained in the phrase "positively
influence".    

Perhaps there are certain questions that, by their nature, force the
researcher into hair-splitting, navel-gazing and other
self-indulgences.  I imagine that a preoccupation with minute but
difficult problems that have little bearing on real-world mathematics, 
for instance, is unlikely to have a positive influence on human
wellbeing.  But, I presume, we can also judge that such preoccupations 
are undesirable from the standpoint of philosophy of mathematics in a
more direct manner.  The extent to which foundational matters affect
the way mathematics itself is done is a good indication of the worth
of the foundational questions we are pursuing.  And, I feel much more
comfortable discussing this question than the more abstruse question
of whether my research has a positive influence on humanity (other
than keeping me off the streets after dark).  Of course, I hope that
my work is worthwhile in the sense you mention, but I hardly feel
competent to defend it on such grounds.

-- 
Jesse Hughes

Position:           Graduate Student
                    Department of Philosophy
                    Carnegie Mellon University
Research interests: Category theory
                    Foundations of Computer Science