[FOM] "Progress" in philosophy

[Robbie Lindauer]

On Mar 10, 2007, at 8:15 AM, Charles Silver wrote:

> 	On the positive side, how about if "we" *define* what we mean by
> philosophy.   (I'd like Harvey to tell us what he means by it.)

It would certainly help some philosophical programs in mathematics  
along if we could define away philosophy altogether.  Philosophy  
could be just a variety of logic, and we could have axioms in  
philosophy and correct dissenters by reference to those axioms.

But before we decide to do this, it's worth remembering the  
philosophical justifications in which almost all of the "great  
mathematicians" indulged, connecting their mathematical work with  
philosophical problems.   Cantor, for instance, regarded his own work  
as existing for the sake of these metaphysical problems.  Cantor's  
mathematical work was an extension in a way of his theology; in many  
ways this theological foundation continues today in the notions of  
Oracle or Super Computer or Super-Advanced Aliens that provide the  
epistemological and ontological foundations when no "real present"  
one can be found.  (Perhaps we are all Santayanists when it comes to  
mathematics, though we "know" no super-decider can exist for the  
mathematical theories we like, its useful to carry on as though one  
does.)

As a result of a refusal to address the epistemological and  
ontological problems facing mathematics head on Philosophically (what  
really qualifies something as an axiom?  what makes a mathematical  
statement true? what makes anyone think we can have the kind of  
knowledge which mathematicians appear to claim to have?  what  
constitutes that as knowledge rather than something else?  Of what  
value is mathematics?) the only reasonable position left is some kind  
of mathematical agnosticism.  That said, the only way, I think, to  
make the program of FOM, for instance, relevant (that is, have  
"general interest"...) is to address the traditional epistemological  
and ontological problems head on and on their native ground -  
philosophically.  Refusal to do this just makes FOM irrelevant for  
those problems of the most general interest (as H. Freidmann put it).

One can make claims of progress in mathematics, then, as long as one  
remains agnostic about results therein having any bearing on these  
overarching epistemic, ontological, political and religious  
questions.  But this kind of agnosticism is itself a philosophical  
position, even the refusal to call it such is an old saw,  
philosophically speaking.

Somewhat relatedly, the question of progress is relative to purpose.   
Progress towards what?  One culture called the advance of the  
Americans across the continent "Progress" in the form of "manifest  
destiny".  Another culture now calls what happened (and continues to  
happen sometimes) "ethnicidal".  Progress is a value-term and what is  
valuable is usually relative to a stated or non-stated purpose.  Some  
forms of mathematics have obvious purpose - one can regard SQL as the  
economic justification of set theory, in a way.  Studies in  
decidability have applications to artificial intelligence, one can  
regard work in that direction as progress if one regards Artificial  
Intelligence as Progress (as I think many people do).  The ability to  
produce a nuclear weapon or nuclear energy depends in a sense on some  
aspects of higher mathematics, and one can regard military  
applications as one of the core justifiers of mathematical research  
and calling what is found there "progress".  If there is an "absolute  
standard of value" and if Mathematics fits under it I have no idea.   
But a decision on that matter will have to be a philosophical  
problem, mathematics is not equipped to deal with this question I  
think.  Certainly putting in place some axioms and working out their  
consequences won't help.

But relative to other problems, the standard "beauty-queen problems"  
in our world, these issues of how to most accurately represent  
financial and personnel data in computers and how to make computers  
"think" like humans and how to destroy civilizations in fell swoop,  
have to be regarded as very low on the importance scale.  (Unless, I  
guess, one is an "Singulitarian" ala R. Kurzweil.) And if we defined  
progress as "creating a world in which there is social and economic  
justice, peace and prosperity" I doubt that the progress that  
mathematics is making toward being able to more accurately predict  
trends in the stock market and create even more powerful super- 
weapons, etc., would be counted as progress in that regard.  Whereas,  
for instance, the work of the (decidedly not mathematical)  
philosopher E. Levinas could certainly be so regarded.


Best Wishes,

Robbie Lindauer