FOM: Philosophy and platonism

[Martin Davis]

I quote from Michael Zeleny's postings which those interested are 
encouraged to read in full:

At 01:06 PM 1/23/00 -0800, Michael Zeleny wrote:

>As Gottlob Frege has taught us, logic is a branch of ethics.  We may
>therefore imagine, mutatis mutandis, Bill Clinton correctly observing
>that ethical issues reach the consciousness of politicians when they
>have impact on political practice, and concluding with an expression
>of his firm belief, by parity of reason, that for work on these issues
>to have importance, they need to be focused in just this manner.  Thus
>we conclusively establish the primacy of utility over truth.


At 02:39 PM 1/24/00 -0800, Michael Zeleny wrote:

>I agree that it is a rank oversimplification to say that Frege taught
>logic to be a branch of ethics.  However expatiating on the essential
>complexity of their interdependencies, well served by Wolfgang Carl's
>book on Frege's theory of sense and reference, need not obscure the
>validity of my simple analogy between the deliverances of pragmatism
>and politics.
>
>The modal aspect of Frege's position is that, which concerns me most
>in our context.  He takes the norms of logic not as teleological, in
>the sense of directions towards the attainment of scientific knowledge
>or the assertion of statements that are certain and universally valid,
>but as apodictic, in the sense of existing unsituatedly, timelessly,
>and independently of relation to any goal.  This view stands in stark
>contrast to the foundational approach of Penelope Maddy, whose stance
>of beginning with what mathematicians actually DO strikes Martin Davis
>as exactly right.  Supposing that the Thousand-Year Reich had lived up
>to its name, we might be living in a world where all research in ZFC
>were suppressed as "Judaische Wissenschaft".  Contrary to the starting
>point delivered under this scenario, I like to think that the facts of
>mathematical foundations would remain invariant under the vagaries of
>political fashion.  It could be objected that the triumph of National
>Socialism lacks the virtue of actuality, demanded by Maddy and Davis.
>Unfortunately, there can be no assurance that foundational mistakes of
>comparable gravity are not perpetrated under the Novus Ordo Seculorum
>in the name of funding efficiency or institutional inertia.  Hence it
>seems to me that by anchoring their fundamental beliefs in the shoals
>of the Whig Interpretation of History, the pragmatists are no more
>entitled to define truth than the politicians are, to define justice.

I consider this political analogy forced and unfortunate. Whatever validity 
Zeleny's foundational views may have, mathematics is the context in which 
it makes sense to talk about them (of which more below). Thus, in the now 
classic debate between Hilbert and Brouwer, they made free use of political 
and social metaphor, but always where it was clear that they were 
discussing mathematical issues. However, now that Zeleny has opened this 
particular Pandora's box, it may be worth a brief look at how Frege's 
ethical principles allegedly "existing unsituatedly, timelessly, and 
independently of relation to any goal" worked out in practice. Frege allied 
himself with Bruno Bauch's extreme right-wing Deutsche Philosophische 
Gessellschaft a movement that enthusiastically welcomed Hitler's rise to 
power. Frege's  notorious 1922 diary combined anti-semitism, a naive hope 
that Germany's troubles could be overcome by the coming of a new leader, 
and a naivete about basic social and economic matters almost beyond belief. 
There is also the case of Heidigger whose devotion as a philosopher to 
"timeless" principles did not prevent him from enthusiastically embracing 
National Socialism.

Of course all of this has little to do with f.o.m., and I wish Zeleny had 
been persuaded to withdraw his flawed analogy.

Returning to f.o.m., the point that Zeleny misses is that the process of 
developing formal and symbolic methods for dealing with genuine 
mathematical issues produces a momentum of its own, a kind of "analytic 
continuation" that impels one beyond the original subject matter. An 
example is the way symbolic algebra pushed the expansion of the number 
system. Others: the extension of the exponential function, divergent series 
... one could go on and on. Today, it is the set-theoretic formalism that 
guides the formulation of new transfinite axioms. Given some property of 
sets of which it has been shown that any set having that property must be 
of such large cardinality that their existence can not be proved in ZFC or 
even some of its extensions, how is one to decide whether the assertion 
that such sets do exist is to be accepted? Are Zeleny's eternal principles 
of the least use here? Pragmatic practice by working mathematicians is the 
ONLY guide we have. Finding unanticipated linear ordering of cardinals 
defined in apparently unrelated ways, unexpected connections with other 
concepts (determinacy), and - most important - the efficacy of such 
assumptions in solving real mathematical problems. This is how (for 
example) the axiom of choice moved from a controversial principle to one 
mathematicians use today without a second thought.

Of course we all aspire to attain as much certainty as we can in 
mathematics, but in  my view (and I believe in Maddy's), we have no other 
technique available  to us but to let the rich development of mathematics 
itself point the way.

Martin




                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
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                        http://www.eipye.com