[FOM] polemic, predicativism, and absolute certainty

[Curtis Franks]

In this forum's recent debate about grounds for restricting mathematical 
methods, Professor Avron discloses an interesting motivation:

 > [F]or thousands of years mathematical propositions
 > were the main (and perhaps the only generally accepted)
 > examples of certainly true propositions. If there is no
 > certain truth even in mathematics then certainly there is
 > no certain truth anywhere else.
 >
 > So for me the most crucial problem of FOM is: is there
 > absolute truth in mathematics, and if there is - what
 > theorems of mathematics can truthfully and safely be taken
 > as meaningful and *certainly true*. Predicativism ...
 > is all about this question.

I'd like to point out that in his earliest descriptions of his 
foundational program (I'll quote from his 1922 Hamburg paper), David 
Hilbert articulated the same motivation--"I should like to regain for 
mathematics the old reputation of incontestable truth"--as an 
*objection* to foundational programs that involve a restriction of 
ordinary mathematical techniques. He even specifically picks out 
Predicativism as a foundational program that jeopardizes the objectivity 
of mathematics:

"[O]ne sees that for the mathematician various methodological 
standpoints exist side by side. The standpoint that Weyl chooses and 
from which he exhibits his vicious circle is not at all one of these 
standpoints; instead it seems to me to be artificially concocted.

Weyl justifies his peculiar standpoint by saying that it preserves the
principle of constructivity, but in my opinion precisely because it ends 
with a circle he should have realized that his standpoint (and therefore 
the principle of constructivity as he conceives it and applies it) is 
not usable, that it blocks the path to analysis."

According to Hilbert, precisely because Predicativism involves a 
restriction of methods, it calls into question the objectivity of 
mathematical truths. For in the first place, since from the restrictive 
standpoint of Predicativism lots of ordinary mathematics appears 
unjustified and even paradoxical, whatever is getting a foundation from 
the Predicativist standpoint isn't the same thing as that body of 
knowledge with a thousands-of-years-old reputation for absolute 
certainty. And in the second place, even the bit of mathematics that can 
be recovered under the aegis of Predicativism gets a foundation only as 
solid as the principles that motivate the Predicativist standpoint. 
Since those aren't natural mathematical principles, but instead are 
principles that arise out of philosophical skepticism, Hilbert thought 
the "security" established by Predicativist foundations was only as good 
as its philosophical roots. And since those roots are contentious (not 
everyone is a Predicativist), this doesn't amount to a demonstration of 
objectivity even for the Predicativist fragment of mathematics.

It is thus ironic that Professor Avron has the same, admirable goal that 
Hilbert expresses: "[T]his is what I require: in mathematical
matters there should be in principle no doubt; it should not be possible 
for half-truths or truths of fundamentally different sorts to exist." 
For according to Hilbert it is the attempt to situate mathematics on 
epistemological grounds like those found in Predicativism that makes 
mathematics appear like a half-truth.

At least it's interesting to see in Hilbert that a commitment (like 
Professor Friedman's) not to restrict mathematical techniques can be 
motivated by the concern about objectivity that Professor Avron shares.

Curtis Franks
Dept of Logic and Philosophy of Science
The University of California, Irvine