[FOM] Mathematical certainty and social activity

[Arnon Avron]

This is the third part of my reply to Davis (and it implicitly
includes partial response to some of the replies I got
to the previous parts).

Martin Davis wrote:

> There are two further points I'd like to make. 
> One is that, in the last analysis mathematics is 
> a social activity,  

Wrong, *certainty* in mathematics is not just a matter
of social activity!
 
  In general, there is in our generation a very dangerous
tendency  to describe  everything as relative and depending
on "social activity". The main reason that I emphasize
the absolute certainty aspect of a very significant
part of mathematics (definitely not the whole of the 
mathematical enterprise, but in my opinion the most important
one) is because the urgent need to fight this illness of 
modern times (and especially its most fatal product:
"post-modernism"). It is my deepest conviction
that there are fundamental principles that are just 
beyond "social activity". Even in questions
of morality (the most important among all philosophical
questions) there are some *absolute* principles. Thus
the Nazi ideology was, is, and always will be evil,
even if most of the people on earth would socially accept it
(this could have happened had the Nazis won WW2!). 

Now mathematics has always been the one great problem of the
"everything is relative" ideology (This is why all post-modernists
are so happy about Goedel incompleteness theorems, even though 
they don't bother to make a real effort to understand them). 
No matter how much some people try to deny it, mathematics does have 
some absolutely certain propositions - and this fact is not
due just to  "social activity". Moreover:
no matter what mathematicians did and do when they were/are 
pursuing goals in their research other than
absolute mathematical certainty, their mathematical  norms 
and ideals were established at the time of the Greeks, and 
the implementation of these norms becomes only *stricter* 
over the years! 

> and as time goes on, methods 
> that achieve desired results without leading to 
> catastrophe will just gain acceptance.

The term "acceptance" is too general. There is
a great difference, for example, between "accepted
for being used for the purposes of Physics",
or "accepted for publication in a mathematical journal",
or "accepted for demonstrating mathematical
proposition with absolute certainty" (and there are
several other relevant notions of "acceptance").
The first two are indeed mainly (but not only!)
a matter of "social activity".
Thus I see no reason why the mathematical tools
used in some scientific theory  should be more certain
or reliable than the parts of that theory which are not purely
mathematical. Accordingly, if it happens in the future
that scientists find some essentially
impredicative (or even inconsistent!) theory as useful for 
making reliable (in the sense of experimental sciences) predictions,
it would be stupid to forbid them to do so on ideological
ground. Such theories can therefore be "accepted" 
for *use in science*.  However, for accepting mathematical 
propositions as absolutely certain  "social activity" will 
never be  enough. Indeed: can you give me an example of a 
mathematical method that was once doubtful, but just because of 
social activity it is now accepted without qualms by practically
every mathematician? (Please do not give AC 
as such an example. It is a counterexample). 

As an example how doubtful method really gains acceptance, 
take the case of the "imaginary" numbers (or even the 
negative numbers).  Mathematicians continued to have strong 
doubts about the mathematical legitimacy of using these 
numbers even when their use was already indispensable from
a practical point of view. However, today nobody with some
mature mathematical knowledge has any problem or doubt
concerning their use (given that the reals are accepted).
This is not due to their gaining "accepted" through practice
or social activity, but to the absolute justification
that has been given to their use, a justification
which meets all mathematical norms of certainty (BTW, the
fact that mathematicians like Gauss devoted time and thought
to find such a justification clearly shows that
"gaining acceptance through social activity" just does
not work when it comes to mathematical rigor).

Arnon Avron