[FOM] The irrelevance of Friedman's polemics and results

[aa@post.tau.ac.il]

On January 21 Friedman sent a very polemical posting to FOM, in which,
ironically, he attacked "polemicists" concerning FOM.

The polemical nature of Friedman's posting becomes apparent
when he writes:

> 4. The polemical nature of this setup becomes apparent because of the
> absence of any convincing explanation as to what is "wrong" with the
> disallowed methods, and the absence of any convincing explanation of what
> "good" mathematics is.

I guess that by "absence of any convincing explanation" Friedman means
"absence of any explanation that might convince Friedman". So this
polemic argument is practically tautological, since I cannot imagine
any explanation whatsoever that might possibly convince Friedman
about points of view he does not already accept...

   Now the real situation is the opposite
of what Friedman is describing. It is Friedman who fails to give
any convincing explanation why the highly dubious methods he accepts
should be accepted. Thus the only "argument" that Friedman
provides for using axioms of strong infinity is that there are
certain arithmetical statements  that cannot "naturally"
be proved without them, and some "core" mathematicians
were kind enough to tell him that they find these statements  interesting
(I am really sorry to say that these efforts of Friedman
to get a "Kosher" stamp from "core" mathematicians look pathetic
to me, especially when I recall that Friedman is the one who
has introduced the very important criterion of g.i.i. into the discussions
in FOM!). So assume that these statements of Friedman  are indeed "natural",
"interesting" "good mathematics" or whatever other attributes Friedman
would like to assign them, and assume that indeed every "natural" proof of them
would require some axiom of strong infinity (or really the assumption
that adding such an axiom to ZF does not lead to contradictions). So what???
Will all these facts make the truth of those statements any more absolutely
certain so to make them entitled to be called  *mathematical theorems* ??
Well - only to the degree that  we can say that the consistency
of the relevant axioms of strong infinity are entitled to be called
mathematical theorems (the truth of which is 100% certain).
Friedman's arguments are just begging the question here, and
(like "proofs" of the existence of God) are able to convince
only those who are already believers!

  Actually, the claim about "absence of any convincing explanation" is the
mild part of Friedman Polemics. Much worse is the grotesque picture
he made of predicativists like myself. From Friedman's message
I have at last understood that I belong to a group of people whose main
joy in life is to look for areas and methods of mathematics that
we might be able to cheerfully reject (for no other reason
except that we find perverse enjoyment in doing so). Unfortunately
for us, because of  Friedman there is no safe place for us anymore
to practice our  favorites activity of attacking mathematics
and its methods. Even the areas we were sure to be safe
for doing our evil (like axioms of strong infinity) are not really
safe for us as long as Friedman is around!

 So Harvey, it is about a time to tell you a secret. Predicativists
like me are not looking at all for areas and methods of mathematics
which we can safely attack. Actually, we are looking to expand as
far as possible  those areas and methods of mathematics
that we can safely *apply*!

  And what do I mean by "safely" here? Well, *my* main concern (and
this has already been hinted above) is ABSOLUTE CERTAINTY. We live in
an era in which a lot of people (as manifested by the disease
called "postmodernism") adore uncertainty and relativism, and deny the
existence of anything absolute: no absolute truth, no absolute
moral values, no absolute beauty. It is very fashionable therefore
these days to deny the existence of certainty even in Mathematics,
with great emphasis on Godel's theorems, the paradoxes of Set theory,
the existence and use of Non-Euclidean Geometries, etc.
(thus the title of one of the books I read when I was young was:
"Mathematics - the loss of Certainty" by M. Kline). There is a good
reason for this: for 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 (at least for me) is all about this question.

   At this point there are a lot of extremely important issues that
should be discussed. But this message is already too long, so I'll
do it some other time. For the time being it suffices to note that
it should be clear to anyone who understands logic that the results
mentioned by Friedman in his latest postings simply
have no significance for predicativists.
The fact that it pays to believe that something is true
(and this is what all the results described by Friedman amount to)
does not makes that something certainly true - unless one is ready
to cheat himself/herself  (which I am not). This does not
mean that these results have no value. Predicativists do not
identify "valuable" with "certain", and there are parts of mathematics
that are definitely less than certain, but still quite valuable.
But whatever value these results of Friedman have, they are
totally irrelevant to the predicativist agenda.

 Thank you, Harvey, for forcing me to explain what are the motivation
and goal of "polimicists" like me.


Arnon Avron




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