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

[joeshipman@aol.com]

Reply to Avron:
I think you are misreading Friedman's purpose.

Friedman is always careful to distinguish "Theorems" from 
"Propositions" in announcing his results, and his "Theorems" are proved 
in very weak systems and are as "absolutely certain" as you would like. 
His "Propositions" are relatively "natural" mathematical statements 
that are equivalent to large-cardinal consistency statements.

He is NOT arguing that the Propositions ought to be accepted as true. 
What he IS claiming is that mainstream mathematicians OUGHT TO BE 
disturbed by the failure of mathematics to settle such natural 
statements, and ought to RESPOND by having a serious discussion about 
what additional axioms ought to be accepted in order to enable progress 
to be made on such natural statements.

What Friedman is criticizing is the determination of most 
mathematicians to regard Godel's Incompleteness phenomenon as a 
curiosity that is not relevant to mathematics as a whole, rather than 
as a challenge to get involved in the METAmathematical pursuit if 
identifying new axioms to be accepted as true. He's not claiming 
that"usefulness" is a reason to accept, for example, the 1-consistency 
of n-Mahlo cardinals as "true"; but it IS a reason for preferring that 
statement to its negation, and the point is you must take SOME stance 
or have no hope of resolving some rather interesting and concrete 
questions.

Between 1800 and 1940, the process of evaluating new mathematical 
axioms and principles of reasoning was a productive part of 
mathematics.  Friedman seems to think that since then, mathematicians 
have mostly abandoned that activity and imagine that they know all the 
principles of correct mathematical reasoning and all the axioms they 
need in order to do their professional work, and he aims to change this.

I see two possible exceptions to the view that mathematicians have 
stopped working on identifying new axioms and principles of reasoning. 
One exception is the introduction of Grothendieck Universes and related 
categorical techniques in the 60's, and another is the use of 
"probabilistic proofs" in computational areas. Unlike the Axioms of 
Infinity, Choice, and Replacement, which were advances that became 
broadly accepted by the mathematical communtiy, neither of these two 
exceptions has been broadly accepted, but both of them have been used 
to expand the collection of mathematical statements generally regarded 
as "true".

-- JS


(Excerpt I am replying to:)
>>   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* 
??