[FOM] The irrelevance of Friedman's polemics and results
joeshipman@aol.com
joeshipman at aol.com
Thu Jan 26 20:59:14 EST 2006
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*
??
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