FOM: Comments on 'Does mathematics need new axioms'

[Torkel Franzen]

  In the little treasure trove of papers that Solomon Feferman gave us
a pointer to (not unlike coming across a hidden chest containing a
Flaming Sword, two red jewels and a scroll with a EW DAR ES spell in
Dungeon Master II) he specifically mentioned 'Does mathematics need
new axioms', so I first looked at this paper.

  It's the text of a lecture aimed at general mathematical audience,
thus semi-popular in character. It's a bit top-heavy, in that the
first twenty-five pages are a very readable and accessible
presentation of developments regarding new axioms from Zermelo via
Godel to Woodin et al, whereas the last five or six pages rather
abruptly turn to a critical discussion of Friedman's approach to new
axioms, and of the Platonistic outlook.

  His basic criticism of Friedman's approach is that his candidates
for combinatorial statements that require Mahlo axioms or stronger
principles for their proof aren't rooted in ordinary mathematical
reasoning. He also observes, more generally, that no previously
formulated open problem from finite combinatorics is known to require
strong principles for its solution. He suggests that the continuum
problem, which is not settled by any proposed axiom, doesn't have any
solution, whereas Goldbach's conjecture etc will most likely be
settled, if at all, by ordinary mathematical reasoning. Finally he
comments on how work in proof theory has more or less established that
only weak formal systems (specifically PA) suffice to formalize the
mathematics used in science.

  The above is not intended as a presentation of the argument
of the lecture, but only as an indication of what understanding of that
argument my further comments are based on.

  The 'wide appeal' of foundations has been mentioned on the list.
Certainly the basic fact that set-theoretical axioms can settle finite
combinatorial problems that are not settled on the basis of any known
evident combinatorial axioms is immediately interesting to
philosophers and others. It's a fact that seems to call for some sort
of philosophical explanation, whether mathematics 'needs' these
axioms or not.

  From a mathematical point of view, an obvious question to ask is if
old or new set-theoretical axioms settle any interesting combinatorial
problems. That is, "do we need them"? The appeal of this question is
not as wide as that of the more general question how the connection
'in principle' between finite combinatorics and infinite structures is
to be understood. As a philosophical outsider to the logical
investigation of this question, I'm content to wait for whatever
Friedman and others come up with. I'm in no position to say whether any
striking combinatorial use - with a 'wide appeal' among mathematicians -
of large cardinals, or other new axioms, will ever appear, but can only
note that none has been presented to me as yet.

  There is one thing, though. At the beginning of the talk, Feferman
notes that the question of the title is "ambiguous in practically
every respect". In the body of the talk he does not, however, raise
the question whether mathematics might "need" new axioms, not in
a logical sense, but in a "more subtle" sense, as Kreisel has it.
He does mention "practical need" towards the end, commenting that
"I see no evidence for the practical need for new axioms to settle
open arithmetical and finite combinatorial problems", and mentions
the solution of the Fermat problem. But isn't the solution of that
problem on the contrary good evidence that there is a practical
need for using lots of abstract mathematics to solve arithmetical
problems, even if later investigations (which as far as I know haven't
yet been carried out for Wiles's proof) show that the problem can
in principle be solved by a proof in a weak arithmetical theory?
Similarly with the closing argument about PA and physical theory:
is there any evidence that PA can be used in physics, not only for
formalization in principle, but for practical reasoning?

  I don't actually have any firm opinions about whether this is so,
but only wish to draw attention to a weak spot in the presentation,
having to do with the ambiguity of "needs".

  Feferman's closing comments in the talk concern a particular point
on which I wholeheartedly agree, namely the naturalness of a
Zermelo-style formulation of comprehension principles and other
schemata. The arithmetical induction principle is, I believe, most
naturally understood neither as a set-theoretical principle, nor as a
schema tied to some particular language, but as the informal principle
that "whatever is true of 0 and is true of n+1 whenever it is true of
n is true of every natural number". On the basis of this informal
principle, we obtain an induction schema for any given language, as
long as we believe we understand what we're talking about in that
language. This does not presuppose that there is any well-defined
totality of things you can say about the natural numbers or of
totalities of natural numbers.