[FOM] Remedial mathematics?
Jeremy Shipley
jeremyrshipley at gmail.com
Mon May 23 22:11:44 EDT 2011
Tim, I wonder if you will indulge a philosophy graduate student who is
ignorant of many things fom by considering to the following (very
humbly and nervously submitted) comment.
I have a fondness for the following (maybe too cute) way of labeling
the positions taken by Frege and Hilbert in their old dispute over the
foundations of geometry, a dispute I have done some thinking and
writing about in the process of developing my philosophical views.
Existentialism (Frege): Existence (ie, intuition of logical and
geometric objects) precedes essence (ie, consistent axiomatic
systemization).
Essentialism (Hilbert): Essence precedes existence.
If I understand, you are suggesting that, according to mathematical
standards, de re intuitions of Z, N, etc may be used to establish
consistency results: i.e., that Frege's existentialism (if not his
logicist reduction of arithmetic) and not Hilbert's essentialism has
carried the day among mathematicians. I am not sure this conclusion
can be drawn from consideration of contexts in which pedagogical
factors are at play, in which "concrete" examples may be given because
they are psychologically prior (in developing the student's "feel" for
the definition) and not necessarily because the examples are prior
ontologically, epistemologically, or logically. In any case, it
strikes me as odd that fom practitioners should appeal to
existentialism, even if it is accepted in ordinary mathematical
reasoning. I would have thought that even if ordinary mathematics is
existentialist, most fom practitioners would be essentialist.
Otherwise why care about consistency proofs in the first place, which
find their relevance, or at least their historical motivation, within
an essentialist program?
-Jeremy
On Mon, May 23, 2011 at 10:57 AM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:
>
> I would normally consider the comments below to be too elementary to post
> to FOM, but in light of some of the recent discussions, I think it may be
> helpful to state the obvious explicitly. The point I wish to make is that
> the consistency of PA is trivial to prove by "ordinary mathematical
> means."
>
> In a first-year graduate course on commutative ring theory, one typically
> begins by stating the axioms for a commutative ring, and then giving some
> examples. The first example everyone gives is Z, the ring of integers.
> The verification that Z satisfies the axioms for a commutative ring is
> usually considered too trivial to discuss in detail.
>
> Perhaps because of its seeming triviality, some people may pass over this
> event without noticing what is going on. In not so many words, the
> lecturer or textbook writer for such a course is *proving the consistency
> of the axioms for a commutative ring*, and doing so by saying that the
> concrete example of Z satisfies all the axioms. Notably, the *existence
> of Z is taken for granted*. There is no caveat in the books saying, "If
> your philosophical prejudices permit you to believe in the existence of
> the ring of integers as a completed infinite totality, then the ring of
> integers is a model of the axioms for a commutative ring." The student is
> not being asked to assume anything other than standard mathematical facts.
>
> I repeat: Standard mathematical practice takes for granted the existence
> of Z and its basic properties without comment or reservation.
>
> Now let us consider PA, the axioms of first-order arithmetic. Since it
> is standard mathematical practice to assume the existence of Z, and a
> fortiori the existence of N, the only question is whether N satisfies the
> axioms of PA. The only axiom that could possibly create an issue is the
> induction axiom. But it is clear that a first-order formula defines a
> precise property of N, on which we can of course perform induction. So N
> is indeed an example of something that satisfies the axioms of PA. (I use
> the word "example" here to underline the analogy with the graduate class
> in commutative ring theory; of course, what is being proved is the
> consistency of PA, but some people seem to have a knee-jerk reaction that
> as soon as the word "consistency" is introduced then we're no longer doing
> mathematics but doing philosophy.)
>
> I've belabored this point because it strikes me that some of the skeptics
> of the consistency of PA aren't fully cognizant of this "trivial" proof of
> the consistency of PA. Indeed, f.o.m. experts may unwittingly encourage
> non-experts to ignore this trivial proof, by focusing on more interesting
> proofs such as Gentzen's. By citing Gentzen's proof, they may create the
> impression that the consistency of PA is, by usual mathematical standards,
> a highly non-trivial fact whose proof demands some sophisticated argument.
>
> In my view, skeptics of the consistency of PA should first of all explain
> why they are dissatisfied with the trivial proof, when under normal
> circumstances they would not only accept it without objection but might
> even regard it as being too simple to bother verifying in detail to a
> class of graduate students. I think this would be more productive then
> jumping ahead to objections to Gentzen's proof prior to addressing the
> elephant in the room.
>
> Some skeptics, like Nelson, of course have an answer---they don't believe
> in N. But this is a pretty radical position, and I think skeptics should
> be forced to acknowledge how radical a position they are taking and how
> much standard mathematical practice they are throwing out with the
> bathwater.
>
> Tim
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--
Jeremy Shipley
Ballard and Seashore Doctoral Research Fellow
Department of Philosophy
The University of Iowa
http://uiowa.academia.edu/JeremyShipley/About
jeremy-shipley at uiowa.edu
jeremyrshipley at gmail.com
847-732-4513
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