FOM: GII/FII

[Moshe' Machover]

I would like to comment on the debate between Harvey Friedman and Lou V.d.
Dries on the issue of General Intellectual Interest (GII).

In the recent exchange about the New Rules for Posting and related matters,
I found myself radically opposed to HF's position; so much so that I am
seriously considering unsubscribing--in fact, this may well be my last
posting to fom.

So I hate to admit it, but I think that on the substantive issue of GII HF
is fundamentally right.

I think he is slightly harming his own case in several ways. For example, I
regard his claim--

>I see a lot of sameness in mathematics, philosophy, computer science, and
>music.  But none of these disciplines is on the right path, and they all
>have serious strengths and weaknesses.  FOM style thought reveals how to
>redirect these fields - and unify them - as well as many more.

--as a bit of hyperbole.  More seriously, I think the expression `*general*
intellectual interest' is somewhat misleading, as it tends to confuse
sociological with substantive yardsticks.  (Hence the pointless exchanges
about what barbers could understand or be interested in.  In any case,
making one's living as a barber is no indication whatever of intellectual
standards.)

I think the debate should really be about whether or not FOM is of
*profound* intellectual interest, and in particular whether it is of more
profound interest than pure maths at large.  (*General* would also be OK if
understood not in a sociological sense but in the sense of largeness of
scope.)

If the question is put in this way, then I think HF is basically (and
clearly) right.

What does pure maths at large study? It studies various structures, or as
Barwise recently reminded us, patterns. (This is *not* meant to be
anything like a full characterization of pure maths.)

What does FOM study?  It studies the nature, limits and limitations of
mathematical thought itself.  Thus it addresses a topic that in its
profundity is comparable to the following:

1. The origin and nature of the universe
2. The nature of life
3. The working of the brain
4. The nature of consciousness
5. The nature and origin of languange

In terms of profundity there is simply no contest between FOM and core
pure maths.

I would like to add a few clarifying remarks, which also bear on other
topics recently discussed in fom postings.

FOM shares its subject matter with the neighbouring discipline of PhOM.
What characterizes the former in contrast with the latter is that it
studies this common subject matter *more mathematico*, in mathematical
fashion. Thus it is mathematical thinking turned on (and in some sense
against) itself. This, BTW, makes it all the more profound intellectually.

In this sense, FOM is truly a branch of maths--a fact that some of us may
resent, but none can deny.

What characterizes mathematics in general is not only, or even mainly, its
subject matter (which can be taken or borrowed from the most diverse
sources) but its unique standard of argument: deductive and (ideally)
conclusive. Thus I disagree with Hersh's claim that

>Like other fields of knowledge, mathematics can survive with moral
>certainty.  Certainty strong enough to justify making important
>choices--but not absolute certainty, not certainty guaranteed to last for
>all time, come what may.

What distinguishes mathematics is, among other things, that it *cannot* in
principle live with anything less than absolute certainty, except as a
temporary and uneasy compromise, which even when unavoidable it regards as
an affront.  (It is this, among other things that makes the study of
mathematical thought so fundamentally important!)  No other field of human
thought can approximate or even strive to the kind of deductive certainty
often achieved and always aimed at in mathematics.

One final remark: when I conceded that math is in some sense socially
constructed, I did not have in mind primarily its subject matter (which, as
I noted above, can come from diverse sources) but its unique standard of
argument: based on deductive logic.  In my view, the norms of deductive
arguments are inculcated in us through social interaction, particularly
dialogue-argument (as claimed by Lorenzen).  Also historically, Szabo'
shows that the origins of matheamtics as a--the--deductive science go back
to dialectic argument.  The question that I then posed was how standards
that are arrived in this way can nevertheless be (as in my view they are)
objective, and ideally absolute. In my view, the question still stands.

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