[FOM] Re: pom/fom. directions

Jon Williamson jon.williamson at kcl.ac.uk
Mon Oct 27 11:28:04 EST 2003

> > Corfield thinks that philosophers of maths should study and understand
> > maths;
> Friedman thinks that philosophers of mathematics should concentrate on
> parts of mathematics for which it is promising to recast them so that they
> attain (a higher level of) general intellectual interest, and/or relate to
> clear philosophical issues.

Friedman writes,

> Before starting on a restatement of my position, let me first say that my
> notion of general intellectual interest does not correspond to items that
> have been successfully or partly successfully popularized. I mean
> quite different than that - but not unrelated - as I shall explain below.

and gives the following paradigmatic cases:

> 1. 1951, 1959. A. Tarski. Decision procedure and completeness of axioms
> elementary Euclidean geometry, real algebra, and complex algebra. A
> general solution to all high school plane geometry problems, as well as
> high school algebra problems involving real numbers, but not integers.
> 2. 1963. PJ Cohen. The unprovability of the axiom of choice from ZF, and
> unprovability of the continuum hypothesis from ZFC. The gii is clear from
> lot of directions. Furthermore, it is closely allied to 1930's
> that have a very well received gii. For very wide audiences, there is the
> story about how the 'rules of the game' that mathematicians have accepted
> for so long are shown to be insufficient for settling some well known
> problems  - raising the issue of an appropriate expansion of the 'rules of
> the game', in the one subject where it was thought that the 'rules of
> will always be the same - mathematics.
> 3. 1970. Y. Matiyasevich, J. Robinson, M. Davis, H. Putnam. No algorithm
> solving all Diophantine equations. Again, closely allied to 1930's
> developments that have a very well received gii. I.e., Turing's model of
> computation, and its robustness. Also, a timeless odyssey in mathematics
> called Diophantine equations.

I certainly believe that these developments were immensely interesting and
important for FOM, as you do and as I'm sure most members of this forum do.
But I fail to see why they count as being of general intellectual interest
(gii). You don't mean as much as `of interest to the general educated
population' by gii, but surely you mean more than `of interest to FOMers'?
(If not then I think you're unlikely to persuade Corfield or indeed any
nonFOMer.) By gii I mean something broader than `of interest to FOMers'.

Although these developments are closely related to pre-1940's FOM
developments that, we agree, were of gii, I don't think that qualifies them
as being of gii themselves. The general interest of the earlier work in FOM
arose from general interest in computability and in logicism and formalism
in the philosophy of maths. By the 1940s logicism and formalism were
generally viewed as having been refuted, and by the 1950s the computer had
taken something like its current form and there was more general awareness
of the limits of computing (and by 1970s there was a backlash against
earlier bold claims about the achievements computers would yield). I think
that explains why the developments you cite, though for FOM perhaps as
important as earlier work, were of less interest outside FOM.

> One interesting comparison would be between 3 and FLT.
> a. FLT and subsequent offshoots are of incomparably greater interest among
> number theorists than 3. Also among mathematicians generally, FLT and
> subsequent offshoots are of greater interest than 3.
> b. However, in terms of gii, 3 ranks significantly higher than FLT and its
> subsequent offshoots. Popular accounts of FLT and subsequent developments,
> even accounts to scholars outside mathematics, rely heavily on the
> historical and personal challenge, along with the fact that a lot of other
> mathematics gets used. However, such an account does not deal directly
> the intellectual content of FLT itself. 3 has the totally honest tie in
> great events from the 1930's of long recognized gii - models of
> and impossibility theorems.

I would say the proof of FLT achieved gii because:
(i) the statement of FLT is simple enough for non-mathematicians to
(ii) it was the culmination of a great story and intellectual challenge
(iii) even mathematicians approved of it because it yielded plenty of
interesting new maths - the proof did involve significant intellectual

Godel's incompleteness theorems achieved gii because:
(i) the lay-person could understand their significance
(ii) they were the culmination of a great intellectual challenge in the
philosophy of maths
(iii) they were mathematically fruitful, setting a whole new agenda in logic
and the foundations of maths.

However 3 did not achieve gii for the reasons suggested above.

> > 2. how should one delimit `traditional philosophical interest'? The
> > philosophical interests of the Pythagorean school? Of Plato? Of Kant? In
> > case, why be so backward-looking?
> I don't care. Anything that makes clear philosophical sense. By
> "traditional" I just meant "makes clear philosophical sense by traditional
> philosophical standards".

"Clear philosophical sense" seems to be much more worthy a goal of
philosophers of maths than the pursuit of gii, which strikes me as a very
nebulous and ephemeral concept. I take it you agree with Corfield then that
questions like
"how do concepts get formed and adopted in maths?"
"can Bayesianism be applied to mathematical beliefs?"
"what is the nature of mathematical research programmes?"
all make clear philosophical sense and are worthy of addressing, whether or
not that involves work in FOM.

all the best,

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