FOM: what sort of foundations
srf1 at st-andrews.ac.uk
Tue Nov 4 16:47:24 EST 1997
Before I begin, let me say a couple of words about myself - I'm a PhD
student working on the philosophy of mathematics trying to find a
principled way of giving an account of structure which coheres with
Frege's semantic analysis of mathematics. If you want to know more, check
out my website: I have some papers there.
I've been having a hard time trying to work out what is being discussed
here. I've always thought of there being at least two notions of
foundation, and tried to keep them separate. One notion is epistemological
- it gives us the idea that we investigate the foundations of mathematics
to find out upon what our mathematics is based, and we hope to secure self
evident and true foundations as a rock upon which to build. This sort of
approach goes hand in hand witha foundationalist view of knowledge - that
things build up froma secure base (eg perceptions, self reflection,
memories and some logic) to give us all of our knowledge. It is the type
of picture of knowledge that philosophers have spent this century taking
to pieces. (It is all supposed to be Descartes' fault).
The second picture, is trying to carry out some sort of conceptual
analysis - to get clearer what our basic notions are, and hopefully to do
something fruitful when we do it.It is driven solely by intellectual
curiosity, not doubt or desire to remove paradox.
It seems to be the second picture that we are really concerned with - even
if it was the first type of enquiry (however spurious the philosophical
motivations which drove it were) which got us started in this business.
If this is the case, why should we be so certain that all mathematics
comes from a commmon way of thinking? For example, the kind of
representation we try to capture doing arithmetic is quite different from
the kind of representation we do when we do group theory. The objects of
arihtmetic appear to be particular objects, i.e. numbers; the elements of
a group seem to be general or arbitrary objects, always open for
instantiation. Sure, we can represent each by way of set theory, and try
to cash out this difference by talking about categoricity, but this does
not seem to be adequate in terms of the phenomenonlogy (if I may use such
a word!) i.e. it does not capture what a mathematician thinks he is doing
when he is doing number theory,a s opposed to when doing group theory.
This was the sort of objection that Poincare raised against trying to
reduce mathematics to logic - he was happy to admit that it could be done,
but sceptical that what you got out at the end would preserve the "felt
quality" of doing matheamtics.
So we seem to be looking for a bunch of (possibly different) things which
will underwrite the different areas of mathematics. (Sure they may turn
out to be the same, but lets not just assume that). What sorts of things
will these be?
We start out with a certain conception of what we are doing - and then try
to express that. For example, we have a notion of what it is we are doing
when we do arithmetic. What makes the Peano axioms a faithful
representaiton of this notion? If we found contradictions in the
expression of such axioms, then we would not stop doing arithmetic - but
look for a better expression of the conception which drives our
arithmetic. We never question this conception, only ever the articulations
of it. Are the axioms good enough? Do they capture what we want, etc.
Take group theory - we know that there was a common thread which became
obvious, linking geomtry, roots of equations and permutations together,
which led Caley to come up with group axioms. What made the axioms right?
We justify that such axioms faithfully capture a more basic notion in an
essentially non-inferential manner. Once we accept the axioms, they then
dictate what counts as a group, and what dopes not - they become normativ,
rather than descriptive; they then reshape our intuitive conception.
SO in this sense, the foundations are those hinges which we accept, those
things we use to open the door to enquiry. We do not question them, but
set them up as dictating what the standards of acceptability will be.
The debate about FOL/ SOL, as well as debates about constructivism and
relevance, all turn on trying to cash out a basic notion of proof or
consequence. Part of the problem is that there is such a barage of
considerations which need to be included, before a formalisation will cut
the mustard. Need there be the same logic in all areas? Would it not be
possible to use a minimal notion across all our mathematical discourses,
and then have that notion beef up in some areas where for example, there
was a greater degree of decidability, or the subject matter has a certain
shape, etc. (There is something like this comes out of considerations of
logic ina category - generally it turns out to be intuitionistic, but in
*Set* and other well behaved categories, it is classical.)
SO sum it all up - I think there are two notions of foundations at play:
the first is epistemological, and generally uninteresting. The second is
conceptual - it relates partly to what people were trying to get at by
talking about barbers and scientific american readers, but it also touches
on the thought that we do not question parts of our conceptual framework,
because they compose our standards of assessment, so cannot themselves be
assessed. We need to take some things for granted, before we can even
begin to do any conceptual enquiry. Once off the ground, then we can go
back and re-examine them, but only in the context of our broader
epistemological goals - which is why I distinguish this notion from the
Cartesian notion of epistemological foundations.
I realise this is very skeletal, but I hope that at least some of you have
some sympathy for this style of approach.
Stephen Ferguson http://www.st-and.ac.uk/~www_spa/STUDENTS/srf1
27, North St, St Andrews, KY16 9PW
Logic and Metaphysics, Univ of St Andrews
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