[FOM] Re: Three Types of Foundational Inquiry

[Harvey Friedman]

Reply to Stidd.

On 10/23/03 11:10 AM, "sean.stidd at juno.com" <sean.stidd at juno.com> wrote:

> 1. What is the best overall theory of mathematics as a whole?
> 
> 2. What are the basic concepts of mathematics? (Perhaps: at various different
> levels of mathematical practice.)
> 
> 3. How should we understand the semantics, epistemology, and metaphysics of
> mathematics?
> 
> I think (FWIW) that you are right to see 1 and 2 (and I would, at least, hold
> out hope for 3) as a unified undertaking, which makes my original assertion
> false. What I think was right about what I was trying to say is that
> sometimes, when 'head butting' arguments break out on the list (each side
> repeating the same points to one another without seeming progress being made),
> it is because people hold incompatible commitments either in general
> philosophy or philosophy of mathematics, that prevent them from acknowledging
> what seem like straightforward points being made on the other side.

You could do some real good if you could distill what these incompatible
commitments are that you speak of in general philosophy or philosophy of
mathematics. 

It will certainly be news to me if you can find some defensible position
held among the people that I argue with on the FOM who are very slow to pay
any attention to the content of my postings. In every case that I can think
of right now  - where they pay no attention to the content of my postings -
there is no defensible rationale behind what they write. And if you can
state just what this is, that would be very interesting news to me.

I probably disagree in various ways with just about any of the approximately
720 (yes 720!) people on the subscriber page. However, most of them do not
post, and most of them would actually pay attention to the content of my
responses. 

So as a test case of your ability to ferret out the essence of
disagreements, look at the next couple of nontechnical postings of mine that
will be seriously critical of some existing postings.

>(For 
> example, the Dummettian intuitionist has semantic views which prevent him or
> her from accepting all of 'core' mathematics, and so prefers a slenderer
> foundation. 

A competent Dummettian intuitionist would put this up front, and the
discussion should then circle around just what core mathematics is or is not
valid, and why, etcetera. There should be no problem with a competent such
person.

By the way, mainstream f.o.m. has many many slender systems, in particular,
lots of intuitionist systems. I don't see any problem I might have with a
competent Dummettian intuitionist. Something productive would undoubtedly
come out of it. 

>Someone who thinks that all concepts need to be articulated in
> something like everyday English in order to be coherent or comprehensible

This bizarre position, going against the basic methodology of all of
mathematics, science, engineering, and technology, could still be explicitly
defended by competent people - e.g., by explicitly showing how mathematics,
science, engineering, technology could be recast this way - and still
thrive. I doubt if anybody can do that - and that is the reason why natural
language is replaced by the useful languages for these enterprises that have
been developed by their practitioners. However, if someone knows nothing
about mathematics, science, engineering, technology, then they will find
such a recasting - even if it were possible, which I doubt - doubly
impossible. They wouldn't even be familiar with the myriad cases where
difficulties arise that cannot be overcome.

I don't know what you think of my "put up or shut up" style on the FOM, but
it certainly isn't symmetric. You notice that I do "put up".

Are you now going to tell us that the very idea that it is OK to "not put
up" and "not shut up" is also something that is a matter of one's point of
view?

>I don't really know if it would be helpful if people were clearer
> about their philosophical commitments up front when that type of head-butting
> dispute breaks out,

Of course it would be helpful. You can play a major role in the FOM if you
could analyze FOM disputes in this way in real time.

This would greatly help me in my garbage collection activities. I am the
head garbage collector on the FOM in terms of recorded volume of activity in
this capacity, and need all the help I can get from reasonable people - I
hope you are one of them.
 
> Your approach to these kind of disputes, which seems to me to be to try to
> turn the conceptual or semantic/epistemic/metaphysical disagreements into
> disagreements that have some kind of fom-theoretical (F1) content, seems to me
> to be a good one.

I am committed to genuine intellectual productivity, as are mathematicians,
scientists, and academics in engineering and technology. This also includes
many philosophers. 

If anyone has any new approach towards semantics/epistemology/metaphysics
relevant to f.o.m., that is productive, you will find me completely
receptive - even participatory.

>But some people will resist that kind of approach because
> they think the answers to questions of type 2 or 3 need somehow to be settled
> before we can have 'real convinction' in our mathematical theories, or that
> certain positions about conceptual or philosophical problems are so obviously
> the right ones that they rule out certain kinds of talk about mathematics from
> the very beginning.

The issue of "real conviction" is productively handled by the enormous and
impressive hierarchy that has been built up by mainstream f.o.m., ranging
from extremely minimal commitments to extremely massive commitments. Sure,
it is true that the absolute extremes, particularly at the minimal end of
the spectrum, are not well studied - perhaps not well formulated. However,
there is nothing preventing them from being well formulated and well studied
according to the experience, knowledge, and methods of mainstream f.o.m.

You may have W in mind here. However, on the FOM, as opposed perhaps to
elsewhere, many people such as myself will apply some intellectual standards
to W ism. 

>On the other hand, demon-world skepticism is still with us
> and (at least to some thoughtful minds) unresolved despite Descartes' efforts,
> whereas analytic geometry is part of the secondary school curriculum.

Standard f.o.m. can also deal productively with demon-world skepticism - or,
for that matter, any issue regarding the nature of mathematical thought.

The great minds from the past have left us a great legacy in the form of
mainstream f.o.m. For many purposes, it is apparently the only game in town.

Harvey Friedman