[FOM] Three Types of Foundational Inquiry

[Sean C Stidd]

One enterprise which both deserves to be called 'foundational' (F1) and
which is utterly dependent on the activity of mathematicians rather than
philosophers is the attempt to come up with a general theory of
mathematics. Given existing mathematics, figure out a deductive theory
which allows one to derive all other results. Given multiple such
theories, compare them; discover relationships between them; recategorize
existing mathematics in terms of different such theories; etc. When
non-foundational mathematical work breaks new ground, use that as a
pair-wise neutral framework (if you can; usually you can't) for testing
rival foundational theories; likewise, when you've broken ground in
foundational work, see if you can get anything out of it in
non-foundational areas of mathematics. This activity is analogous to
foundational work in physics, in which physical theories of maximum
generality (quantum mechanics, general relativity, maybe string theory)
are sought which will allow one to derive all other results in terms of
the core non-foundational (phenomenological particle physics and
astrophysics, electrostatics and electrodynamics, solid-state, p-chem,
etc.) areas. Developments in each push on the other in analogous ways.
Not much for philosophers to do here.

Another kind of work, which also has some claim to the title
'foundational' (F2) and to which philosophers (at least, mathematically,
logically, and perhaps historically (psychologically?!) very
well-informed philosophers) could make a contribution, is the analysis of
the basic concepts and principles of reasoning in play in actual advanced
mathematical and scientific theorizing. In the 21st century this kind of
work has become frighteningly hard, but it's still there to be done.
Possibly a philosophically astute mathematician would do better here too
though.

A third sort of work which also has some claim to the title
'foundational' (F3), which I like but which leaves many mathematicians
and some philosophers bored or irritated, involves examining the
ontological, epistemic, and semantic
structures/commitments/implications/presuppositions of various
mathematical theories and/or fragments of 'natural' mathematical
language. The answers to virtually all these questions are
underdetermined by the mathematical theories of which they are asked, at
least insofar as those theories are understood as self-contained calculi.
Work in these areas requires a reasonably good but not
professional-in-foundations-in-sense-1 mastery of logic, set theory,
model theory, and a whole lot of knowledge of philosophy.

There are in my view important connections between these three subjects,
and there are points at which they shade off into one another, but they
are not the same kind of study.

This list tends to be tolerant of F3 up to a point, but mostly is
interested in F3 for its potential heuristic and/or theoretical value for
F1. Likewise, when F2 work is done on set theory and other foundational
theories, the list tends to have some interest; when F2 work is done on
non-F1 theories, some list members get very irritated. F1 work is
generally immediately recognized as of value and interest and as
'belonging' here.

Assuming these sociological observations are correct, then we have a
clear sense based on past history of what the list is: it's a tolerant F1
list, in the sense that concerns not directly part of F1 but relevant to
it are kosher, but participants will tend to want such concerns to 'pay
their way' in terms of F1, and if they don't, they'll probably come to
regard them as irritating, off-topic, grandstanding, etc.. 

At the risk of being torn to pieces by the competing parties, I will now
venture a thesis: I think that some of the arguments which have broken
out here in recent months come from participants either (a) applying
valid standards from one of the three areas to another, where their
status is murkier or (b) not taking into account the dominant interest of
many list participants (and many of the most famous list participants in
particular) in F1.