[FOM] Three Types of Foundational Inquiry

Harvey Friedman friedman at math.ohio-state.edu
Sat Oct 11 05:34:18 EDT 2003

Reply to Stidd. This is an interesting attempt to distinguish three kinds of
"foundational" activity, with the suggestion that f.o.m. researchers
generally consider or value only some of the three.

I want to adhere to a unified view of f.o.m. To be persuaded otherwise, I
will at least need clarification, as indicated below.

On 10/10/03 6:01 PM, "Sean C Stidd" <sean.stidd at juno.com> wrote:

> 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.

I take this to refer to, say, the development of ZFC and significant
fragments of ZFC, and understand their relationships in terms of
conservative extension results, relative consistency, interpretations, etc.
I take this to include reverse mathematics.

> 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;

I am less clear about what this refers to. For example, there is a body of
long discarded "work" using infinitesimals. This work was interpreted in a
rather interesting way by Abraham Robinson (nonstandard analysis), and also
has been subjected to a more direct proof theoretic analysis (myself,
Avigad, Sommer, etc.). Here again multiple systems are compared in
interesting ways through conservative extension results, relative
consistency, interpretations, etc.

There is the work in the first half of the 20th century in descriptive set
theory, which ran into a block at fairly low levels of the projective
hierarchy, and also went out of fashion in favor of the inexorable trend in
20th century mathematics towards concreteness. Explanations for the impasses
where given by set theorists through various independence results starting
with Godel.

>likewise, when you've broken ground in
> foundational work, see if you can get anything out of it in
> non-foundational areas of mathematics. ... Developments in each push on the
>other in analogous ways.

For example, large cardinals break new ground in foundational work. There
was a big, successful, effort to get something out of it in descriptive set
theory - in the projective hierarchy. (Martin, Steel, Woodin, Kechris,
Steel, Solovay, and others, names given in totally random order).

The rejection of the projective hierarchy by the mathematics community as
not concrete ensures the *historical* importance of getting something out of
large cardinals in concrete mathematics. (The *intrinsic* importance of
doing this is obvious).

> Not much for philosophers to do here.

I agree that there is

not much for philosophers to do here ALONE.

But I don't agree with your assessment without the word "ALONE".

> 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.

Do you want to emphasize the word


here? I think that the hard early question in this sort of thing is to
identify where in "advanced" mathematics, there is a real prospect of saying
something of substance and quality here that is not essentially mere
informed commentary.

Reverse mathematics at least has the potential to be a kind of universal
setup for being able to state at least some deep and significant features of
"advanced" mathematics, that goes far beyond commentary. However, some
additional advances I think are needed in RM in order to really be able to
do this properly. I think that this can, and will be done, but it hasn't

Also, John Baldwin could comment on the extent to which applied model theory
can do this, or can hope to do this.

But as I said previously on the FOM, in general I think that the scope of
f.o.m. in terms of dealing with more and more significant features of actual
mathematics, is moving along as an entirely appropriate, deliberate, speed.
Forced acceleration will undoubtedly prove nonproductive.
> 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.

This is the one that I am least clear about. Please give some examples of
this kind of work, especially since you indicate that you work in this, or
at least "like" it.
> 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.

I would prefer a unified view of f.o.m. E.g., for F1 and F2, I think that a
unified view is workable. Such a unified view does not preclude the drawing
of conclusions about things being premature, things being only commentary,
> 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. 

I would like some examples.

>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.

Give an example of F2 work in non-F1 theories.
> 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..

Since I want to adhere to a unified view of f.o.m., I need some examples.
For example, I would like to explain what you might call "not paying its way
in terms of F1" in other terms like "being descriptive", "being anecdotal",
"being observational", "being commentary", 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.

This needs examples.

Harvey Friedman

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