FOM: Foundational issue

[Harvey Friedman]

I have been interviewing a variety of mathematicians outside mathematical
logic about foundational issues raised by recent work. Attached is a copy
of a letter I have sent to them.

***********

Thanks to all of my nonlogician friends for your patient feedback about the
logical issues that seem to be emerging in normal mathematics (in the
setting of sets of integers). Any further comments from the perspective of
the working mathematician would be greatly appreciated.

Specifically, those logical issues arising from the demonstrably necessary
and sufficient use of additional (proposed) axioms for mathematics for
proving theorems of a down to earth universally accessible and attractive
character, which form part of an extensive theory that cuts accross
virtually every area of mathematics. A tentative name for this theory is

*Boolean relation theory*

I think that the conceptual issues make perfectly good sense before any
formal announcement of the details of this theory. In fact, the conceptual
issues have been around for some time, without a truly convincing real
world down to earth normal mathematical context. The expectation is that
Boolean relation theory provides such a context.

After talking to many mathematicians, it has become clear that a common
reaction to such a development would be to view any proposed new axiom
being used as merely an extra hypothesis in theorems. I.e., if XXX is the
proposed new axiom, then write

THEOREM (XXX). Blah blah blah.

Specifically, several mathematicians felt that this new situation would not
be much different from the present situation where we often see

THEOREM (Riemann hypothesis). Blah blah blah.

>From this point of view, many of you predicted, in essence, that the
mathematics community

i) would not be compelled to engage in a discussion of why X should be
accepted as a new axiom. E.g., they don't engage in a discussion of whether
RH should be accepted as a new axiom;

ii) would not be compelled to engage in a discussion of objective truth in
mathematics; e.g., comparing it to that in other subjects.

I have some comments on your reactions that are much clearer than what I
was able to enunciate during our conversation.

The first point is that in the course of the history of mathematics, we
have seen the persistent use of

THEOREM (Axiom of Choice). Blah blah blah.

Nowadays, use of the Axiom of Choice is not mentioned, since the Axiom of
Choice has long since attained the status of a new axiom. On the other
hand, it seems pretty obvious that the Riemann Hypothesis is not going to
attain such a status. An important philosophical question is: why?

So for some XXX, we still see (XXX), where for other XXX, we never see
(XXX). The question is: what is going to be the fate of the "large
cardinal" assumptions used in Boolean relation theory? Are they forever
going to be enclosed in parentheses, or are they going to get the preferred
treatment of, say, the Axiom of Choice?

As normally stated, these "large cardinal" assumptions assert the existence
of a well ordering satisfying certain combinatorial properties. It is well
known that such a well ordering must be of enormous size - incomparably
larger than anything that has appeared as yet in normal mathematics.

So even if (XXX) is always used for XXX = the "large cardinal" assumption,
then mathematicians will be facing the unprecedented prospect of
manipulating objects of a totally different character than any objects
previously manipulated by them in the course of doing normal mathematics.

[Of course, the force of this discussion is predicated on the expectation
that Boolean relation theory will be accepted as an important and
attractive area of normal mathematics. This remains to be tested; e.g., by
the appearance of an AMS Classification number for Boolean relation theory
by the end of this decade].

In the course of manipulating such unprecedented mathematical objects for
the development of Boolean relation theory, many mathematicians will
undoubtedly feel sufficiently uncomfortable as to inquire whether

1) such objects really do exist;
2) for mathematicians who are not comfortable with 1): does the use of such
objects lead to a contradiction?

Presently, mathematicians are logically "coddled." In the course of doing
normal mathematics, mathematicians use only a minimal fragment of the usual
axioms of set theory (say ZFC), and these axioms - particularly the minimal
fragment that is used in normal mathematics - are particularly intuitively
clear and natural. So issues 1) and 2) are no longer serious issues for the
working mathematician.

But the large cardinal assumptions used for Boolean relation theory
apparently cannot be viewed as intuitively clear and natural in the same
way - at least on the basis of the mathematical intuitions of present day
working  mathematicians. This is apprarent since the working mathematician
has not previously encountered such foreign objects, let alone manipulated
them in the course of doing normal mathematics.

What are the factors surrounding a decision by the general mathematical
community as to whether to accept the relevant large cardinal assumptions
as axioms - manifested by not mentioning them as hypotheses in theorems as
is the case with the Axiom of Choice?

For example, will persistent successful uses of them in the anticipated far
ranging extensive development of Boolean relation theory - and perhaps
beyond - be sufficient to elevate their status as new axioms? Or will
mathematicians instead fixate on perceived differences they have with the
usual axioms of mathematics - the ones that currently are used in normal
mathematics or the bigger set that is formalized by ZFC?

The large cardinal axiom used for Boolean relation theory can be replaced
by other assumptions of a somewhat different character. However, the large
cardinal assumption referred to here is the only assumption sufficient for
the development of Boolean relation theory (in its present embryonic
development) that is presently being offered up with the status of a new
axiom for mathematics by the set theory community.

For instance, the large cardinal assumption can be replaced by an
assumption which involves "only" real valued functions on the space of all
subsets of the closed unit interval. This is remarkably concrete when
compared to the large cardinal axiom used for Boolean relation theory.

This assumption is that

*there exists a countably additive probability measure on all subsets of
the closed unit interval that extends Lebesgue probability measure*

This assumption is obviously couched in terms that are familiar to
analysts. In fact, it is known that this assumption is far stronger than
the large cardinal assumption used in Boolean relation theory - in the
subtle sense that it is known to have - theoretically -  far more normal
mathematical consequences.

However, one serious obstacle to the adoption of such an axiom is that
there appears to be no way to appropriately describe the probability
measure that is asserted to exist. However, this is not a fatal objection
to its adoption, since the Axiom of Choice also has this attribute.
Nevertheless, the large cardinal assumption used in Boolean relation theory
is not subject to this objection.

And we repeat that the set theorists are not proposing adoption of the
existence of this measure as a new axiom for mathematics, but are proposing
the relevant large cardinal assumption referred to earlier as a new axiom
for mathematics.

FOOTNOTE: How much of the Axiom of Choice is actually used in normal
mathematics? The standard answer is - countable dependent choice. In fact,
for really normal mathematics at the normality level of, say, Boolean
relation theory, it is known that the Axiom of Choice can be dispensed with
completely. END.