FOM: Does Mathematics Need New Axioms?

[Harvey Friedman]

This is a reply to Steel 10:46AM, 5/10/00.

I wrote

>>View A. (Most closely associated with John Steel). For the purposes of
>>the
>>main question, we should ignore distinctions between different kinds of
>>mathematics for various reasons.

Steel responded

>       If any part of mathematics needs new axioms, then mathematics needs
>new axioms. Of course, the importance of the need relates to the
>importance of the areas in need.

This inference depends on an uninteresting notion of "mathematics needs."

Would Steel say

"If any person in the U.S. needs universal health care coverage, then the
U.S. needs health care coverage"?

or

"If 'do measureable cardinals exist' needs new axioms to answer, then
mathematics needs new axioms"?

This is not what any significant number of mathematicians mean by
"mathematics needs new axioms."

Thus the question begs to be modified to become more interesting. For instance:

Does normal mathematics need new axioms?
Should the mathematics community as a whole adopt new axioms?
Should new axioms of mathematics be part of the curriculum?
etcetera

I mainly touch on the first version.

I wrote

>>Firstly, it doesn't make sense to me to disucss the main question without
>>considering the views and attitudes of the mathematics community. After
>>all, who is going to adopt new axioms?

Steel responded

>      Is the question what it would be rational to accept, or what will
>sell? I don't think we need to be doing market research on the attitudes
>of what Harvey calls "normal" or "core" mathematicians.

Understanding and investigating and contemplating and taking seriously the
overwhelming pervasive views of normal mathematicians is in no way shape or
form "market research" - at least not in the derogatory sense I think Steel
is suggesting. There can be no doubt that legitimate intellectual
perspectives, views, and attitudes are behind these strong views of normal
mathematicians, which can in fact be understood and analyzed and above all,
learned from.

In the end, one may well find *some* aspects that definitely do not appear
to be explicable in rational terms, but that can only be properly
ascertained after very careful and painstaking consideration of precisely
what these perspectives, views, and attitudes are.

Ever since I came into f.o.m. in the late 60's, I became acutely aware that
the attitudes and perspectives of normal mathematicians were at very strong
odds with what I saw in mathematical logic. I felt the same way with
respect to the attitudes and perspectives of philosophers vis a vis
mathematical logicians. This was so obviously true that I came to the
conclusion early in my career that mathematical logic was very seriously on
the wrong track - full of missed opportunities and unproductive illusions.
And I acted strongly on those perceptions.

In fact, the missed opportunities and illusions appeared to me to be so
extreme as to threaten the very survival of the subject. I now believe that
the subject will survive, but only after a struggle to "reinvent" itself -
which is actually starting to happen in earnest on several fronts.

But part of the serious damage that has been done to the subject has
emananted from this counterproductive and ill considered attitude that the
perspective of the overwhelming majority of normal mathematicians is
somehow so fundamentally misguided so as to be utterly irrelevant to the
development of mathematical logic. Nothing could be further from the truth
- and nothing can be more dangerous to its development and survival.

>I wonder what Harvey
>himself believes concerning the rationality of accepting large cardinal
>hypotheses. Was it rational to do this in, say, 1990?

The normal mathematician still has absolutely no rational reason to
seriously reconsider the axioms of mathematics - and as far as I know, none
of them are in fact seriously reconsidering the axioms of mathematics.

However, this situation is going to radically change in the near future -
but only as a result of long term research that resulted partly in an
intense consideration and understanding and appreciation of the
perspectives and attitudes of normal mathematicians.

>   If set theory needs new axioms, then mathematics needs new axioms.

This implication only holds in an uninteresting sense. If one adheres to
this implication, then the interesting question becomes, e.g.,

does normal mathematics need new axioms?

I repeat what I said earlier: obviously answering

do measureable cardinals exist?

needs new axioms. Does this mean that mathematics needs new axioms?

I wrote

>>Secondly, the distinctions are not only reasonably well defined, but also
>>are intrinsically important. There are apparently fixed fundamental views
>>and attitudes towards mathematical abstraction and generality that have
>>began in earnest in the 60's and have been strengthened since then.
>
>   How can you know views which "began in earnest in the 1960's" are
>fixed and fundamental? We need a much larger frame of reference if we
>are to discuss the foundations of all mathematics for all time.

I can know about this particularly fundamental view - about mathematical
abstraction and generality - In many convincing ways.

1. During the last 2500 years, for almost all of the time, mathematical
abstraction and generality of the kind we are talking about was never
considered, or considered legitimate. Then, starting primarily with Cantor,
we went through a period where it became  accepted as valid, and a few
mathematicians of high reputation became engaged in its development. But
after a few decades, such mathematical abstraction and generality for its
own sake (i.e., not in service of the concrete) has fallen into virtually
unanimous disrepute - even though it is still regarded as being
mathematically valid.

I will be more precise about the position of the mathematics community
regarding abstraction and generality at the end of this posting in an
APPENDIX - as there are some subtle nuances.

2. Even during this heyday for abstraction and generality from about, say,
1920 through 1960, the overwhelming portion of the most celebrated
developments by the overwhelming number of the most respected normal
mathematicians did not rely on the type of abstraction and generality we
are talking about. E.g., look at the work of Hilbert, Poincare, Weyl,
Brouwer, Pontryagin, Kolmogorov, Lefshetz, Morse, Whitney, Atiyah,
etcetera.

3. From an a priori point of view, it makes complete sense. After all,
mathematics is based on perhaps three main traditions: the
arithmetic/algebraic/geometric tradition, the science/engineering
applications tradition, the computational tradition. All of these
traditions are grounded in concreteness, where abstraction and generality
are evaluated solely in terms of how they serve the purposes of the
concrete. When abstraction and generality stop aiding the concrete, when
they create their own diffictulties, when there are no natural examples -
then the interest in the abstraction and generality dies, and normal
mathematicians go after more concrete formulations.

In fact, it is inconceivable to me that this can ever change - it is just
as ingrained in normal mathematics as much as is rigorous proof. The only
thing that is going to make normal mathematicians get seriously interested
in delicate complications concerning abstract, general formulations is not
for their own sake - but if they are necessary for concrete investigations.

Of course, that is exactly what is about to happen.

>   Let me add that I think applications are very important. The more
>concrete and pervasive, the better. Boolean relation theory may turn out
>to be a great success story for large cardinals; I hope so. If it does,
>then it may be the first application of large cardinals in obtaining some
>systematic, interesting, non-metamathematical consequences which are
>concrete enough to be "absolute", i.e. Sigma^1_2 or simpler. That would
>be an important development.

In a subsequent posting, Steel has corrected this impression that Boolean
relation may be the first ... I had many earlier results at this "absolute"
level through Borel diagonalization in the 1980's.

I can only agree with this paragraph, but it does hide a disagreement. That
without connections with the concrete, large cardinals were destined to be
thrown into the great ashcan of history - e.g., with absolutely no hiring
of specialists in large cardinals starting in the near future. The subject
of large cardinals probably will not only be saved, but sharply rise in
prominence (from near zero).

However, I should add that there is no good reason to believe that the
currently popular research directions in large cardinals today will play
any role in any groundbreaking developments that may occur of the type we
are talking about. However, at some point in the future, even the current
specialists and their students will get involved in the new developments -
even if they cling to the entirely wrong idea that their speciality did not
need to be saved in this way from extinction.

>   Nevertheless, we had strong reasons to believe that mathematics needs
>large cardinal axioms by 1970, if not before. By 1990, the evidence was
>overwhelming.

There were no reasons to believe that normal mathematics needs large
cardinal axioms - or any new axioms - by 1970. Nor by 1990. In fact, none
even yet, except it looks like that's about to change.

APPENDIX
Role of generality in mathematics

1. Abstraction and generality is embraced by normal mathematicians only
when it aides in concrete investigations.

This aiding in concrete investigations comes about by either making the
mathematics simpler or through neccessity. E.g., Lang's Algebra is written
in terms of abstract, general, notions such as arbitrary group or arbitrary
field.

But when that abstraction and generality creates its own difficulties -
usually of a set theoretic nature - then it no longer serves it original
purpose, and the normal mathematician backs off, focusing attention on more
concrete formulations. These difficulties created by strong abstraction and
generality are normally regarded as pathology by the normal mathematician -
and the detailed study of pathology is generally regarded by the normal
mathematician as among the lowest possible froms of mathematical activity.

2. A key sign of abstraction and generality of the kind that will be
readily dispensed with the moment it causes trouble - is the lack of
interesting examples.

E.g., there is a lack of interesting, varied, groups or fields of high
cardinality.  Another context: there is a lack of mathematically
interesting and mathematically varied sets of real numbers or functions of
a real variable that are not Borel. In fact, there is a lack of
mathematically interesting and mathematically varied functions of a real
variable that are not piecewise real analytic.

3. Normal mathematicians can easily sense that there is an unusual degree
of freedom in the objects being constructed.

Witness arbitrary sets of reals, or arbitrary functions from reals to
reals. Generally, normal mathematicians are only comfortable with countably
many degrees of freedom. Some hard nosed normal mathematicians are only
comfortable with finitely many degrees of freedom.

The perspective of countably many degrees of freedom is most appropriately
represented by complete separable metric spaces and Borel sets and Borel
functions in and between complete separable metric spaces.

Let me give you an example of the attitude that only finitely many degrees
of freedom are really interesting. It is an excerpt from a conversation I
had with a very famous mathematician recently called MATH. I am contacting
him for permission to quote him, but I don't want to hold up this posting
for that.

HMF: What do you think of arbitrary functions from Z into Z of several
variables?

MATH: Much too general. Too much pathology.

HMF: What do you think of integral piecewise linear functions from Z into Z
of several variables? Finitely many pieces, of course.

MATH: Well, far better, but still somewhat peculiar.

HMF: What do you think of linear functions from Z into Z of several
variables restricted to an integral halfplane?

MATH: Even better - good. You have to do something sensible with them, of
course.

HMF: What do you think of linear functions from Z into Z of several variables?

MATH: (Smile). Well, perfect. You still have to do something sensible with
them.

HMF: What do you think of functions from a finite subset of Z^k into Z?

MATH: Also perfect. But again you have to do something sensible.

*****************

An obvious question is: surely the reals are fine, and they are based on
infinitely many degrees of freedom.

But the correspondence between, say, functions on Z of several variables
and real numbers is too bizarre to transfer the interest in the latter to
the former.

Also, the conception of real numbers by many normal mathematicians is not
quite Dedekind cuts or Cauchy sequences - although this is the standard way
of completely formalizing it that is in current use. There is a kind of
geometric picture which is not really set theoretic in nature. So this
makes it also dubious to try to infer from acceptance and interest in real
numbers to, say, arbitrary functions of several variables on Z.