FOM: Mathematics as objective subjectivity: ten theses.

[Solomon Feferman]

The following is part of an answer I have been trying to work out for some
time to my questions: What makes mathematics such a distinctive body of
thought?  What is it about its conceptual content and verificational
structure that distinguishes it?  How can it be objective in its judgments
if it is subjective in its source [as I am convinced it is]?  How can
mathematical ideas be shared?

My first attempt to answer this in public was made in a talk I gave under
the title "Mathematics as objective subjectivity" for the Philosophy Dept.
Colloquium at Columbia University on Dec. 9, 1977.  The text of that
lecture, with some elaborative notes, has been circulated to a few of my
colleagues but never published, because it never reached a form that I was
fully satisfied with.  Nevertheless I still hew to much of what I said
there, and think it worthwhile to quote a list of 10 theses from the talk
which bear reference to recent discussions in this list, particularly by
Machover, Tragesser and Hersh. I am not trying here to respond directly to
those postings.  I am also not prepared to defend fully each and every one
of the following theses, but thought it better to quote them as I gave
them in 1977, not as I might try to revise them now.  

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1. Mathematics consists in reasoning about more or less clearly and
coherently groups of objects which exist only in our imagination.

2. The reasoning of mathematics is logical, but mathematics is not the
same as logic since logic concerns the nature of correct reasoning applied
to any subject matter, whether or not that is clearly conceived.

3. The general concepts of property, of relation, of rule and of operation
are pre-mathematical, lying alongside of logic.

4. Basic mathematical conceptions are not of objects in isolation but of
structures which consist of certain kinds of objects interconnected by
relations and operations.  These have partial but universal correlates in
everyday experience.

5. The contents of our imagination can be communicated to others; the
features of the imagination can be delineated and scrutinized.  Under
examination, what is private and subjective becomes public and objective.  
The _objectively subjective_ is that which can be communicated and
confirmed in this way.

6. While reasoning is our only known path to secure mathematical truth,
there are objective questions of truth and falsity.

7. Some mathematical conceptions are clearer than others; which these are
is both person-dependent and historical-time-dependent.  In particular,
the individual imagination can be expanded and refined by training.

8. One should distinguish those concepts which are directly present to
the imagination from those which are derived by reflection and thus
reduced to the former.

9. In the development of mathematics there has been a continual effort to
replace less clear concepts by clearer ones.  This has been accomplished
both by the process of reduction and by their elimination in favor of
alternative concepts. 

10. This process of conceptual clarification is still necessarily in
progress.  In particular, there are basic concepts of current mathematics
which are insufficiently clear.  It is the present task of the foundations
of mathematics to seek means of clarifying these concepts and, failing
that, to search for viable alternatives.

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You can imagine why my alternative title for this was "Mathematics in the
imagination".  Naturally, there have been many objections to the central
use of the word 'imagination' here.  What I mean is how things are
conceived.  Here are the paradigm examples.  (This is not be quotation
from the original text, where these were elaborated.)

I. The positive integers are conceived within the structure of objects
obtained from an initial object by unlimited iteration of its adjunction,
e.g. 1, 11, 111, 1111, .... , under the operation of successor.

II. Euclidean geometry is conceived to be about our conception of
perfectly fine points, perfectly straight lines, perfectly flat planes,
related by incidence, betweeness, equidistance, etc.

III. Set theory is supposed to be about the cumulative hierarchy,
conceived as the transfinite iteration of the power set operation. At
base that depends on the conception of the totality of arbitrary subsets
of any given set under the membership relation.

In I, the operations of addition, multiplication, etc. are derived by
reflection on the basic structure directly evident to us.  Thus the
meaning of numbers defined, e.g., by iterated exponentiation is not
directly evident to us but is only accessed through its reduction to the
bare-bones structure.

II is less clear than I, and III less clear than II.  Clarity aside, in
each case some principles are immediately evident and can serve as axioms.  
Surprisingly much follows from few such statements.  This is another
example of why "a little bit goes a long way" in mathematics, and how it
is we can reach firm conclusions about objects (e.g. sets) with an
indefinite extent.  For me, the conception I is the clearest of all
mathematical concepts, and statements about the positive integers which
may involve concepts derived by reflection on this structure have an
objective truth value which is independent of our means to determine it. 
That is not the case in III.

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I hope this helps better situate my own views.  Comments are of course
welcome.

Sol Feferman