FOM: Ontology of Mathematics, revisited

[Todd Wilson]

In my [10 Jul 2000, 15:14:40], I responded to a post of Jan Mycielski
[10 Jul 2000, 10:54:06] concerning his "rational philosophy" of
mathematics, which I had summarized thus:

> - mathematical objects are real in the sense that they exist as
>   configurations in the brains of mathematicians, but differ from
>   other "real" objects in not having external referents.
> 
> - only finitely many such mental configurations can exist at any one
>   time (indeed, from the beginning of time up to the present), and
>   they constitute the totality of the subject matter of (pure)
>   mathematics.

I listed a couple of problems that I had with this view, the main one
being how to explain the consistency mathematicians experience when
studying mathematical objects, if these objects are essentially
private; I illustrated my questions with an analogy involving
mathematicians wandering a maze; and I concluded with the following
statement:

> I would like to believe that mathematical ontology had a neurophysical 
> explanation, but I don't see how to get beyond these questions without 
> invoking some kind of extra-mental structure.

This post generated a lot of private email correspondence, not to
mention a paper I received priority mail from Paris, and I am grateful
to all of my correspondents for sharing their thoughts (and paper) on
this subject with me; my views have been refined considerably as a
result of this correspondence.  At the suggestion of our moderator,
Stephen Simpson, I am bringing this discussion back onto FOM by
posting a summary of these views, and I welcome responses to them by
my original correspondents (whom I have intentionally not named, out
of respect for the privacy of email), as well as by others interested
in the subject.

- The main unanswered question is the regularity of nature; we don't
  offer any explanation for this.

- Our brains are hard-wired to recognize and respond to regularities
  (i.e., invariances) in nature.  Because our own mental reactions to
  stimuli are themselves stimuli, and so on, we are capable of
  recognizing and responding to ever higher orders of regularity.

- Mathematics objects, for a given mathematician, consist of the
  results of conceptualizing these (meta-)regularities, in line with
  our mental habit of introducing summary symbols to stand, as objects
  or referents, for these regularities in further cognition.  In this
  sense, mathematical objects are entirely mental constructs, yet they
  are born completely from the vast evolutionary interactions (over
  millions of years) of our {per,con}ceptual hardware with the
  regularities of nature, and are thus inextricably bound up with the
  latter.

- Mathematical objects are distinguished from other conceptual objects
  in that they objectify maximal levels of invariance (or, dually,
  comprise minimal levels of variance).  This accounts for their great
  stability, contextual independence, and applicability.  Moreover,
  these aspects of mathematical objects, along with their evolutionary
  origin, go a long way toward explaining why mathematics is so
  effective in the physical sciences.

- The tremendous overlap in *potential* mathematical ontology between
  mathematicians is explained by the physiological similarities of
  their {per,con}ceptual hardware and the similarities in the uses to
  which it is put.  Education accounts for the great overlap in
  *actual* ontology and the ability of mathematicians to communicate
  effectively about (their private but essentially identical)
  mathematical objects.  We would expect the mathematics of alien
  cultures to agree largely with ours, with the greatest agreement
  coming on those aspects relevant to our common survival and
  reproductive strategies.

Even more speculatively, I would like to suggest that the views
summarized above can be reconciled with traditional Platonist and
formalist views along the following lines:

- When the Platonist speaks about ideals (non-physical reality), we
  can interpret his (or her) utterances to refer to the common mental
  configurations that are mathematical objects.  He is not wrong to
  say that these objects are real, because they do have a kind of
  physical embodiment in terms of mental configurations of
  mathematicians, nor is he wrong to call them non-physical, in the
  sense that they are abstractions.  The strong feeling he has about
  the independence of these objects from human whims, desires, etc.,
  is explained by the independence of our shared mental hardware from
  the uses to which we put it (at least to the extent that we believe
  in free will!).

- The formalist may be said to be agnostic regarding the physicality
  of mathematical objects, instead preferring to focus attention on a
  *different kind* of embodiment of those same objects:  the symbolic
  properties of formal systems.  She (or he) may not have an
  explanation for why formal systems can display the same kinds of
  regularities as nature can (preferring agnosticism here, as well),
  but she finds empirically that they do, and that is enough.

In producing such terse renderings of complex philosophical issues and
positions, I have no doubt oversimplified them, but I hope that these
renderings are provocative enough to bring about more discussion of
these issues on FOM.

-- 
Todd Wilson
Computer Science Department
California State University, Fresno