Glen wrote, in relevant part, "Like mathematicians, maybe we have to ultimately
commit to the
ontological status of our parsing methods?" I wish to question the implicit
assumption that
mathematicians _do_ (or even _ought to_) "ultimately commit to the ontological
status" of
_anything_ in particular.
I wrote (some time ago, and not here) something I will still stand by. It
appears at the
beginning of a me-authored chapter in a me-edited book, "Qualitative
Mathematics for the
Social Sciences: Mathematical models for research on cultural dynamics"; the
"our" and "we" in
the first sentence refer to me and my coauthor in an introductory chapter, not
to me-and-a-
mouse-in-my-pocket. (Note that I am a mathematician, _not_ a social scientist,
and only very
occasionally a mathematical modeler of any sort.) I have edited out some
footnotes, etc., but
in return have expanded some of the in-line references {inside curly braces}.
===begin===
In our Introduction (p. 17) we quoted "three statements, by mathematicians
{Ralph Abraham;
three guys named Bohle-Carbonell, Booß, Jensen, who I'd not heard of before
working on the
book; and Phil Davis} on mathematical modeling". Here is a fourth.
(D) Mathematics has its own structures; the world (as we perceive and cognize
it) is, or
appears to be, structured; mathematical modeling is a reciprocal process in
which we
_construct/discover/bring into awareness_ correspondences between mathematical
structures and
structures `in the world´, as we _take actions that get meaning from, and give
meaning to,_
those structures and correspondences.
Later (p. 24 ff.) we briefly viewed modeling from the standpoint of
"evolutionary
epistemology" in the style of Konrad Lorenz (1941) {Kant´s doctrine of the a
priori in the
light of contemporary biology}. In this chapter, I view modeling from the
standpoint
informally staked out by (D), which I propose to call "evolutionary ontology."
My discussion
is sketchy (and not very highly structured), but may help make sense of this
volume and
perhaps even mathematical modeling in general.
Behind (D) is my conviction that there is no need to adopt any particular
ontological
attitude(s) towards "structures", in the world at large and/or in mathematics,
in order to
proceed with the project of modeling the former by the latter and drawing
inspiration for
the latter from the former. It is, I claim, possible for someone simultaneously
to adhere to a
rigorously `realist´ view of mathematics (say, naïve and unconsidered
Platonism) and to take
the world to be entirely insubstantial and illusory (say, by adopting a crass
reduction of the
Buddhist doctrine of Maya), _and still practice mathematical modeling in good
faith_ if not
with guaranteed success. Other (likely or unlikely) combinations of attitudes
are (I claim)
just as possible, and equally compatible with the practice of modeling.
I have the impression that many practitioners, if polled (which I have not
done), would
declare themselves to be both mathematical `formalists´ and physical
`realists´. I also have
the impression that a large, overlapping group of practitioners, observed in
action (which
I have done, in a small and unsystematic way), can reasonably be described to
_behave_ like
thoroughgoing ontological agnostics. Mathematical modeling _as human behavior_
is based, I am
claiming, on acts of pattern-matching (or Gestalt-making)-which is to say,in
other language,
on creation/recognition/awareness of `higher order structures´ relating some
`lower order
structures´-that one performs (or that occur to one) independently of one´s
ontological
stances. That is not all there is to it, as behavior; but that is its basis.
===end===
To take Glen's question in (perhaps) a different direction, I note that Imre
Lakatos also used
the word "ultimate" about mathematicians, as follows: "But why on earth have
`ultimate´ tests,
`final authority´? Why foundations, if they are admittedly subjective? Why not
honestly admit
mathematical fallibility, and try to defend the dignity of fallible knowledge
from cynical
scepticism, rather than delude ourselves that we can invisibly mend the latest
tear in the
fabric of our "ultimate" intuitions?" As I have learned from Nick, Peirce is
also committed to
the defense of "the dignity of fallible knowledge" (at least, I *think* I've
learned that from
Nick; but I might be wrong...).
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