[FRIAM] Abduction

[lrudolph at meganet.net]

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...).