Arnold, Patrick, list,
Although I myself write none too accurately when I use the word "interior"
regarding a Klein bottle, here is a case where accuracy really is needed.
Peirce said that mathematics is the science which _draws_ necessary
conclusions, as opposed to its being a science _of_ necessary conclusions. The
science _of_ reasoning, necessary and otherwise, he called logic and placed it
in philosophy. Peirce says that the science which _draws_ necessary conclusions
is mathematics and includes (indeed begins with) mathematics _of_ logic.
I agree that it's quite impoverishing to regard mathematics as primarily a
study of calculation, which would basically be to say that mathematics is all
algebra ("algebra" in the sense of "theory of calculation"). Insofar as the
ordering in a structure has special pertinence to logic and, in particular, is
what is relevant in determining the applicability of mathematical induction to
the set with said structure, and insofar as ordered structure is the basic kind
of structure involved in structures of ranking, preference, etc., it appears
that ordered structures are the mathematical structures with the most special
relevance to elucidation of the phenomenon of rational animals.
I recall in high school that treatment of ordered structures, measure &
enumeration, and topology & graph theory, ranged from minimal to zero. Even
recently I was initially uncertain whether Marty's lattice amounted,
technically, to a partially ordered set or whether it was some "other" sort of
not-entirely-ordered set.
Dieudonne in his Encyclopedia Britannica 15th Edition article on maths
discussed math in terms of "structures of order," "structures of group"
(including abstract algebra and much geometry), and "structures of space"
(including topology). He somewhere says, however, that Bourbaki (the group
which he often represented) probably hadn't paid enough attention to the
"combinatorial" aspects of mathematics. Of course, I don't know whether he was
referring to enumerative combinatorics, measure theory, etc., or (though I
somewhat doubt it) in the sense of "formal, finitely presented properties of
the inscriptions of the ambient formal language" (I'm uncertain of how to
translate that into English)
http://publish.uwo.ca/~jbell/foundations%20of%20mathematics.pdf .
Best, Ben
Arnold wrote,
AS: Now, I guess what I am getting at here is that the more one begins to grasp
the history of both math and logic through the lens of Peirce's undoubted
mastery of both (however idiosyncratic some of his inferences from history may
appear to some), the more one should be led to take a wider view of both. As
the `science of necessary reasoning', the discipline (as in self-control)
required for mathematical inquiry seems to me to indicate that there should be
no reason why one can't undertake the study of the diagrammatic forms of
necessary reasoning about human experience in a non-computational way. Peirce
treats the foundations of mathematics as a form of relational reasoning (which,
I am led to understand, runs counter to the modern mathematical tradition; I
won't debate that because I am no mathematician, but am never the less
fascinated by the potential arcaneness of the topic). At 3.562 he essays an
accessible account of this relational foundation (the CP source consists of
material left out of an article in an educational journal of 1898), and anybody
with some familiarity of anthropological field methods will immediately
recognize a relation that lies at the core of ALL possible experience: the
relation of sequence. Surely there can be no continuity in human affairs, the
basis upon which one could say we make all those judgements and inferences we
call `experience', without a sequence of generations, which Peirce very
accessibly shows has proprties that are quite mathematical.
AS: I won't take this further for now, because I suspect I'm going to start
blathering on without getting all my ducks in a row first. But I guess that
what I wanted to suggest to Patrick and the List is that the "trend in our
time" need not be accepted as fatalistically as all that. It does, after all,
represent perhaps 100-150 years' of debate in a tradition going back maybe 2500
years or more (I mean: how long ago did the distinction between
naturwissenschaften and geisteswissenchaften enter the conversational lexicon
of academia?). Maybe I'm overly optimistic, but surely it won't take that long
for the fashion to fade away?
Cheers
Arnold Shepperson
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