Irving,
Do you think that your "theoretical - computational" distinction and
likewise Pratt's "creator - consumer" distinction between kinds of
mathematics could be expressed in terms of Peirce's "theorematic -
corollarial" distinction? That identification seems not without
issues but still pretty appealing to me, but maybe I've missed
something. (For readers unfamiliar with Peirce's way of
distinguishing theormatic from corollarial, see further below where
I've copied my Wikipedia summary with reference links in the
footnotes.)
Peirce at least once said that theorematic deduction is peculiar to
mathematics, though he didn't say that it was peculiar to pure
mathematics. He tended to regard probability theory as mathematics
applied in philosophy, and I don't recall him saying that (at its
theoretical level) probability theory tends to draw mainly
corollarial conclusions. He also allowed of theorematic deduction,
when needed, in the formation of scientific (idioscopic) predictions.
Obviously some pretty deep math has been and continues to be inspired
by problems in special sciences, e.g., in 1990 Ed Witten won a Fields
Medal from the International Union of Mathematics for math that he
developed for string theory.
In case like those of Newton, Leibniz, Hamilton, Witten, etc., one
can say that they were doing theorematic math for computational use
in special sciences, but should we say that mathematical physics in
general is a theorematic, or mathematically theoretical, area? The
question seems still more acute as to probability theory and the
'pure'' maths of information. I've seen it said that probability
theory can be considered a mathematical application of enumerative
combinatorics and measure theory, and that the laws of information
have turned out to have corresponding group-theoretic pinciples. It
seems hard not to call nontrivial areas like probability theory and
such information theory "theorematic," yet they are traditionally
regarded as "applied." Bourbaki's Dieudonné in his math
classifications article in (I think) the 15th edition of Encyclopedia
Britannica complained that the term "applied" mixes trivial and
nontrivial aras of math together.
What I'm wondering is whether the pure-applied distinction would tend
to re-assert itself (in cases like that of measure and enumeration
vs. probability theory) as "theorematic pure mathematics" and
"theorematic applied mathematics," or some such. I've noticed, about
these mathematically nontrivial areas of "applied" mathematics, that
they tend to pay special attention to total populations, universes of
discourse, etc., and to focus on structures of alternatives and
implications, among cases (or among propositions, or whatever), often
with regard to the distribution or attribution of characters to
objects. They seem to be "sister sciences" (to use the old-fashioned
phrase) - John Collier once said at peirce-l that among probability
theory, such information theory, and mathematical logic, he found
that he could base any two of them on the remaining third one. (But
Peirce classified mathematics of logic as the first of three
divisions of pure mathematics.) How, if this subject interests you,
do you think one might best capture the difference between these
something-like-applied yet mathematically nontrivial areas, and
so-called 'pure' mathematics?
Best, Ben (summary of Peirce views on corollarial vs. theorematic
appears below)
Charles Sanders Peirce held that the most important division of
kinds of deductive reasoning is that between corollarial and
theorematic. He argued that, while finally all deduction depends in
one way or another on mental experimentation on schemata or
diagrams,[1] still in corollarial deduction "it is only necessary to
imagine any case in which the premisses are true in order to perceive
immediately that the conclusion holds in that case," whereas
theorematic deduction "is deduction in which it is necessary to
experiment in the imagination upon the image of the premiss in order
from the result of such experiment to make corollarial deductions to
the truth of the conclusion."[2] He held that corollarial deduction
matches Aristotle's conception of direct demonstration, which
Aristotle regarded as the only thoroughly satisfactory demonstration,
while theorematic deduction (A) is the kind more prized by
mathematicians, (B) is peculiar to mathematics,[1] and (C) involves
in its course the introduction of a lemma or at least a definition
uncontemplated in the thesis (the proposition that is to be proved);
in remarkable cases that definition is of an abstraction that "ought
to be supported by a proper postulate.".[3]
1.. 1 a b Peirce, C. S., from section dated 1902 by editors in the
"Minute Logic" manuscript, Collected Papers v. 4, paragraph 233,
quoted in part in "Corollarial Reasoning" in the Commens Dictionary
of Peirce's Terms, 2003-present, Mats Bergman and Sami Paavola,
editors, University of Helsinki.
2.. 2 Peirce, C. S., the 1902 Carnegie Application, published in
The New Elements of Mathematics, Carolyn Eisele, editor, also
transcribed by Joseph M. Ransdell, see "From Draft A - MS L75.35-39"
in Memoir 19 (once there, scroll down).
3.. 3 Peirce, C. S., 1901 manuscript "On the Logic of Drawing
History from Ancient Documents, Especially from Testimonies', The
Essential Peirce v. 2, see p. 96. See quote in "Corollarial
Reasoning" in the Commens Dictionary of Peirce's Terms.
----- Original Message -----
From: Irving
To: <PEIRCE-L@LISTSERV.IUPUI.EDU>
Sent: Wednesday, March 07, 2012 8:32 AM
Subject: Re: [peirce-l] Mathematical terminology, was, review of
Moore's Peirce edition
About two and a half weeks ago, Garry Richmond wrote (among other
things), in reply to one of my previous posts:
> You remarked concerning an "older, artificial, and somewhat
inaccurate" terminological distinction between "practical or applied
on the one hand and pure or abstract on the other." In this context
one finds Peirce using "pure", "abstract" and "theoretical" pretty
much interchangeably, while I agree that "theoretical" is certainly
"newer" and I can see why you think it is less artificial and
inaccurate than the other two. But on the other side of the
distinction, while "practical" seems a bit antiquated, "applied"
appears to me quite accurate and legitimate. My question then is
simply this: what is the terminology used today in consideration of
this distinction? Is it, as I'm assuming, "theoretical" and
"applied"? Further, are there other important distinctions which
aren't aspects or sub-divisions of these two terms? Where, for
example, would you place Peirce's "mathematics of logic", which he
characterizes as the "simplest mathematics" including a kind of
mathematical valency theory (to use Ken Ketner's language of monadic,
dyadic, and triadic relations "retrospectively" analyzed as
tricategorial). A more fundamental question: is there a place for
this kind of 'valental' (Ketner) thinking in contemporary mathematics
or logic?
The characterization which I propounded obviously mirrors to a
considerable extent the medieval distinction between logica utens and
logica docens. The reason that I regard such distinctions between the
"older, artificial, and somewhat inaccurate" terminological
distinction between "practical or applied on the one hand and pure or
abstract on the other" is that the history of mathematics
demonstrates that much of what we think of as applied mathematics was
not particularly created for practical purposes, but turned out in
any case to have applications, whether in one or more of the
mathematical sciences or for other uses, but from intellectual
curiosity, that is, for the sake of illuminating or extending some
aspect of a mathematical system or set of mathematical objects, just
to see where [else] they might lead, what other new properties can be
discovered; and as many examples in the history of mathematics in the
other direction, that new fields of mathematics were developed for
the sake of solving a particular problem or set of problems in, say
physics or astronomy, that led to the development of "abstract" or
"theoretical" systems. One might point to numerous particular aspects
of work, e.g., in real analysis that grew out of dissatisfaction with
Newton's fluxions or Leibniz's infinitesimals in their ability to
deal with problems in terrestrial mechanics or in celestial
mechanics. As a separate mathematical problem, there is the issue of
functions which are everywhere continuous but nowhere differentiable,
which lead Weierstrass to his work in formalizing the theory of
limits in terms of the epsilon-delta notation. And Cantor's work in
set theory emerged specifically as an attempt to provide a
mathematical foundation for Weierstrass's real analysis. The
"peculiarly behaving" functions of Jacobi and Weierstrass turned out
also to be applicable; the motion of a planar pendulum (Jacobi), the
motion of a force-free asymmetric top (Jacobi), the motion of a
spherical pendulum (Weierstrass), and the motion of a heavy symmetric
top with one fixed point (Weierstrass). The problem of the planar
pendulum, in fact, can be used to construct the general connection
between the Jacobi and Weierstrass elliptic functions. Another
example: group theory, as a branch of algebra, was used by Felix
Klein as a way of organizing geometries according to their rotation
properties; but group theory itself arose from the work of Abel,
Cayley, and others, to deal with generalizations of algebra, in
particular in their efforts to solve Fermat's Last Theorem and to
determine whether quintic equations have unique roots. The
application by Heisenberg and Weyl of group theory to quantum
mechanics, makes group theory, in this respect at least, applicable,
as well as "pure". This is why I suggest that a more useful
distinction is between theoretical and computational rather than
"pure" and "applied".
It was, I think Vaughn Pratt who very recently (in a post to FOM)
proposed that the distinction between "pure" and "applied" be
replaced by a more reliable and compelling characterization in terms
of the consumers of mathematics; between those who create mathematics
and those who do not create, but make use of, mathematics. Given this
fluidity between "theory" and "practice" -- and one can find numerous
examples of mathematicians who were also physicists, e.g. Laplace,
even Euler, I think it would be beneficial to adopt Pratt's "creator"
and "consumer" distinction. A notable example of the latter would be
Einstein, who, with the help of Minkowski, applied the Riemannian
geometry to classical mechanics to provide the mathematical tools
that allowed formulation of the theory of relativity as requiring a
four-dimensional, curved space.
> You mention the two "conflicting definitions" of mathematics and
offer an extraordinarily helpful passage of Hans Hahn's to the effect
that mathematicians generally concern themselves with "how a proof
goes" while the logician sets himself the task of examining "why it
goes this way". Besides arguing that "we should do well to understand
necessary reasoning as mathematics" (EP2:318), Peirce also states
that theoretical mathematics is a "science of hypotheses" (EP2:51),
"not how things actually are, but how they might be supposed to be,
if not in our universe, then in some other" (EP2:144).
I would now say that "conflicting" was far too strong and too
negative a characterization of Hahn's remark. But I would continue to
argue that mathematicians who are not logicians and mathematical
logicians who are mathematicians still vary in their conception of
what constitutes a proof in mathematics, if not of what mathematics
is; namely, that the "'working' mathematician" is concerned primarily
with cranking out theorems, whereas the logician is primarily
concerned with the inner workings of the procedures used in deriving
or deducing theorems. It is most unlikely, however, that the person
who attempts to prove theorems without some essential understanding
of "why they [the proofs] go this way, rather than that way or that
other way" will develop into an original mathematician, but will
remain a consumer, capable of carrying out computations, but most
unlikely capable of creating any new mathematics. (One is reminded
here of all those miserable school teachers who, teaching -- or, more
accurately, attempting to teach -- mathematics, could not explain to
their students what they were doing or why they were doing it, but
probably relied on rote memory . and the teacher's manual.)
This is another reason for preferring to distinguish, if distinguish
we must, between theoretical and computational over the older,
Aristotelian, distinction of pure and applied mathematics.
> I believe that your discussion of Peirce's remarks (which Fiske
commented on) add this hypothetical dimension to theoretical
mathematics. You wrote that there is "a three-fold distinction, of
the creative activity of arriving at a piece of mathematics, the
mathematics itself, and the elaboration of logical arguments whereby
that bit of mathematics is established as valid." For the moment I am
seeing these three as forming a genuine tricategorical relationship,
which I'd diagram in my trikonic way, thus:
>
> Theoretical mathematics:
>
> (1ns) mathematical hypothesis formation (creative abduction--that
"piece of mathematics")
> |> (3ns) argumentative proof (of the validity of the mathematics)
> (2ns) the mathematics itself
> Does this categorial division make any sense to you? I'm working
on a trichotomic (tricategorial) analysis of science as Peirce
classified it, but I'm challenged in the areas of mathematics as well
as certain parts of what Peirce calls "critical logic", or, "logic as
logic" (the second division of logic as semeiotic, sandwiched between
semeiotic grammar and rhetoric/methodeutic, all problematic terms for
contemporary logic, I'm assuming). I certainly don't want to create
tricategorial relations which don't exist, so would appreciate your
thoughts in this matter.
Sounds okay to me, but that is question perhaps better dealt with by
someone more familiar with Peirce's understanding of category theory
and his tri-categorical conceptions. Incidentally, I remember ages
ago reading Emil Fackenheim's _The Religious Dimension in Hegel's
Thought_, which, as I recall, presented the thesis that Hegel's
triadism was an abstractification (or "philosophization") and
secularization of the religious idea of the Trinity. Does anyone
propound the view that Peirce's triadism is something similar?
Irving H. Anellis, Ph.D.
Visiting Research Associate
Peirce Edition, Institute for American Thought
902 W. New York St.
Indiana University-Purdue University at Indianapolis
Indianapolis, IN 46202-5159
USA
URL: http://www.irvinganellis.info
----- Original Message ---------
Date: Sat, 18 Feb 2012 19:17:55 -0500
From: Gary Richmond
Reply-To: Gary Richmond
Subject: Mathematical terminology, was, review of Moore's Peirce edition
To:Irving Anellis, PEIRCE-L@LISTSERV.IUPUI.EDU
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