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: <[email protected]>
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, [email protected]
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