Ben Udell asked:

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?

Given that Peirce wrote at MS L75:35-39 that: "Deduction is only of value in tracing out the consequences of hypotheses, which it regards as pure, or unfounded, hypotheses. Deduction is divisible into sub-classes in various ways, of which the most important is into corollarial and theorematic. Corollarial deduction is where 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. Ordinary syllogisms and some deductions in the logic of relatives belong to this class. 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. The subdivisions of theorematic deduction are of very high theoretical importance. But I cannot go into them in this statement." the answer to the question would appear to be: no. Whereas Peirce's characterization of theorematic and corrolarial deduction would seem, on the basis of this quote, to have to do with whether the presumption that the premises of a deductive argument or proof are true versus whether they require to be established to be true, and seems more akin, at least peripherally, to the categorical/hypothetical status of the premises, the distinctions "theoretical - computational" which I suggested and likewise Pratt's "creator - consumer" are not at all about the deriving theorems or the what is assumed about the truth of the premises. Rather the distinction between creator-theoretician vs. consumer-practitioner is a distinction in which the former is concerned (in the main) to develop new mathematics on the basis of the mathematics that has already been established, whereas the consumer practitioner borrows and utilizes already established mathematics for purposes other than establishing new mathematical results. The example which I cited, of Riemann and Minkowski vs. Einstein is applicable here. Riemann expanded known mathematical results regarding three-dimensional geometries to n-dimensional geometries (Riemann manifolds) and contributed to the development of non-Euclidean geometries, and Minkowski starting from non-Euclidean geometries, in particular parabolic and hyperbolic, arrived at his "saddle-shaped" space, and Minkowski taught Einstein the mathematics of Riemannin and Minkowski geometry, who used it to work out the details of relativity, but, unlike Riemann or Minkowski, did not create any new mathematics, just utilized the already given mathematics of Riemann and Minkowski to mathematically solve a particular problem in physics. I think most would agree with the proposition that Einstein was a physicist, rather than a mathematician, albeit unassailably a mathematical physicist, who employed already established mathematics and mathematical equations to advance physics, and along those same lines, I think most would likewise agree with the proposition that Einstein was not a mathematician. This does not, of course, take away from his status as a physicist. By the same token, Newton can be credited as both a mathematician, for his fluxional caculus as well as a physicist, although his invention -- and I would not want to get into the Newton-Leibniz battle here -- of the calculus was developed in large measure for the purpose of doing physics. But the fact that Newton (although he used geometry rather than the calculus in the mathematics of the Principia) obtained the fluxional calculus in part to advance mathematics (a major advance over Cavalieri's ponderous method of indivisibles, and in part to work out and express mathematically the laws of gravity and of terrestrial and celestial mechanics, illustrates that a theoretical/ applied distinction is somewhat artificial as compared with the "theoretical - computational" distinction and "creator - consumer" distinction. ----- Message from bud...@nyc.rr.com --------- Date: Wed, 7 Mar 2012 14:41:08 -0500 From: Benjamin Udell <bud...@nyc.rr.com> Reply-To: Benjamin Udell <bud...@nyc.rr.com> Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition To: PEIRCE-L@LISTSERV.IUPUI.EDU

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 --------------------------------------------------------------------------------- You are receiving this message because you are subscribed to the PEIRCE-L listserv. To remove yourself from this list, send a message to lists...@listserv.iupui.edu with the line "SIGNOFF PEIRCE-L" in the body of the message. To post a message to the list, send it to PEIRCE-L@LISTSERV.IUPUI.EDU

----- End message from bud...@nyc.rr.com ----- Irving H. Anellis 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 --------------------------------------------------------------------------------- You are receiving this message because you are subscribed to the PEIRCE-L listserv. To remove yourself from this list, send a message to lists...@listserv.iupui.edu with the line "SIGNOFF PEIRCE-L" in the body of the message. To post a message to the list, send it to PEIRCE-L@LISTSERV.IUPUI.EDU