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
Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition
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,
Re: [peirce-l] Proemial: On The Origin Of Experience
Dear Steven, That's what I increasingly thought after re-reading your thread-commencing post again after sending my post about it. You did not think the things that you at times had seemed to me to think. It was really about stylistics and word choice. In one case I noted that you had not literally said that which you somehow seemed to me to say, - instead you had indeed said the thing that made more sense - you had not said, as I somehow had thought, that a certain _discovery_ would impact the human species and the universe, instead you spoke of the discovery of _something_ that would impact the human species and the universe, and that thing was something on the order of nature's plan. How did I go astray? Impacting us sounds like something that a _discovery_ would do, not something that _nature's plan_ would do. Nature's plan does something deeper than that, it plans or plots us. I suppose that one could speak of something with radical significance for the human species and the universe. Well, maybe I'm too sleepy to make suggestions right now. Now, you have a right to expect a reader to attend to what you actually say and not just to vague impressions of what you say. But when one writes a book blurb, it's best to write it in extra-hard-to-misconstrue ways, as if the reader may be a bit groggy, like I am right now! Best, Ben - Original Message - From: Steven Ericsson-Zenith To: PEIRCE-L@LISTSERV.IUPUI.EDU Cc: Benjamin Udell Sent: Wednesday, March 07, 2012 8:40 PM Subject: Re: [peirce-l] Proemial: On The Origin Of Experience Dear Ben, I appreciate your very useful response. I said the entire species and that the universe could not proceed, not the entire universe. So I would not expect the impact to fill the eternal moment, only localized parts. Similarly, I would hesitate to suggest that the entire mass/energy complex of the world could eventually be structured to become a single organism. It seems implausible 'though it is perhaps worth some consideration equally as a theme for a Science Fiction novel or as a potential solution to the dark-energy problem (I do, after all, propose a weak universe effect that may, I suppose, accumulate at very large scales to increase thinning edge-wise expansion). Your points, however, are well taken. If it continues in its current form I should define more clearly what I mean by proceed. For example: ... the universe itself could not proceed, could not further evolve beyond the stage that we represent ... Thanks. With respect, Steven -- Dr. Steven Ericsson-Zenith Institute for Advanced Science Engineering http://iase.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
