### Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

Method liegt der eigentliche Zauber der Infinitesimalrechnung. Theodor Ziehen defined logicism in his Lehrbuch der Logik auf postivischer Grundlage mit Berücksichtigung der Geschichte der Logik [1920, p. 173] to mean that there is an objective realm of ideal entities, studied by logic and mathematics, and he numbered on that account Lotze, Windelband, Husserl, and Rickert among those adhering to logicism. Having said that: as I wrote in the FOM back in May 2011, I recall that, many years ago (probably some time in the early or mid-1980s), Reuben Hersh gave a colloquium talk in the mathematics department at the University of Iowa. I don't recall the specifics of that talk, but in its general tenor it went along the lines that, in their workaday world. most mathematicians are Platonists, working as though the mathematical structures with which they are working and which are the subject of theorems exist, whereas, on weekends, they deny the real existence of mathematical entities. In the description for Reuben Hersh's What Is Mathematics Really? (Oxford U. Press, 1997), Hersh's position is described (in part) as follows: Platonism is the most pervasive philosophy of mathematics. Indeed, it can be argued that an inarticulate, half-conscious Platonism is nearly universal among mathematicians. The basic idea is that mathematical entities exist outside space and time, outside thought and matter, in an abstract realm. ...In What is Mathematics, Really?, renowned mathematician Reuben Hersh takes these eloquent words and this pervasive philosophy to task, in a subversive attack on traditional philosophies of mathematics, most notably, Platonism and formalism. Virtually all philosophers of mathematics treat it as isolated, timeless, ahistorical, inhuman. Hersh argues the contrary, that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. Mathematical objects are created by humans, not arbitrarily, but from activity with existing mathematical objects, and from the needs of science and daily life. Hersh pulls the screen back to reveal mathematics as seen by professionals, debunking many mathematical myths, and demonstrating how the humanist idea of the nature of mathematics more closely resembles how mathematicians actually work. At the heart of the book is a fascinating historical account of the mainstream of philosophy--ranging from Pythagoras, Plato, Descartes, Spinoza, and Kant, to Bertrand Russell, David Hilbert, Rudolph Carnap, and Willard V.O. Quine--followed by the mavericks who saw mathematics as a human artifact, including Aristotle, Locke, Hume, Mill, Peirce, Dewey, and Lakatos. ... - Message from eugene.w.halto...@nd.edu - Date: Tue, 13 Mar 2012 17:09:42 -0400 From: Eugene Halton eugene.w.halto...@nd.edu Reply-To: Eugene Halton eugene.w.halto...@nd.edu Subject: RE: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition To: PEIRCE-L@LISTSERV.IUPUI.EDU PEIRCE-L@LISTSERV.IUPUI.EDU Dear Irving, A digression, from the perspective of art. You quote probability theorist William Taylor and set theorist Martin Dowd as saying: The chief difference between scientists and mathematicians is that mathematicians have a much more direct connection to reality. This does not entitle philosophers to characterize mathematical reality as fictional. Yes, I can see that. But how about a variant: The chief difference between scientists, mathematicians, and artists is that artists have a much more direct connection to reality. This does not prevent scientists and mathematicians to characterize artistic reality as fictional, because it is, and yet, nevertheless, real. This is because scientist's and mathematician's map is not the territory, yet the artist's art is both. Gene Halton -Original Message- From: C S Peirce discussion list [mailto:PEIRCE-L@LISTSERV.IUPUI.EDU] On Behalf Of Irving Sent: Tuesday, March 13, 2012 4:34 PM To: PEIRCE-L@LISTSERV.IUPUI.EDU Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition Ben, Gary, Malgosia, list It would appear from the various responses that. whereas there is a consensus that Peirce's theorematic/corollarial distinction has relatively little, if anything, to do with my theoretical/computational distinction or Pratt's creator and consumer distinction. As you might recall, in my initial discussion, I indicated that I found Pratt's distinction to be somewhat preferable to the theoretical/computational, since, as we have seen in the responses, computational has several connotations, only one of which I initially had specifically in mind, of hack grinding out of [usually numerical] solutions to particular problems, the other generally thought of as those parts of mathematics taught in catch-all undergrad courses that frequently go by the name of Finite Mathematics

### Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

...@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 Malgosia, Irving, Gary, list, I should add that this whole line of discussion began because I put the cart in front of the horse. The adjectives bothered me. Theoretical math vs. computational math - the latter sounds like of math about computation. And creative math vs. what - consumptive math? consumptorial math? Then I thought of theorematic vs. corollarial, thought it was an interesting idea and gave it a try. The comparison is interesting and there is some likeness between the distinctions. However I now think that trying to align it to Irving's and Pratt's distinctions just stretches it too far. And it's occurred to me that I'd be happy with the adjective computative - hence, theoretical math versus computative math. However, I don't think that we've thoroughly replaced the terms pure and applied as affirmed of math areas until we find some way to justly distinguish between so-called 'pure' maths as opposed to so-called 'applied' yet often (if not absolutely always) mathematically nontrivial areas such as maths of optimization (linear and nonlinear programming), probability theory, the maths of information (with laws of information corresponding to group-theoretical principles), etc. Best, Ben - Original Message - From: Benjamin Udell To: PEIRCE-L@LISTSERV.IUPUI.EDU Sent: Monday, March 12, 2012 1:14 PM Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition Malgosia, list, Responses interleaved. - Original Message - From: malgosia askanas To: PEIRCE-L@LISTSERV.IUPUI.EDU Sent: Monday, March 12, 2012 12:31 PM Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition [BU] Yes, the theorematic-vs.-corollarial distinction does not appear in the Peirce quote to depend on whether the premisses - _up until some lemma_ - already warrant presumption. BUT, but, but, the theorematic deduction does involve the introdution of that lemma, and the lemma needs to be proven (in terms of some postulate system), or at least include a definition (in remarkable cases supported by a proper postulate) in order to stand as a premiss, and that is what Irving is referring to. [MA] OK, but how does this connect to the corollarial/theorematic distinction? On the basis purely of the quote from Peirce that Irving was discussing, the theorem, again, could follow from the lemma either corollarially (by virtue purely of logical form) or theorematically (requiring additional work with the actual mathematical objects of which the theorem speaks). [BU] So far, so good. [MA] And the lemma, too, could have been obtained either corollarially (a rather needless lemma, in that case) [BU] Only if it comes from another area of math, otherwise it is corollarially drawn from what's already on the table and isn't a lemma. [MA] or theorematically. Doesn't this particular distinction, in either case, refer to the nature of the _deduction_ that is required in order to pass from the premisses to the conclusion, rather than referring to the warrant (or lack of it) of presuming the premisses? [BU] It's both, to the extent that the nature of that deduction depends on whether the premisses require a lemma, a lemma that either gets something from elsewhere (i.e., the lemma must refer to where its content is established elsewhere), or needs to be proven on the spot. But - in some cases there's no lemma but merely a definition that is uncontemplated in the thesis, and is not demanded by the premisses or postulates but is still consistent with them, and so Irving and I, as it seems to me now, are wrong to say that it's _always_ a matter of whether some premiss requires special proof. Not always, then, but merely often. In some cases said definition needs to be supported by a new postulate, so there the proof-need revives but is solved by recognizing the need and conceding a new postulate to its account. [MA] If the premisses are presumed without warrant, that - it seems to me - does not make the deduction more corollarial or more theorematic; it just makes it uncompleted, and perhaps uncompletable. [BU] That sounds right. Best, Ben - 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

### Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

Dear Irving, A digression, from the perspective of art. You quote probability theorist William Taylor and set theorist Martin Dowd as saying: The chief difference between scientists and mathematicians is that mathematicians have a much more direct connection to reality. This does not entitle philosophers to characterize mathematical reality as fictional. Yes, I can see that. But how about a variant: The chief difference between scientists, mathematicians, and artists is that artists have a much more direct connection to reality. This does not prevent scientists and mathematicians to characterize artistic reality as fictional, because it is, and yet, nevertheless, real. This is because scientist's and mathematician's map is not the territory, yet the artist's art is both. Gene Halton -Original Message- From: C S Peirce discussion list [mailto:PEIRCE-L@LISTSERV.IUPUI.EDU] On Behalf Of Irving Sent: Tuesday, March 13, 2012 4:34 PM To: PEIRCE-L@LISTSERV.IUPUI.EDU Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition Ben, Gary, Malgosia, list It would appear from the various responses that. whereas there is a consensus that Peirce's theorematic/corollarial distinction has relatively little, if anything, to do with my theoretical/computational distinction or Pratt's creator and consumer distinction. As you might recall, in my initial discussion, I indicated that I found Pratt's distinction to be somewhat preferable to the theoretical/computational, since, as we have seen in the responses, computational has several connotations, only one of which I initially had specifically in mind, of hack grinding out of [usually numerical] solutions to particular problems, the other generally thought of as those parts of mathematics taught in catch-all undergrad courses that frequently go by the name of Finite Mathematics and include bits and pieces of such areas as probability theory, matrix theory and linear algebra, Venn diagrams, and the like). Pratt's creator/consumer is closer to what I had in mind, and aligns better, and I think, more accurately, with the older pure (or abstract or theoretical) vs. applied distinction. The attempt to determine whether, and, if so, how well, Peirce's theorematic/corollarial distinction correlates to the theoretical/computational or creator/consumer distinction(s) was not initially an issue for me. It was raised by Ben Udell when he asked me: 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? I attempted to reply, based upon a particular quote from Peirce. What I gather from the responses to that second round is that the primary issue with my attempted reply was that Peirce's distinction was bound up, not with the truth of the premises, but rather with the method in which theorems are arrived at. If I now understand what most of the responses have attempted to convey, the theorematic has to do with the mechanical processing of proofs, where a simple inspection of the argument (or proof) allows us to determine which inference rules to apply (and when and where) and whether doing so suffices to demonstrate that the theorem indeed follows from the premises; whereas the corollarial has to do with intuiting how, or even if, one might get from the premises to the desired conclusion. In that case, I would suggest that another way to express the theorematic/corollarial distinction is that they concern the two stages of creating mathematics; that the mathematician begins by examining the already established mathematics and asks what new mathematics might be Ben Udell also introduces the issue of the presence of a lemma in a proof as part of the distinction between theorematic and corollarial. His assumption seems to be that a lemma is inserted into a proof to help carry it forward, but is itself not proven. But, as Malgosia has already noted, the lemma could itself have been obtained either theorematically or corollarially. In fact, most of us think of a lemma as a minor theorem, proven along the way and subsequently used in the proof of the theorem that we're after. I do not think that any of this obviates the main point of the initial answer that I gave to Ben's question, that neither my theoretical/computational distinction nor Pratt's creator and consumer distinction have anything to do with Peirce's theorematic/corollarial distinction. In closing, I would like to present two sets of exchanges; one very recent (actually today, on FOM, with due apologies to the protagonists, if I am violating any copyrights) between probability theorist William Taylor (indicated by '') and set theorist Martin Dowd (indicated by ''), as follows: More seriously, any freshman philosopher encounters the fact

### Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

Irving, all, In my previous post I said that I would include the full Peirce quotes, but for the first Peirce quote I included only the portion included in the Commens Dictionary. For the full quote (CP 4.233), go here: http://books.google.com/books?id=3JJgOkGmnjECpg=RA1-PA193lpg=RA1-PA193dq=%22Mathematics+is+the+study+of+what+is+true+of+hypothetical+states+of+things%22 - Original Message - From: Benjamin Udell To: PEIRCE-L@LISTSERV.IUPUI.EDU Sent: Tuesday, March 13, 2012 6:11 PM Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition Irving, Gary, Malgosia, list, Irving, I'm sorry that I gave you the impression that I think that a lemma is something helpful but unproven inserted into a proof. I mean a theorem placed in among the premisses to help prove the thesis. Its proof may be offered then and there, or it may be a theorem from (and already proven in) another branch of mathematics, to which the reader is referred. At any rate it is as Peirce puts it a demonstrable proposition about something outside the subject of inquiry. The idea that theorematic reasoning often involves a lemma comes not from me but from Peirce. Theorematic reasoning, in Peirce's view, involves experimentation on a diagram, which may consist in a geometrical form, an array of algebraic expressions, a form such as All __ is __, etc. I don't recall his saying anything to suggest that theorematic reasoning is particularly mechanical. I summarized Peirce's views in a paragraph in my first post on these questions, and I'll reproduce it, this time with the full quotes from Peirce. He discusses lemmas in the third quote. 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 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.: How it can be that, although the reasoning is based upon the study of an individual schema, it is nevertheless necessary, that is, applicable, to all possible cases, is one of the questions we shall have to consider. Just now, I wish to point out that after the schema has been constructed according to the precept virtually contained in the thesis, the assertion of the theorem is not evidently true, even for the individual schema; nor will any amount of hard thinking of the philosophers' corollarial kind ever render it evident. Thinking in general terms is not enough. It is necessary that something should be DONE. In geometry, subsidiary lines are drawn. In algebra permissible transformations are made. Thereupon, the faculty of observation is called into play. Some relation between the parts of the schema is remarked. But would this relation subsist in every possible case? Mere corollarial reasoning will sometimes assure us of this. But, generally speaking, it may be necessary to draw distinct schemata to represent alternative possibilities. Theorematic reasoning invariably depends upon experimentation with individual schemata. We shall find that, in the last analysis, the same thing is true of the corollarial reasoning, too; even the Aristotelian demonstration why. Only in this case, the very words serve as schemata. Accordingly, we may say that corollarial, or philosophical reasoning is reasoning with words; while theorematic, or mathematical reasoning proper, is reasoning with specially constructed schemata. (' Minute Logic', CP 4.233, c. 1902) 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

### Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

Malgosia, Irving, Gary, list, I should add that this whole line of discussion began because I put the cart in front of the horse. The adjectives bothered me. Theoretical math vs. computational math - the latter sounds like of math about computation. And creative math vs. what - consumptive math? consumptorial math? Then I thought of theorematic vs. corollarial, thought it was an interesting idea and gave it a try. The comparison is interesting and there is some likeness between the distinctions. However I now think that trying to align it to Irving's and Pratt's distinctions just stretches it too far. And it's occurred to me that I'd be happy with the adjective computative - hence, theoretical math versus computative math. However, I don't think that we've thoroughly replaced the terms pure and applied as affirmed of math areas until we find some way to justly distinguish between so-called 'pure' maths as opposed to so-called 'applied' yet often (if not absolutely always) mathematically nontrivial areas such as maths of optimization (linear and nonlinear programming), probability theory, the maths of information (with laws of information corresponding to group-theoretical principles), etc. Best, Ben - Original Message - From: Benjamin Udell To: PEIRCE-L@LISTSERV.IUPUI.EDU Sent: Monday, March 12, 2012 1:14 PM Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition Malgosia, list, Responses interleaved. - Original Message - From: malgosia askanas To: PEIRCE-L@LISTSERV.IUPUI.EDU Sent: Monday, March 12, 2012 12:31 PM Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition [BU] Yes, the theorematic-vs.-corollarial distinction does not appear in the Peirce quote to depend on whether the premisses - _up until some lemma_ - already warrant presumption. BUT, but, but, the theorematic deduction does involve the introdution of that lemma, and the lemma needs to be proven (in terms of some postulate system), or at least include a definition (in remarkable cases supported by a proper postulate) in order to stand as a premiss, and that is what Irving is referring to. [MA] OK, but how does this connect to the corollarial/theorematic distinction? On the basis purely of the quote from Peirce that Irving was discussing, the theorem, again, could follow from the lemma either corollarially (by virtue purely of logical form) or theorematically (requiring additional work with the actual mathematical objects of which the theorem speaks). [BU] So far, so good. [MA] And the lemma, too, could have been obtained either corollarially (a rather needless lemma, in that case) [BU] Only if it comes from another area of math, otherwise it is corollarially drawn from what's already on the table and isn't a lemma. [MA] or theorematically. Doesn't this particular distinction, in either case, refer to the nature of the _deduction_ that is required in order to pass from the premisses to the conclusion, rather than referring to the warrant (or lack of it) of presuming the premisses? [BU] It's both, to the extent that the nature of that deduction depends on whether the premisses require a lemma, a lemma that either gets something from elsewhere (i.e., the lemma must refer to where its content is established elsewhere), or needs to be proven on the spot. But - in some cases there's no lemma but merely a definition that is uncontemplated in the thesis, and is not demanded by the premisses or postulates but is still consistent with them, and so Irving and I, as it seems to me now, are wrong to say that it's _always_ a matter of whether some premiss requires special proof. Not always, then, but merely often. In some cases said definition needs to be supported by a new postulate, so there the proof-need revives but is solved by recognizing the need and conceding a new postulate to its account. [MA] If the premisses are presumed without warrant, that - it seems to me - does not make the deduction more corollarial or more theorematic; it just makes it uncompleted, and perhaps uncompletable. [BU] That sounds right. Best, Ben - 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

### Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

Jason, all, If I had bothered to search on computational mathematics I would have found that the potential ambiguity that worried me is already actual, as you clearly show. Do you think that the phrase computative mathematics is too close to the phrase computational mathematics for comfort? I hope not, but please say so if it is. Problem is, the applied in applied mathematics is used in various ways that, as Dieudonné of the Bourbaki group pointed out in his Britannica article (15th edition I think), jumbles trivial and nontrivial areas of math together, and has all too many, umm, applications. One area of pure math X may be _applied_ in another area of math Y, whih is to say that Y is the guiding research interest. If on the other hand Y is applied in X, then that's to say that X is the guiding research interest. And both X and Y remain areas of 'pure' math. Then there are areas of so-called 'applied' but often nontrivial math like probability theory. Then there are applications in statistics and in the special sciences. Then there applications in practical/productive sciences/arts. And of course, sometimes theoretical or 'pure' math is developed specifically for a particular application. (All in all, we won't be able to get rid of the term applied, but in some cases we may be find an alternate term with the same denotation in the given context). Best, Ben - Original Message - From: Khadimir To: Benjamin Udell Cc: PEIRCE-L@listserv.iupui.edu Sent: Monday, March 12, 2012 2:14 PM Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition This latest post caught my attention. Since my first degree was a B.S. in computational mathematics, I thought that I would weigh-in. One can make the distinctions as follows, beginning with pure vs. applied mathematics. I will give a negative definition, since I am not so skilled with the Peircean terminology used so far; applied mathematics is the use of mathematics as a formal, ideal system to specific problems of existence. For instance, consider the use of statistical confidence intervals to solve problems in manufactoring relating to the rate of production of defective vs. non-defective goods. Pure mathematics is not bound by existent conditions, but pure becomes applied when used in that context. Hence, I am treated applied mathematics as an informal, existential constraint that alters the purpose and use of pure mathematics. Computational mathematics is for the most part a subset of applied mathematics, which focuses on how to adapt computational formulas so that they may be run or run more efficiently on a given computation system, e.g., a binary computer. Computational mathematics, then, is primarily focused on formulas and computation of said formulas, which is to be more specific about the limits that make it an applied mathematic. I offer this as a different viewpoint, one coming from where the distinction has practical effects. Jason H. On Mon, Mar 12, 2012 at 12:47 PM, Benjamin Udell bud...@nyc.rr.com wrote: Malgosia, Irving, Gary, list, I should add that this whole line of discussion began because I put the cart in front of the horse. The adjectives bothered me. Theoretical math vs. computational math - the latter sounds like of math about computation. And creative math vs. what - consumptive math? consumptorial math? Then I thought of theorematic vs. corollarial, thought it was an interesting idea and gave it a try. The comparison is interesting and there is some likeness between the distinctions. However I now think that trying to align it to Irving's and Pratt's distinctions just stretches it too far. And it's occurred to me that I'd be happy with the adjective computative - hence, theoretical math versus computative math. However, I don't think that we've thoroughly replaced the terms pure and applied as affirmed of math areas until we find some way to justly distinguish between so-called 'pure' maths as opposed to so-called 'applied' yet often (if not absolutely always) mathematically nontrivial areas such as maths of optimization (linear and nonlinear programming), probability theory, the maths of information (with laws of information corresponding to group-theoretical principles), etc. Best, Ben - Original Message - From: Benjamin Udell To: PEIRCE-L@LISTSERV.IUPUI.EDU Sent: Monday, March 12, 2012 1:14 PM Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition - 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

### Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

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

### Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

Irving wrote, quoting Peirce MS L75:35-39: 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. [...] 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 [...] I would disagree with this reading of the Peirce passage. It seems to me that the distinction he is making is, rather, between (1) the case where the conclusion can be seen to follow from the premisses by virtue of the logical form alone, as in A function which is continuous on a closed interval is continuous on any subinterval of that interval (whose truth is obvious without requiring us to imagine any continuous function or any interval), and (2) the case where the deduction of the conclusions from the premisses requires turning one's imagination upon, and experimenting with, the actual mathematical objects of which the theorem speaks, as in A function which is continuous on a closed interval is bounded on that interval. -malgosia - 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

### 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

, 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

### Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

Thanks for the thoughtful reply, Gary! The issue you raise about how deduction and induction should be categorised is an interesting one. I had always thought of deduction as falling clearly under secondness, due to the compulsion involved. But you are right to note that in theorematic deduction the mind is not passive but active, and that this form of reasoning was very important to Peirce. I don't see how one might interpret induction as secondness though. Though a *misplaced* induction may well lead to the secondness of surprise due to error. H... Cheers, Cathy On Thu, Feb 23, 2012 at 1:49 PM, Gary Richmond gary.richm...@gmail.com wrote: Cathy, list, When I first read your remark suggesting that the birth, growth and development of new hypostatic abstractions should be in the position of 3ns rather than argumentative proof of the validity of the mathematics as I had earlier abduced, I thought this might be another case of the kind of difficulty in assigning the terms of 2ns and 3ns in genuine triadic relations which had Peirce, albeit for a very short time in his career, associating 3ns with induction (while before and after that time he put deduction in the place of 3ns as necessary reasoning--I have discussed this several times before on the list and so will now only refer those interested to the passage, deleted from the 1903 Harvard Lectures--276-7 in Patricia Turrisi's edition--where Peirce discusses that categorial matter). I think his revision of his revision to his original position may have been brought about by the clarification resulting from thinking of abduction/deduction/induction beyond critical logic (where they are first analyzed as distinct patterns of inference), then in methodeutic where a complete inquiry--in which hypothesis formation is 1ns, the deduction of the implications of the hypothesis for testing is 3ns, and, finally, the actual inductive testing is 2ns--provides a kind of whetstone for categorial thinking about these three. (Yet, even in that 1903 passage he remarks that he will leave the question open.) Be that as it may, I am beginning to think that you are clearly on to something and that that transforming of a predicate into a relation which we call hypostatic abstraction certainly ought to be in the place of 3ns. Re-reading parts of Jay Zeman's famous and fine article on hypostatic abstraction further strengthened that opinion. See: http://web.clas.ufl.edu/users/jzeman/peirce_on_abstraction.htm Zeman writes: It is hypostatic or subjectal abstraction that Peirce is interested in; a hint as to why he is interested in it is given in his allusions in these passages to mathematical reasoning [. . .] Jaakko Hintikka has done us the great service of bringing to our attention and tying to contemporary experience one of Peirce's central observations about necessary—which is to say mathematical—reasoning: this is that nontrivial deductive reasoning, even in areas where explicit postulates are employed, always considers something not implied in the conceptions so far gained [in the particular course of reasoning in question], which neither the definition of the object of research nor anything yet known about could of themselves suggest, although they give room for it. As is well known, Peirce calls this kind of reasoning theorematic (in contrast to corollarial reasoning) because it introduces novel elements into the reasoning process in the form of icons, which are then 'experimented upon in imagination.' Zeman quotes Hintikka to the effect that Peirce himself seems to have considered a vindication of the concept of abstraction as the most important application of his discovery [of the theorematic/corollarial distinction] and then remarks that Peirce would indeed have agreed that the light shed on necessary reasoning by this distinction helps greatly to illuminate the role of abstraction. . . See, also: EP2:394 where Peirce comments that it is hypostatic abstraction that leads to the generalizality of a predicate and, of course, what is general is 3ns. In short, I think you are quite right Cathy to have suggested that correction of my categorial assignments. As Peirce notes near the end of the Additament to the Neglected Argument, hypothetic abstraction concerns itself with that which necessarily would be *if* certain conditions were established (EP2:450). Best, Gary On 2/21/12, Catherine Legg cl...@waikato.ac.nz wrote: Gary wrote: For the moment I am seeing these three as forming a genuine tricategorial 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 [...] Wouldn't argumentative proof be the 2ness, and the 3ness would be something like the birth, growth and

### Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

1. Hypothesis (Abduction) 2. Induction 3. Deduction But isn't it also the case that we can mix firsts, seconds and thirds if we think it appropriate. As in Terms Propositions Symbols. Best, S *ShortFormContent at Blogger* http://shortformcontent.blogspot.com/ On Fri, Mar 2, 2012 at 12:25 PM, Catherine Legg cl...@waikato.ac.nz wrote: Thanks for the thoughtful reply, Gary! The issue you raise about how deduction and induction should be categorised is an interesting one. I had always thought of deduction as falling clearly under secondness, due to the compulsion involved. But you are right to note that in theorematic deduction the mind is not passive but active, and that this form of reasoning was very important to Peirce. I don't see how one might interpret induction as secondness though. Though a *misplaced* induction may well lead to the secondness of surprise due to error. H... Cheers, Cathy On Thu, Feb 23, 2012 at 1:49 PM, Gary Richmond gary.richm...@gmail.com wrote: Cathy, list, When I first read your remark suggesting that the birth, growth and development of new hypostatic abstractions should be in the position of 3ns rather than argumentative proof of the validity of the mathematics as I had earlier abduced, I thought this might be another case of the kind of difficulty in assigning the terms of 2ns and 3ns in genuine triadic relations which had Peirce, albeit for a very short time in his career, associating 3ns with induction (while before and after that time he put deduction in the place of 3ns as necessary reasoning--I have discussed this several times before on the list and so will now only refer those interested to the passage, deleted from the 1903 Harvard Lectures--276-7 in Patricia Turrisi's edition--where Peirce discusses that categorial matter). I think his revision of his revision to his original position may have been brought about by the clarification resulting from thinking of abduction/deduction/induction beyond critical logic (where they are first analyzed as distinct patterns of inference), then in methodeutic where a complete inquiry--in which hypothesis formation is 1ns, the deduction of the implications of the hypothesis for testing is 3ns, and, finally, the actual inductive testing is 2ns--provides a kind of whetstone for categorial thinking about these three. (Yet, even in that 1903 passage he remarks that he will leave the question open.) Be that as it may, I am beginning to think that you are clearly on to something and that that transforming of a predicate into a relation which we call hypostatic abstraction certainly ought to be in the place of 3ns. Re-reading parts of Jay Zeman's famous and fine article on hypostatic abstraction further strengthened that opinion. See: http://web.clas.ufl.edu/users/jzeman/peirce_on_abstraction.htm Zeman writes: It is hypostatic or subjectal abstraction that Peirce is interested in; a hint as to why he is interested in it is given in his allusions in these passages to mathematical reasoning [. . .] Jaakko Hintikka has done us the great service of bringing to our attention and tying to contemporary experience one of Peirce's central observations about necessary—which is to say mathematical—reasoning: this is that nontrivial deductive reasoning, even in areas where explicit postulates are employed, always considers something not implied in the conceptions so far gained [in the particular course of reasoning in question], which neither the definition of the object of research nor anything yet known about could of themselves suggest, although they give room for it. As is well known, Peirce calls this kind of reasoning theorematic (in contrast to corollarial reasoning) because it introduces novel elements into the reasoning process in the form of icons, which are then 'experimented upon in imagination.' Zeman quotes Hintikka to the effect that Peirce himself seems to have considered a vindication of the concept of abstraction as the most important application of his discovery [of the theorematic/corollarial distinction] and then remarks that Peirce would indeed have agreed that the light shed on necessary reasoning by this distinction helps greatly to illuminate the role of abstraction. . . See, also: EP2:394 where Peirce comments that it is hypostatic abstraction that leads to the generalizality of a predicate and, of course, what is general is 3ns. In short, I think you are quite right Cathy to have suggested that correction of my categorial assignments. As Peirce notes near the end of the Additament to the Neglected Argument, hypothetic abstraction concerns itself with that which necessarily would be *if* certain conditions were established (EP2:450). Best, Gary On 2/21/12, Catherine Legg cl...@waikato.ac.nz wrote: Gary wrote: For the moment I am