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" in Memoir 19 (once there, scroll down): 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. (Carnegie Institute Application, from Draft A - MS L75.35-39, 1902) 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. "This appears to be in harmony with Kant's view of deduction, namely, that it merely explicates what is implicitly asserted in the premisses. This is what is called a half-truth. Deductions are of two kinds, which I call corollarial and theorematic. The corollarial are those reasonings by which all corollaries and the majority of what are called theorems are deduced; the theorematic are those by which the major theorems are deduced. If you take the thesis of a corollary, -- i.e. the proposition to be proved, and carefully analyze its meaning, by substituting for each term its definition, you will find that its truth follows, in a straightforward manner, from previous propositions similarly analyzed. But when it comes to proving a major theorem, you will very often find you have need of a _lemma_, which is a demonstrable proposition about something outside the subject of inquiry; and even if a lemma does not have to be demonstrated, it is necessary to introduce the definition of something which the _thesis_ of the theorem does not contemplate. In the most remarkable cases, this is some abstraction; that is to say, a subject whose existence _consists_ in some fact about other things. Such, for example, are operations considered as in themselves subject to operation; lines, which are nothing but descriptions of the motion of a particle, considered as being themselves movable; _collections_; _numbers_; and the like. When the reform of mathematical reasoning now going on is complete, it will be seen that every such supposition ought to be supported by a proper postulate. At any rate Kant himself ought to admit, and would admit if he were alive today, that the conclusion of reasoning of this kind, although it is strictly deductive, does not flow from definitions alone, but that postulates are requisite for it. ('On the Logic of Drawing History from Ancient Documents, Especially from Testimonies', EP 2:96, 1901) Best, Ben ----- Original Message ----- From: Irving To: PEIRCE-L@LISTSERV.IUPUI.EDU Sent: Tuesday, March 13, 2012 4:33 PM 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 that there are >> fundamental differences between physical reality and mathematical reality. > Quite so. And one of these is noted by Hilbert (or maybe Hardy, > anyone > help?) >- > "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. > Quite so; but philosophers tend to have a powerful sense of entitlement. the other, in Gauss's famous letter November 1, 1844 to astronomer Heinrich Schumacher regarding Kant's philosophy of mathematics, that: "you see the same sort of [mathematical incompetence] in the contemporary philosophers.... Don't they make your hair stand on end with their definitions? ...Even with Kant himself it is often not much better; in my opinion his distinction between analytic and synthetic propositions is one of those things that either run out in a triviality or are false." ----- Message from bud...@nyc.rr.com --------- Date: Mon, 12 Mar 2012 13:47:10 -0400 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 > 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://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