Dear Ben, lists - Thanks for two mails. The first largely resumes parts of my chapter and indeed Peirce's basic ideas of theorematicity - although it is not entirely correct that P saw his distinction as relative to intellect so that which is corollarial to a grownup will be theorematic to a child. Peirce insisted that theorematicity consists in the addition of something new to a problem - an additional object, manipulation, abstraction, perspective, etc. If once you know which such addition to add in the single case, the remaining problem becomes corollarial, but that has nothing to do with the intellect of the reasoner. A stupid but well-informed person may repeat Euclid's theorematic proof of the angular sum of the triangle, while an uninformed genius may be unable to find that proof - and still the proof requires the addition of auxiliary lines to the triangle no matter what.
But you address other important things. It seems as if, to some degree, Peirce without saying it assumes something like the discovery/justification distinction. When saying mathematical reasoning is deductive, this seems to be a justification claim merely, because in the actual procedure of searching for the proof of a theorem, Peirce realizes there may be an abductive trial-and-error phase, particularly in the theorematic cases where it is not evident which new element to add to your problem (is there also something akin to an inductive phase in mathematical proofmaking, e.g. when mathematicians compare and evaluate their result with respect to its potential effects in other areas of math?). So even if mathematics is the science that proceeds by deductive reasoning, there are non-deductive phases in it (discovery), even if the results are deductively valid (justification). My idea with Wegener's map was, of course, to find a theorematic example from applied math. In such a case both the mathematical formalism (here, approximately Euclidean geometry) and the basic assumptions of the material field (geology) must be part of the status quo to which a new element, manipulation, principle etc. should be added. The transformation making the two continent coasts meet is trivial in the Euclidean sense, but the change in underlying geological ontology (from the axiom that continents are eternally stable to the axiom that they float on the surface of the earth) indeed requires the addition of a new idea. In some sense, the radicality of this new idea is eased by the triviality of the transformation in purely geometrical terms. What prompted the idea of generalizing the mathematical notions of corollarial/theorematic to the applied sciences, of course, is Peirce's classificaiton of the sciences where math is number one, implying that all other sciences wihtout exception use mathematical structures - but simultaneously that generalization cannot take place without introdcuding basic principles of those "lower" sciences, thereby modifying the corr/theor. distinction to some degree because it now has to involve ontological assumptions regarding positive knowledge. But still I think it makes good sense. Best F Frederik, lists, I'm dissatisfied with my previous post in this thread, I feel like I've missed the forest for the trees. While I'm not convinced that there's a theorematic applied deduction in the Wegener example, still, the idea of continental drift is not merely a simplifying explanation of the fit between continental coastlines, it's also an idea that anybody would call nontrivial. It involves a complex new idea, and, if true (as it turned out to be), would foreseeably be a basis and foundation for much further discovery. Its nontriviality doesn't give it intrinsic abductive merit in the way that its plausibility does, but said nontriviality still makes it something to be prized if it pans out (as it did). But so far, that's the nontriviality of prospective discoveries, what about a nontriviality of how one got to the abduction of continental drift? I'm trying to think of some parallelism between its abduction and theorematic deduction, so as to analogize the idea of abductive nontriviality to deductive theorematicity. Roughly, something involving nontrivial changes of standing beliefs about geology, changes equivalent to the idea of continental drift. Well, when even I think I'm talking too much, it's time I call it a day. Best, Ben : Franklin, lists, I think that Frederik is largely assuming Peirce's terminology. Peirce uses the words 'schema' and 'diagram' pretty much interchangeably. Here are some key quotes on which Frederik is basing his discussion of the theormatic-corollarial distinction. http://www.commens.org/dictionary/term/corollarial-reasoning I once did a summary (footnoted with online links) of key points (at least as they seemed to me at the time); here it is with a few adjustments of the links: Peirce 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] Peirce, C. S., from section dated 1902 by editors in the "Minute Logic" manuscript, Collected Papers v. 4, paragraph 233, quoted only in part http://www.commens.org/dictionary/entry/quote-minute-logic-chapter-iii-simplest-mathematics in "Corollarial Reasoning" in the Commens Dictionary of Peirce's Terms, 2003–present, Mats Bergman and Sami Paavola, editors, University of Helsinki. FULL QUOTE: https://archive.org/stream/TheWorldOfMathematicsVolume3/Newman-TheWorldOfMathematicsVolume3#page/n366/mode/1up in The World of Mathematics, Vol. 3, p. 1776. [2] Peirce, C. S., the 1902 Carnegie Application, published in The New Elements of Mathematics, Carolyn Eisele, editor, quoted in "Corollarial Reasoning" http://www.commens.org/dictionary/entry/quote-carnegie-institution-correspondence-4 in the Commens Dictionary of Peirce's Terms, also transcribed by Joseph M. Ransdell, see "From Draft A - MS L75.35-39" in Memoir 19 http://www.iupui.edu/~arisbe/menu/library/bycsp/l75/ver1/l75v1-06.htm#m19<http://www.iupui.edu/%7Earisbe/menu/library/bycsp/l75/ver1/l75v1-06.htm#m19> (once there, scroll down). [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 http://www.commens.org/dictionary/entry/quote-logic-drawing-history-ancient-documents-especially-testimonies-logic-histor-5 in "Corollarial Reasoning" in the Commens Dictionary of Peirce's Terms. The introduction of an idea beyond the explicit conditions of a problem and not contemplated in the thesis to be proved is precisely a 'complexifying' step. One might think of it as a leveraging of imagination to deepen understanding, by which vague remark I'm trying to get at the idea that such complexity is very different from the tedious complication of hundreds or thousands of trivial computations, computations that need to be done sometimes even in pure mathematics, where it is known as 'brute force'. Tedious computation used to be done by people called 'computers' up until computing machines came into use; part of Peirce's burden at the Coast Survey was that there came a time when he had to do his own tedious, lengthy computations and, worse, he found that his computing power was no longer what it was when he was younger; errors crept in. In CP 4.233 (again https://archive.org/stream/TheWorldOfMathematicsVolume3/Newman-TheWorldOfMathematicsVolume3#page/n366/mode/1up) in "The Essence of Mathematics", Peirce says, [....] 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. [....] The above is an example of why I keep talking about complexity in the sense of nontriviality. A theorem in the old sense, that is, as opposed to a corollary, is a proposition whose proof requires, at least as a practical matter, some 'complexifying', active new-idea-adding experimentation of theorematic reasoning. Such reasoning does not just add steps and operations, but incorporates ideas in ways that enrich the understanding, make 'new gestalts', to borrow some lingo that may sound hokey today. The mathematical theorem's nontriviality is its character of being a mathematical theorem in the sense of not being a mathematical corollary; it's such a theorem's non-corollarity. The theorem's nontiviality reflects, is, in a sense, the needed theorematicity of its proof, and for that very reason it reflects also the prospect of its occasioning in turn further theorematic proofs of further theorems, whatever they might be, as opposed to mere corollaries; its all about deepened understandings, as opposed to merely additional tidbits, soever multitudinous, of information. It takes nontrivia to make nontrivia. It's true that I bring in a methodology-of-inquiry perspective in addition to the critique-of-arguments perspective taken by Peirce in analyzing theorematic and corollarial reasonings. But I think that it does matter in understanding the role of theorematic reasoning in mathematics, and in relating the ideas of theorematic and corollarial reasonings to the common parlance (at least what I've been told of it) of mathematicians, where 'nontriviality', 'depth', 'fecundity' are prized characters of proven propositions. It's not that the theorematic deduction brings something to light while the corollarial deduction brings nothing to light. It's a matter of degree as you say; indeed what seems theorematic to a schoolchild may well seem corollarial to a mathematician. Peirce generally discusses reasoning and inquiry in the context of discovery rather than in the context of justification, as Frederik pointed out; and we never entirely depart the context of discovery even when we're focused on justification. Anyway, corollarial reasoning that is not manifestly redundant (redundant like 'pq, ergo p') does provide some jot of novelty or nontriviality; the categorical syllogisms (such as All A is B, all B is C, ergo all A is C) are deductive forms designed to assure some modicum of novelty in corollarial conclusions; and massive, brute-force corollarial computation may bring things to light that we couldn't find otherwise (it still plays a big role in the proof of the four-color theorem). What Peirce says is that sometimes corollarial deduction won't suffice, and that then theorematic deduction is needed in order to bring something to light. Whew. I'm not sure I've addressed all in your post, but I'll let it stand for now and retract who knows what tomorrow. Best, Ben On 4/19/2015 5:12 PM, Franklin Ransom wrote: ---------- Forwarded message ---------- From: Franklin Ransom <pragmaticist.lo...@gmail.com<mailto:pragmaticist.lo...@gmail.com> > Date: Sun, Apr 19, 2015 at 5:11 PM Subject: Re: [biosemiotics:8342] Re: [PEIRCE-L] Natural Propositions, Ch. 10: Corollarial and Theorematic Experiments with Diagrams To: biosemiot...@lists.ut.ee<mailto:biosemiot...@lists.ut.ee> Ben, lists, Thank you, Ben, for a post that is (clearly) on topic. Frederik notes, in the fourth definition of theorematic reasoning, that it involves schemata rather than words. Actually, he qualifies this claim, noticing that Peirce says even words are schemata, but rather simple schemata. Theorematic reasoning typically involves then complicated schemata. It is really a matter of degree or gradation though, as corollarial reasoning typically involves simpler schemata and theorematic reasoning typically involves complicated schemata, relative to each other. In the text, p.276-7, Frederik seems to associate schemata with diagrams, so that corollarial reasoning makes less use of diagrams and theorematic reasoning makes greater use of diagrams. If I recall correctly, this is all that is really mentioned about complexity or complication. Otherwise, there is the discussion in the chapter regarding the possibility that some theorematic reasoning, using a different logic system (by this, meaning a different set of axioms and rules), may be reworked as corollarial reasoning, because not needing to include something new or foreign to the premises and conclusion as the other logic system would have required. I believe that is in p.280-3. As I understand it, what Frederik takes to be most essential is the introduction of something new or foreign to the reasoning, and not so much the relative simplicity or complexity of the reasoning. This is probably due to the flexibility of some reasonings as being capable of classification under either head, depending upon the logic system at work. With respect to nontriviality or depth, this isn't really discussed in the chapter. The point of the chapter is less about the value of theorems than it is about explaining what theorematic diagrammatic reasoning is and what its significance is. In fact, the significance seems to be less about the importance of theorematic reasoning in mathematics and more about the importance of theorematic reasoning for epistemology, i.e. for knowledge whether of the scientific sort or of the everyday sort. My concern about corollarial reasoning is that, since corollarial reasoning does involve experimentation, what should be the point of experimentation if nothing unnoticed or hidden ever appeared as a result? I don't doubt that theorematic reasoning is better for the purpose, I just don't think that it's a hard-and-fast line to be drawn between theorematic and corollarial reasoning. Perhaps my concern would be better answered though if it were made clearer what the role of these reasonings is in the context of scientific method, which would allow for a clearer account of the Holm example. -- Franklin On Sun, Apr 19, 2015 at 2:05 PM, Benjamin Udell <bud...@nyc.rr.com<mailto:bud...@nyc.rr.com> > wrote: Franklin, lists, I agree with Jon, thanks for your excellent starting post. You wrote, [....] Why can't corollarial reasoning, which involves observation and experimentation, reveal unnoticed and hidden relations? After all, on p.285-6, Frederik mentions the work of police detective Jorn "Old Man" Holm and his computer program, which Frederik describes as a "practical example of corollarial map reasoning" (p.285). In this example, Holm uses the corollarial reasoning to reveal information about the whereabouts of suspects. Doesn't the comparison of the map reasoning with suspects' testimony end up revealing unnoticed and hidden relations? There's a distinction that some make between complexity and mere complication. Corollarial reasonings may accumulate mere complications until the result becomes hard to see, although it involves little if any complexity in, more or less, the sense of depth or nontriviality. I don't know whether there's a theorematic approach to Jørn Holm's diagrammatization that would show its result in a nontrivial aspect, and anyway its diagrammatic, pictorial presentation already leaves one in no doubt that a pattern is revealed. A good example involving alternate proofs that seem corollarial and theorematic is the Monty Hall problem, a popular puzzle based in probability theory. I remember reading an essentially corollarial proof of the answer, and seeing a round diagram that showed how alternatives lead inevitably to the conclusion in the diagram's center. The answer to the Monty Hall problem remains, however, notoriously counter-intuitive to people; the essentially corollarial but multi-step proof - in words, even with the round diagram - often leaves people with nagging vague doubts. They get that it must be true but they feel that they don't fully get the problem, they keep re-examining the problem, wondering whether it was well disambiguated, etc. (it describes an actual standard scenario on a popular TV game show). But the problem's answer has also a proof that deserves to be called theorematic (even if it is not very much so) since it involves varying the conditions of the problem, adding things not contemplated in the thesis, going a little deeper into the mathematical possibilities. One increases the number of doors in the scenario from 3 to 10. With 10 doors, the basically the same solution makes obvious sense, then one reduces the number doors from 10 to 9 to 8, etc. down to 3, and sees that the basic solution does not change at all; people get satisfied (for whatever that's worth). It has become hard to avoid running into that proof if one searches the Internet for "Monty Hall problem". I also vaguely remember a geometric problem involving the fitting of circles, shown to me by a roommate during college; he was dissatisfied with a particular usual proof, he wanted a proof that gave more satisfactory understanding, and it turned out to be more imaginative and, as I'd call it now, theorematic. Nontriviality or depth of a result should not be confused with mere complication and lengthiness of a proof; take the Pythagorean theorem, which is considered both deep and not very hard to prove. The nontriviality or depth of a theorem consists not in the difficult complication of proving it but in its favorability as a bridge to further nontrivial lessons or, to put it less recursively, its favorability for use as a basis for further proofs almost as if it were another postulate even though it is entailed by the postulates and axioms already granted. It's a place where one can come to rest for a while and set up camp; if I were to coin a word dedicated to expressing it I'd say 'basatility'. Likewise the nontriviality or depth (apart from mere complication as distinguished from complexity) of a proof of a theorem is properly its favorability as a basis for further lessons. (I'm not sure that there is much difference between 'depth' and 'power' of a theorem or a proof.) The nontrivial or deep is more or less _difficult _ (which is a usual connotation especially of 'nontrivial') since, of course, it requires some corresponding depth or or nontriviality of understanding and perspective. (One should distinguish such depth, complexity, etc., of theorems and proofs also from the logical complexity that a fact or datum, as a relation or complexus of relations, possesses; I mean such 'complexity' as quantified and characterized by valence, transitivity or intransitivity, etc. This is likewise as one distinguishes the novelty or new aspect of a deductive conclusion from Shannonesque quantity of information or 'newsiness'.) Best, Ben ----------------------------- PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu<mailto:peirce-L@list.iupui.edu> . To UNSUBSCRIBE, send a message not to PEIRCE-L but to l...@list.iupui.edu<mailto:l...@list.iupui.edu> with the line "UNSubscribe PEIRCE-L" in the BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm .
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