---------- Forwarded message ---------- From: Franklin Ransom <[email protected]> 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: [email protected]
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 <[email protected]> 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 > > >
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