Ben, lists, It looks like Ben's post was sent to Peirce-L, but not the biosemiotics list-serv. For biosemiotics list members, please see below for the post to which I am responding.
"I think that Frederik is largely assuming Peirce's terminology. Peirce uses the words 'schema' and 'diagram' pretty much interchangeably." Yes, Ben, I would have guessed as much. In connection with this, I wonder whether Peirce would have said Kant's schematization in the Critic of Pure Reason is in fact a diagrammatization of some sort. As to the rest of what you had to say, I don't find myself really in any disagreement. I see that Frederik mentioned that whether something is corollarial or theorematic is not relative to a person's intelligence. I would suppose you meant that a mathematician would have a much more advanced logic system available to think about then the (average) schoolchild, in which case your remark would make sense. While Frederik is right to point out that only whether something new or foreign is introduced is what makes the reasoning theorematic (I believe I mentioned in a previous post that this is what is signficant for Frederik, and not so much the complexity of the schema), it is also true that what logic system one is using will affect what counts as corollarial reasoning and what as theorematic; that is, it will affect whether the something "new" or "foreign" is really new or foreign to the system, and the typical schoolchild probably has a much simpler logic system to work with than the typical mathematician. Actually, it just struck me that I mentioned that the complexity is not so important for Frederik's account, and you have continued discussing complexity to show how it is important. I'm sorry, I didn't mean to imply that non-triviality is unimportant. It does get mentioned in the text at some point, but does not play a prominent role, not nearly as much as the point that something new or foreign must be introduced into the reasoning. Though, I do wonder somewhat whether non-triviality is connected to a theorem not being easily absorbed into a logic system that could turn what was originally a theorematic reasoning into a purely corollarial reasoning. Perhaps the newer or more foreign the idea, the more nontrivial and fecund it may turn out to be? -- Franklin On Mon, Apr 20, 2015 at 4:18 PM, Benjamin Udell <[email protected]> wrote: > 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 > (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 <[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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