[PEIRCE-L] Conflict between deduction and discovery in mathematics
List, I am new to Peirce's work, but I am curious to know if this is a valid interpretation of his work. I would also like to thank you for your answers to my previous post. In various places, Peirce states that all deduction is diagrammatic. For example, in a letter to William James (1909, L224), he affirms that he "first found, and subsequently 'proved', that every deduction involves the observation of a diagram (...) and having drawn the diagram, one finds the conclusion to be represented by it" (NEM 3:869, his emphasis). I believe there is no risk in taking this thesis to mean that diagrammaticity is a necessary feature of deductive reasoning, even if it is not always explicit. In other words, diagrammatic reasoning is not a product of our cognitive limitations, nor is it merely another type of valid reasoning. What this statement does not say is why “every deduction involves the observation of a diagram.” However, in PAP (1906), Peirce offers an argument that seems to ask this question. First, he states that “necessary reasoning makes its conclusion evident” (NEM 4:317). He then states that this evidence "consists in the fact that the truth of the conclusion is perceived in its full generality, and in the generality the how and why of the truth is perceived" (Ibid.). He then considers the evidence that can be furnished by the three kinds of signs, symbols, indices and diagrams. Finally, he concludes that diagrams are the only signs that "literally show, as a percept shows the perceptual judgment to be true, that a consequence does follow, and more marvellous yet, that it would follow under all varieties of circumstances accompanying the premisses" (NEM 4:318). The argument seems to rely on a distinction between what Peirce calls "following a rule of thumb" and "reasoning properly called". Previously, to present the argument in PAP, Peirce makes this distinction as follows: "[I]t is necessary to distinguish reasoning, properly so called, where the acceptance of the conclusion in the sense in which it is drawn, is seen evidently to be justified, from cases in which a rule of inference is followed because it has been found to work well, which I call following a rule of thumb, and accepting a conclusion without seeing why further than that the impulse to do so seems irresistible. In both cases, there might be a sound argument to defend the acceptance of the conclusion; but to accept the conclusion without any criticism or supporting argument is not what I call reasoning." (NEM 4:314) The citation is in accordance with other citations from Peirce, such as this one, in which he makes a similar distinction between "reason" and the work of a machine. "[I]t has been shown that all possible general conclusions can be arranged in a serial order and as soon as anybody wished to defray the not extravagant cost, the specifications will be ready for a machine that will actually turn new theorems from a given set of premises, one after another, as long as they continue to have any interest. But though a machine could do all that, and thus accomplish all that many an eminent mathematician accomplishes, it still cannot properly be called a reasoning machine, any more than the sort of a man I have in view can be called a reasoner. It does not reason; it only proceeds by a rule of thumb." (NEM 3:115, n. d., Our Senses as Reasoning Machines) Based on this argument, we can conclude that diagrams are necessary for deduction according to Peirce, because they are the only signs that furnish the evidence that properly called reasoning requires. This does not mean that it is impossible to draw conclusions from premises without the use of diagrams. A logical machine could do this, but such a process would not be considered proper reasoning. It is a blind process that is analogous to following a rule of thumb. Furthermore, this is not psychologism, although it involves a subject who sees that the inference is justified. In PAP Peirce also mention that “there are at least two other entirely different lines of argumentation each very nearly, and perhaps quite, as conclusive as the above, though less instructive, to prove that all necessary reasoning is by diagrams. One of these shows that every step of such an argumentation can be represented, but usually much more analytically, by Existential Graphs. Now to say that the graphical procedure is more analytical than another is to say that it demonstrates what the other virtually assumes without proof. Hence, the Graphical method, which is diagrammatic, is the sounder form of the same argumentation. The other proof consists in taking up, one by one, each form of necessary reasoning, and showing that the diagrammatic exhibition of it does it perfect justice" (NEM 4:319). However, he does not follow these other two lines of argumentation in the mentioned text. I am not sure if I am correct about this interpretation. I apreciate your feedbak. Thank you very much in advance for your
Re: [PEIRCE-L] Conflict between deduction and discovery in mathematics
Jon, list, I thank you very much for your answer. As you suggest, I believe that Peirce's answer to the problem lies in his notion of theorematic deduction. However, I'm having trouble understanding what he means by that. For example, I am confounded by the meaning of this citation. "It was because those logicians who were mathematicians saw that the notion that mathematical reasoning was as rudimentary as that was quite at war with its producing such a world of novel theorems from a few relatively simple premisses, as for example it does in the theory of numbers, that they were led,-first Boole and DeMorgan, afterwards others of us, -to new studies of deductive logic, with the aid of algebras and graphs. The non-relative logic having soon been exhausted, we went into the study of the logic of relatives, first the dyadic, and subsequently I, almost alone, into polyadic relations. These studies threw a great deal of light upon logic; but still they did not really explain mathematical reasoning, until I opened up the subject of abstraction. It now appears that there are two kinds of deductive reasoning, which might, perhaps, be called explicatory and ampliative. However, the latter term might be misunderstood; for no mathematical reasoning is what would be commonly understood by amp/iative, although much of it is not what is commonly understood as explicative. It is better to resort to new words to express new ideas. All readers of mathematics must have felt the great difference between corollaries and major theorems, although these words are not sharply distinguished. It is needless to say that the words come to us, not from Euclid, but from the editions of Euclid's elements. The great body of the propositions called corollaries (all but 27 in the whole 13 books) are due to commentators, and are of an obvious kind. Kant's characterization of all deductive reasoning is true of them: they are mere explications of what is implied in previous results. The same is true of a good many of Euclid's own theorems; probably the numerical majority of the whole 369 of them are of this character. But many of them are of a different nature. We may call the two kinds of deduction corol/arial and theorematic." (NEM 4:1, 1901) Here, Peirce first gives some hints about the history of the problem. He then puts his own contribution in this context, acknowledging the limits of his studies of polyadic logic. Finally, he affirms that the problem arises when deduction is reduced to Kant's characterization. Nevertheless, he conjectures that there are in fact two kinds of deductions, which are explicative and "ampliative". This can eventually throw light on the problem by explaining how deductions can be both certain and novel. However, within what framework should Peirce's reference to Kant's characterization of deductive reasoning be interpreted: the new logic of relations or syllogistic logic? Why did Peirce claim that his own studies on the logic of polyadic relations did not yet fully explain mathematical reasoning? Is it because the logic of relatives cannot explain some inferential steps, for example, the introduction of abstractions or the construction in Euclid's propositions? Or is it because we cannot find premises that can transform every proof into a corollarial or explicative proof? Or is there another reason? Additionally, I do not fully understand the relation between the notion of theorematic deduction and Peirce's thesis about the diagrammatic character of all deduction. Here, I suspect there is some important clue to understanding Peirce's argument. Thank you again for your time. Best regards, Matias 2023-08-19 13:04 GMT-03:00, Jon Alan Schmidt : > Matias, List: > > Although I cannot offer "any information that traces the history of this > problem" as requested, I can suggest Peirce's own explanation of it. > > CSP: 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. (CP 7.204, 1901) > > > See also NEM 4:1-12 (1901), which begins with the second quotation below; > CP
Re: [PEIRCE-L] Conflict between deduction and discovery in mathematics
Ben, Phyllis, Thank you both for your answers. I appreciate your insights. Ben, I will check out the Gilman article you mentioned. I didn't know about it, but it sounds like it could be helpful. I believe that Peirce's answer to the paradox lies in his notion of theorematic deduction. However, I'm also having trouble understanding what he means by that. I'm hoping that the Gilman article will shed some light on this. Furthermore, I think it would be helpful to put his answer in perspective, taking into account the history of the problem and the subsequent development of logic. Best regards, Matias El sáb, 19 de ago de 2023, 09:24, Ben Udell escribió: > > > > > > > > * I just found B.I. Gilman's article at Google Books. The whole article > was accessible to me here in the USA. > https://books.google.com/books?id=dPhl9SLIU54C=PA38=PA38 > <https://books.google.com/books?id=dPhl9SLIU54C=PA38=PA38> I'll try > to see (not immediately!) what to think of it. Best, Ben On 8/19/2023 7:22 > AM, Ben Udell wrote: Matias, Phyllis, One does often start with guessing, > retroduction, etc., in trying to solve a mathematical problem, be the > problem trivial or deep. However this guesswork or the like is usually not > formalized in publications. Occasionally a mathematician publishes a > mathematical conjecture, and some have been pretty important. One of > Peirce's students Benjamin Ives Gilman whom Peirce got published in Studies > in Logic (1883) > https://archive.org/details/studiesinlogic00gilmgoog/page/n15/mode/2up?ref=ol=theater > <https://archive.org/details/studiesinlogic00gilmgoog/page/n15/mode/2up?ref=ol=theater> > did not make a career in logic but did author a published (1923) article > "The Paradox of the Syllogism Solved by Spatial Construction" Mind, New > Series, Vol. 32, No. 125 (Jan., 1923), pp. 38-49 (12 pages) Published By: > Oxford University Press https://www.jstor.org/stable/2249497 > <https://www.jstor.org/stable/2249497> and I've meant to get hold of it and > read it because the general question interests me. Peirce thought highly of > Gilman; and Gilman in that article may reflect, explicitly or implicitly, > Peirce's views on novelty in deduction. Gilman claimed to have solved the > problem! It certainly is a problem. Who would bother with explicit, > deliberately weighed deduction if it did not produce conclusions with > aspects at least mildly surprising or with at least a jot of depth, > nontriviality? It's an instance of a broader paradox. Induction actually > (as opposed to seemingly like some deduction) adds claims; in Peirce's > later view it should conclude with verisimilitude a.k.a likelihood > http://www.commens.org/dictionary/term/verisimilitude > <http://www.commens.org/dictionary/term/verisimilitude> - which, as far as > I can tell, is to say that it ought to seem UNsurprising despite going > beyond the premises, or as Peirce put it, resemble the facts already in the > premises. Similar remarks can be made about abductive inference. I tend > to think that all reasoning depends for its value in part on > characteristics that resist being exactly quantified or exactly defined and > which are in some sort of tension, some sort of counterbalance, with the > inference mode's distinctive or definitive entailment-related structure. > I've noticed that the question of "seeing" or "not seeing" deductive > implications is sometimes discussed as the question of logical omniscience > and the lack thereof, for example by Sergei Artemov and Roman Kuznets in > "Logical omniscience as infeasibility", Annals of Pure and Applied Logic, > Volume 165, Issue 1, January 2014, Pages 6-25 > https://www.sciencedirect.com/science/article/pii/S0168007213001024 > <https://www.sciencedirect.com/science/article/pii/S0168007213001024> . > Best, Ben On 8/18/2023 9:08 PM, Phyllis Chiasson wrote: * > > * Wouldn't this be true for all of nature versus the all of discovery? > Discovery is human and therefore retroductive (as are "newspapers and great > fortunes"). Nature is. On Fri, Aug 18, 2023, 4:14 PM Matias > wrote: * > > * Dear list members, I am trying to contextualize Peirce's reference to > the long-standing conflict between the notion of mathematical reasoning and > the novelty of mathematical discoveries. I would appreciate any information > that traces the history of this problem. Here are two citations in which > Peirce mentions such a conflict: "It has long been a puzzle how it could be > that, on the one hand, mathematics is purely deductive in its nature, and > draws its conclusions apodictically, while on the other hand, it presents > as rich and apparently unending a series of surprising discoveries as
[PEIRCE-L] Conflict between deduction and discovery in mathematics
Dear list members, I am trying to contextualize Peirce's reference to the long-standing conflict between the notion of mathematical reasoning and the novelty of mathematical discoveries. I would appreciate any information that traces the history of this problem. Here are two citations in which Peirce mentions such a conflict: "It has long been a puzzle how it could be that, on the one hand, mathematics is purely deductive in its nature, and draws its conclusions apodictically, while on the other hand, it presents as rich and apparently unending a series of surprising discoveries as any observational science. Various have been the attempts to solve the paradox by breaking down one or other of these assertions, but without success." (Peirce, 1885, On the Algebra of Logic, p. 182) "It was because those logicians who were mathematicians saw that the notion that mathematical reasoning was as rudimentary as that was quite at war with its producing such a world of novel theorems from a few relatively simple premisses, as for example it does in the theory of numbers, that they were led,--first Boole and DeMorgan, afterwards others of us, -to new studies of deductive logic, with the aid of algebras and graphs." (NEM 4:1) I know that I am asking a basic question, but thank you for your time. Best regards, Matías A. Saracho _ _ _ _ _ _ _ _ _ _ ► 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 . ► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to l...@list.iupui.edu with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the message and nothing in the body. More at https://list.iupui.edu/sympa/help/user-signoff.html . ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and co-managed by him and Ben Udell.
