Matias, List: In the quoted passage, Peirce suggests "that there are two kinds of deductive reasoning, which *might, perhaps*, be called explicatory and ampliative" (emphases mine). However, he immediately adds that "no mathematical reasoning is what would be commonly understood by ampliative," and goes on to say that "Kant's characterization of all deductive reasoning" as *strictly *explicative is also incorrect. He ultimately proposes instead calling "the two kinds of deduction *corollarial *and *theorematic*" (emphases in original).
MAS: Why did Peirce claim that his own studies on the logic of polyadic relations did not yet fully explain mathematical reasoning? He states in the quoted passage, "These studies [of the logic of polyadic relations] 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." He elaborates elsewhere, as follows. CSP: Another characteristic of mathematical thought is the extraordinary use it makes of abstractions. ... Look through the modern logical treatises, and you will find that they almost all fall into one or other of two errors, as I hold them to be; that of setting aside the doctrine of abstraction (in the sense in which an abstract noun marks an abstraction) as a grammatical topic with which the logician need not particularly concern himself; and that of confounding abstraction, in this sense, with that operation of the mind by which we pay attention to one feature of a percept to the disregard of others. The two things are entirely disconnected. The most ordinary fact of perception, such as "it is light," involves *precisive *abstraction, or *prescission*. But *hypostatic *abstraction, the abstraction which transforms "it is light" into "there is light here," which is the sense which I shall commonly attach to the word abstraction (since *prescission *will do for precisive abstraction) is a very special mode of thought. It consists in taking a feature of a percept or percepts (after it has already been prescinded from the other elements of the percept), so as to take propositional form in a judgment (indeed, it may operate upon any judgment whatsoever), and in conceiving this fact to consist in the relation between the subject of that judgment and another subject, which has a mode of being that merely consists in the truth of propositions of which the corresponding concrete term is the predicate. ... Abstractions are particularly congenial to mathematics. Everyday life first, for example, found the need of that class of abstractions which we call *collections*. Instead of saying that some human beings are males and all the rest females, it was found convenient to say that mankind consists of the male *part *and the female *part*. The same thought makes classes of collections, such as pairs, leashes, quatrains, hands, weeks, dozens, baker's dozens, sonnets, scores, quires, hundreds, long hundreds, gross, reams, thousands, myriads, lacs, millions, milliards, milliasses, etc. These have suggested a great branch of mathematics. Again, a point moves: it is by abstraction that the geometer says that it "describes a line." This line, though an abstraction, itself moves; and this is regarded as generating a surface; and so on. So likewise, when the analyst treats operations as themselves subjects of operations, a method whose utility will not be denied, this is another instance of abstraction. Maxwell's notion of a tension exercised upon lines of electrical force, transverse to them, is somewhat similar. These examples exhibit the great rolling billows of abstraction in the ocean of mathematical thought; but when we come to a minute examination of it, we shall find, in every department, incessant ripples of the same form of thought, of which the examples I have mentioned give no hint. (CP 4.234-235, 1902) He also says later in the manuscript that you quoted, "Theorematic reasoning, at least the most efficient of it, works by abstraction; and derives its power from abstraction" (NEM 4:11, 1901). In an alternate version of the same text, he says that "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" (EP 2:96, 1901). MAS: 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. I refer you again to CP 4.233 (1902) and CP 4.612-616 (1908), the first of which includes the following explanation. CSP: 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. In short, "A Theorematic Deduction is one which, having represented the conditions of the conclusion in a diagram, performs an ingenious experiment upon the diagram, and by the observation of the diagram, so modified, ascertains the truth of the conclusion" (EP 2:298, 1903). Daniel Campos offers a thorough explanation of this in his 2010 book chapter, "The Imagination and Hypothesis-Making in Mathematics: A Peircean Account" ( https://www.academia.edu/373131/The_Imagination_and_Hypothesis-Making_in_Mathematics_A_Peircean_Account), presenting Peirce's own example of Euclid's Fifth Proposition (*Pons Asinorum*) as an illustrative example. A book chapter that I recently co-authored with Joseph Dauben and List moderator Gary Richmond, "Peirce on Abduction and Diagrams in Mathematical Reasoning" ( https://doi.org/10.1007/978-3-031-03945-4_25), might also provide further clarification. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Tue, Aug 22, 2023 at 9:32 AM Matias <matias....@gmail.com> wrote: > 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 ampliative, > 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 corollarial 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 >
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