John S, List,
Here is a passage where Peirce seems to agree, at least in part, with the way Hamilton characterizes "pure logic". The remarks he makes about this account suggest that pure logic is being understood in a manner that is parallel to the way Peirce characterizes "pure mathematics. §2. PURE . . . . Pure Logic, a phrase often used, but to which no distinct conception can be attached. The following explanation by Hamilton (Lectures on Logic, App. I) is as good an explanation as can be given: "The doctrine which expounds the laws by which our scientific procedure should be governed, in so far as these lie in the forms of thought, or in the conditions of the mind itself, which is the subject in which knowledge inheres--this science may be called formal, or subjective, or abstract, or pure logic. The science, again, which expounds the laws by which our scientific procedure should be governed, in so far as these lie in the contents, materials, or objects about which logic is conversant--this science may be called material, or objective, or concrete, or applied logic." Perhaps we may say that pure logic is a logic deduced from hypotheses (which some will look upon as axioms) without any inquiry into the observational warrant for those hypotheses. (CP 2.544) . . . . Pure probation, or proof, is proof by deduction from hypotheses, or axioms, without any inquiry into the observational warrant for those premisses. Such is the usual reasoning of geometry. (CP 2.545) It appears to me that there are points of contrast between the account of pure mathematics that John offers and what Hamilton and Peirce seem be saying here. In particular, the idea the mathematics is free to "begin with any assumptions whatever and derive all possible conclusions." The main contrast I see is that Peirce suggests pure mathematics--where pure deductive logic is one order of such inquiry--is constrained to those postulates that are purely formal and ideal in character. As such, Peirce asks in "The Logic of Mathematics, an attempt to develop my categories from within" for an adequate explanation of the reason why pure mathematics is constrained to only three types of postulates--those dealing with idealized formal systems that are discrete and finite, discrete and infinite, and those that are truly continuous. The answer, he suggests, is found in the phenomenological account of the categories of experience because all mathematical hypotheses are drawn from the formal elements in such experience. It is an interesting question as to why some formal systems of logic seem to fit into the order of mathematics that deals with systems of hypothesis that posit formal relations that are discrete and finite, while other systems of logic seem to fit into the order of mathematics that deals with truly continuous systems. --Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________ From: John F Sowa <[email protected]> Sent: Sunday, September 16, 2018 9:04:54 AM To: [email protected] Subject: Re: [PEIRCE-L] Categories and Modes of Being Jon AS, Auke, Gary F, and Kirsti, This thread started with what I thought was an uncontroversial diagram that summarized Peirce's classification of the sciences. Your questions, objections, and citations have been very helpful in forcing me to fill in the gaps, to correct some points, and to state the issues more clearly. But I suspect that the difference between pure mathematics and applied mathematics has been a stumbling block that has caused this thread to drag on interminably: 1. Pure mathematics consists of all possible theories that begin with any assumptions whatever and derive all possible conclusions. 2. The set of all consistent theories specify everything in Peirce's "universe of possibilities" and the set of all theorems in them specify everything in the "universe of necessity". 3. Therefore, when Peirce said that every science depends on mathematics, he meant that every theory applied to any subject whatever is a copy of some theory of pure mathematics. The only change is to make the labels point to something actual instead of merely possible. JAS > Peirce said here that /deductive /logic is "the science of drawing > necessary conclusions," but he elsewhere (and repeatedly) also > recognized /inductive /and /retroductive /logic as having their own > validity, despite drawing conclusions that are /not /necessary. I agree. For De Morgan, Peirce, and modern logicians, formal logic has been identified with deductive logic. > All three fall under Critic, the middle branch of the Normative > Science of Logic as Semeiotic; I agree. But I would add that every theory, including theories of induction and abduction, are copies of theories of pure mathematics (possibly with some change of names or labels). > Peirce, logic was already "a normative science" in 1902. OK. I stand corrected about the dates. > CSP: The logician does not care particularly about this or that > hypothesis or its consequences, except so far as these things may > throw a light upon the nature of reasoning. (CP 4.239; 1902) > > JFS: That is the opposite of the normative logician, who cares > very much about hypotheses and their consequences. > > JAS: That seems like a clear misreading of the quote. In context, > Peirce was talking about mathematical hypotheses and their > consequences, which need not (and often do not) have any bearing > on Reality whatsoever. No disagreement: We both read that quote in exactly the same way. AvB > I can't recall having written... I apologize for making a mistake in attribution. AvB > Putting Emil Post in the Speculative Grammar domain, is not quite > what I expected and I see no bridge at all. Formal grammar is the modern name for the pure mathematical theory. Peirce used the name 'speculative grammar' for the application to normative logic. GF > abduction and induction — the generation and empirical testing of > hypotheses — are the important types of argument for all positive > sciences (including both cenoscopy and idioscopy). I certainly agree. GF > This is quite different from saying that all theories can be stated > in formal-logical terms, because those terms are not indexically > anchored to the objects of scientific theories, as inductive logic is. I agree. Pure mathematical theories have no indexical anchors to anything actual. But their copies, when relabeled for any application, are anchored to the subject matter. By the way, modern mathematicians consider two theories that differ only in the choice of names or labels to be identical. I believe that Benjamin and Charles made the same assumption. KM > English has more and more become the new Latin of scientists and > scholars. Thus perpetuating a new uniformity within ways of thinking. Uniformity is the death of creativity. Benjamin taught Charles Latin, Greek, and mathematics from a very early age. I believe that was critical for his ability to discover, relate, and synthesize a wide range of diverse ideas and points of view. KM > Peirce named Tetens Kant's teacher... and especially pointed out > that he used his concept of FEELING in the same sense that it was > used and developed by Tetens. That's important. I did a bit of googling, and I think Tetens may have influenced Peirce's highly generalized view of quasi-mind. KM > no Peircean should do the error of taking the mind as the same as > consciousness, whatever one may mean by that. Husserlian philosophy > was designed as a philosophy of consciousness, it aimed to answer to > the modern question of knowledge, not the ancient or medieval one. That would be a reason why Peirce replaced the term 'phenomenology' with 'phaneroscopy'. Peirce's version of phenomenology seems to be identical with his phaneroscopy. But unlike Husserl, Peirce considered "present to the mind" to include the extremely important unconscious influences: *present* does not imply *aware*. John
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