John S, List, The question of what kinds of activities counts as mathematical inquiry and what kinds of systems count as mathematical systems is a question that is answered (on Peirce's account), first, by looking at the purposes that properly guide the inquiries, and second by looking at the methods that should be used to accomplish those purposes. The character of the diagrams that are constructed on the basis of the precepts is a tertiary concern.
When it comes to mathematics "I mean by its general hypothesis the substance of its postulates and axioms, and even of its definitions, should they be contaminated with any substance, instead of being the pure verbiage they ought to be. We have to make choice, then, between a division of mathematics according to the matter of its hypotheses, or according to the forms of the schemata of which it avails itself." (CP 4.246) We could ask the further question of where the mathematical study of the systems drawn from games such as chess, go, checkers and tic-tac-toe fit within its major branches. In order to answer this question, Peirce indicates that the division should proceed on the following grounds: Let us, then, divide mathematics according to the nature of its general hypotheses, taking for the ground of primary division the multitude of units, or elements, that are supposed; and for the ground of subdivision that mode of relationship between the elements upon which the hypotheses focus the attention. CP 4.248 As such, the question of whether or not one or another kind of game counts as mathematics is determined, on Peirce's account, mainly by the purposes that guide the formation and use of the general hypotheses that serve as the starting points for the more deductive parts of inquiries. What is more, the primary division between the mathematical study of these games is made first on the grounds of the kinds of multitudes that are involved and second on the grounds of the kinds of relations that hold between the elements. So, we should ask: where does the mathematical study of games such as chess, go, checkers and tic-tac-toe fit within the main branches of mathematics. Which are the study of finite and discrete systems, which are the study of denumerable or first-abnumerable infinite and discrete systems, and which are the study of higher order abnumerable infinite and continuous systems? Let us apply Peirce's distinction between the practical arts and sciences to this classification of mathematical systems. In a sense, the counting, measuring and calculating that was done for many centuries by ancient Chinese, Egyptian, and Indian mathematicians was certainly mathematical in character, but some of these activities (like much of our own) is really mathematics as a practical art. Doing mathematics in a more scientific spirit requires, it seems, an understanding of the purposes that govern the activities and the methods that should be employed. The main purpose of scientific inquiry in mathematics is the same as the guiding purpose of all scientific inquiry which is to conduct experiments that are designed to help us learn the truth concerning general questions about which we are currently experiencing real doubt. (CP 8.115) In "The Logic of Mathematics, an attempt to develop my categories from within," Peirce asks three questions: 1. 1. what are the different systems of hypotheses from which mathematical deduction can set out, 2. 2. what are their general characters, 3. 3. why are not other hypotheses possible, and the like? The answers to these questions--including especially the third--will help us sort out what kinds of purposes and methods are really mathematical as sciences and which are not. When it comes to the character of systems of hypotheses that lie at the basis of the scientific study of games such as chess, go, checkers and tic-tac-toe, what is necessary in order for the mathematical deduction of theorems to proceed? In his discussions of how mathematical inquiries often get started, Peirce points out that someone such as an ancient Egyptian property tax collector, a Chinese astronomer, or an Indian engineer asks a practical question that couldn't be answered by the normal means, so the mathematicians were called in for assistance. At this point, the mathematician went to work constructing diagrams that could serve as little "models" of the problem. In order for the models to be of any service, they had to be simpler than the real problem. Once the "models" and the rules for manipulating the diagrams were established, the mathematicians tries to discover what follows--necessarily--from those diagrams and rules. At this point, the mathematician is guided by a very different purpose than the tax collector, astronomer or engineer because the mathematician cares, first and foremost, about the purpose of finding the truth about what really does or doesn't follow consistently from the idealized hypotheses. It is a different question altogether--having a different purpose--when we ask what can be learned about the actual world from the results of the mathematical inquiries concerning this idealized systems. So, I am drawn to the conclusion that most people who play games such as chess and go are not engaged in mathematical inquiry--at least not in a scientific spirit. They just want to win the game. Perhaps I am being uncharitable to most people who play these games, or perhaps I'm being too much of a stickler about what is requisite to be guided by the proper aims and methods, but this is the way I interpret the texts cited above. --Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________________ From: John F Sowa <[email protected]> Sent: Thursday, March 9, 2017 7:09 AM To: Jerry LR Chandler; Peirce-L; Clark Goble Cc: Benjamin Udell; Frederik Stjernfelt; Jeffrey Brian Downard; Jeffrey Goldstein; Jon Alan Schmidt; Ahti-Veikko Pietarinen Subject: Re: [PEIRCE-L] Truth as Regulative or Real; Continuity and Boscovich points. Jerry, Clark, list, In my response to Jeff B.D., I was defending the claim that board games are versions of mathematics. But I definitely do *not* restrict math to board games or to set-theoretic models. Jerry > Many mathematicians reject set theory as a foundation for mathematics Yes. Peirce did and so do I. The four board games I cited illustrate diagrammatic reasoning. But those diagrams use only discrete set theory. Peirce also considered continuous diagrams, and so do I. I would also allow diagrams for any mathematical structures anyone might propose or discover -- including quantum-mechanical diagrams. >>> JFS >>> Thanks for the reference. On page 134, Béziau makes the >>> following point, and Peirce would agree: >> JYB >> Universal logic is not a logic but a general theory of different >> logics. > Jerry > Analyze this quote. Is [JYB] saying anything more beyond > a contradiction of terms? Peirce's semiotic is a general theory of all kinds of sign systems. Those systems include, as special cases, all natural languages and all versions of formal logic. I agree with Montague that the underlying semantics of NLs and formal logics are essentially the same, but I would add that formal logics are weaker than NLs. I interpreted JYB as saying that universal logic is a theory about logics in the same sense that CSP's semiotic is a theory about logics. But JYB's notion of universal logic is weaker than CSP's semiotic. >> JYB >> This general theory is no more a logic itself than is >> meteorology a cloud. > Jerry > What the hell is this supposed to mean? Merely an ill-chosen metaphor? My interpretation of JYB: Universal logic is to any particular logic as meteorology is to clouds. Jerry > Chemical isomers are not mathematical homomorphisms because of the > intrinsic nature of chemical identities. Thus, this reasoning is > not relevant to the composition of Boscovichian points. I would not impose any restrictions on the kinds of diagrams or the mappings that define similarity. If you can define a Boscovichian diagram for chemistry, I believe that Peirce's notion of diagrammatic reasoning could accommodate that diagram. Implication: Instead of defining a special kind of logic for every kind of subject matter, I would just change the kinds of diagrams -- quantum mechanical diagrams, Boscovichian diagrams, or whatever mathematical structures anyone might discover or imagine. JLRC > Semiotics is not, in my view, a foundation for logic which is > grounded on antecedent and consequences. That is a Fregean view of logic, not a Peircean view. For his Begriffsschrift, Frege chose implication, negation, and the universal quantifier as his primitives. For his algebraic logic, Peirce started with Boolean algebra and added quantifiers. But he later switched to existential graphs. The early version distinguished Alpha (Boolean) from Beta (which added the line of identity). But he later started with relational graphs (existence and conjunction) and added ovals for negation. For beginning students, Boolean algebra is too abstract. It just represents an NL sentence with a single letter like 'p'. Peirce's relational graphs are a better starting point because they can be translated to and from actual NL sentences. As a pedagogically sounder approach, I follow Peirce's later tutorials (circa 1909). See the first 25 slides of http://www.jfsowa.com/talks/egintro.pdf Note slides 3 and 4 which come from Peirce's own intro in MS 145. In slide 8, I discuss one of CSP's examples that has a direct mapping to and from RDF -- the basic notation for the Semantic Web. Many people believe RDF is a good starting point for logic. I hate the RDF notation, but I use the comparison to show semantic webbers how a real logic can be defined on top of something like RDF. Also note CSP's rules of inference (slide 25). They are grounded in the need to preserve truth (as determined by endoporeutic). And they apply equally well to Kamp's Discourse Representation Structures, which Kamp designed for NL semantics. Note slide 31, which presents two *derived rules of inference* that are implied by the rules in slide 25. These derived rules emphasize generalization and specialization. I believe that it is more appropriate to say that logic is a theory of generalization and specialization. That includes implication as a special case (p implies q iff p is more specialized than q). There is much more to say, some of which I say in the slides http://www.jfsowa.com/talks/ppe.pdf . See slides 39 to 60. In particular, note slide 59 about Turing oracles. Clark > The problem with the game theoretical view of mathematics is > the question of realism. I'm not sure what you mean by "game theoretical view". There are three options, with some similarities among them: 1. The idea that games like chess are mathematical systems. 2. The point that Peirce's endoporeutic may be characterized as an example of Hintikka's game theoretical semantics. 3. Wittgenstein's debate with Turing. (I prefer LW's side.) Clark > there’s a difference between how we use the language of > mathematics and what the objects of mathematics are. That > is what are the relationship between the game and reality. The issues of nominalism vs. realism are orthogonal to all three of these kinds of games. Clark > I think we have to think through carefully what sort of game we > are playing if we’re going to use that as our metaphor. Yes. But I believe that both nominalists and realists could adapt any of the three "game theoretical views" to metaphors that are compatible with their ways of thinking and talking. Summary: What I'm trying to emphasize is the fundamental importance of diagrammatic reasoning for logic, mathematics, language, science, and everyday life. The model-theoretic semantics used to define truth in formal logics is a special case of diagrammatic reasoning. John
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