Hi John S, List,
You say: For that matter, chess, go, and bridge are just as mathematical as any other branch of mathematics. They have different language games, but nobody worries about unifying them with algebra or topology. Peirce characterizes mathematics as a science in terms of the character of the hypotheses from which the inquiries proceed. He uses tic-tac-toe in the Elements of Mathematics as an example of how to take a kid's game, and then to examine it in a mathematical spirit. Does this make the game a part of mathematics? I would think not, for the simple reason that pure mathematics starts from a set of hypotheses of a particular sort , and it does not seem obvious to me that these games are grounded on such hypotheses. Those games could be studied in terms of such a formal system of hypotheses, but they need not be. What is more, the playing of those games does not need to a science that deduces theorems from hypotheses. They can be played on the basis of hunches, where the goal is simply to win and not to prove anything of a more general sort. Let's consider a further example based on your own work involving the analysis of the geometric proofs in Euclid's Elements in terms of the Existential Graphs. One thing that seems to be essential to the proofs is the statement of the postulates as hypotheses. As such, the analysis of the arguments should start by the introduction of such postulates on the sheet of assertion--and then following the precepts articulated in the hypotheses as the diagrams are constructed. Lacking an explicit statement of the hypotheses, we can only rely on unstated assumptions as unanalyzed common notions. Those will often suffice for practical purposes, but they won't suffice for developing mathematics as a pure science. The scientific pursuit of such inquiries requires explicitly stated hypotheses, where all of the matter is removed from the conceptions and all that remains are the formal relations between the idealizations, otherwise it will not be possible to settle the question of whether or not specific conclusions follow deductively--much less build systems of theorems from such meager starting points. --Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________ From: Edwina Taborsky <[email protected]> Sent: Tuesday, March 7, 2017 8:54 AM To: Jerry LR Chandler; Peirce List; John F Sowa Cc: Benjamin Udell; Frederik Stjernfelt; Jeffrey Brian Downard; Jeffrey Goldstein; Jon Alan Schmidt; Ahti-Veikko Pietarinen Subject: Re: Re: [PEIRCE-L] Truth as Regulative or Real; Continuity and Boscovich points. John Sowa - very nice outline of 'thinking', which is, as you say, diagrammatic. And as you say, independent of any language or notation. The ability of the human species to 'symbolize', i.e., to transform that diagrammatic reasoning into symbols was certainly a massive evolutionary capacity. BUT, we must acknowledge that this transformation is just that, a transformation, and can mislead, mistransform from the one to the other. Then, we become rigid and 'stick to our words' and our 'symbolic meanings' and ignore the vitality of the diagram. I think that the triadic semiosis, with that mediative process, is a key factor in helping to prevent such rigidity. Edwina Taborsky -- This message is virus free, protected by Primus - Canada's largest alternative telecommunications provider. http://www.primus.ca On Tue 07/03/17 9:56 AM , John F Sowa [email protected] sent: Jerry, We already have a universal foundation for logic. It's called "Peirce's semiotic". JLRC > the mathematics of the continuous can not be the same as the > mathematics of the discrete. Nor can the mathematics of the > discrete become the mathematics of the continuous. They are all subsets of what mathematicians say in natural languages. In Wittgenstein's terms, they are "language games" that mathematicians play with a subset of NL semantics. It's irrelevant whether they use special symbols or words like 'set', 'integral', 'derivative' ... For that matter, chess, go, and bridge are just as mathematical as any other branch of mathematics. They have different language games, but nobody worries about unifying them with algebra or topology. I believe that Richard Montague was half right: RM, Universal Grammar (1970). > There is in my opinion no important theoretical difference between > natural languages and the artificial languages of logicians; indeed, > I consider it possible to comprehend the syntax and semantics of > both kinds of languages within a single natural and mathematically > precise theory. But Peirce would say that NL semantics is a more general version of semiotic. Every version of formal logic is a disciplined subset of NL (ie, one of Wittgenstein's language games). JLRC > I am simply saying that the thought processes of the scientific > community (and my thought processes) did not stop on April 19, 1914. Peirce would certainly agree. He said that building on the foundations he laid "would be a labor for generations of analysts, not for one" (MS 478). The 20th c logicians who ignored Peirce were on the wrong track. Many of them haven't yet reached the 14th c. Peirce was far ahead of the 20th c because he did his homework. JLRC > For a review of recent advances in logic, see > http://www.jyb-logic.org/Universallogic13-bsl-sept.pdf<parse.php?redirect=http%3A%2F%2Fwww.jyb-logic.org%2FUniversallogic13-bsl-sept.pdf>, > 13 QUESTIONS ABOUT UNIVERSAL LOGIC. Thanks for the reference. On page 134, Béziau makes the following point, and Peirce would agree: > Universal logic is not a logic but a general theory of different > logics. This general theory is no more a logic itself than is > meteorology a cloud. JYB, p. 137 > we argue against any reduction of logic to algebra, since logical > structures are differing from algebraic ones and cannot be reduced > to them. Universal logic is not universal algebra. Peirce would agree. JYB, 138 > Universal logic takes the notion of structure as a starting > point; but what is a structure? Peirce's answer: a diagram. Mathematics is necessary reasoning, and all necessary reasoning involves (1) constructing a diagram (the creative part) and (2) examining the diagram (observation supplemented with some routine computation). What is a diagram? Answer: an icon that has some structural similarity (homomorphism) to the subject matter. JYB, 138 > structuralism as we understand it is something still larger that > includes linguistics, mathematics, psychology, and so on... > what concerns us are not so much historical and sociological > considerations about the development of structuralism, but rather > the issue of the ultimate view of structuralism as underlying > mathematical structuralism and universal logic. If you replace 'structuralism' with 'diagrammatic reasoning', Peirce would agree. JYB, 145 > Some wanted to go further and out of the formal framework, namely > those working in informal logic or the theory of argumentation. > The trouble is that one runs the risk of being tied up again in > natural language. See my comment above about Montague, Wittgenstein, and Peirce. Universal logic (diagrammatic reasoning) is *independent of* any language or notation. The differences between the many variants are the result of drawing different kinds of diagrams for sets, continua, quantum mechanics, etc. (Note Feynman diagrams.) Whatever the reasoning stuff may be, it would support NL-like reasoning as a more general version of the 20th c kinds of logic. I develop these points further in the following lecture on Peirce's natural logic: http://www.jfsowa.com/talks/natlogP.pdf<parse.php?redirect=http%3A%2F%2Fwww.jfsowa.com%2Ftalks%2FnatlogP.pdf> See also "Five questions on epistemic logic" and the references cited there: http://www.jfsowa.com/pubs/5qelogic.pdf<parse.php?redirect=http%3A%2F%2Fwww.jfsowa.com%2Fpubs%2F5qelogic.pdf> John
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