List, Robert: It may be useful to add a few comments that may be helpful for the comity of this group.
Higher education in the sciences is radically different from eduction in mathematics. I believe that my own personal experience is typical for most, but not all, scientists. Mathematical terminology is intermingled and entangled with material terminology freely and with abandon. This entanglement of meanings reaches it zenith in the engineering disciplines. Even though tacit recognition is given to the distinct meaning of mathematical symbols, the logic used to ground scientific theories, including chemistry and biology become synonymous with mathematical terms. The antecedent “maths-scientific" beliefs, formed under strict mentorships, from the early formative years are difficult to re-formulate in later years because they encompass both the ontology and epistemology of the scientific belief systems. C P Snow (1958) diagnosed the grounding issues. The situation has not changed much today. For me personally, it required about three decades to fully separate the meanings of mathematical symbols for numbers from biochemical symbols for numbers (with heritable internal relational structures.). Admittedly, a good fraction of this effort was devoted to separating the sin-signs from the legi-signs because the ontology of chemistry is tightly intermingled with the epistemology of chemistry. CSP often attempted to semantically express this fact by differentiating the meaning of the term “mathematics” from the term “logic”. Unfortunately, the semantics of set theory, without a hint of natural semiosis, as promoted by Whitehead and Russell, compelled the development of proof structures for formal logics and indeed the natural semantics of computer science. It could have been otherwise! CSP grounded his diagrammatic logics on the logical diagrams of chemistry (relevance logics) which we now know to be vastly more perplex than the Venn diagrams of Boolean logic. N,B. Recall the nature of arithmetic calculations has not changed in centuries, only the philosophical interpretations of mathematics and the structures of proof (Skolemization of logical semantic symbols). Cheers Jerry > On Aug 17, 2021, at 11:39 AM, robert marty <robert.mart...@gmail.com> wrote: > > Dear Jon Alan, > > When we put the last lines of CP 3.559 before your eyes, do you look away? > > "… Thus, the mathematician does two very different things: namely, he first > frames a pure hypothesis stripped of all features which do not concern the > drawing of consequences from it, and this he does without inquiring or caring > whether it agrees with the actual facts or not; and, secondly, he proceeds to > draw necessary consequences from that hypothesis" (CP 3.559) > > Why does Peirce write this? Because it is obvious that the famous mathematics > of which you say that ADT "explicitly affirms the dependence of phaneroscopy > (and every other positive science) on mathematics for certain principles, > including formal deductive logic" [emphasize mine ], are for him pure > artifacts. Indeed, he does not exhibit any of them, and neither do you. They > are empty argumentation factors, "elements of language without denotation," > like "unseen characters" are in the theater (sorry, I have to repeat myself). > Thanks to them, one can sing the great merits of ghosts without risking being > contradicted to better exclude realities, like every mathematical object. > > Moreover, Peirce (mathematician) wrote this makes sense: how to recognize > "mathematical principles" and abstract them from complex phanerons if one > does not have them, either in one's mind or if one does not have the capacity > to construct them? > "At the same time, it frequently happens that the facts, as stated, are > insufficient to answer the question that is put. Accordingly, the first > business of the mathematician, often a most difficult task, is to frame > another simpler but quite fictitious problem (supplemented, perhaps, by some > supposition), which shall be within his powers, while at the same time it is > sufficiently like the problem set before him to answer, well or ill, as a > substitute for it." (CP 3.559, again) > > But maybe it is "tribalistic" to remind it? > > Regards, > Robert Marty > Honorary Professor; Ph.D. Mathematics; Ph.D. Philosophy > fr.wikipedia.org/wiki/Robert_Marty > <https://fr.wikipedia.org/wiki/Robert_Marty> > https://martyrobert.academia.edu/ <https://martyrobert.academia.edu/> > > > > Le mar. 17 août 2021 à 17:38, Jon Alan Schmidt <jonalanschm...@gmail.com > <mailto:jonalanschm...@gmail.com>> a écrit : > John, List: > > JFS: They show that De Tienne has misunderstood the role of mathematics in > Peirce's philosophy. > > On the contrary, those three quotations show that anyone accusing André of > hostility toward mathematics and mathematicians has completely misunderstood > his point. He explicitly affirms the dependence of phaneroscopy (and every > other positive science) on mathematics for certain principles, including > formal deductive logic. Nevertheless, he rightly distinguishes pure > mathematics as the science which draws necessary conclusions about strictly > hypothetical states of things from applied mathematics as an integral part of > every other science, including phaneroscopy. We cannot count on pure > mathematicians to help figure out what goes on in experience, because they > only formulate and explicate a pure hypothesis "without inquiring or caring > whether it agrees with the actual facts or not" (CP 3.559, 1898). > > JFS: In the second sentence, the phrase "rest of us", which is intended to > exclude mathematicians, is extremely insulting to Peirce and the many > mathematicians quoted in ppe.pdf. > > There is no reason to take this remark by André so personally. As with his > hyperbolic statement on slide 23--"The world could stop existing, but to pure > mathematicians that would at most be an inconvenience"--he is clearly > referring here only to the idealization of someone who never inquires or > cares about actual facts. Peirce was indeed a mathematician, but he was not > only a mathematician, and he was certainly not a pure mathematician in this > extreme sense. > > JFS: Diagrams are the form of mathematics where the mathematicians and the > people who claim they know nothing about mathematics share common ground. > > I agree--for Peirce, all necessary reasoning is mathematical reasoning, and > all mathematical reasoning is diagrammatic reasoning, so all necessary > reasoning is diagrammatic reasoning. > > CSP: For mathematical reasoning consists in constructing a diagram according > to a general precept, in observing certain relations between parts of that > diagram not explicitly required by the precept, showing that these relations > will hold for all such diagrams, and in formulating this conclusion in > general terms. All valid necessary reasoning is in fact thus diagrammatic. > (CP 1.54, c. 1896) > > CSP: All necessary reasoning is strictly speaking mathematical reasoning, > that is to say, it is performed by observing something equivalent to a > mathematical diagram ... (EP 2:36, 1898) > > CSP: ... I declare that all necessary reasoning, be it the merest verbiage of > the theologians, so far as there is any semblance of necessity in it, is > mathematical reasoning. Now mathematical reasoning is diagrammatic. (CP > 5.148, EP 2:106, 1903) > > Nevertheless, the distinction between mathematics as a hypothetical science > and all the positive sciences must be carefully maintained. Accordingly, I > believe that what Peirce says in the following passage about metaphysics and > metaphysicians is equally applicable to phaneroscopy and phaneroscopists. > > CSP: Metaphysicians have always taken mathematics as their exemplar in > reasoning, without remarking the essential difference between that science > and their own. Mathematical reasoning has for its object to ascertain what > would be true in a hypothetical world which the mathematician has created for > himself,--not altogether arbitrarily, it is true, but nevertheless, so that > it can contain no element which he has not himself deliberately introduced > into it. All that his sort of reasoning, therefore, has to do is to develop a > preconceived idea; and it never reaches any conclusion at all as to what is > or is not true of the world of existences. The metaphysician, on the other > hand, is engaged in the investigation of matters of fact, and the only way to > matters of fact is the way of experience. ... It follows, that deductive, or > mathematical, reasoning, although in metaphysics it may oftener "take the > stage" than in the drama of special research, yet after all, has precisely > the same rôle to enact, and nothing more. All genuine advance must come from > real observation and inductive reasoning. (CP 8.110, c. 1900) > > Regards, > > Jon Alan Schmidt - Olathe, Kansas, USA > Structural Engineer, Synechist Philosopher, Lutheran Christian > www.LinkedIn.com/in/JonAlanSchmidt > <http://www.linkedin.com/in/JonAlanSchmidt> - twitter.com/JonAlanSchmidt > <http://twitter.com/JonAlanSchmidt> > On Sun, Aug 15, 2021 at 11:37 PM John F. Sowa <s...@bestweb.net > <mailto:s...@bestweb.net>> wrote: > Robert, List, > > I strongly agree with your approach, and I would like to add three quotations > by Peirce (copied below). They show that De Tienne has misunderstood the > role of mathematics in Peirce's philosophy. > > But I am not claiming that ADT does not understand Peirce, People were doing > mathematical thinking for thousands of years before anyone knew they were > doing mathematics. What they were doing is diagrammatical reasoning, which > creative mathematicians, especially Peirce, have always known is the > foundation for mathematics. > > For quotations that emphasize that point, see the first 10 slides of a talk I > presented at a Peirce session at an APA meeting in April 2015 and extended > for a workshop hosted by Zalamea in Bogota: Peirce, Polya, and Euclid: > Integrating logic, heuristics, and geometry, http://jfsowa.com/talks/ppe.pdf > <http://jfsowa.com/talks/ppe.pdf> > > In the first sentence of ADT's slide 25 (see the attached file ADT25.jpg), he > belittles Peirce's life's work: "we cannot count on mathematicians to help > figure out what goes on in experience." > > That is contrary to all three quotations by CSP. There are indeed some > mathematicians (pedantic, non-creative ones) whose guidance would be > unreliable. But Peirce, Polya, Euclid, Archimedes, Einstein, and others > quoted in ppe.pdf aren't among them. > > In the second sentence, the phrase "rest of us", which is intended to exclude > mathematicians, is extremely insulting to Peirce and the many mathematicians > quoted in ppe.pdf. > > In the third sentence, the question "how do we transition" is answered by > Peirce: use diagrams! Diagrams are the form of mathematics where the > mathematicians and the people who claim they know nothing about mathematics > share common ground. > > John > _____________________________________ > > Three quotations by Peirce: > > Phaneroscopy... is the science of the different elementary constituents of > all ideas. Its material is, of course, universal experience, -- experience I > mean of the fanciful and the abstract, as well as of the concrete and real. > Yet to suppose that in such experience the elements were to be found already > separate would be to suppose the unimaginable and self-contradictory. They > must be separated by a process of thought that cannot be summoned up > Hegel-wise on demand. They must be picked out of the fragments that > necessary reasonings scatter; and therefore it is that phaneroscopic research > requires a previous study of mathematics. (R602, after 1903 but before 1908) > > My trichotomy is plainly of the family stock of Hegel’s three stages of > thought, -- an idea that goes back to Kant, and I know not how much further. > But the arbitrariness of Hegel's procedure, utterly unavoidable at the time > he lived, -- and presumably, in less degree, unavoidable now, or at any > future date, -- is in great measure avoided by my taking care never to miss > the solid support of mathematically exact formal logic beneath my feet.... > (R318, 1907, p. 37) > > The little that I have contributed to pragmatism (or, for that matter, to any > other department of philosophy), has been entirely the fruit of this > outgrowth from formal logic, and is worth much more than the small sum total > of the rest of my work, as time will show. (CP 5.469, R318, 1907) > _ _ _ _ _ _ _ _ _ _ > ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON > PEIRCE-L to this message. 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