Different people have different ways of thinking and talking. That is important, because the world is so complex, so diverse, and so dynamic that no single method could comprehend and describe it all. Peirce's method of diagrammatic thinking, which is the foundation for his logic and philosophy, is more fundamental than thinking in words.
For Peirce, words are necessary, but imperfect methods of communication. For example, his 76 definitions of the word 'sign' do not imply 76 different meanings. The multiplicity of definitions and "outlandish" terminology in the Commens dictionary shows his lifelong struggle to map his diagrammatic insights to words. Phenomenology, phaneroscopy, or phenoscopy is the first stage of analyzing and interpreting the phaneron in diagrams. It depends on the three branches of mathematics (formal logic, discrete math, and continuous math) to derive and classify the elements and patterns of elements. The patterns are possibilities (hypotheses or guesses) whose probability is evaluated by the normative sciences. For background, see the three appendices below: (1) quotations by Peirce about diagrammatic reasoning; (2) quotations by other mathematicians; and (3) quotations by Peirce about formal, mathematical methods. For details, see Frederik Stjernfelt's "Diagrammatology: An Investigation on the Borderlines of Phenomenology, Ontology, and Semiotics". Stjernfelt goes into great detail about the mathematical foundations. He shows that Peirce and Husserl, despite completely different terminology, had developed closely related theories. http://frederikstjernfelt.dk/Peirce/Diagrammatology.%202007.pdf Husserl, by the way, had a PhD in mathematics and a strong background in logic. Both Peirce and Husserl were influenced by Hegel, and both of them used mathematics to develop a better foundation for phenomenology. As Peirce wrote, "the arbitrariness of Hegel's procedure... 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) John --------------------------------- Appendix 1: Quotations about diagrammatic reasoning These quotations by Peirce are discussed in "Natural logic is diagrammatic reasoning about mental models" and related to current research in cognitive science. See http://jfsowa.com/pubs/natlog.pdf All necessary reasoning without exception is diagrammatic. That is, we construct an icon of our hypothetical state of things and proceed to observe it. This observation leads us to suspect that something is true, which we may or may not be able to formulate with precision, and we proceed to inquire whether it is true or not. For this purpose it is necessary to form a plan of investigation, and this is the most difficult part of the whole operation. We not only have to select the features of the diagram which it will be pertinent to pay attention to, but it is also of great importance to return again and again to certain features. (EP 2:212) The word diagram is here used in the peculiar sense of a concrete, but possibly changing, mental image of such a thing as it represents. A drawing or model may be employed to aid the imagination; but the essential thing to be performed is the act of imagining. Mathematical diagrams are of two kinds; 1st, the geometrical, which are composed of lines (for even the image of a body having a curved surface without edges, what is mainly seen by the minds eye as it is turned about, is its generating lines, such as its varying outline); and 2nd, the algebraical, which are arrays of letters and other characters whose interrelations are represented partly by their arrangement and partly by repetitions. If these change, it is by instantaneous metamorphosis. (NEM 4:219) We form in the imagination some sort of diagrammatic, that is, iconic, representation of the facts, as skeletonized as possible. The impression of the present writer is that with ordinary persons this is always a visual image, or mixed visual and muscular... This diagram, which has been constructed to represent intuitively or semi-intuitively the same relations which are abstractly expressed in the premisses, is then observed, and a hypothesis suggests itself that there is a certain relation between some of its parts -- or perhaps this hypothesis had already been suggested. In order to test this, various experiments are made upon the diagram, which is changed in various ways. (CP 2.778) Diagrammatic reasoning is the only really fertile reasoning. If logicians would only embrace this method, we should no longer see attempts to base their science on the fragile foundations of metaphysics or a psychology not based on logical theory. (CP 4.571) --------------------------------- Appendix 2: Related quotations by other mathematicians The following quotations are discussed in "Peirce, Polya, and Euclid: Integrating Logic, Heuristics, and Geometry" and compared to closely related comments by Peirce. http://jfsowa.com/talks/ppe.pdf Archimedes: "Eureka!" Shouted as he jumped out of his bathtub. Leonhard Euler: "The properties of the numbers known today have been mostly discovered by observations... long before their truth has been confirmed by rigid demonstrations." Paul Halmos: "Mathematics -- this may surprise or shock some -- is never deductive in its creation. The mathematician at work makes vague guesses, visualizes broad generalizations, and jumps to unwarranted conclusions. He arranges and rearranges his ideas, and becomes convinced of their truth long before he can write down a logical proof... the deductive stage, writing the results down, and writing its rigorous proof are relatively trivial once the real insight arrives; it is more the draftsmans work not the architects." Albert Einstein: "The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be voluntarily reproduced and combined... The abovementioned elements are, in my case, of visual and some of muscular type. Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the mentioned associative play is sufficiently established and can be reproduced at will." --------------------------------- Appendix 3: Quotations by Peirce about formal, mathematical methods 1898; We pretend that the [existential] graph is a general description of a certain recognized state of things. We only pretend that it is so; for our purpose is merely to study formal logic, and the graph is a mere specimen of an assertion for whose matter we care nothing. In the contents of consciousness we recognize three sorts of elements, Firstness, Secondness, Thirdness. (R339, 11 June 1898) 1902: Accordingly by regarding logic as a science of signs or formal semeiotic, and in the main as a science of symbols, or formal symbolic, we accurately cover its subject matter, and at the same time insure ourselves against all risk of being led astray into psychology. (R 425:117-118, 1902) 1903: For every symbol is a living thing, in a very strict sense that is no mere figure of speech. The body of the symbol changes slowly, but the meaning inevitably grows, incorporates new elements and throws off old ones. (CP 2.222). 1903: Phenomenology ascertains and studies the kinds of elements universally present in the phenomenon. (CP 1.186, 1903) 1905: The other doctrine of mine which Royce attacks, as remarkably shows how unscientific his training has been. He attacks my one-two-three doctrine in the very field where it is most obviously defensible, that of formal logic. (Letter to William James, August 1905) c 1906: 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) 1907: My trichotomy is plainly of the family stock of Hegels 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) CSP: 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) In a footnote to CP 4.240, Peirce added "'Formal logic' is also used, by Germans chiefly, to mean that sect of Logic, which makes Formal Logic pretty much the whole of Logic." Since Whitehead and Russell also adhered to that "sect", the term 'formal logic' means any version of logic that uses some precisely defined notation, linear or graphic.
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