Jon AS,
Peirce's definition of mathematics is consistent with the mainstream of mathematical thinking since antiquity. There are many versions of the philosophy of mathematics, but Peirce's version is still at the forefront of modern research. JAS> anyone is free to disagree with Peirce's definition of mathematics. Nobody does. Peirce's three branches (Formal logic, discrete mathematics, and continuous mathematics) are still the backbone of math today, but there have been some additions. There are, however, controversies between nominalists and realists. Peirce adopted Aristotle's answer to Plato: The mathematical forms (or patterns) are possibilities, which occur in actuality only when embodied. But he went one step further: he defined three universes of discourse, each of which is real in the sense that they exist independent of what anybody may think about them: mathematics is the universe of all possibilities; actuality is the universe of everything that exists in space and time; the necessitated is the universe of all the laws that govern actuality, including any aspect of it, no matter how small or insisngificant. JAS> Mathematical reasoning is always and only necessary reasoning about "creations of the mind," Yes. But every science, including phaneroscopy, is a creation of the mind. The elements of phaneroscopy -- the categories, hypoicons, and all their possible combinations are the result of interpreting experience in the phaneron in terms of mathematical patterns. Without formal logic and other branches of mathematics, phaneroscopy is limited to a Hegelian style of speculation: JAS> but the hard facts of experience require probable reasoning, which the normative science of logic prescribes as induction. There is a mathematical/deductive aspect to the latter, to be sure--"All mathematical reasoning, even although it relates to probability, is of the nature of necessary reasoning" (CP 7.180, EP 2:82, 1901)--but this is true of all reasoning in every science. That quotation by Peirce contradicts what you said. Peirce's extensive writings about probability are purely deductive mathematics. Since phaneroscopy comes before normative science, it depends only on pure maathematics, especially formal logic. Note the following quotations: CSP: Phenomenology is that branch of scBience which is treated in Hegels Phenomenologie des Geistes (a work far too inaccurate to be recommended to any but mature scholars, though perhaps the most profound ever written) in which the author seeks to make out what are the elements, or, if you please, the kinds of elements, that are invariably present in whatever is, in any sense, in mind. (EP 2:267, 1903) Note that Peirce considered formal logic essential to everything he wrote that is of any significance: 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) CSP: 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 Hegels 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.... (EP 2:428, R318, 1907) Without mathematics, especially formal logic, phaneroscopy cannot go beyond Hegel. John
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