List, Michael, Kirsti, John: On Nov 12, 2014, at 11:47 AM, Michael DeLaurentis wrote:
I don’t find any such distinction, implicit or explicit, in Peirce’s late writings. Motivated by your assertions, I re-read 4.172 and later paragraphs, searching for distinctions between CSP logic and set theory logic. In contrast to your assertion, I certainly find numerous critical philosophic distinctions between CSP logic and Cantorian/Russellian logic with respect to inquiry into the mathematics/logic of the continuum. Although a large number of texts could be cited, availability of time and energy restrict my rhetoric principally to 4.172 to 4.176. 1. 4.173 introduces with the notion of a collection. A collection is a consequence of "bring or gather together", parts of a whole. CSP bases his notion of relation on collections as parts of a whole. It requires activity to bring together a collection. Thus, CSP is grounding his argument, among other mathematical concepts, on the theory of numbers, the collectability of numbers, and the antecedent parts being brought together to construct a whole. This is clearly distinct from Cantor / Russell views which pre-supposes a geometric line. 2. 4.174 (and 4.172) introduces the notion of a relative of a part versus the relative of a whole, drawing on the statistical example in 4.172 and the concept of a unit of a partition of a role of a pair of dice. Each role of the pair of die generates a relative value among all possible roles of the pair of six-sided die, exactly 36. This is clearly distinct from Cantor / Russell views. 3. 4.175 "But when the units lose there individual identity because the collection exceeds every positive existence of the universe, the word multitude ceases to be applicable. I will take the word multiplicity to mean the greatness of any collection discrete or continuous." I infer from this, in light of 4.172-175, that individual identity is related to parts of a whole such that parts, as units, can be collected into whole, generating the NOUN, collection. The "bringing together" of a collection is of the nature of a sublation. The quality of the collection, is, presumable for CSP, a matter of sensory experience, as one perceives from the usage of the term "because" in this sentence, inferring causality. (And qualities are an aspect of sensory experiences, are they not?) This is clearly distinct from Cantor / Russell views of memberships and classes. Yes, set theory, as a dominant force in modern mathematics, has ignored the logical basis of CSP notion of multitude and his terminology for distinguishing between parts and wholes, points and lines, and sensory experiences. But, CSP’s philosophy expressed in 4.172-4.175 is consistent with many aspects of chemical logic; modern mathematics is not consistent with chemical logic for very specific reasons of the non-transitivity of the mathematics of chemical sublations of individual identities. Non-transitivity is illustrated, for example, by the handedness of chemical isomers.) I conclude that although many many aspects of CSP logic and set theory logic are consistent with one another, the distinction between them (modes of constructions) at the rhetorical and semantic levels differ in mathematically profound ways. The basic conundrum of the nature of distinction between discrete and continuous mathematics remains alive and open. Indeed, a very active subfield of mathematics is the Brouwer School of intuitionism. ( http://en.wikipedia.org/wiki/Intuitionistic_logic ) Parenthetically (or perhaps metaphorically) I conclude that studying CSP texts without an in-depth knowledge of the state of the science in the 2nd half of the 19 Th Century is like attempting to solve a crossword puzzle with only the superficial "across" clues. The depth of his thought corresponds with knowledge of mathematics and the natural sciences and the natural propositions in his time, that is, the "down" clues. Extending the metaphor, the sensory experiences of the American cultural milieu of the late 19 Th Century are interwoven into the very fabric of CSP's text. Cheers, Jerry (BTW, Thanks to Gary F. for suggesting a puzzle analogy for hermeneutics.)
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