List, Jeff: (NB: This message contains technical arguments that may be incomprehensible to non-technical readers.)
Your paragraph (see below) is mathematically precise, well almost so because of the profound restriction that is placed on the rhetorical space of its applicability may exclude/obscure some of CSP's meanings. It has taken a few days to parse the meaning of your arguments in relation to CSP writings. The restriction becomes profound when you limit the nature of antecedent chains to "ALL regularities". CSP does NOT restrict the triadic triad, qualisign, sinsign, legisign, icon, index, symbol, rhema, dicisign, argument, to regularity of inference chains. A mathematic chain, (as either a short exact chain or a long exact chain as used in topology), defines a linear sequence of points/symbols/ numbers. The meaning of mathematical term "chain" is an analogy because the visual images of a real (mechanical, visible) chains with increasing number of links is an appropriate analogy. The extension from "chains" to "inference chains" is problematic for the natural sciences, which differ from mathematics. The natural sciences differ from mathematics by the intrinsic irregularities (that is, non-regular) of form. CSP's "triadic triad" does not demand "regular chains". The distinction between regularity and irregularity is clear from the sub-triad "indices, rhema and dicisign." as well as CSP's beta-graphs, which do not require the concept of chains. Is it possible that CSP's definitions were framed to account for both regular and irregular chains? Two facts suggest that this possibility exists. One fact is the well-known phenomena of many possible branches of chemical isomers where the number of possible isomers depends on the length of the chains of each branch (as well as the quantities of valences of each atom). CSP used this fact in his development of the rhetoric linking icons to rhema and rhema to indices. The second fact is less well known but was well known to CSP. This is fact that the "handedness" of molecules (measurable by the relations of light to matter) is a function of the icon of the molecule. These relations are highly irregular, not predictable, and dependent on the branching of the irregular chains of indices of the atoms. So, Jeff, my question to you is: How do these facts influence your beliefs about the relationships between mathematics and CSP notion of inferences? Cheers Jerry Postscript: By the way, Jeff, these arguments are related to logical problems in solving higher-order polynomials. On Sep 25, 2014, at 11:05 AM, Jeffrey Brian Downard wrote: > Based on this understanding of inference and argument, we have reason to > adopt the following as regulative ideas in our methodeutic: > 3) all regularities—wherever they are found--may be conceived as inference > chains. Those inference chains that are no longer evolving in their embodied > regularities no longer appear to be changing towards some end and may, at > that point in time, be conceived simply as mechanical causes. They can be > thought of as mechanical because the inferences all seem to have a > demonstrative form and the synthetic inferences that involve continued growth > do not appear to be present. Those regularities that are still evolving may > reasonably be supposed to involve something like an end (more or less > determinate), where the end is one of the things that may be undergoing > evolution. Those processes that are merely finious, but not entirely final, > may be conceived as inference chains where the resting points are > representamens that may have a structure that is similar in some respects to > a dicisign. Those inference chains that appear to embody final causes may be > conceived as thoughts that are directed towards some natural end, where the > resting points of the chains are dicisigns. When the interpretants of this > process involve only the lower degrees of self-control, we may reasonably > suppose that there is a quasi-mind that serves as interpreter. Consequently, > we may conceive of all regularities as inference chains where the evolution > of those regularities involve greater and lesser degrees of control over the > processes that are involved in the modification and adaptation of those > regularities in relation to ends that are more or less determinate, where > those ends themselves have something like a life history—some are dying or > are dead, and some are alive are growing. > > These regulative ideas are put to use in metaphysics as Peirce tries to > articulate the assumptions we should adopt as we seek to explain all of the > basic kinds of things that call out for explanation in the special sciences. > The goal is to keep the door of inquiry open by avoiding any explanatory move > that will make it impossible to explain what needs to be explained.
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