From the preface: "The purpose of this book is to make Peirce's philosophy of mathematics more readily available to contemporary workers in the field, and to students of his thought. Peirce's philosophical writings on mathematics are reasonably well represented—despite the shortcomings detailed in [J. W. Dauben, Centaurus 38 (1996), no. 1, 22–82; MR1368047 (pp. 28–39)]—in [C. S. Peirce, Collected papers, Edited by Charles Hartshorne and Paul Weiss. 6 vols. I: Principles of philosophy. II: Elements of logic. III: Exact logic. IV: The simplest mathematics. V: Pragmatism and pragmaticism. VI: Scientific metaphysics, Belknap Press, Cambridge, MA, 1960; MR0110632], much more so in [C. S. Peirce, The new elements of mathematics, Vol. I, Mouton, The Hague, 1976; MR0532685; Vol. II; MR0532686; Vol. III/1; MR0532687; Vol. III/2; MR0532688; Vol. IV; MR0532689]. The Chronological Edition of Peirce's Writings [Writings of Charles S. Peirce. Vol. 1, a chronological edition, Indiana Univ. Press, Bloomington, IN, 1982; MR1403570; Vol. 2, 1984; MR1403566; Vol. 3, 1986; MR1403567; Vol. 4, 1986; MR1403568; Vol. 5, 1993; MR1403569; Vol. 6, 2000; MR1961048] will surely set the gold standards, in this area as in others, of comprehensiveness and textual scholarship; it will also provide the annotations and other auxiliary apparatus whose absence so greatly diminishes the usability of the earlier editions for all but the most seasoned Peirceans. But the Chronological Edition [op. cit.] has only just reached the last quarter century of Peirce's life, when most of the central texts on mathematics (and nearly all of those in the present volume) were written. And its very comprehensiveness will make it difficult for those with a particular interest in the philosophy of mathematics to find their way to what they really need.
"Hence this book. It does not pretend to be a comprehensive selection, even of Peirce's most important philosophical writings on mathematics. It seeks rather to be a selection of major texts that is comprehensive enough to sere as a serious introduction to this philosophy of mathematics, sufficient in itself for those whose primary interests lie elsewhere, and a stepping-stone for specialists to more advanced investigations. If it helps to turn some of its readers into specialists, so much the better; for the secondary literature on the mathematical aspects of Peirce's thought makes on more than a good beginning, and much remains to be done.
"Precious little of that literature is due to philosophers of mathematics who come to Peirce for insight into the living problems of their discipline. I will argue in the introduction that some of Peirce's deepest insights into mathematics cannot be fully appreciated in isolation from his larger philosophical system (which is not to say that one must buy fully into the system in order to appropriate the insights). This creates something of a dilemma for a volume such as this one; for the system is complicated and unfinished. Anything like a thorough exposition would overwhelm the primary content, a redundancy for much of its intended audience, and a stumbling block for the rest. My imperfect solution has been to provide a very brief overview in the introduction, to be supplemented by more detailed piecemeal explanations in the headnotes to the individual selections. Those who feel the need of a more thorough systematic survey will do well to consult Nathan Houser's introductions to [C. S. Peirce, The essential Peirce. Vol. 1 (1867–1893), Indiana Univ. Press, Bloomington, IN, 1992; MR1188412; Vol. 2 (1893–1913), 1998; MR1625634], and Cheryl Misak's to [C. Misak (ed.), The Cambridge companion to Peirce, Cambridge Univ. Press, Cambridge, 2004]. These works provide convenient points of entry into Peirce's own writings and into the secondary literature.''
The PDF was originally entitled "Beings of Reason.book". That reminds me of John of St. Thomas (João Poinsot, O.P.)'s Tractatus de Signis.
cf. my answer (and comments) to the Philosophy StackExchange question "Is Mathematics considered a science?"
cf. Jeff Kalb's Music and Measurement: An the Eidetic Principles of Harmony and Motion (St. Cecilia's Feast Day, 2016)
Do you agree that
the signs that signify better are those whose "sign-vehicles" ( representamen s) are more dissimilar to what they signify
?
This would seem to explain why mathematics, a tool for describing nature, is effective at constructing physical theories.
Yes, I tend to agree, and do like the quote, because ‘analogicity’ can range from several analogous contact points to nearly “nothing”, as long as the comparison maintains some non-arbitrary rational basis. In other words, hinting may suffice, providing the hint is rationally sound (what I call signifying non-arbitrarily).
However the limit (to my view) of the strict application of the quote to the mathematics-physics relationship is also found in Peirce’s understanding of what a representamen is from the viewpoint of its essentially triadic nature. The “Sign” no longer has any being or real signifying function if dissociated from the “Object” and the “Interpretant” (a view “post-real” mere representational theory rejects both in theory and practice). It is so in Peirce’s view because the proper function of a “Sign” (which also corresponds to its being) is to be the medium for the communication of a represented objective element thus conceived and cognized in terms of another “object”). This is the reason Peirce talks about the “two objects of a sign” (a little like there are two uses of reason), positing a clear cognitive as small as semiotic difference between the “Mediate Object” and the “Immediate Object”. And mathematical signs, for various reasons, tend to remain cognitively and semiotically elusive (not to mention that they certainly are not all equal in their referential signifying proportion to objective elements). Their status as signs properly speaking is therefore equivocal (and contemporary physics unfortunately does not help its mathematical tools to regain their ability to act on behalf of Peircean’s “Dynamical Objects”).
- §8 "Dichotonic Mathematics" (PDF pp. 102ff.) contains his definitions of mathematical terms such as axiom, postulate, and definition.
- §10 "Prolegomena to an Apology for Pragmaticism" (PDF pp. 123ff.). The first full ¶ of intro. p. xxxvi (PDF p. 37) summarizes that Peirce believed mathematics is done by experimentation on mathematical diagrams, exactly how a chemist does experiments.
- §14 "The Logic of Quantity" (PDF pp. 152ff.) is a good refutation of Kant's analytic/synthetic dichotomy. Peirce appears to think mathematics is an experimental science.
- §15 "Recreations in Reasoning" (PDF pp. 158ff.) is a, "Along with selection 10, […] mathematico-philosophical treatment of the natural numbers", "one of the strongest pieces of evidence in support of the claim that Peirce held to a kind of mathematical structuralism." Peirce defines a number as a “meaningless vocable”, an indexical sign.
Peirce, who interestingly considers mathematics an experimental science, answers Einstein's (and Wigner's) question about the "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" very insightfully and succinctly in §10 "Prolegomena to an Apology for Pragmaticism" (PDF pp. 123ff. of Philosophy of Mathematics: Selected Writings). (cf. first full ¶ of intro. p. xxxvi (PDF p. 37) for a brief summary of how Peirce believed mathematics is done by experimentation on mathematical diagrams, much as how a chemist does experiments with chemicals!)
In §15 "Recreations in Reasoning" (PDF pp. 158ff.), Peirce defines a number as a "meaningless vocable" (thus an indexical sign).
The editor, Matthew Moore, thinks §10 and §15 are "the strongest pieces of evidence insupport of the claim that Peirce held to a kind of mathematical structuralism." What exactly is "mathematical structuralism"? Does it have anything to do with "structural realism"?