Nov 112012

So this is to announce that Charles S. Peirce on the Logic of Number has been pub­lished by Docent Press. It was originally written over 30 years ago.

Charles S. Peirce on the Logic of Number

Here are some extracts from the Introduction:

In 1881 the American philosopher Charles S. Peirce published a remark­able paper in The American Journal of Mathematics called “On the Logic of Number.” Peirce’s paper was a watershed in nineteenth cen­tury mathematics; it contained the first successful axiom system for the natural numbers. Since scholarship has traditionally attributed priority in this regard to the axiom systems of Richard Dedekind, in 1888, and Giuseppe Peano, in 1889, we will show that Peirce’s axiom system is actually equivalent to these better known systems.

It is not generally known that Peirce’s 1881 paper provided the first abstract formulation of the notions of partial and total linear order, that it introduced recursive definitions for arithmetical operations, nor that it proposed the first general definition of cardinal num­bers in terms of ordinals.

Peirce was probably America’s greatest philosopher, and his interest in the foundations of mathematics was closely tied to his main philosophical concerns. Some of his most characteristic philosophical positions – his synechism and his phenomenological categories – bear the direct imprint of his research into the theory of sets and transfinite numbers. Peirce’s 1881 paper, in particular, is important for understanding his view of the nature of mathematics and its relation to deductive logic. It was published concurrently — in the same issue of AJM — with his father’s famous definition of mathematics as the science which draws necessary con­clusions.

In the course of tracing out the implications of Peirce’s 1881 paper, we address the problem of locating his mature philosophy of mathematics vis-à-vis the traditional triad of logicism, formalism, and intuitionism. Although we show that Peirce’s view had similarities to and differences from all three, his understanding of mathematics was essentially sui generis.  Perhaps the most characteristic aspect of Peirce’s approach is that he did not conceive mathematics to require any sort of epistemological foundation, whether in logic, intuition, or by means of constructive completeness proofs. This is why Peirce, in his scheme of categories, characterized mathematics as a First. “There is no more satis­factory way of assuring ourselves of anything,” Peirce said, “than the mathematical way of assuring ourselves of mathematical theorems.”


Nov 172010

In a letter to William James on November 25, 1902, Peirce spoke of “the completely developed system, which all hangs together and cannot receive any proper presen­tation in fragments,” and he went on to describe synechism as: “the keystone of the arch.”[1] Now synechism, according to Peirce, is just “that tendency of philosophical thought which insists upon the idea of continuity.”[2] Hence, in order to make sense of Peirce’s synechism, and its role in his “completely developed system”, it is essential first to understand what Peirce meant by the idea of continuity.

Peirce was far from reticent on the topic:

If I were to attempt to describe to you in full all the scientific beauty and truth that I find in the principle of continuity, I might say in the simple language of Matilda the Engaged, “the tomb would close over me e’er the entrancing topic were exhausted” . . .[3]

Yet, even though much of Peirce’s writing was devoted to this idea, there is not much in the secondary literature on his technical definitions of continuity.[4] In this paper we will show how these definitions changed as Peirce’s thinking on con­ti­nu­ity evolved. This should be valuable not only to scholars expressly concerned with Peirce’s work in the foundations of mathematics, but also to those mainly interested in other aspects of his thought.

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  1. Charles Hartshorne and Paul Weiss, eds. The Collected Papers of Charles Sanders Peirce, vols I-VI, Belknap Press, Harvard, 1931-1935, and Arthur W. Burks, ed. vols. VII, VIII, Belknap Press, Harvard, 1938,  8.255-257. The Collected Papers will be referenced by the conventional volume and paragraph number. []
  2. 6.169 []
  3. 1.171 []
  4. Some important works are Murray G. Murphey, The Development of Peirce’s Philosophy, Harvard U. Press, 1961, and George A. Benedict, The Concept of Continuity in Charles Peirce’s Synechism, Ph.D. Dissertation, SUNY Buffalo, 1973. We have found Murphey’s book invaluable, although we disagree with him on important points. We also look forward to the appearance of Carolyn Eisele’s edition of Peirce’s mathematical writing. []
Apr 172010

benjamin_peirce_1857Benjamin Peirce, the father of Charles Sanders Peirce, taught mathe­matics and astronomy at Harvard from 1831 until his death in 1880. He was probably the leading American mathe­ma­ti­cian of his time. He is best known in the annals of mathe­ma­tics for his pioneering Linear Asso­ci­a­tive Alge­bra in 1870, and for his proof, as a young man, that there is no odd perfect num­ber with fewer than four distinct prime factors. [1] Benjamin pub­lished over a dozen other mathe­matical works, in­clu­ding his well-known System of Ana­lytical Mechanics in 1855. He helped to cre­ate a modern science curri­culum at Harvard, and was an important force behind the profes­sion­al­iza­tion of mathe­matics and sci­ence educa­tion in America. [2]

Benjamin’s personality made a powerful impression on those who encountered him. Lawrence Lowell, president of Harvard from 1909 to 1933, described him as follows:

Looking back over the space of fifty years since I entered Harvard College, Benjamin Peirce still impresses me as having the most massive intellect with which I have ever come into close contact, and as being the most profoundly inspiring teacher that I ever had. His personal appearance, his powerful frame, and his majestic head seemed in harmony with his brain. [3]

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Feb 262009

Peter Smith, of Logic Matters, has noticed a new Stanford Encyclopedia of Philosophy entry, Set Theory: Constructive and Intuitionistic ZF.  Constructive and intuitionistic set theories result from the rejection of the law of excluded middle, and effectively restrict set theoretical ontology to poten­tially infinite sets:

The shift from classical to intuitionistic logic, as well as the requirement of predicativity, reflects a conflict between the classical and the constructive view of the universe of sets. This also relates to the time-honoured distinction between actual and potential infinity. According to one view often associated to classical set theory, our mathematical activity can be seen as a gradual disclosure of properties of the universe of sets, whose existence is independent of us. This tenet is bound up with the assumed validity of classical logic on that universe. Brouwer abandoned classical logic and embarked on an ambitious programme to renovate the whole mathematical landscape. He denounced that classical logic had wrongly been extrapolated from the mathematics of finite sets, had been made independent from mathematics, and illicitly applied to infinite totalities.

In a constructive context, where the rejection of classical logic meets the requirement of predicativity, the universe is an open concept, a universe “in fieri”. This coheres with the constructive rejection of actual infinity (Dummett 2000, Fletcher 2007). Intuitionism stressed the dependency of mathematical objects on the thinking subject. Following this line of thought, predicativity appears as a natural and fundamental component of the constructive view. If we construct mathematical objects, then resorting to impredicative definitions would produce an undesirable form of circularity. We can thus view the universe of constructive sets as built up in stages by our own mathematical activity and thus open-ended. [SEP]

This article might interest our BA Seminar students, as well as students in Programming Languages who have recently encountered Curry-Howard Isomorphism — the correspondence between intuitionistic logic and CLK.


Feb 042009

We have had some rather vigorous discussions on platonism in our BA Seminar. Recently the discussion centered on the existence of “natural kinds” — the question, for instance, of whether biological species are arbitrary distinctions or grounded in reality. In a famous passage from the Phaedrus, Plato talks about dividing things into forms “following the objective articulation; we are not to attempt to hack off parts like a clumsy butcher. . .” (265e)  Most of our students, it seems, tend to be nominalists rather than realists [which is not meant to imply that they are clumsy butchers]. Thinking about platonism reminded me of the following cartoon — linked in a comment to our GRE post. It is from a nice site, xkcd: A Webcomic of Romance, Sarcasm, Math and Language.


Aug 152008

Raymond Smullyan is 89 years old, and lives across the Hudson in the Catskill mountains. A distinguished mathematician, logician, and philosopher, he has written over 20 books which have been translated into more than 17 languages. Smullyan is the Oscar Ewing Professor Emeritus of Philosophy at Indiana University and played a prominent role in the history of modern logic. In 1957 he wrote an influential paper for the Journal of Symbolic Logic, called “Languages in Which Self-Reference is Possible,” showing that Gödel incom­pleteness holds for many formal systems more elementary than those considered by Gödel. Georg Kreisel described Smullyan’s Theory of Formal Systems as “the most elegant exposition of the theory of recursively enumerable (r.e.) sets in existence.” Smullyan is probably best-known, though, for his popular collections of logic puzzles. When my children were growing up, they spent many hours with The Lady or the Tiger. My own favorite is To Mock a Mockingbird, which is about combinatory logic and the lambda calculus, one of the foundations of computer science.

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