Detached Ideas

From xkcd.

Benjamin Peirce, the father of Charles Sanders Peirce, taught mathematics and astronomy at Harvard from 1831 until his death in 1880. and was probably the leading American mathe­matician of his time. He is best known, in the history of mathematics, for his Linear Asso­ci­ative Algebra of 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 also published over a dozen other mathema­tical texts and treatises, including his well known A System of Analytical Mechanics in 1855. He helped to create a modern science curriculum at Harvard, and was an importent force in the professionalization of mathematics and science education across America. [2]

Benjamin’s personality left 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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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 CL_K.

–Paul