Polynomizing: Logic Inference in Polynomial Format and the Legacy of Boole

Polynomizing is a term that intends to describe the uses of polynomiallike representations as a reasoning strategy and as a tool for scientific heuristics. I show how proof-theory and semantics for classical and several non-classical logics can be approached from this perspective, and discuss the assessment of this prospect, in particular to recover certain ideas of George Boole in unifying logic, algebra and the differential calculus. 1 From finite and hard to infinite and smooth One of the most fascinating episodes of the history of Mathematics, which is nowadays almost considered to be a triviality, is the discovery of the polynomial representation (by infinite series) of numerical functions. One can situate this historical point in the western historiography, although variants of his methods were already known before in Europe and in China and India as well, around the English mathematician Brook Taylor (1685 1731) and his book Methodus incrementorum directa et inversa, of 1715, which led to the development of the Taylor’s and MacLaurin’s expansions. Surprisingly, however, the importance of Taylor’s discovery remained unrecognized until 1772, when J. L. Lagrange realized its relevance and proclaimed it to be “the principal foundation of differential calculus”. Any infinitely differentiable function f(x), under certain circumstances, can be rewritten as an infinite polynomial series in the neighborhood of a base point x0: f(x) = α0(x0)+α1(x0) · (x−x0)+α2(x0) · (x−x0)+ . . . αn(x0) · (x−x0)+ . . .