Natural mathematics

Natural mathematics is the polemical position which asserts that arguments striving for internal consistency, inherently lead to paradox. Therefore, it is necessary to arbitrarily insert into arguments the idea that mathematics is an inherent human activity. For adherents of natural mathematics, this sacrifices any possibility that arguments may have logical content. However, they feel that the gain is to avoid paradox. For an extended treatment of the natural mathematics polemical position, see P. Maddy, NATURALISM IN MATHEMATICS (1996).

History
Natural mathematics is at least as old as Artistotle, who felt argumentation was challenged by Zeno's paradox. It is as recent as intuitionism, formalism and logicism, which were developed at the turn of the century in order to cope with the supposed paradoxes of set theory. See Alejandro Garciadiego, BERTRAND RUSSELL AND THE ORIGINS OF THE SET-THEORETIC 'PARADOXES' (1992). In the worlds of mathematics and physics, natural mathematics is overwhelmingly favored, so strongly favored that most practitioners are unaware that they are even using it.

Application
The most famous current use of natural mathematics is the relativity of simultaneity. In his train experiment, Einstein (see RELATIVITY), writes that at a crucial stage of the formulation of the notion, point M "naturally" coincides with point M'. This is a statement outside the argument--it is neither an assumption nor a deduction. But it IS a complete application of natural mathematics. Einstein enthusiastically adopted the expression of natural mathematics found in Poincaré, SCIENCE AND HYPOTHESIS (1902), which, it turns out, was written in response to set theory controversies of the 1890s. Einstein called his own brand of natural mathematics, "practical geometry."

Controversy
Today, natural mathematics is under assault by new work on the history of set theory. Many of the set-theoretic 'paradoxes' are no longer considered to be so, thus undermining the program of natural mathematics. See I. Grattan-Guinness, THE SEARCH FOR MATHEMATICAL ROOTS (2000). In the various disciplines where there are theories using it, the criticism of dominant theories seems to take the form of exposing the natural mathematics roots of the arguments. See, for example, Peter Woit, NOT EVEN WRONG (2006).

Advocates of natural mathematics mainly count on there being some logical content to the various known paradoxes. There is, for example, still controversy about whether Zeno's paradox has any logical content at all. The idea is that if there is even one paradox rattling around in argumentation, all argumentation is implicated, thus making vital the natural mathematics program.

Main Menu

See Also

Main Menu