Oct 10, 2012
Elegi Menggapai 'The Foundations of Mathematics'
Yogyakarta State University
The search for foundations of mathematics is in line with the search for philosophical foundation in general. The aspects of the foundation of mathematics can be traced through the tread of philosophical history and mathematics history as well. Hersh, R. elaborates that the foundations of mathematics 1 have ancient roots; the philosophers behind Frege are Hilbert, Brouwer, Immanuel Kant. The philosopher behind Kant is
Gottfried Leibniz; the philosophers behind Leibniz are Baruch Spin¬oza and Rene Descartes. The philosophers behind all of them are Thomas Aquinas, Augustine of Hippo, Plato, and the great grandfather of foundationism-Pythagoras.
The term "foundations of mathematics" 2 is sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory and model theory. The search for foundations of mathematics is however also the central question of the philosophy of mathematics: on what ultimate basis can mathematical statements be called "true"? Hersh R. describes that , the name "foundationism" was invented by a prolific name-giver, Imre Lakatos. Gottlob Frege, Bertrand Russell, Luitjens Brouwer and David Hilbert, are all hooked on the same delusion that mathematics must have a firm foundation; however, they differ on what the foundation should be. The currently foundation of mathematics 3 is, characteristically as more formalistic approach, based on axiomatic set theory and formal logic; and therefore, all mathematical theorems today can be formulated as theorems of set theory.
Hersh R. (1997) writes that the truth of a mathematical statement, in the current view, is then nothing but the claim that the statement can be derived from the axioms of set theory using the rules of formal logic. However, this formalistic 4 approach does not explain several issues such as: why we should use the axioms we do and not some others, why we should employ the logical rules we do and not some other, why "true" mathematical statements appear to be true in the physical world. The above mentioned notion of formalistic truth could also turn out to be rather pointless; it is perfectly possible that all statements, even contradictions, can be derived from the axioms of set theory. Moreover, as a consequence of Gödel's second incompleteness theorem, we can never be sure that this is not the case.
1 Hersh, R., 1997, “What is Mathematics, Really?”, London: Jonathan Cape, p. 91
2 …….., “Foundations of mathematics”, Retrieved 2004 < http://www.wikipedia.org/>
3 Hersh, R., 1997, “What is Mathematics, Really?”, London: Jonathan Cape, p.91
4 Ibid. p. 256