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“Mathesis Universalis na nasze czasy: Wkład Fregego, Cantora i Gödla

100%

Marciszewski W.

nr 4(190)

455-471

The phrase "Mathesis Universalis" (MU) denotes the project of unifying the whole of knowledge with the help of mathematical methods; under this title the project has appeared at the eve of modernity, having been somehow anticipated by antiquity and middle ages. Its main proponents were Descartes and Leibniz. Leibniz's approach is radically formalistic, being thereby tractable for a machine, while Descartes's is decidedly antiformalistic. The article focuses on Leibniz's project as one being continued in modern science. Its crucial idea that MU should operate through an universal symbolism and a logical calculus is being nowadays realized with respect to the whole of mathematics. The fact that this gets accomplished is owed to those devices, to wit an ingenious notation and a logical calculus, which have been created firstly by Gottlob Frege (1879). An essential contribution is due to Georg Cantor as the author of the theory dealing with powers of sets. This theory introduces the distinction of countably infinite sets and those having the power of continuum. This makes it possible to prove the two propositions fundamental for contemporary MU, one due to Gödel, stating the undecidability of natural arithmetic, the other stating the undecidability of logic. The latter has been demonstrated by Alan Turing. He proved that the programs able to decide about the values of a definite function (i.e. to compute the values) form a countable set, while the set of problems to be decided possesses the power of continuum. Hence there are problems which are not decidable in a machine-like manner what implies, in turn, undecidability of logic. An analogous result of Gödel is accompanied by his very important statement that the limitations of mechanical decidability are just relative to the current state of a formalized system, and can get overcome by creative constructing new devices: primitive notions, axioms and methods of research. Thus the science enjoys a perspective of incessant progress. Such a dynamic vision is a feature to essentially distinguish the modern MU version from the classical one having been static.