Anti-Platonic Logic and Mathematics

In this paper, I show that we can absolutely eliminate notions of apriority (axioms, theorems) in symbolic logic and mathematics and build up formal theories using only notions of aposteriority. The new kind of logic and mathematics is said to be anti-Platonic. Both satisfy basic presuppositions of unconventional computing. 1. Platonism in Logic and Mathematics According to Platonism, mathematical entities, which we obtain from primary terms and axioms, are eternal and unchanging. This means that, on the one hand, they exist in fact and, on the other hand, they are abstract and have supranatural properties instead of spatiotemporal or causal ones. Plato says about as follows: The sphere of the intelligible will also have two divisions,— one of mathematics, in which there is no ascent but all is descent; no inquiring into premises, but only drawing of inferences. In this division the mind works with figures and numbers, the images of which are taken not from the shadows, but from the objects, although the truth of them is seen only with the mind’s eye; and they are used as hypotheses without being analysed. Whereas in the other division reason uses the hypotheses as stages or steps in the ascent to the idea of good, to which she fastens them, and then again descends, walking firmly in the region of ideas, and of ideas only, in her ascent as well as descent, and finally resting in them. ‘I partly understand,’ he replied; ‘you mean that the ideas of science are superior to the hypothetical, metaphorical conceptions of geometry and the other arts or sciences, whichever is to be the name of them; and the latter conceptions you refuse to make subjects of pure intellect, because they have no first principle, although when resting on a first principle, they pass into the higher sphere.’ You understand me very well, I said. And now to those four divisions of knowledge you may assign four corresponding faculties — pure intelligence to the highest sphere; active intelligence to the second; to the third, faith; to the fourth, the perception of shadows — and the clearness of the several faculties will be in the same ratio as the truth of the objects to which they are related... (The Republic (Jowell translation). Book VII). The main question of mathematical ontology and epistemology, whether mathematics is discovered or invented, was answered by Plato and Platonists rigorously. They claimed that mathematical entities exist indeed as abstract eternal objects that were discovered, not invented. According to this claim, the mathematical realm is considered as plenitudinous. Such an approach is regarded by Marc Balaguer as full-blooded Platonism. In his words, the pure Platonistic point of view is expressed as follows: “all the mathematical objects (logically possibly) could exist actually do exist” (Balaguer). Formally: x [(x is a mathematical object & x is logically possible)  x exists] This approach was fully embodied in modern logic and mathematics. Since David Hilbert, logicians and mathematicians have been sure that if a mathematical entity is correctly defined within axiomatic method, then it really exists as eternal object: An example of this way of setting up a theory can be found in Hilbert's axiomatization of geometry. If we compare Hilbert's axiom system to Euclid's, ignoring the fact that the Greek geometer fails to include certain [necessary] postulates, we notice that Euclid speaks of figures to be constructed, whereas, for Hilbert, system of points, straight lines, and planes exist from the outset. Euclid postulates: One can join two points by a straight line; Hilbert states the axiom: Given any two points, there exists a straight line on which both are situated. “Exists” refers here to existence in the system of straight lines (Bernays). The world of eternal objects that has been assumed in modern logic and mathematics was considered by Gottlob Frege as das dritte Reich (the third world) that exists independently on the physical world (the first world) and our subjective impressions, fictions and imaginations (the second world). Das dritte Reich contains all interpersonal senses (Sinne): Die Gedanken sind weder Dinge der Außenwelt noch Vorstellungen. Ein drittes Reich muß anerkannt werden. The thoughts neither are things of physical world nor impressions. A third world should be accepted (Frege 1918/19). However, we could prove that modern logic and mathematics may be established without das dritte Reich, i.e. without axiomatic method and Platonic presuppositions. In the beginning, let us show, where exactly there are Platonic assumptions in modern logic and mathematics in order to avoid them.