Vol 8, No 1, 2009 pp. 61-68
UDC 510.21
KANT, SCHELLING … OR CREATIVISM
or
On a question in the philosophy of mathematics
Milan D. Tasić
University of Niš, Teacher's Training Faculty, Vranje, Serbia
E-mail: tmild@ptt.yu
In the philosophy of mathematics, as in its a meta-domain, we find that the words as: consequentialism, implicativity, operationalism, creativism, fertility, … grasp at most of mathematical essence and that the questions of truthfulness, of common sense, or of possible models for (otherwise abstract) mathematical creations, i.e. of ontological status of mathematical entities etc. - of second order.
Truthfulness of (necessary) succession of consequences from causes in the science of nature is violated yet with Hume, so that some traditional footings of logico-mathematical conclusions should equally be fallen under suspicion in the last century. We have in mind, say, strict-material implication which led the emergence of relevance logics, or the law of excluded middle that denied intuitionists i.e. paraconsistent logical systems where the contradiction is allowed, as well as the quantum logic which doesn't know, say, the definition of implication etc. Kant's beliefs miscarried hereafter that number (arithmetic) and form (geometry) would bring a (finite) truth on space and time, when they revealed relative and curveted, just as it is contradictory essentially understanding of basic phenomena in the nature: of light as an unity of wave – particle, or that both ''exist'' and ''don't exist'' numbers as powers of sets between א 0 and c (the independence of continuum hypothesis) etc. Mathematical truths are "truths of possible worlds", in which we have only to believe that they will meet once recognizable models in reality. At last, we argue in favor of thesis that a possible representing "in relief" of mathematical entities and relations in the "noetic matter" (Aristotle) would be of a striking heuristic character for this science.
Key words: Consequentialism, implicativity, truthfulness, possible worlds, noetic matter.
KANT, ŠELING, … ILI KREATIVIZAM
ili
O osnovnom pitanju u filozofiji matematike
U filozofiji matematike, kao jedne meta-oblasti njene, mi nalazimo da reči kao implikativnost, konsekvencijalizam, operacionalizam, kreativizam, plodotvornost, … zahvataju najviše od matematičke suštine, a da su pitanja istinitosti, zdravog razuma, ili mogućih modela za (inače apstraktne) matematičke tvorevine – drugorazredna. Istinitost (nužnog) sleđenja posledica iz uzroka u nauci o prirodi narušena je još s Hjumom, da bi nekolika tradicionalna uporišta logičko-matematičkog zaključivanja bila jednako dovedena pod sumnju u prošlom stoleću. Imamo u vidu, recimo, protivnost striktna – materijalna implikacija koja je dovela do nastanka relevantnih logika, ili zakon isključivog trećeg koji su poricali intuicionisti, odnosno paraneprotivurečne logičke sisteme gde se protivurečnost dopušta, jednako kao i kvantnu logiku koja ne poznaje, recimo, odredbu implikacije itd. Potom, izjalovila su se i Kantova uverenja da će broj (aritmetika) i oblik (geometrija) doneti (konačnu) istinu o prostoru i vremenu, onda kada su se vreme i prostor "pokazali" relativnim i iskrivljenim, kao što je protivurečno i suštinsko razumevanje osnovnih fenomena u prirodi: svetlosti kao ''jedinstva'' talasa-čestice, ili to da i ''postoje'' i ''ne postoje'' brojevi kao moći skupova između א 0 i c (nezavisnost hipoteze kontinuuma) itd. Istine u matematici su ''istine mogućih svetova'', za koje treba samo verovati da će naići jednom na prepoznatljive modele u stvarnosti. Najzad mi argumentišemo u prilog teze da bi moguće ''reljefno'' predstavljanje matematičkih entiteta i odnosa u ''umnoj materiji'' (Aristotel) bilo od izrazito heurističkog karaktera po ovu nauku.
Ključne reči: Konsekvencijalizam, implikativnost, kreativizam, istinitost, mogući svetovi, umna materija