Bertrand Russell paid considerable attention to the problem of universals throughout his long life. One of main factors which contributed to Russell’s rejection of Hegelian philosophy (which is commonly viewed as a beginning of analytic philosophy) was rejection of so-called internal relations theory, according to which relations reduce to properties of relata or of the whole composed of them. For Russell relations were examples of indispensable universals. Russell is also famous for developing the similarity argument for realism: if we want (...) to get rid of universals by reducing them to sets of objects similar in a certain respect, we have to accept similarity as a genuine universal, for otherwise we are threatened by a vicious regress. The paper contains a presentation of evolution of Russell’s thought regarding universals and a defense of similarity argument against criticism of Michał Hempoliński, according to which causal explanations are sufficient to explain the similarity of objects. It is argued that such explanations are insufficient, as they do not apply to, for example, fundamental particles, like electrons, and the view that there are no fundamental (indivisible) particles is threatened by another vicious regress. (shrink)

W artykule przedstawione i analizowane są trzy kontrprzykłady dla tezy, że wszystkie zdania matematyczne są konieczne, tzn. koniecznie prawdziwe lub koniecznie fałszywe: argument z przygodnych relacyjnych własności empirycznych, argument z własności wynikających z konwencjonalnych reprezentacji i argument z relatywizacji do modelu. Pierwsze dwa argumenty poddają się łatwemu podważeniu, do zakwestionowania trzeciego natomiast potrzebne jest przyjęcie dość silnego stanowiska realistycznego w teorii mnogości.

This paper contains a criticism of mathematical constructivism, i.e. the class of views in the philosophy and foundations of mathematics according to which only constructive notions and methods of proof should be allowed in mathematics. Three main arguments are deployed against such view and its philosophical background. Firstly, an argument from pluralism: constructivism often appeals to intuitive evidence as the root of mathematics, effectively excluding large parts of classical, ab-stract mathematics. But appeals to «intuition» are utterly subjective and unstable, which (...) results in multitude of incompatible constructivist systems of mathematics and makes any criticism toward classical mathematics as «non-constructive» unsubstanti-ated. Secondly, an argument which shows that epistemological arguments, deployed by many constructivists against intelligibility of classical mathematics, are unsound, and moreover, consistent appeal to such arguments leaves constructivists in no position to avoid the menace of ultrafinitism. Thirdly, it is argued that constructivism faces a dilemma whether to consider mathematical truth as what is actually proved, or what is provable, and that this dilemma is unsolvable in a satisfactory way, because the first horn of the dilemma is highly counter-intuitive or absurd, and the second one is impossible to square with constructivist views. (shrink)