Book Review

The main purpose of the paper concerns the question of the existence of hard mathematical facts as truth-makers of mathematical sentences. The paper defends the standpoint according to which hard mathematical facts do not exist in semantic models of mathematical theories. The argumentative line in favour of the defended thesis proceeds as follows: (i) slingshot arguments supply us with some reasons to reject various ontological theories of mathematical facts; (ii) there are two ways of blocking these arguments: through the rejection of the principle of extensionality for individual terms (Ext) or through the rejection of the principle of Wittgenstein; (iii) the first way cannot be accepted because it leads to the practice of softening mathematical facts; (iv) the second way, called fine-graining facts, cannot be accepted because it also results in the practice of softening facts. (v) Hence, only soft mathematical facts (SMFs) can be introduced into semantic models of mathematical theories. Because they should be rather interpreted as mental representations of mathematical objects, they no longer satisfy the semantic role of truth-makers in relation to mathematical sentences. The argument formulated in the paper is based on the analysis of mathematical extensions of the basic non-Fregean logic. Sixty-two definitions of potential fact-identity connectives are also presented. The paper's conclusion may be interpreted as undermining the situational paradigm of building semantics for mathematical languages.