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: slingshot arguments supply us with some reasons to reject various ontological theories of mathematical facts; there are two ways of blocking these arguments: through the rejection of the (...) principle of extensionality for individual terms or through the rejection of the principle of Wittgenstein; the first way cannot be accepted because it leads to the practice of softening mathematical facts; the second way, called fine-graining facts, cannot be accepted because it also results in the practice of softening facts. Hence, only soft mathematical facts 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. (shrink)

'You ask me about the idiosyncrasies of philosophers? There is their lack of historical sense, their hatred of even the idea of becoming, their Egyptianism. They think they are doing a thing honour when they dehistoricize it, sub specie aeternitatis — when they make a mummy of it.'Nietzsche, Twilight of the Idols.

is true, there is a truth-maker (e.g., a fact) with which it is identical and the truth of the former consists in its identity with the latter. The theory is best understood as a reaction to the correspondence theory, according to which the relation of truth-bearer to truth-maker is correspondence. A correspondence theory is vulnerable to the nagging suspicion that if the best we can do is make statements that merely correspond to the truth, then we inevitably fail to capture (...) the reality they are about and thus fall short of the truth we aim at. An identity theory is designed to overcome this suspicion. (shrink)

The paper presents a new argument supporting the ontological standpoint according to which there are no mathematical facts in any set theoretic model of arithmetical theories. It may be interpreted as showing that it is impossible to construct fact-arithmetic. The importance of this conclusion arises in the context of cognitive science. In the paper, a new type of slingshot argument is presented, which is called hyper-slingshot. The difference between meta-theoretical hyper-slingshots and conventional slingshots consists in the fact that the former (...) are formulated in the semantic meta-language of mathematical theories without the use of the iota-operator or the lambda-operator as the abstractor, whereas the latter require for their expression at least one of these non-standard term-operators. Hyper-slingshots implement simpler language tools in comparison with those used in conventional slingshots. (shrink)

Pretheoretically, ‘all believers are immortal’ is about all believers, but B is not about any unbeliever. Similarly, ‘all mortals are unbelievers’ is not about any immortal, but M is about all mortals. But B and M are logically equivalent universal generalizations, so arguably they are about exactly the same objects; by, they are about those mortals who are unbelievers, contradicting. If one responds by giving up, is there still a sense in which B treats unbelievers differently from believers? I argue (...) that there is. B is uninformative about unbelievers but informative about believers, in the following sense: for any object o, the information that B provides only about o—namely, ‘o is a believer only if o is immortal’—is entailed by ‘o is an unbeliever’ but not by ‘o is a believer’. (shrink)

constructive empiricism asserts that it is not for science to reach a verdict on whether a theory is true or false, if the theory is about unobservable entities; science's only interest here, says Van Fraassen, is to discover whether the theory is ‘empirically adequate’. However, if a theory is soley about observables, empirical adequacy and truth are said to ‘coincide’, here discovering the theory's truth value is an appropriate scientific goal. Constructive empiricism thus rests an epistemological thesis on a semantical (...) distinction. This paper critically examines the notion of aboutness and its epistemological significance. * I am grateful to members of my philosophy of science seminar during Spring, 1983 for useful discussion and to the two anonymous referees of this journal for providing helpful suggestions. I also thank the University of Wisconsin, Madison Graduate School for support in the form of a Romnes Faculty Fellowship. (shrink)