Wojciech Krysztofiak

In the paper there are presented main assumptions underlying the construction of theoretic models of mental processes of numeral reference in mathematical practice which comprises such abilities as counting, solving story-tasks, estimating cardinalities and comparing magnitudes. Numerals are understood as any expressions which enable mind to refer to numbers, cardinalities and magnitudes. The main research question formulated in the article sounds: What cognitive processes do there occur in the mind during execution of various numeral acts of reference?

The paper presents a new argument supporting the ontological standpoint according to which there are no mathematical facts in any set theoretic model (world) constructed on the grounds of second order arithmetical theories founded upon set theory. Slingshot arguments directed against facts, situations and other propositional entities are usually blocked by rejecting one of the main inference rules used in various versions of this argument. Authors distinguish two types of main inference rules used in these variants: (i) for non-propositional expressions: iota-conversion rules (ι-conv), iota-substitution rules (ι-subs), lambda-conversion principle (λ-conv); and (ii) for propositional expressions: the principle of substitutivity for logical equivalents (PSLE). Even if strategies for defending facts by the rejection of one of the mentioned rules are accepted, it may be shown that the acceptance of the requirement that mathematical facts (situations or truth-makers) differing with regard to their constituents are different facts, leads to the contradiction in meta-theories of set theoretic models for first order arithmetical theories. In the paper, a new type of slingshot argument is presented, which may be 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 name-forming lambda-operator, whereas the latter require for their expression at least one of these non-standard term-operators. Furthermore, in hyper-slingshots PSLE is not used, whereas in conventional slingshots, PSLE plays a crucial inferential role. Hyper-slingshots implement simpler language tools in comparison with those used in conventional slingshots

In the paper there is presented the argument for the situational paradigm of theory of language. In comparison to the nominativistic semantics it is argued in the paper that the propositional semantics is a better tool of explaining various speech acts in which the principle of compositionality is collapsed. In the paper there are also described referential mechanisms of metaphorical speech acts. In accordance with these mechanisms the metaphorical status of speech acts is determined by the collapse of Fregean referential compositionality of speech acts. In the paper there are constructed two notions of compositionality. First of them may be called classical Fregean concept of compositionality which establishes a semantic correlate of a complex speech act on a functional basis of semantic correlates of constituents of a complex speech act. According to the second notion of compositionality, correlates of constituents of a complex speech act are recursively deter-mined by a correlate of a complex speech act. Mataphorical speech acts possess such a property that correlates of their constituent expressions are determined in processes of de-coding by recursive compositionality functions. Maximal metaphors are speech acts in which recursive as well as Fregean compositionality collapse. In the paper it is argued that the traditional nominativistic semantics is not able to explain communicational functioning of metaphorical speech acts. That is why the conclusion of the paper resolves itself to the thesis that the situational semantic paradigm which should base on recursive notion of compositionality, is better than the nominativistic paradigm

The paper presents a new model of the structure of basic arithmetical representa-tions encoded in minds which enable them to solve simple story-tasks. According to the dominating paradigm the process of acquiring basic counting abilities culminates in encoding the exact number line in mind. This linear number representation enables the mind to solve simple story-tasks which do not require any mathematical mastery knowledge comprising laws, definitions and theorems. Some researchers try to show that the process of encoding the exact number line stems from transformations of the approximate number line (the mental number line) whereas others model this process as being dependent on the linguistic and logical resources of mind