Stipulative character of Hume's principle

The paper analyzes neo-logicism of C. Wright and B. Hale. The neo-Fregean methodology is based on the introduction of the concept of number by using the abstraction principle - the Hume’s principle. Abstraction principles have a distinguished status: they are a special kind of stipula-tion.Wright derives the basic laws of arithmetic from a stipulation governing the concept of number, without providing an explicit definition of the individual numbers or of «immediately precedes» on the basis of a purely logical vocabulary. There are challenges related to the use of stipulative definitions in the neo-Fregean philosophy of mathematics, in the first place, the so-called «bad company objection». The first difficulty is that the use of the principles of abstraction, for example, Axiom V Frege, can lead to a contradiction. In response, Wright and Hale demonstrate that the presence of erroneous principles of abstraction does not mean that methodology is irremediably flawed. The second difficulty is next. Given the fact that some abstraction principles are inconsistent with each other, it is necessary to supplement the account with a criterion capable of governing the choice of one abstraction over another. Wright and Hale suggest as a solution to this difficulty the use of the criterion of «conserva-tism», according to which an abstraction principle is acceptable if it does not constrain the cardinality of concepts with whose introduction it is not explicitly concerned. However, this solution is also problematic. Wright and Hale maintains that the existence of numbers is something discovered rather than stipulated, while holding that our a priori knowledge of their necessary existence is derived from a principle truth is a matter of stipulation. They’re trying to show that Hume's principle merely lays down partial satisfaction conditions for the relation of numerical identity. The main difficulty of the neo-Fregean methodology, indeed, is with transition from the characterization of the truth of Hume's principle as a simple stipulation to its deployment as a truth having significant existential implications. Presently, these discussions are still ongoing and decisions are pending.

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