Certainty and error in mathematics: Deductivism and the claims of mathematical fallibilism (W. V. Quine, Imre Lakatos, Bertrand Russell).

摘要

This project has two goals: (1) to analyze the claims of mathematical fallibilism in order to show that they are less controversial than their usual statement implies; (2) to resurrect deductivism with modifications from its premature burial. For the purposes of this project fallibilism is the disjunctive claim that mathematical proofs are insufficient either (a) to justify mathematical knowledge where knowledge has a certainty-clause or (b) to establish conclusively any mathematical truths. The first clause contrasts with the traditional view that mathematical knowledge is certain. The second clause reinforces this uncertainty by undermining the demonstrative force of proofs. Five arguments for fallibilism are considered. First, (Chapter 2) the distinction between pure and applied math leads to an uncontroversial form of mathematical fallibilism. Moreover, this distinction is shown to fit well with deductivism. Second, (Chapter 3) mathematics employs non-deductive methods. Non-deductive procedures are also, prima facie, difficult to reconcile with deductivism. The fallibility entailed by non-deductive methods is both limited and uncontroversial. In terms of deductivism, old-style Russellian deductivism is abandoned in favor of a more general notion of deductive proof. Third, (Chapter 4) there is the possibility of an infinite regress for mathematical justification. For Lakatos the regress originates in the various prospects for mathematical foundations. His favored theory, quasi-empiricism, is justified by comparison to foundational approaches. Modified deductivism is shown to halt the infinite regress as well. Fourth, (Chapter 4) the fact that informal arguments---i.e., arguments not valid in terms of their logical form---are nearly ubiquitous in mathematics is prima facie evidence in favor of quasi-empiricism. Deductivism is shown to be consistent with informal arguments. Fifth, (Chapter 5) Quine's naturalism is shown to lead to fallibilism. However, when properly understood this fallibility is shown to be uncontroversial. Moreover, although Quine never officially endorses anything but naturalism, it is shown that naturalism is both consistent with and aided by deductivism. Finally, (Chapter 6) the modifications to deductivism are unified and the claims of fallibilism are restated in their uncontroversial forms.