Certainty and error in mathematics: Deductivism and the claims of mathematical fallibilism

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dc.contributor.advisorGrandy, Richard E.en_US
dc.creatorDove, Ian J.en_US
dc.date.accessioned2009-06-04T08:29:30Zen_US
dc.date.available2009-06-04T08:29:30Zen_US
dc.date.issued2004en_US
dc.description.abstractThis 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.en_US
dc.format.extent219 p.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.callnoTHESIS PHIL. 2004 DOVEen_US
dc.identifier.citationDove, Ian J.. "Certainty and error in mathematics: Deductivism and the claims of mathematical fallibilism." (2004) Diss., Rice University. <a href="https://hdl.handle.net/1911/18625">https://hdl.handle.net/1911/18625</a>.en_US
dc.identifier.urihttps://hdl.handle.net/1911/18625en_US
dc.language.isoengen_US
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.en_US
dc.subjectMathematicsen_US
dc.subjectPhilosophyen_US
dc.titleCertainty and error in mathematics: Deductivism and the claims of mathematical fallibilismen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentPhilosophyen_US
thesis.degree.disciplineHumanitiesen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophyen_US
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