De Toffoli, Silvia. Groundwork for a Fallibilist Account of Mathematics
2021, The Philosophical Quarterly, 71(4).
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Added by: Fenner Stanley TanswellAbstract: According to the received view, genuine mathematical justification derives from proofs. In this article, I challenge this view. First, I sketch a notion of proof that cannot be reduced to deduction from the axioms but rather is tailored to human agents. Secondly, I identify a tension between the received view and mathematical practice. In some cases, cognitively diligent, well-functioning mathematicians go wrong. In these cases, it is plausible to think that proof sets the bar for justification too high. I then propose a fallibilist account of mathematical justification. I show that the main function of mathematical justification is to guarantee that the mathematical community can correct the errors that inevitably arise from our fallible practices.
Jenkins, Carrie. What can we know a priori?
2014, Neta, Ram (ed.), Current Controversies in Epistemology. London: Routledge. 11-22.
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Added by: Jie GaoAbstract: Michael Devitt has been developing an influential two-pronged attack on the a priori for over thirteen years. This attack does not attempt to undermine the coherence or significance of the distinction between the a priori and the a posteriori, but rather to answer the question: 'What Can We Know A Priori?' with: 'Nothing'. In this paper I explain why I am dissatisfied with key extant responses to Devitt's attack, and then take my own steps towards resisting the attack as it appears in two recent incarnations. Devitt aims firstly to undermine the motivation for believing in any a priori knowledge, and secondly to provide reasons directly against believing in any. I argue that he misidentifies the motivations available to the a priorist, and that his reasons against believing in the a priori do not take account of all the options. I also argue that his attempt to combine the two prongs of the attack into an abductive argument for his anti-a priorist position does not succeed.
Comment: Suitable for an upper-level undergraduate courses or master courses on epistemology. It is good for teachings on topics of a priori knowledge.
Jenkins-Ichikawa, Carrie. Knowledge of Arithmetic
2005, British Journal for the Philosophy of Science 56: 727-747.
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Added by: Jamie CollinAbstract: The goal of the research programme I describe in this article is a realist epistemology for arithmetic which respects arithmetic's special epistemic status (the status usually described as a prioricity) yet accommodates naturalistic concerns by remaining funda- mentally empiricist. I argue that the central claims which would allow us to develop such an epistemology are (i) that arithmetical truths are known through an examination of our arithmetical concepts; (ii) that (at least our basic) arithmetical concepts are accurate mental representations of elements of the arithmetical structure of the inde- pendent world; (iii) that (ii) obtains in virtue of the normal functioning of our sensory apparatus. The first of these claims protects arithmetic's special epistemic status relative, for example, to the laws of physics, the second preserves the independence of arithmetical truth, and the third ensures that we remain empiricists.
Comment: Useful as a primary or secondary reading in an advanced undergraduate course epistemology (in a section on a priori knowledge) or an advanced undergraduate course on philosophy of mathematics. This is not an easy paper, but it is clear. It is also useful within a teaching context, as it provides a summary of the influential neo-Fregean approach to mathematical knowledge.
Yap, Audrey. The Logical Syntax of Prejudice: Oppression and the Constitutive A Priori
2024, In R. Cook and A. Yap (eds.), Feminist Philosophy and Formal Logic. University of Minnesota Press
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Added by: Franci Mangraviti and Viviane FairbankAbstract: I argue that a thoroughgoing naturalized epistemology can easily underestimate the extent to which certain background assumptions will infl uence arguments. Instead, then, I suggest that we can borrow a conceptual tool from neo-Kantian philosophy of science, namely the constitutive a priori. This idea originates in neo-Kantian philosophers who understood, in light of Einsteinian physics, that Kantian views about the a priority of space were untenable. Frameworks that adopt some version of a constitutive a priori take certain propositions to play the role of a priori principles, without granting them the universality or necessity that such principles traditionally hold. I will argue that thinking of certain views or values as having the status of constitutive a priori principles can help us understand what would be required for an epistemic agent to change them, and thus illustrate the extent to which they are resistant to being dislodged by evidence.
Comment: available in this Blueprint
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Comment (from this Blueprint): De Toffoli makes a strong case for the importance of mathematical practice in addressing important issues about mathematics. In this paper, she looks at proof and justification, with an emphasis on the fact that mathematicians are fallible. With this in mind, she argues that there are circumstances under which we can have mathematical justification, despite a possibility of being wrong. This paper touches on many cases and questions that will reappear later across the Blueprint, such as collaboration, testimony, computer proofs, and diagrams.