Thinking about mathematics generates intriguing philosophical questions. For instance, what is mathematical truth about? That is, is mathematical truth about something we construct, or is mathematical truth instead about something independent of us? Are there mathematical truths in the first place? Furthermore, how do we attain mathematical knowledge? For that matter, how do we gain beliefs about mathematics at all? In answering these questions, I defend standard platonism about mathematicsas the best account of mathematical truth and knowledge. In doing so, I explain why we should accept standard platonism as opposed to naturalised platonism or plenitudinous platonism. I also examine the various epistemological and ontological problems that confront standard platonism, and I present responses to these worries. I then discuss an account that aims to explain the reliability of mathematical inquiry and show how Paul Benacerraf’s famous condition for explaining reliability fails. Thus, I reject a causal account of the generation and reliability of our mathematical beliefs. In light of this rejection, I propose an explanation that depends upon inference and the resources of mathematics itself. Finally, I consider a selection of standard platonism’s foremost rivals. This list includes other platonist views such as Penelope Maddy’s naturalised platonism, W. V. O. Quine’s naturalised platonism, plenitudinous platonism, and set-theoretic reductionism. But I also examine competing non-platonist accounts such as nominalism, intuitionism, and structuralism. In surveying these positions, I aim to show that the various rivals of platonism in mathematics either face worse problems or else imply the very platonism they aim to avoid.

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