Platonic Relations and Mathematical Explanations

Some scientific explanations appear to turn on pure mathematical claims. The enhanced indispensability argument appeals to these ‘mathematical explanations’ in support of mathematical platonism. I argue that the success of this argument rests on the claim that mathematical explanations locate pure mathematical facts on which their physical explananda depend, and that any account of mathematical explanation that supports this claim fails to provide an adequate understanding of mathematical explanation.

Mathematical Entities without Objects. On Realism in Mathematics and a Possible Mathematization of (Non)Platonism: Does Platonism Dissolve in Mathematics?

By looking at three significant examples in analysis, geometry and dynamical systems, I propose the possibility of having two levels of realism in mathematics: the upper one, the one of entities; and a subordinated ground one, the one of objects. The upper level (entities) is more the one of ‘operations’, of mathematics in action, of the dynamics of mathematics, whereas the ground floor (objects) is more dedicated to culturally well-defined objects inherited from our perception of the physical or real world. I will show that the upper level is wider than the ground level, therefore foregrounding the possibility of having in mathematics entities without underlying objects. In the three examples treated in this article, this splitting of levels of reality is created directly by the willingness to preserve different symmetries, which take the form of identities or equivalences. Finally, it is proposed that mathematical Platonism is – in fine – a true branch of mathematics in order for mathematicians to avoid the temptation of falling into the Platonist alternative ‘everything is real’/‘nothing is real’.

Metaphysics and the Mathematical Diagram

Accounts of geometry are caught between the demands of history and philosophy, and are difficult to reduce to either. In a profoundly influential move, Plato used geometrical proof as one means of bootstrapping his Theory of Forms and what came to be called metaphysics, and the emergence of ontological modes of thinking. This has led to a style of thinking still common today that gets called ‘mathematical Platonism’. By contrast, the sheer diversity of mathematical practices across cultures and time has been adduced to claim their historical contingency, which has recently prompted Ian Hacking to question why there is philosophy of mathematics at all. The different roles assigned to geometrical diagrams in these debates form the focus of this chapter, which analyses in detail the contrasting discussions of diagrams, and of the linearization and spatialization of thinking, by Plato (especially Meno and the Republic), by the cognitive historian Reviel Netz, the media theorist Sybille Krämer, and the anthropologist Tim Ingold.

Theism, Explanation, and Mathematical Platonism

Dan Baras has recently argued for the claim that Theistic Mathematical Platonism (TMP) fares no better than Mathematical Platonism (MP) with respect to explaining why our mathematical beliefs are correlated with mind-independent mathematical truths. In this paper I argue that, insofar as TMP provides a proximate or local explanation for this truth-tracking correlation whereas MP fails to offer any corresponding explanation, Baras’s claim that TMP fares no better than MP with respect to explaining this correlation is false.

Avicenna against Mathematical Platonism

In this paper I investigate Avicenna’s criticisms of the separateness of mathematical objects and of the view that they are principles for natural things. These two theses form the core of Plato’s view of mathematics; i.e., mathematical Platonism. Surprisingly, Avicenna does not consider his arguments against these theses as attacks on Plato. This is because his understanding of Plato’s philosophy of mathematics differs from both Plato’s original view and what Aristotle attributes to Plato.

The uncanny accuracy of God's mathematical beliefs

I show how mathematical platonism combined with belief in the God of classical theism can respond to Field's epistemological objection. I defend an account of divine mathematical knowledge by showing that it falls out of an independently motivated general account of divine knowledge. I use this to explain the accuracy of God's mathematical beliefs, which in turn explains the accuracy of our own. My arguments provide good news for theistic platonists, while also shedding new light on Field's influential objection.

Gödel, Kurt (1906–78)

The greatest logician of the twentieth century, Gödel is renowned for his advocacy of mathematical Platonism and for three fundamental theorems in logic: the completeness of first-order logic; the incompleteness of formalized arithmetic; and the consistency of the axiom of choice and the continuum hypothesis with the axioms of Zermelo–Fraenkel set theory.