Up to and including Plato’s generation, the concept of prime number isn’t found. Plato, who never mentions the primes, generates 2 and 3 in Parmenides, and with them the even and odd, and what is taken to be all other kinds of number. That project suggests the challenge of generating all numbers by Plato’s methods – a challenge which cannot be met because of the primes. I argue that this brought about the formation of the concept of prime number, probably (...) in the Academy, and that the attempt to develop the Parmenides’ ideas led to the discovery that there are infinitely many primes. (shrink)

This essay shows how the Epicureans could have anticipated time dilation measurements precisely as our standard model predicts and precisely as we measure it today. Specifically, a mathematical equivalent of the velocity Lorentz transformation can be derived from the Epicurean atomist doctrine of isotakheia, which states that all (Epicurean) atoms have equal speed. The derivation is brief and classical, and it requires no mathematical concepts that would be foreign to the ancient Greeks. The significance of this derivation is addressed.

According to the Standard Definition ‘a fallacious argument, as almost every account from Aristotle onwards tells you, is one that seems to be valid but is not so’ ( Hamblin 1970 : 12). Scholars take this definition to be problematic in part because ‘appearances can vary from person to person, thus making the same argument a fallacy for the one who is taken in by the appearance, and not a fallacy for the one who sees past the appearances’ ( Hansen (...) 2020 : §4.2). We argue that Aristotle's treatment of fallacy is not subjective in the way that critics of the Standard Definition allege. (shrink)

Plotinus argues that materialism cannot explain reflexive cognition. He argues that mere bodies cannot engage in the self-reflexive activity of both cognizing some content and being cognitively aware of cognizing this content. Short of outright denying the cognitive unity underlying this phenomenon of self-awareness, materialism is in trouble. However, Plotinus bases his argument on the condition that material bodies are capable of a spatial unity at most, and while this condition has purchase on ancient materialists, it would be rejected today. (...) After considering objections based on our contemporary understanding of a material body, I amend Plotinus' argument to focus instead on the core materialist commitment to external efficient-causal relations. This amended argument regains its urgency for materialism, while retaining the spirit of Plotinus' view. (shrink)