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Euclid’s definitions of point, line, and straightness allow a range of mathematical and philosophical interpretation. Historically, however, these definitions may not have been in the original text of the Elements at all. Regardless,

How should axioms be justified? By appeal to intuition, or sensory perception? Or are axioms legitimated merely indirectly, by their logical consequences? Plato and Aristotle disagreed, and later Newton disagreed even more.

Euclid’s Elements, read backwards, reduces complex truths to simpler ones, such as the Pythagorean Theorem to the parallelogram area theorem, and that in turn to triangle congruence. How far can this reductive process be taken,

The Pythagorean Theorem might have been used in antiquity to build the pyramids, dig tunnels through mountains, and predict eclipse durations, it has been said. But maybe the main interest in the theorem was always more theoretical.

Greek geometry is written in a style adapted to oral teaching. Mathematicians memorised theorems the way bards memorised poems. Several oddities about how Euclid’s Elements is written can be explained this way.

Proof-oriented geometry began with Thales. The theorems attributed to him encapsulate two modes of doing mathematics, suggesting that the idea of proof could have come from either of two sources: attention to patterns and relations that emerge from exp...

In ancient Mesopotamia and Egypt, mathematics meant law and order. Specialised mathematical technocrats were deployed to settle conflicts regarding taxes, trade contracts, and inheritance. Mathematics enabled states to develop civil branches of governm...

The Greek islands were geographically predisposed to democracy. The ritualised, antagonistic debates of parliaments and law courts were then generalised to all philosophical domains, creating a unique intellectual climate that put a premium on adversar...

What did 17th-century mathematicians such as Newton and Huygens think of Galileo? Not very highly, it turns out. I summarise my case against Galileo using their perspectives and a mathematical lens more generally.

Our picture of Greek antiquity is distorted. Only a fraction of the masterpieces of antiquity have survived. Decisions on what to preserve were made by in ages of vastly inferior intellectual levels. Aristotelian philosophy is more accessible for medio...