Rationalism says mathematical knowledge comes from within, from pure thought; empiricism that it comes from without, from experience and observation. Rationalism led Kepler to look for divine design in the universe, and Descartes to reduce all

Euclid inspired Gothic architecture and taught Renaissance painters how to create depth and perspective. More generally, the success of mathematics went to its head, according to some, and created dogmatic individuals dismissive of other branch

Philosophical movements in the 17th century tried to mimic the geometrical method of the ancients. Some saw Euclid—with his ruler and compass in hand—as a “doer,” and thus characterised geometry as a “maker’s knowledge.” Others got into a feud

The use of diagrams in geometry raise questions about the place of the physical, the sensory, the human in mathematical reasoning. Multiple sources of evidence speak to how these dilemmas were tackled in antiquity: the linguistics of diagram co

Euclid spends a lot of time in the Elements constructing figures with his ubiquitous ruler and compass. Why did he think this was important? Why did he think this was better than a geometry that has only theorems and no constructions? In fact,

The etymology of the term “postulate” suggests that Euclid’s axioms were once questioned. Indeed, the drawing of lines and circles can be regarded as depending on motion, which is supposedly proved impossible by Zeno’s paradoxes. Although wheth

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, the subtle

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. Their philosophies c

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, and what should

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. Euclid’s pr

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. Transcript Greek geometry is oral

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

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

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

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. Transcript I’m going to conclu

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 f

What was Galileo’s great innovation in science? To give practical experience more authority than philosophical systems? To insist on mechanical as opposed to teleological or supernatural explanations of natural phenomena? To take mathematical p

Was Galileo “the father of modern science” because he was the first to unite mathematics and physics? Or the first to base science on data and experiments? No. Galileo was not the first to do any of these things, despite often being erroneously

Galileo’s sentencing by the Inquisition was avoidable. The Church had no interest in prosecuting mathematical astronomers, but since Galileo had so little to contribute in that domain he foolishly got himself involved with Biblical interpretati

Galileo thought comets were an atmospheric phenomenon, not physical bodies in outer space. How could he be so wrong when all his colleagues got it right? Perhaps because his theory was a convenient excuse for not doing any mathematical astronom

Telescopic observations of Venus provided evidence for the Copernican view of the solar system. But was Galileo the first to see this, as he claims? Or did he steal the idea from a colleague and lie about having made the observations months bef

Galileo thought sunspots were one of the three best arguments for heliocentrism. He was wrong. Transcript The early days of telescopic astronomy were exhilarating. Listen to this anecdote by Kepler. He is writing in 1610, right after the appear

The telescope offered a shortcut to stardom for Galileo. We offer some fun cynical twists on the standard story. Transcript The year is 1609. What a time to be alive. In London you can go to the theatre and catch the fresh new play Macbeth. In

Galileo is credited with defeating Ptolemaic earth-centered astronomy, but most mathematical astronomers had already abandoned this theory long before Galileo. Transcript Does the earth move around the sun, or is it the other way around? Copern

Two thousand years before Galileo, Greek astronomers argued that the heavenly bodies revolve around the sun. Their reasoning involved sophisticated mathematics and sound physical considerations. Transcript Is the earth the center of the univers

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