Opinionated History of Mathematics

Latest Episodes:

Maker’s knowledge: early modern philosophical interpretations of geometry May 10, 2021

Show notes

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 about what to do when Euclid was at odds with Aristotle. Descartes thought Euclid’s axioms should … Continue reading Maker’s knowledge: early modern philosophical interpretations of geometry

“Let it have been drawn”: the role of diagrams in geometry Mar 10, 2021

Show notes

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 construction, the state of drawings in the oldest extant manuscripts, commentaries of philosophers, and implicit assumptions in … Continue reading “Let it have been drawn”: the role of diagrams in geometry

Why construct? Jan 20, 2021

Show notes

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, constructions protect geometry from foundational problems to which it would otherwise be … Continue reading Why construct?

Created equal: Euclid’s Postulates 1-4 Dec 10, 2020

Show notes

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 whether these postulates correspond to ruler and compass or not is debatable, especially since Euclid seems to restrict himself … Continue reading Created equal: Euclid’s Postulates 1-4

That which has no part: Euclid’s definitions Nov 03, 2020

Show notes

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 subtlety of defining fundamental concepts such as straightness is best seen by considering the geometry not only of a flat plane … Continue reading That which has no part: Euclid’s definitions

What makes a good axiom? Oct 04, 2020

Show notes

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 can be seen as rival interpretations of Euclid’s Elements. Transcript What kinds of axioms do we want in our geometry? How … Continue reading What makes a good axiom?

Consequentia mirabilis: the dream of reduction to logic Sep 08, 2020

Show notes

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 be its ultimate goals? Some have advocated that the axiomatic-deductive program in mathematics is best seen in … Continue reading Consequentia mirabilis: the dream of reduction to logic

Read Euclid backwards: history and purpose of Pythagorean Theorem Jul 30, 2020

Show notes

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 proof of the Pythagorean Theorem is perhaps best thought of not as establishing the truth of the … Continue reading Read Euclid backwards: history and purpose of Pythagorean Theorem

Singing Euclid: the oral character of Greek geometry Jun 21, 2020

Show notes

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 geometry. Mathematicians memorised theorems the way bards memorised poems. Euclid’s Elements was almost like a song book or … Continue reading Singing Euclid: the oral character of Greek geometry

First proofs: Thales and the beginnings of geometry May 15, 2020

Show notes

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 explorative construction and play, or the realisation that “obvious” things can be demonstrated using formal definitions and … Continue reading First proofs: Thales and the beginnings of geometry