Opinionated History of Mathematics

Intellectual Mathematics

History of mathematics research with iconoclastic madcap twists

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20 episodes
Updated Jul 15, 2025

Episodes20

Newest first

Jul 15, 2025 · 26 min
Death of Archimedes
Archimedes’s emblematic death makes sense psychologically and embodies a rich historical picture in a single scene. Transcript Archimedes died mouthing back at an enemy soldier: “Don’t disturb my circles.” Or that’s how the story goes.
Dec 30, 2024 · 56 min
Torricelli’s trumpet is not counterintuitive
There is nothing counterintuitive about an infinite shape with finite volume, contrary to the common propaganda version of the calculus trope known as Torricelli’s trumpet. Nor was this result seen as counterintuitive at the time of its discovery in th...
Nov 29, 2023 · 1 hr 27 min
Did Copernicus steal ideas from Islamic astronomers?
Copernicus’s planetary models contain elements also found in the works of late medieval Islamic astronomers associated with the Maragha School, including the Tusi couple and Ibn al-Shatir’s models for the Moon and Mercury.
Jul 23, 2023 · 1 hr 16 min
Operational Einstein: constructivist principles of special relativity
Einstein’s theory of special relativity defines time and space operationally, that is to say, in terms of the actions performed to measure them. This is analogous to the constructivist spirit of classical geometry. Transcript Oh no,
Oct 11, 2022 · 52 min
Review of Netz’s New History of Greek Mathematics
Reviel Netz’s New History of Greek Mathematics contains a number of factual errors, both mathematical and historical. Netz is dismissive of traditional scholarship in the field, but in some ways represents a step backwards with respect to that traditio...
May 20, 2022 · 32 min
The “universal grammar” of space: what geometry is innate?
Geometry might be innate in the same way as language. There are many languages, each of which is an equally coherent and viable paradigm of thought, and the same can be said for Euclidean and non-Euclidean geometries.
Feb 20, 2022 · 30 min
“Repugnant to the nature of a straight line”: Non-Euclidean geometry
The discovery of non-Euclidean geometry in the 19th century radically undermined traditional conceptions of the relation between mathematics and the world. Instead of assuming that physical space was the subject matter of geometry,
Nov 17, 2021 · 30 min
Rationalism 2.0: Kant’s philosophy of geometry
Kant developed a philosophy of geometry that explained how geometry can be both knowable in pure thought and applicable to physical reality. Namely, because geometry is built into not only our minds but also the way in which we perceive the world.
Sep 18, 2021 · 43 min
Rationalism versus empiricism
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,
Jul 10, 2021 · 33 min
Cultural reception of geometry in early modern Europe
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 branches of le...
May 10, 2021 · 49 min
Maker’s knowledge: early modern philosophical interpretations of geometry
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.
Mar 10, 2021 · 51 min
“Let it have been drawn”: the role of diagrams in geometry
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 constructi...
Jan 20, 2021 · 1 hr 18 min
Why construct?
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,
Dec 10, 2020 · 41 min
Created equal: Euclid’s Postulates 1-4
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.
Nov 3, 2020 · 43 min
That which has no part: Euclid’s definitions
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,
Oct 4, 2020 · 35 min
What makes a good axiom?
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.
Sep 8, 2020 · 35 min
Consequentia mirabilis: the dream of reduction to logic
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,
Jul 30, 2020 · 41 min
Read Euclid backwards: history and purpose of Pythagorean Theorem
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.
Jun 21, 2020 · 40 min
Singing Euclid: the oral character of Greek geometry
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.
May 15, 2020 · 42 min
First proofs: Thales and the beginnings of geometry
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...

Opinionated History of Mathematics

Death of Archimedes

Torricelli’s trumpet is not counterintuitive

Did Copernicus steal ideas from Islamic astronomers?

Operational Einstein: constructivist principles of special relativity

Review of Netz’s New History of Greek Mathematics

The “universal grammar” of space: what geometry is innate?

“Repugnant to the nature of a straight line”: Non-Euclidean geometry

Rationalism 2.0: Kant’s philosophy of geometry

Rationalism versus empiricism

Cultural reception of geometry in early modern Europe

Maker’s knowledge: early modern philosophical interpretations of geometry

“Let it have been drawn”: the role of diagrams in geometry

Why construct?

Created equal: Euclid’s Postulates 1-4

That which has no part: Euclid’s definitions

What makes a good axiom?

Consequentia mirabilis: the dream of reduction to logic

Read Euclid backwards: history and purpose of Pythagorean Theorem

Singing Euclid: the oral character of Greek geometry

First proofs: Thales and the beginnings of geometry