Opinionated History of Mathematics

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 s

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 trumpe

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

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

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

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 s

“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 ass

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 b

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 Kepl

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, acco

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 thu

“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 h

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 bet

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,

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 hav

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 Arist

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

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

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 Elemen

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 fr