Opinionated History of Mathematics
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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…
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…
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…
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…
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…
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…
“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…
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…
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…
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…
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…
“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…
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…
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…
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…
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…
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…
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…
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…
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:…
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Opinionated History of Mathematics has published 40 episodes since November 2018, covering topics in History, Mathematics.
Opinionated History of Mathematics is currently declining with new episodes every 2 months. Average episode length is 35m.
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