Opinionated History of Mathematics

Jaksot

• Death of Archimedes 15.07.2025 26min

  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. Is this fact or fiction? We have third-hand accounts at best so there is plenty of room for doubt. But … Continue reading Death of Archimedes
• Torricelli’s trumpet is not counterintuitive 30.12.2024 56min

  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 the 17th century, contrary to many commonplace historical narratives. Transcript Torricelli’s trumpet is not counterintuitive. Your … Continue reading Torricelli’s trumpet is not counterintuitive
• Did Copernicus steal ideas from Islamic astronomers? 29.11.2023 1t 27min

  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. On this basis many historians have concluded that Copernicus must have gotten his hands on these Maragha ideas somehow or other, even … Continue reading Did Copernicus steal ideas from Islamic astronomers?
• Operational Einstein: constructivist principles of special relativity 23.07.2023 1t 16min

  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.
• Review of Netz’s New History of Greek Mathematics 11.10.2022 52min

  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 tradition. I argue against Netz’s dismissal of many anecdotal historical testimonies as fabrications, and his “ludic proof” … Continue reading Review of Netz’s New History of Greek Mathematics
• The “universal grammar” of space: what geometry is innate? 20.05.2022 32min

  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. As our native language is shaped by experience, so might our “native geometry” be. Yet substantive innate conceptions may be a precondition for any linguistic or spatial thought to be possible at all, as Chomsky said for language and Kant for geometry. Just as language learning requires singling out, from all the sounds in the environment, only the linguistic ones, so Poincaré articulated criteria for what parts of all sensory data should be regarded as pertaining to geometry.
• “Repugnant to the nature of a straight line”: Non-Euclidean geometry 20.02.2022 30min

  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, mathematicians elaborated numerous alternative geometries abstractly and formally, distancing themselves from reality and intuition.
• Rationalism 2.0: Kant’s philosophy of geometry 17.11.2021 30min

  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. In this way, Kant solved the applicability problem of classical rationalism, albeit at the cost of making our perception of the world around us inextricably subjective. Kant’s theory also showed how rationalism, and philosophy generally, could be reconciled with Newtonian science, with which it had been seen as embarrassingly out of touch. In particular, Kant’s perspective shows how Newton’s notion of absolute space, which had seemed philosophically repugnant, can be accommodated from an epistemological point of view.
• Rationalism versus empiricism 18.09.2021 43min

  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 mechanical phenomena to contact mechanics and all curves in geometry to instrumental generation. Empiricism led Newton to ignore the cause of gravity and dismiss the foundational importance of constructions in geometry.
• Cultural reception of geometry in early modern Europe 10.07.2021 33min

  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 learning. Some thought the uncompromising rigour of Euclid went hand in hand with totalitarianism in political and spiritual domains, while others thought creative mathematics was inherently free and liberal.
• Maker’s knowledge: early modern philosophical interpretations of geometry 10.05.2021 49min

  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 be justified via theology.
• “Let it have been drawn”: the role of diagrams in geometry 10.03.2021 51min

  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 mathematical proofs.
• Why construct? 20.01.2021 1t 18min

  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 susceptible, such as inconsistencies, hidden assumptions, verbal logic fallacies, and diagrammatic fallacies.
• Created equal: Euclid’s Postulates 1-4 10.12.2020 41min

  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 to a “collapsible” compass in Proposition 2. Furthermore, why did Euclid feel the need to postulate that “all right angles are equal”? Perhaps in order to rule out non-flat surfaces such as cones.
• That which has no part: Euclid’s definitions 03.11.2020 43min

  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 but also of curved surfaces.
• What makes a good axiom? 04.10.2020 35min

  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.
• Consequentia mirabilis: the dream of reduction to logic 08.09.2020 35min

  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 purely logical terms, but this perspective leaves some fundamental challenges unresolved.
• Read Euclid backwards: history and purpose of Pythagorean Theorem 30.07.2020 41min

  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 theorem but as breaking the truth of the theorem apart into its constituent parts to analyse what makes it tick. Euclid’s Elements as a whole can be read in this way, as a project of epistemological analysis.
• Singing Euclid: the oral character of Greek geometry 21.06.2020 40min

  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.
• First proofs: Thales and the beginnings of geometry 15.05.2020 42min

  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 proof by contradiction.