Opinionated History of Mathematics

Opinionated History of Mathematics episodes:

• 11.29.2023

  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. On this basis many historians have concluded...

  
• 07.23.2023

  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.

  
• 10.11.2022

  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 tradition. I...

  
• 05.20.2022

  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. As our native...

  
• 02.20.2022

  “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, mathematicians elaborated numerous alternative geometries abstractly and...

  
• 11.17.2021

  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...

  
• 09.18.2021

  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, and Descartes to reduce all mechanical phenomena to contact...

  
• 07.10.2021

  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 learning. Some thought the...

  
• 05.10.2021

  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.” Others got into a feud about...

  
• 03.10.2021

  “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 construction, the...

  
• 01.20.2021

  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...

  
• 12.10.2020

  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. Although whether these postulates correspond...

  
• 11.03.2020

  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, the subtlety of defining fundamental concepts such...

  
• 10.04.2020

  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. Their philosophies can be seen as rival interpretations...

  
• 09.08.2020

  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, and what should be its...

  
• 07.30.2020

  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. Euclid’s proof of...

  
• 06.21.2020

  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.

  
• 05.15.2020

  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 explorative construction...

  
• 03.29.2020

  Societal role of geometry in early civilisations

  In ancient Mesopotamia and Egypt, mathematics meant law and order. Specialised mathematical technocrats were deployed to settle conflicts regarding taxes, trade contracts, and inheritance. Mathematics enabled states to develop civil branches of government instead of relying on force and violence....

  
• 02.16.2020

  Why the Greeks?

  The Greek islands were geographically predisposed to democracy. The ritualised, antagonistic debates of parliaments and law courts were then generalised to all philosophical domains, creating a unique intellectual climate that put a premium on adversarialism and pure reason. This style...

  
• 01.11.2020

  The mathematicians’ view of Galileo

  What did 17th-century mathematicians such as Newton and Huygens think of Galileo? Not very highly, it turns out. I summarise my case against Galileo using their perspectives and a mathematical lens more generally.

  
• 12.03.2019

  Historiography of Galileo’s relation to antiquity and middle ages

  Our picture of Greek antiquity is distorted. Only a fraction of the masterpieces of antiquity have survived. Decisions on what to preserve were made by in ages of vastly inferior intellectual levels. Aristotelian philosophy is more accessible for mediocre minds...

  
• 10.28.2019

  More things Galileo didn’t do first

  What was Galileo’s great innovation in science? To give practical experience more authority than philosophical systems? To insist on mechanical as opposed to teleological or supernatural explanations of natural phenomena? To take mathematical physics as our best window into the...

  
• 09.21.2019

  Galileo was the first to … what exactly?

  Was Galileo “the father of modern science” because he was the first to unite mathematics and physics? Or the first to base science on data and experiments? No. Galileo was not the first to do any of these things, despite...

  
• 08.15.2019

  Galileo and the Church

  Galileo’s sentencing by the Inquisition was avoidable. The Church had no interest in prosecuting mathematical astronomers, but since Galileo had so little to contribute in that domain he foolishly got himself involved with Biblical interpretation. His scriptural interpretations not only...

  
• 07.07.2019

  Galileo’s theory of comets is hot air

  Galileo thought comets were an atmospheric phenomenon, not physical bodies in outer space. How could he be so wrong when all his colleagues got it right? Perhaps because his theory was a convenient excuse for not doing any mathematical...

  
• 06.02.2019

  Phases of Venus

  Telescopic observations of Venus provided evidence for the Copernican view of the solar system. But was Galileo the first to see this, as he claims? Or did he steal the idea from a colleague and lie about having made the...

  
• 05.04.2019

  Blemished sun

  Galileo thought sunspots were one of the three best arguments for heliocentrism. He was wrong.

  
• 04.06.2019

  The telescope

  The telescope offered a shortcut to stardom for Galileo. We offer some fun cynical twists on the standard story.

  
• 03.09.2019

  Heliocentrism before the telescope

  Galileo is credited with defeating Ptolemaic earth-centered astronomy, but most mathematical astronomers had already abandoned this theory long before Galileo.

  
• 02.11.2019

  Heliocentrism in antiquity

  Two thousand years before Galileo, Greek astronomers argued that the heavenly bodies revolve around the sun. Their reasoning involved sophisticated mathematics and sound physical considerations.

  
• 01.18.2019

  Galileo’s theory of tides

  Galileo dismissed the notion that the moon influences the tides as “childish” and “occult.” Instead he argued that tides are a kind of sloshing due to the motion of the earth. This very poor theory is inconsistent with several of...

  
• 12.27.2018

  Why Galileo is like Nostradamus

  Galileo committed scores of errors in his physics. These are bad in themselves and also undermine Galileo’s claim to credit for the things he did get right. Transcript Nostradamus published a famous book of prophesies in 1555. Some people like...

  
• 12.10.2018

  Galileo’s errors on projectile motion and inertia

  Galileo gets credit he does not deserve for the parabolic nature of projectile motion, the law of inertia, and the “Galilean” principle of relativity. In reality, his treatments of all of these matters were riddled with errors and fundamental misunderstandings....

  
• 11.29.2018

  The case against Galileo on the law of fall

  Galileo is praised for his work on falling bodies, but his arguments were dishonest and his trifling discoveries were not new.

  
• 11.21.2018

  Galilean science in antiquity?

  Ancient Greek scientists studied the dynamics of falling bodies. Were “Galileo’s” discoveries anticipated in these treatises that have since been lost? This question leads to a bigger one regarding relativism versus universalism in the history of thought.

  
• 11.21.2018

  Mathematics versus philosophy, then and now

  Divergent interpretations of Galileo’s alleged greatness cut across disciplinary divides: mathematics versus philosophy, science versus humanities. Understanding Galileo means dealing with these fundamental tensions.

  
• 11.21.2018

  Galileo bad, Archimedes good

  Galileo's bumbling attempts at determining the area of the cycloid suggests a radical new interpretation of his scientific opus. Archimedes's work on floating bodies is an example of excellent Greek science that has not been sufficiently appreciated.