Opinionated History of Mathematics

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 … <a href="https://intellectualmathematics.com/blog/death-of-archimedes/" class="more-link">Continue reading <span class="screen-reader-text">Death of Archimedes</span></a>

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 … <a href="https://intellectualmathematics.com/blog/torricellis-trumpet-is-not-counterintuitive/" class="more-link">Continue reading <span class="screen-reader-text">Torricelli’s trumpet is not counterintuitive</span></a>

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 … <a href="https://intellectualmathematics.com/blog/did-copernicus-steal-ideas-from-islamic-astronomers/" class="more-link">Continue reading <span class="screen-reader-text">Did Copernicus steal ideas from Islamic astronomers?</span></a>

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, we are chained to a wall! Aaah! This is going to mess up our geometry big time. Remember what … <a href="https://intellectualmathematics.com/blog/operational-einstein-constructivist-principles-of-special-relativity/" class="more-link">Continue reading <span class="screen-reader-text">Operational Einstein: constructivist principles of special relativity</span></a>

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” … <a href="https://intellectualmathematics.com/blog/review-of-netzs-new-history-of-greek-mathematics/" class="more-link">Continue reading <span class="screen-reader-text">Review of Netz’s New History of Greek Mathematics</span></a>

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 … <a href="https://intellectualmathematics.com/blog/the-universal-grammar-of-space-what-geometry-is-innate/" class="more-link">Continue reading <span class="screen-reader-text">The “universal grammar” of space: what geometry is innate?</span></a>

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. Transcript The mathematician has only one nightmare: to … <a href="https://intellectualmathematics.com/blog/repugnant-to-the-nature-of-a-straight-line-non-euclidean-geometry/" class="more-link">Continue reading <span class="screen-reader-text">“Repugnant to the nature of a straight line”: Non-Euclidean geometry</span></a>

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 … <a href="https://intellectualmathematics.com/blog/rationalism-2-0-kants-philosophy-of-geometry/" class="more-link">Continue reading <span class="screen-reader-text">Rationalism 2.0: Kant’s philosophy of geometry</span></a>

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 … <a href="https://intellectualmathematics.com/blog/rationalism-versus-empiricism/" class="more-link">Continue reading <span class="screen-reader-text">Rationalism versus empiricism</span></a>

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, … <a href="https://intellectualmathematics.com/blog/cultural-reception-of-geometry-in-early-modern-europe/" class="more-link">Continue reading <span class="screen-reader-text">Cultural reception of geometry in early modern Europe</span></a>

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 … <a href="https://intellectualmathematics.com/blog/makers-knowledge-early-modern-philosophical-interpretations-of-geometry/" class="more-link">Continue reading <span class="screen-reader-text">Maker’s knowledge: early modern philosophical interpretations of geometry</span></a>

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 … <a href="https://intellectualmathematics.com/blog/let-it-have-been-drawn-the-role-of-diagrams-in-geometry/" class="more-link">Continue reading <span class="screen-reader-text">“Let it have been drawn”: the role of diagrams in geometry</span></a>

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 … <a href="https://intellectualmathematics.com/blog/why-construct/" class="more-link">Continue reading <span class="screen-reader-text">Why construct?</span></a>

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 … <a href="https://intellectualmathematics.com/blog/created-equal-euclids-postulates-1-4/" class="more-link">Continue reading <span class="screen-reader-text">Created equal: Euclid’s Postulates 1-4</span></a>

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 … <a href="https://intellectualmathematics.com/blog/that-which-has-no-part-euclids-definitions/" class="more-link">Continue reading <span class="screen-reader-text">That which has no part: Euclid’s definitions</span></a>

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. Transcript What kinds of axioms do we want in our geometry? How … <a href="https://intellectualmathematics.com/blog/what-makes-a-good-axiom/" class="more-link">Continue reading <span class="screen-reader-text">What makes a good axiom?</span></a>

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 … <a href="https://intellectualmathematics.com/blog/consequentia-mirabilis-the-dream-of-reduction-to-logic/" class="more-link">Continue reading <span class="screen-reader-text">Consequentia mirabilis: the dream of reduction to logic</span></a>

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 … <a href="https://intellectualmathematics.com/blog/read-euclid-backwards-history-and-purpose-of-pythagorean-theorem/" class="more-link">Continue reading <span class="screen-reader-text">Read Euclid backwards: history and purpose of Pythagorean Theorem</span></a>

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. Transcript Greek geometry is oral geometry. Mathematicians memorised theorems the way bards memorised poems. Euclid’s Elements was almost like a song book or … <a href="https://intellectualmathematics.com/blog/singing-euclid-the-oral-character-of-greek-geometry/" class="more-link">Continue reading <span class="screen-reader-text">Singing Euclid: the oral character of Greek geometry</span></a>

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 … <a href="https://intellectualmathematics.com/blog/first-proofs-thales-and-the-beginnings-of-geometry/" class="more-link">Continue reading <span class="screen-reader-text">First proofs: Thales and the beginnings of geometry</span></a>

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. Mathematics enabled complex economies in which people could count on technically competent administration and an … <a href="https://intellectualmathematics.com/blog/societal-role-of-geometry-in-early-civilisations/" class="more-link">Continue reading <span class="screen-reader-text">Societal role of geometry in early civilisations</span></a>

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 of thought proved ideal for mathematics. Transcript Why the Greeks, of all people? Why did … <a href="https://intellectualmathematics.com/blog/why-the-greeks/" class="more-link">Continue reading <span class="screen-reader-text">Why the Greeks?</span></a>

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. Transcript I’m going to conclude my case against Galileo with this final episode on this subject. Here’s a little anecdote I found … <a href="https://intellectualmathematics.com/blog/the-mathematicians-view-of-galileo/" class="more-link">Continue reading <span class="screen-reader-text">The mathematicians’ view of Galileo</span></a>

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 than advanced mathematics and science. Hence this simpler part of Greek intellectual achievement was eagerly … <a href="https://intellectualmathematics.com/blog/historiography-of-galileos-relation-to-antiquity-and-middle-ages/" class="more-link">Continue reading <span class="screen-reader-text">Historiography of Galileo’s relation to antiquity and middle ages</span></a>

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 fundamental nature of reality as opposed to just a computational tool for a small set … <a href="https://intellectualmathematics.com/blog/more-things-galileo-didnt-do-first/" class="more-link">Continue reading <span class="screen-reader-text">More things Galileo didn’t do first</span></a>

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 often being erroneously credited with these innovations. Transcript Galileo is “the father of modern science,” … <a href="https://intellectualmathematics.com/blog/galileo-was-the-first-to-what-exactly/" class="more-link">Continue reading <span class="screen-reader-text">Galileo was the first to … what exactly?</span></a>

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 got him into hot water: they are also scientifically unsound and blatantly inconsistent with his … <a href="https://intellectualmathematics.com/blog/galileo-and-the-church/" class="more-link">Continue reading <span class="screen-reader-text">Galileo and the Church</span></a>

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 astronomy of comets. We also discuss his unsavoury ways of dealing with data in the case … <a href="https://intellectualmathematics.com/blog/galileos-theory-of-comets-is-hot-air/" class="more-link">Continue reading <span class="screen-reader-text">Galileo’s theory of comets is hot air</span></a>

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 observations months before? Transcript Galileo and the phases of Venus: it’s a plot that mirrors … <a href="https://intellectualmathematics.com/blog/phases-of-venus/" class="more-link">Continue reading <span class="screen-reader-text">Phases of Venus</span></a>

Galileo thought sunspots were one of the three best arguments for heliocentrism. He was wrong. Transcript The early days of telescopic astronomy were exhilarating. Listen to this anecdote by Kepler. He is writing in 1610, right after the appearance of Galileo’s first telescope reports. Here’s what Kepler says: “My friend the Baron Wakher von Wachenfels … <a href="https://intellectualmathematics.com/blog/blemished-sun/" class="more-link">Continue reading <span class="screen-reader-text">Blemished sun</span></a>

The telescope offered a shortcut to stardom for Galileo. We offer some fun cynical twists on the standard story. Transcript The year is 1609. What a time to be alive. In London you can go to the theatre and catch the fresh new play Macbeth. In Amsterdam you can make a quick buck trading in … <a href="https://intellectualmathematics.com/blog/the-telescope/" class="more-link">Continue reading <span class="screen-reader-text">The telescope</span></a>

Galileo is credited with defeating Ptolemaic earth-centered astronomy, but most mathematical astronomers had already abandoned this theory long before Galileo. Transcript Does the earth move around the sun, or is it the other way around? Copernicus worked out the right answer long before Galileo was even born, as did the best Greeks mathematicians thousands of … <a href="https://intellectualmathematics.com/blog/heliocentrism-before-the-telescope/" class="more-link">Continue reading <span class="screen-reader-text">Heliocentrism before the telescope</span></a>

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. Transcript Is the earth the center of the universe? Or does it orbit around the sun? The Greeks had some interesting ideas about this. For example, the earth is basically … <a href="https://intellectualmathematics.com/blog/heliocentrism-in-antiquity/" class="more-link">Continue reading <span class="screen-reader-text">Heliocentrism in antiquity</span></a>

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 his own scientific principles. Transcript At the end of his famous Dialogue, Galileo lists what … <a href="https://intellectualmathematics.com/blog/galileos-theory-of-tides/" class="more-link">Continue reading <span class="screen-reader-text">Galileo’s theory of tides</span></a>

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 to praise him for having predicted the future. Allegedly he foresaw all kinds of things … <a href="https://intellectualmathematics.com/blog/why-galileo-is-like-nostradamus/" class="more-link">Continue reading <span class="screen-reader-text">Why Galileo is like Nostradamus</span></a>

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. Transcript Pick up a rock and throw it in front of you. It makes a … <a href="https://intellectualmathematics.com/blog/galileos-errors-on-projectile-motion-and-inertia/" class="more-link">Continue reading <span class="screen-reader-text">Galileo’s errors on projectile motion and inertia</span></a>

Galileo is praised for his work on falling bodies, but his arguments were dishonest and his trifling discoveries were not new. Transcript In 1971, Apollo 15 astronauts conducted a famous experiment on the moon. Here’s a bit of the original recording: “In my left hand I have a feather. In my right hand a hammer. … <a href="https://intellectualmathematics.com/blog/the-case-against-galileo-on-the-law-of-fall/" class="more-link">Continue reading <span class="screen-reader-text">The case against Galileo on the law of fall</span></a>

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. Transcript Quiz! Who said the following: “The study of mechanics is eagerly pursued by all those interested in … <a href="https://intellectualmathematics.com/blog/galilean-science-in-antiquity/" class="more-link">Continue reading <span class="screen-reader-text">Galilean science in antiquity?</span></a>

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. Transcript Those who can’t do, teach. I’m sure you have heard this saying. It sums up Galileo’s role in the history of scientific thought, in my opinion. Galileo’s books are “Science … <a href="https://intellectualmathematics.com/blog/mathematics-versus-philosophy-then-and-now/" class="more-link">Continue reading <span class="screen-reader-text">Mathematics versus philosophy, then and now</span></a>

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. Transcript Galileo is the most overrated figure in the history of science. That’s the thesis of Season 1 … <a href="https://intellectualmathematics.com/blog/galileo-bad-archimedes-good/" class="more-link">Continue reading <span class="screen-reader-text">Galileo bad, Archimedes good</span></a>