### Opinionated History of Mathematics

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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.

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 and play, or the realisation that “obvious” things can be demonstrated using formal definitions and proof by contradiction.

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. Mathematics enabled complex economies in which people could count on technically competent administration and an objective justice system.

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 of thought proved ideal for mathematics.

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.

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 than advanced mathematics and science. Hence this simpler part of Greek intellectual achievement was eagerly pursued, while technical works were neglected and perished. The alleged predominance of an Aristotelian worldview in antiquity is an illusion created by this distortion of sources. The “continuity thesis” that paints 17th-century science as building on medieval thought is doubly mistaken, as it misconstrues both ancient science and Galileo’s role in the scientific revolution.

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 fundamental nature of reality as opposed to just a computational tool for a small set of technical problems? No, none of the above. All of these things had been old hat for thousands of years.

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 often being erroneously credited with these innovations.

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 got him into hot water: they are also scientifically unsound and blatantly inconsistent with his own science.

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 astronomy of comets. We also discuss his unsavoury ways of dealing with data in the case of double stars and the rings of Saturn.

#### Latest Episodes

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.

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 and play, or the realisation that “obvious” things can be demonstrated using formal definitions and proof by contradiction.

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. Mathematics enabled complex economies in which people could count on technically competent administration and an objective justice system.

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 of thought proved ideal for mathematics.

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.

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 than advanced mathematics and science. Hence this simpler part of Greek intellectual achievement was eagerly pursued, while technical works were neglected and perished. The alleged predominance of an Aristotelian worldview in antiquity is an illusion created by this distortion of sources. The “continuity thesis” that paints 17th-century science as building on medieval thought is doubly mistaken, as it misconstrues both ancient science and Galileo’s role in the scientific revolution.

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 fundamental nature of reality as opposed to just a computational tool for a small set of technical problems? No, none of the above. All of these things had been old hat for thousands of years.

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 often being erroneously credited with these innovations.

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 got him into hot water: they are also scientifically unsound and blatantly inconsistent with his own science.

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 astronomy of comets. We also discuss his unsavoury ways of dealing with data in the case of double stars and the rings of Saturn.