Opinionated History of Mathematics Intellectual Mathematics

 Mathematics

Cracking tales of historical mathematics and its interplay with science, philosophy, and culture. Revisionist history galore. Contrarian takes on received wisdom. Implications for teaching. Informed by current scholarship. By Dr Viktor Blåsjö.

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 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
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
Prooforiented 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 17thcentury 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.