Opinionated History of Mathematics
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 i
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 dog
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.
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 constructi...
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,
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.
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,
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.
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,
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.