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 to a “collapsible” compass in Proposition 2. Furthermore, why did Euclid feel the need to postulate that “all right angles are equal”? Perhaps in order to rule out non-flat surfaces such as cones.

Created equal: Euclid’s Postulates 1-4

History of mathematics research with iconoclastic madcap twists

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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 … Continue reading Did Copernicus steal ideas from Islamic astronomers?

Did Copernicus steal ideas from Islamic astronomers?

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.

Operational Einstein: constructivist principles of special relativity

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” … Continue reading Review of Netz’s New History of Greek Mathematics

Review of Netz’s New History of Greek Mathematics

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 may be a precondition for any linguistic or spatial thought to be possible at all, as Chomsky said for language and Kant for geometry. Just as language learning requires singling out, from all the sounds in the environment, only the linguistic ones, so Poincaré articulated criteria for what parts of all sensory data should be regarded as pertaining to geometry.

The “universal grammar” of space: what geometry is innate?

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.

Created equal: Euclid’s Postulates 1-4

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