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 though no direct evidence for such transmission has been found. Let us consider the evidence as to whether Copernicus plagiarized these Arabic sources or not. See PDF slides for figures and references ..read more

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” theory. Transcript A new book just appeared: A New History of Greek Mathematics, by Stanford Professor Reviel Netz, Cambridge University Press. Let’s do a book review. It will be a critical review. The main theme will be the science ..read more

A strangely petty review by Victor Pambuccian of my article on Greek operationalism has appeared in zbMATH. Its supposed critiques are as follows. “Although the reader is told that the concept [of operationalism] will be defined in 2.2.1, one doesn’t find a definition there (except for a quotation from P. W. Bridgman [The logic of modern physics. New York: Macmillan (1927)], which is supposed to explain the ‘core principle of operationalism’), and needs to read the entire paper for bits and pieces about what operationalism is.” In 2.2.1 I cite and discuss in some detail the definition of opera ..read more

In 2014 and 2018 I challenged the common view that Copernicus stole ideas from Maragha astronomers. My arguments were discussed by Nikfahm-Khubravan & Ragep (2019), who are believers in the “influence thesis” that I was challenging. Some people have asked me what I think of their take. Nikfahm-Khubravan & Ragep do not try to refute me on any significant point. The agree with my main point, e.g.: “as Blåsjö has recently shown, Swerdlow based his assessment on a misunderstanding of what Copernicus was saying regarding the behavior of the Mercury model” (35). In other words, they agree th ..read more

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 ar ..read more

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 claim to have proved something that later turns out to be false. There are thousands of theorems in Greek geometry, and every last one of them is correct. That’s what mathematical ..read more

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 the cost of making our perception of the world around us inextricably subjective. Kant’s theory also showed how rationalism, and philosophy generally, could be reconciled with Newtonian science, with which it had been seen as embarrassingly out of touch. In particu ..read more

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 cause of gravity and dismiss the foundational importance of constructions in geometry. Transcript Here’s a fundamental problem in the philosophy of mathematics. You can sit in an isolated room, in an arm chair, and prove th ..read more

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, while others thought creative mathematics was inherently free and liberal. Transcript Gothic architecture is known for its pointed arches. Unlike round arches like a classical Roman aqueduct for example. Those are se ..read more

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 be justified via theology. Transcript Everybody has seen the painting The School of Athens, the famous Renaissance fresco by Rafael. It shows all the great thinkers of antiquity engaged in lively intellectual activity. Plato and Aristotle are debating the re ..read more