25

By way of throat-clearing, what is Occam's razor? John Baez has a useful essay giving the history and some examples. William of Ockham's original formulation was Entities should not be multiplied unnecessarily. In other words, don't assume the existence of something unless there is good evidence for it. Again quoting Baez, "In physics we use the razor to ...


18

According to Mayo, Popper did not designate statistical tests implementing his logic of falsification, or as Hilborn and Mangel put it "Popper supplied the philosophy, and Fisher, Neyman and colleagues supplied the statistics", see references in Quinn and Keough's Experimental Design and Data Analysis for Biologists (Ch. 3). Popper viewed probability ...


17

Let's start with everyone's first go-to source: Wikipedia. Right in the long introduction at the top lies the passage Carl Friedrich Gauss (1777–1855) referred to mathematics as "the Queen of the Sciences". Benjamin Peirce (1809–1880) called mathematics "the science that draws necessary conclusions". That's a good start. Later on, we find that Many ...


15

In a word, yes, but only interpreting "western" broadly. The tradition of precise and systematic astronomical observations comes from ancient Babylonians, i.e. from Mesopotamia, and even Alexandria, where many scientists worked during Hellenistic times, is in the middle east. John Philoponus (c. 490 – 570) working there wrote things like "but this [view of ...


15

Let me clarify a couple of things. No student of Pythagoras discovered irrational numbers, although this is a common misconception, Pythagoreans and even Euclid did not associate numbers with geometric points or segments, the only numbers available were positive integers. Instead they had magnitudes of different dimensions (segments, areas, volumes), and ...


12

Eratosthenes of Cyrene did an experiment that confirmed that the Earth was roughly spherical, and estimated its circumference around 200 BC. Note that the lunar eclipse observation mentioned by Gregory Grant suggests that the Earth is round in 2-dimensions (an Earth shaped like a flat disc is consistent with this observation), but does not provide ...


12

Although many problems that we now reduce to polynomial equations were solved since time immemorial early occurences are coached in verbal and/or geometric terms, and polynomials are not treated as separate items. For early occurences of geometric problems that lead (today) to quadratic equations see The origin of quadratic equation in actual practice. The ...


12

I do not agree with the assumptions made in the title and in main text of this question, namely that Lakatos rejects the Euclidean methodology and exposition of mathematics. In the way I read it, Lakatos is putting his fingers on a rather different topic. Many people, while describing mathematics as a science, tend to confuse the Euclidean exposition with ...


11

In the 1960s the mathematical structure of Turing degrees was conjectured to be rather simple and homogeneous. This was consistent with what was known at the time. It later turned out that the opposite is true in a sense: the Turing degrees are as complicated as can be. Details in Ambos-Spies and Fejer, History of degree theory.


11

The area of knowledge separates itself from philosophy as soon as a reliable method of obtaining exact knowledge in this area is invented. Thus mathematics separated from philosophy at its very beginning. In astronomy, there was an area covered by exact knowledge (based on observations) and another, speculative part. As exact knowledge expanded, the ...


11

It is a strange idea that scientific laws can be only expressed with algebraic means. The Greek did discover several scientific laws. The oldest one is attributed to Pythagoras himself: it relates the length of the string to its pitch. This seems to be the oldest scientific law ever discovered. More laws were discovered in Hellenistic times: the law of ...


11

The idea, yes, Aryabhata speculated about something like that as early as c. 500 AD, Brahmagupta called it gurutvākarṣaṇ. So did Kepler, at about the same time as Ahmad Baba al Massufi (late 1500-s), and much less vaguely. Russo even ascribes the idea to Hipparchus (c.150 BC), although this is far fetched. Even the inverse square law for gravity predates ...


10

Air or more generally medium resistance was not yet treated as a separate effect in Aristotle's time. Nor was there a clear idea of motion in a vacuum, in fact most ancient Greek philosophers, including Aristotle, did not believe that vacuum exists. So he had to explain phenomena as they are observed, resistance and all, and without the benefit of ...


10

A person's life and behavior are always shaped by a number of factors, not (in general) just one. I think it highly unlikely that any one of the three points you bring up is responsible for Grothendieck's success. At the same time, all of them may have contributed to his life. The point that I find singular about Alexander Grothendieck is his overarching ...


10

One has to distinguish static pressure from the dynamic pressure in a flow. Static pressure was understood to some extent by Archimedes and later Hellenistic writers like Hero and Ctesibius. At least their texts that we know do not contain mistakes: all that they wrote was essentially correct. Dynamic pressure was very poorly understood before the modern ...


10

It goes back at least to Aristotle's De Anima, Book II, ch. 7-11 (these five chapters being respectively devoted to sight, hearing, smell, taste, and touch). This is perhaps where it started, since Aristotle was an incorrigible cataloguer of all human experience, be it either sensory or intellectual. That there are no more than five senses was subsequently ...


9

Some people would still call mathematics a science. (V. I. Arnold is a notable example). The distinction became commonly accepted in the first half of the 20-s century, but the process was slow and it was different in different cultures. For example, in the Soviet universities the degree in mathematics is still called "Doctor of physical and mathematical ...


9

The Assayer & Redondi's "G3" Michele Camerota's 2008 Complete Dictionary of Scientific Biography entry on Galileo describes this theory in a section entitled "Atomism and the Eucharist": Atomism and the Eucharist. In section 48 of The Assayer (1623), Galileo set forth a theory of knowledge based on a sharp distinction between “objective” and “...


8

Leibniz and Newton never thought of manifolds outside of ambient space as far as I know, Newton is credited with forging the concept of absolute space, and Leibniz with reducing it to a relational fiction in the style of Aristotle. The most influential philosopher of science in 19-th century was Kant, specifically his "Copernican revolution" of ...


8

The modern conflict is not so much between physics and philosophy, as between physics and "half" of philosophy, the continental philosophy. Scientists, and physicists in particular, see culturally relativistic attitudes of continental philosophy as leading to ignorance and outright hostility to science. Sokal and Bricmont's Fashionable Nonsense gives an idea ...


8

Problem: classical geometry is not happy with infinitesimals Newton is systematically trying to avoid basing calculus on infinitesimal geometric quantities. We can see this from how he emphasizes that his method is consistent with the "ancient" standard of rigor: To institute an Analysis after this manner in finite Quantities... is consonant to the ...


8

You do not say what field of mathematics you are working in, and perhaps there are signs of separation there. Overall however, lively interaction between mathematics and physics is alive and well. John Baez has a blog This Week's Finds in Mathematical Physics, that is full of contemporary examples of it, so does Terence Tao. Nature, a leading journal in ...


8

To not be measured is not to have any behavior. No dynamic is induced on any other system in the universe by this object (Rosen 1978). Objects without behavior do not exist. Their behavior is that which makes some perceptions objective not subjective things (Hutton 1794). What is measured but not found different is double counted; it does not exist in ...


8

It's hard to put a precise date without generating debate, but I'd put forward that Europe took the lead at some point between the early 14th to late 15th century. Several processes were in full motion around then: Medieval Scientific Dynamism Where there was indeed a dearth of books and literate scholars in the Early Middle Ages, science and technology ...


7

The Vienna circle, and more broadly logical positivism that it promoted, was the key link connecting philosophy of classical physics in 19-th century, positivism and neo-Kantianism of the Marburg school, to the modern philosophy of science post the discoveries of relativity theory and quantum mechanics. There is a direct line of tradition from Kant, Mach and ...


7

Perhaps, the most insightful analysis (possibly to this day) of indeterminism in classical mechanics and its implications was given by Joseph Boussinesq, best known for his work on solitons, in a book long essay Reconciliation of Mechanical Determinism with Moral Freedom (1878). His ideas were based on the general theory of solutions to differential ...


7

Some centuries before Mermin, Leibniz in the 17th century was seeking a solution to some of the denominational quarrels that were plaguing his generation by envisioning a calculus ratiocinator that would make it possible for the quarreling parties to "sit down and calculate". A hypothetical science he envisioned was called Mathesis Universalis and included ...


7

Mermin has a thorough analysis[1] and traces the phrase to himself in a 1989 Physics Today column [2] & makes a strong case that the numerous attributions to Feynman are mistaken. [1] Could feynman have said this? / Mermin, Physics Today, 2004 [2] What's Wrong with this Pillow? / Mermin, Physics Today, 1989


7

Gauss in Disquisitiones Arithmeticae (1799) does indeed express something close to what is now called mathematical formalism and structuralism. He writes: "What is calculated (in the sense of things already counted) are not substances (thinkable objects for themselves), but relations between two objects counted two by two... The mathematician abstracts ...


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