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diffxxJun 2024

I happen to be reading Poincare's Science and Hypothesis right now and he introduced a way of defining the square root of two that I found enlightening. In less articulate form: consider two sets, one of which contains all numbers whose square is less than two and one of which contains all numbers whose square is greater than two. The square root of two is then the symbolic name for the element that divides those two sets.

Poincare says it better in the book though.

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photochemsynFeb 2023

GH Hardy (1940) discusses this whole issue in a readable manner:

https://archive.org/details/AMathematiciansApology-G.h.Hardy...

He discusses two relatively simple proofs, the first being Euclid's proof of the infinity of the primes, the second being Pythagoras's proof of the irrationality of the square root of two:

> "Euclid’s theorem tells us that we have a good supply of material for the construction of a coherent arithmetic of the integers. Pythagoras’s theorem and its extensions tell us that, when we have constructed this arithmetic, it will not prove sufficient for our needs, since there will be many magnitudes which obtrude themselves upon our attention and which it will be unable to measure: the diagonal of the square is merely the most obvious example. The profound importance of this discovery was recognized at once by the Greek mathematicians. They had begun by assuming (in accordance, I suppose, with the ‘natural’ dictates of ‘common sense’) that all magnitudes of the same kind are commensurable, that any two lengths, for example, are multiples of some common unit, and they had constructed a theory of proportion based on this assumption."

> "Pythagoras’s discovery exposed the unsoundness of this foundation, and led to the construction of the much more profound theory of Eudoxus which is set out in the fifth book of the Elements, and which is regarded by many modern mathematicians as the finest achievement of Greek mathematics. The theory is astonishingly modern in spirit, and may be regarded as the beginning of the modern theory of irrational number, which has revolutionized mathematical analysis and had much influence on recent philosophy."

Another good discussion is Poincare's "Science and Hypothesis" (1902) in which he asserts that one defining feature of all mathematics is the self-consistency of arguments, which is the freedom from contradiction (a view all philosophy would hopefully adopt, vs. say, the expediency of political behavior and nation-state propaganda).

https://archive.org/details/scienceandhypoth00poinuoft/page/...

> "To sum up, the mind has the faculty of creating symbols, and it is thus that it has constructed the mathematical continuum, which is only a particular system of symbols. The only limit to its power is the necessity of avoiding all contradiction; but the mind only makes use of it when experiment gives a reason for it."

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russdillSep 2022

If you read the literature of the time, the issue was considered as largely understood. I would highly recommended science and hypothesis by Poincare (1905). You'll see a lot of concepts and equations that you might have assumed came from special relativity or even general relativity.

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rramadassMay 2026

You jest but are right.

There is nothing new in the article and has already been covered well by some of the greatest Scientists/Mathematicians. We must be careful that articles/papers like these are not used by the anti-scientific crowd to promote their talking points and agendas.

Notably, Henri Poincare (https://en.wikipedia.org/wiki/Henri_Poincar%C3%A9 and https://henripoincarepapers.univ-nantes.fr/en/) wrote three philosophy of science books; viz. 1) Science and Hypothesis 2) The Value of Science and 3) Science and Method.

These were published together under the apt title, The Foundations of Science which is available here - https://www.gutenberg.org/files/39713/39713-h/39713-h.htm and here (ebook versions) - https://archive.org/details/foundationsscie01poingoog

Details of the works;

1) Science and Hypothesis (1902) - https://en.wikipedia.org/wiki/Science_and_Hypothesis

2) The Value of Science (1904) - https://en.wikipedia.org/wiki/The_Value_of_Science

3) Science and Method (1908) - pdf at https://henripoincarepapers.univ-nantes.fr/chp/hp-pdf/hp1914... At the minimum read this completely.

See also;

a) History of Scientific Method - https://en.wikipedia.org/wiki/History_of_scientific_method

b) Scientific Method - https://en.wikipedia.org/wiki/Scientific_method

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ParacompactFeb 2026

I am not a scientific historian, or even a physicist, but IMO relativity has a weak case for being a completely novel discovery. Critique of absolute time and space of Newtonian physics was already well underway, and much of the methodology for exploring this relativity (by way of gyroscopes, inertial reference frames, and synchronized mechanical clocks) were already in parlance. Many of the phenomena that relativity would later explain under a consistent framework already had independent quasi-explanations hinting at the more universal theory. Poincare probably came the closest to unifying everything before Einstein:

> In 1902, Henri Poincaré published a collection of essays titled Science and Hypothesis, which included: detailed philosophical discussions on the relativity of space and time; the conventionality of distant simultaneity; the conjecture that a violation of the relativity principle can never be detected; the possible non-existence of the aether, together with some arguments supporting the aether; and many remarks on non-Euclidean vs. Euclidean geometry.

https://en.wikipedia.org/wiki/History_of_special_relativity

Now, if I had to pick a major idea that seemed to drop fully-formed from the mind of a genius with little precedent to have guided him, I might personally point to Galois theory (https://en.wikipedia.org/wiki/Galois_theory). (Ironically, though, I'm not as familiar with the mathematical history of that time and I may be totally wrong!)

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bcravenNov 2024

My understanding was that it was generally agreed to be related to concurrent mathematical breakthroughs in higher-dimensional space:

_"Picasso was particularly struck by Poincaré's advice on how to view the fourth dimension, which artists considered another spatial dimension. If you could transport yourself into it, you would see every perspective of a scene at once. But how to project these perspectives on to canvas? Poincaré's suggestion in Science and Hypothesis was to do so one at a time, showing each in succession. Picasso disagreed. He wanted to depict them all at once."_

https://www.theguardian.com/science/blog/2012/jul/17/henri-p...

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