Hidden philosophy of the Pythagorean theorem

In Plato’s dialogue, the Timaeus, we are presented with the theory that the cosmos is constructed out of right triangles.

This proposal Timaeus makes after reminding his audience [49Bff] that earlier theories that posited “water” (proposed by Thales), or “air” (proposed by Anaximenes), or “fire” (proposed by Heraclitus) as the original stuff from which the whole cosmos was created ran into an objection: if our world is full of these divergent appearances, how could we identify any one of these candidates as the basic stuff? For if there is fire at the stove, liquid in my cup, breathable invisible air, and temples made of hard stone — and they are all basically only one fundamental stuff — how are we to decide among them which is most basic?

A cosmos of geometry

However, if the basic underlying unity out of which the cosmos is made turns out to be right triangles, then proposing this underlying structure — i.e., the structure of fire, earth, air, and water — might overcome that objection. Here is what Timaeus proposes:

“In the first place, then, it is of course obvious to anyone, that fire, earth, water, and air are bodies; and all bodies have volume. Volume, moreover, must be bounded by surface, and every surface that is rectilinear is composed of triangles. Now all triangles are derived from two [i.e., scalene and isosceles], each having one right angle and the other angles acute… This we assume as the first beginning of fire and the other bodies, following the account that combines likelihood with necessity…” [Plato. Timaeus 53Cff]

A little later in that dialogue, Timaeus proposes further that from the right triangles, scalene and isosceles, the elements are built — we might call them molecules. If we place on a flat surface equilateral triangles, equilateral rectangles (i.e., squares), equilateral pentagons, and so on, and then determine which combinations “fold-up,” Plato shows us the discovery of the five regular solids — sometimes called the Platonic solids.

Three, four, and five equilateral triangles will fold up, and so will three squares and three pentagons.

If the combination of figures around a point sum to four right angles or more, they will not fold up. For the time being, I will leave off the dodecahedron (or combination of three pentagons that makes the “whole” into which the elements fit) to focus on the four elements: tetrahedron (fire), octahedron (air), icosahedron (water), and hexahedron (earth).

Everything is a right triangle

Now, to elaborate on the argument [53C], I propose to show using diagrams how the right triangle is the fundamental geometrical figure.

All figures can be dissected into triangles. (This is known to contemporary mathematicians as tessellation, or tiling, with triangles.)

Inside every species of triangle — equilateral, isosceles, scalene — there are two right triangles. We can see this by dropping a perpendicular from the vertex to the opposite side.

Inside every right triangle — if you divide from the right angle — we discover two similar right triangles, ad infinitum. Triangles are similar when they are the same shape but different size.

And thus, we arrive at Timaeus’ proposal that the right triangle is the fundamental geometrical figure, in its two species, scalene and isosceles, that contain within themselves an endless dissection into similar right triangles.

Now, no one can propose that the cosmos is made out of right triangles without a proof — a compelling line of reasoning — to show that the right triangle is the fundamental geometrical figure. Timaeus comes from Locri, southern Italy, a region where Pythagoras emigrated and Empedocles and Alcmaon lived. The Pythagoreans are a likely source of inspiration in this passage but not the other two. What proof known at this time showed that it was the right triangle? Could it have been the Pythagorean theorem?

Pythagorean theorem goes beyond squares

We now know that there are more than 400 different proofs of the famous theorem. Does one of them show that the right triangle is the basic geometrical figure? Be sure, it could not be a² + b² = c² because this is algebra, and the Greeks did not have algebra! A more promising source — the proof by similar right triangles — is the proof preserved at VI.31.

Notice that there are no figures at all on the sides of the right triangle. (In the above figure, the right angle is at “A.”) What the diagram shows is that inside every right triangle are two similar right triangles, forever divided.

Today, the Pythagorean theorem is taught using squares.

But, the Pythagorean theorem has nothing to do with squares! Squares are only a special case. The theorem holds for all figures similar in shape and proportionately drawn.

So, why the emphasis on squares? Because in the ancient Greek world proportional-scaling was hard to produce exactly and hard to confirm, and the confirmation had to come empirically. But squares eliminate the question of proportional scaling.

Pythagoras and the philosophy of cosmology

We have an ancient report that upon his proof, Pythagoras made a great ritual sacrifice, perhaps one hundred oxen. What precisely was his discovery that merited such an enormous gesture?

Could this review help us to begin to understand the metaphysical meaning of the hypotenuse theorem — namely, that what was being celebrated was not merely the proof that the area of the square on the hypotenuse of a right triangle was equal to the sum of the areas of the squares on the other two sides, but moreover, was the proof that the fundamental figure out of which the whole cosmos was constructed was the right triangle?

Prof. Robert Hahn has broad interests in the history of ancient and modern astronomy and physics, ancient technologies, the contributions of ancient Egypt and monumental architecture to early Greek philosophy and cosmology, and ancient mathematics and geometry of Egypt and Greece. Every year, he gives “Ancient Legacies” traveling seminars to Greece, Turkey, and Egypt. His latest book is The Metaphysics of the Pythagorean Theorem.

LINK ORIGINAL: https://bigthink.com/thinking/hidden-philosophy-pythagorean-theorem/#Echobox=1637200606

Quantum Mechanics, the Mind-Body Problem and Negative Theology

Credit: Alamy

Scientists and philosophers should keep trying to solve realitie’s deepest riddles while accepting that they are unsolvable. By John Horgan – Dec. 2015

Here’s how I distinguish science from philosophy. Science addresses questions that can be answered, potentially, through empirical investigation. Examples: What’s the best way to defeat Covid-19? What causes schizophrenia, and how should it be treated? Can nuclear power help us overcome climate change? What are the causes of war, and how can we end it?

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5 frases de Epicteto para dejar de sufrir

Epicteto nos dejó varias frases que nos permiten reflexionar sobre aquello por lo que sufrimos.

Epicteto fue un filósofo griego que durante gran parte de su vida fue esclavo en Roma. No se sabe a ciencia cierta en qué fecha fue liberado, pero sí que adquirió un gran renombre. Quizás porque, a pesar de haber sido un esclavo, se centró en la búsqueda de la felicidad y el bienestar. Por esta razón, hemos querido recoger algunas frases de Epicteto que le ayudaron en sus peores momentos y que nos pueden ayudar a nosotros.

La corriente filosófica que siguió Epicteto fue la estoica. De ahí que la felicidad de la que él hablaba se conozca como eudaimonia, la cual se alcanza a través de la realización de actividades congruentes con valores profundos y con el establecimiento de un compromiso pleno.

Veamos a continuación qué podemos extraer de sus frases para gestionar nuestro sufrimiento.

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Ser livre não é libertar-se dos outros, mas de si mesmo, de acordo com Epicteto

Buscamos a liberdade quando um cego procura suas chaves. Como resultado, não é estranho que acabemos imbuídos em uma busca malsucedida, condenada previamente ao fracasso, sem perceber que o que buscamos fora está dentro. Sem perceber que a liberdade externa não é nem mais nem menos do que a medida em que conseguimos conquistar a liberdade interna, parafraseando Gandhi.

Os estóicos podem nos dar uma lição incrível sobre liberdade, especialmente Epicteto, que foi escravo em Roma durante parte de sua vida. Para esse filósofo, a verdadeira liberdade não significa apenas livrar-se das correntes da sociedade, suas normas e pressões a que somos submetidos pelos outros, mas nos libertar das correntes que às vezes nos impomos. E esse é um conceito que agora, mais do que nunca, devemos internalizar.

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¿Qué es el estoicismo?

La filosofía de 2.000 años de antigüedad que se usa para sobrevivir al caos

En 1965 durante la guerra entre Estados Unidos y Vietnam, el piloto de la armada estadounidense James Stockdale recibió un disparo mientras volaba sobre el país enemigo.

El joven no sabía que pasaría siente años como prisionero de guerra de los vietnamitas. Y que un filósofo que había vivido en Grecia en el siglo I d.C. se convertiría en su gran maestro y amigo, ayudándole a soportar sufrimientos inimaginables.

El filósofo se llamaba Epicteto y su filosofía, el estoicismo.

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La Sabiduría de Este Filósofo Te Ayudará a Llevar una Vida Feliz


Marco Aurelio fue un Emperador Romano del año 161 al 180. Conocido como el último de los “Cinco Emperadores Buenos,” Aurelio no nació dentro del poder, más bien fue adoptado (su padre biológico murió cuando él tenía tres años de edad). Los historiadores se refieren a él como un gobernante bondadoso, quien fue increíblemente leal a su labor de servir al Imperio Romano.

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