MATHEMATICAL PHYSICS VS PHILOSOPHY: Hegel, Pythagorean triples, Spinors and Clifford Algebras – Daniel Parrochia, 2020.

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It is well known that Hegel had a bad opinion of mathematics. Even if, over time, and under the pressure of facts (notably the expansion of differential calculus and Gauss’s arithmetic research) this opinion has changed, it remains that his initial view was negative. This has never been more clearly expressed than in the famous text of the preface to the Phenomenology of Spirit (1807) which takes for example the famous demonstration by Euclid of the Pythagorean theorem.”

Considering this demonstration, Hegel states the following thesis:

1. The process of mathematical proof does not belong to the ob-ject(*); it is a function that takes place outside the matter in hand.

(*) The English word ‘object’ is the translation of the German word Gegenstand, which literally means ‘what is posed in front’. This is why the French translator of the Phenomenology of Spirit B. Bourgeois suggests writing ‘object’ with a hyphen, in order to recall that this word ‘designates the content that the spirit, splitting up inside its primary unity (the soul) opposes objects to itself, to become properly consciousness’. (…)

2. In mathematics, construction and proof contain, no doubt, true propositions, but the content, for Hegel, is ‘false’. (…) <Thus we find negativity of content coming in here too, a negativity which would have to be called falsity, just as much as in the case of the movement of the notion where thoughts that are taken to be fixed pass away and disappear.>¹.”

¹ Tradução minha abaixo.

Assim, encontramos a negatividade do conteúdo sobrevindo nesta proposição, uma negatividade que deverá ser chamada falsidade, bem como no caso do movimento da noção em que pensamentos que são tomados como fixos passam e desaparecem.”

(continuação da tese hegeliana)

3. The real defect of this kind of knowledge affects its process of knowing as much as its material. As to that process, one does not see any necessity in the construction. An external purpose controls it. Concerning the material, it only consists of space and numerical units (das Eins)¹.”

¹ Literalmente “os Uns”. Em alemão a pronúncia é idêntica a “daseins”, vir-a-ser no plural, devires (existências ou seres no sentido mais dinâmico).

4. For all that, philosophy has nothing to do with mathematics. (…) <does not attain to essential opposition or unlikeness; and hence involves no transition of one opposite element into its other, no qualitative, immanent movement, no self-movement.> [H.]

5. (…) <It does not consider, for example, the relation of line to surface, and when it compares the diameter of a circle with its circumference, it runs up against their incommensurability, i.e. a relation in terms of the notion, an infinite element, that escapes mathematical determination.>

6. Even applied mathematics does not take in account true concrete realities.” (Aplica relações conceituais – meramente abstratas – expressas em fórmulas)

7. (…) The abstract or unreal is not [philosophy’s] element and content, but the real, what is self-establishing, has life within itself, existence in its very notion.” (O todo do movimento, do processo, constitui a positividade) “This movement includes, therefore, within it the negative factor as well, the element which would be named falsity if it could be considered one from which we had to abstract. The element that disappears has rather to be looked at as itself essential, not in the sense of being something fixed, that has to be cut off from truth and allowed to lie outside it, heaven knows where” “Appearance is the process of arising into being and passing away again, a process that itself does not arise and does not pass away, but is per se, and constitutes reality and the life-movement of truth. <The truth is thus the bacchanalian revel, where not a member is sober>

A verdade é, assim, o êxtase bacanaliano, em que nenhum membro é sóbrio”

8. In consequence, mathematics cannot be a useful model for philosophy. <It is not difficult to see that the method of propounding a proposition, producing reasons for it and then refuting its opposite by reasons too, is not the form in which truth can appear. (…) Hence it is peculiar to mathematics and must be left to mathematics, which, as already indicated, takes for its principle the relation of quantity, a relation alien to the notion, and gets its material from lifeless space, and the equally lifeless numerical unit.>

9. Generally speaking, Hegel protests against any schematizing formalism. However, it will allow himself to make use of triplicity <now that the triplicity, adopted in the system of Kant… has been raised to its significance as an absolute method> so that true form is thereby set up in its true content, and the conception of science has come to light.”

9. Genericamente falando, H. protesta contra qualquer formalismo esquematizante. E no entanto, esse mesmo formalismo esquematizante facultará a H. o uso da triplicidade <agora que a triplicidade, adotada no sistema de Kant …. foi elevada a sua significância como um método absoluto>, [!] de modo que a forma verdadeira é doravante estabelecida em seu verdadeiro conteúdo, e o conceito de ciência veio à luz.”

Hegel’s criticism against mathematical thought, which was already beginning to meet limits in its time, is no longer in season today (see Larvor 1999, 24). But it has, in fact, never really been admissible.” carece de fundamentação

Some of the Euclid’s proofs of the famous proposition I. 47 in the Elements (see [Euclid 56], 349) are classified as:

1. Proofs by rearrangement (see, for example, Heath’s proof as reported in [Euclid 56], 354-355 or in [Benson 99], 172-173)).

2. Proofs by dissection without rearrangement (like Einstein’s proof (see [Schroeder 12], 3-4)).

3. Proofs using similar triangles (already known in the Antiquity).”

Algumas das provas de Euclides da famosa proposição I.47 nos Elementos são classificadas como:

1. Provas por rearranjo (vd., p.ex., a prova de Heath como relatada em (…) Benson 1999, 172-3).

2. Provas por dissecção sem rearranjo (como a prova de Einstein, vd. Schroeder 1212, 3-4).

3. Provas utilizando triângulos análogos (já conhecidas na Antiguidade).”

In all these proofs, the initial triangle is either divided into other triangles, or inserted into a more complex figure in which it disappears or, let’s say, only occupies an inessential place.”

We know that there are in fact hundreds of proofs of Pythagoras’ theorem, not to say thousands. In his famous book The Pythagorean proposition, Elisha Scoot Loomis presents a collection of 370 proofs, grouped into the 4 following categories: Algebraic (109 proofs), Geometric (255), Quaternionic (4); and those based on mass and velocity, Dynamic (2). This author even asserts that the number of algebraic proofs is limitless – as is also the number of geometric proofs (see Loomis 1968, viii).

For many of these proofs, Hegel’s reasoning does not hold water.”

For example, in a certain number of ‘algebraic’ proofs, the triangle is not dismembered, but multiplied.”


As we can see, the theorem can be proved algebraically using 4 copies of a right triangle with sides a; b and c, arranged inside a square with side c (…)




c² = a² + b².

This proof (or similar proof) would have already been known from the Hindu mathematician Bhaskara (12th century) and would not be much different from much older proof, which can be found in the Chinese classic Zhoubi Suanjing (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven), which gives a reasoning for the (3, 4, 5) triangle. In China, it is called the ‘Gougu theorem’

But we can also make use of certain advances in mathematics, which have occurred since antiquity, for example, differential calculus, moreover known from Hegel.”

The triangle ABC is a right triangle, as shown in the upper part of the diagram, with BC the hypotenuse. At the same time the triangle lengths are measured as shown, with the hypotenuse of length y, the side AC of length x and the side AB of length a, as seen above.

If x is increased by a small amount dx by extending the side AC slightly to D, then y also increases by dy. These form 2 sides of a triangle, CDE, which (with E chosen so CE is perpendicular to the hypotenuse) is a right triangle approximately similar to ABC. Therefore, the ratios of their sides must be the same, that is, dy/dx = x/y.” etc. etc. y² = x² + C . . . “The constant can be deduced from x = 0, then we can pose y = a and obtain the equation: y² = x² + a².

Maybe one would say that this is more of an intuitive proof than a formal one. But it can be made more rigorous if proper limits are used in place of dx and dy.” O que ele não entende é que intuitiva ou formal, ambas são apenas provas matemáticas

In all these demonstrations, the triangle is by no means dismembered and Hegelian criticism does not apply. But we can go even further by showing that in reality, what is in question behind Pythagoras’ statement refers to synthetic physico-mathematical structures much deeper than the simple figure of the triangle, which is only an appearance.”

In fact, what Hegel has not seen – maybe he could not – is that the most important in the Pythagorean formula is not the triangle in itself but the relation between the 3 quantities a; b and c, which constitute what we call now a «triple», and in our case a «Pythagorean triple».”

It is also possible that an intuitive knowledge of Pythagoras’ relationship would be much older than Chinese mathematics, since it could have its roots in ancient Mesopotamia and, beyond, in the Egypt of the pyramids. As Thom has shown, the cuneiform tablet known as Plimpton 322 from Mesopotamia enlists 15 Pythagorean triples and is dated for almost 2000 BCE. The second pyramid of Giza is based on the 3-4-5 triangle quite perfectly and was build before 2500 BCE. <It has also been argued that many megalithic constructions include Pythagorean triples> (see Kocik 2007).”

In a previous work, I quoted a lecture pronounced by Trautman 1990 in Belgium in 1987. This text explained that the Pythagorean equation, in the interpretation of Diophante, enveloped in itself an extraordinarily modern synthetic notion, the notion of spinor.”

But let us remain, for the moment, inside the Euclid’s Elements, even if the existence of Pythagorean triples, that is triples of natural numbers (a; b; c) satisfying:

a² + b² = c²;

has been known, in fact, for thousands of years.”

Spinors: excedem completamente o meu métier, então sequer me darei ao trabalho de insertar imagens!… Só vou continuar citando o que pode ser considerado “história da matemática” e compreensível para nós de humanas!

Though Cartan 1938 described this kind of structure long before, the explicit notion of ‘spinor’ appears in 1931 in Physical Review.”

* * *

So, as it appears, we are very far from the simple geometric characterization of the right triangle, to which Hegel’s understanding was limited.

But we can go further.”

* * *

In conclusion, behind Pythagoras theorem and the demonstration of the right triangle, exists a very deep rational organization with complex synthetic structures like Clifford algebras, rotations in space and spinors. Who could contest that there are here, with those structures, dialectical syntheses (geometric algebra), self-movement and ‘life’ which place mathematics far beyond the (fairly negative) view that Hegel had of it?”

Parrochia é matemático, então ele não entendeu que a “prova” hegeliana não pode ser “provada” matematicamente. O que se pode provar em matemática é conteúdo do campo matemático; o que H. objetava é que provas matemáticas tenham qualquer conotação metafísica; portanto, H. continua com razão nesse tocante (afinal é impossível “desprová-lo” não-metafisicamente). O que Parrochia chama de ‘vida’ é um estereótipo barroco-jornalístico: da perspectiva da filosofia, por mais que suas equações sejam belíssimas, continua não havendo “vida” em sua “dialética geométrica euclidiana/pós-euclidiana”, porque faltou-lhe entender o sentido da expressão viva empregada pelo filósofo alemão. “Negativo” também parece ter sido entendido por ele como “depreciativo”, “o que faz mal, o que é de qualidade baixa”. Não é esse o sentido do negativo em Hegel, mas ele deve ser buscado na lógica aristotélica do princípio de não-contradição, e guarda relação com o que a filosofia posterior denomina de “nada”, “império do niilismo”, etc., sempre em relação com o Ser.

Não sou sequer partidário de Hegel e sua teleologia hoje está morta, mas tenho de defendê-lo quando uma suposta crítica a seus postulados parte de um ponto de vista injusto pré-hegeliano mesmo. A diferença entre o filósofo de primeira linha e o mero “diletante” é que não se deve assumir a priori o significado de uma expressão (vida ou, neste caso, mais detalhadamente no que ele crítica em Hegel como sendo “o trabalho do negativo”) pelo dicionário, o que é outra ingenuidade para-além da crítica a um filósofo que ainda filosofava ‘em sistema’, e só quem leu os filósofos da linguagem do séc. XX e se deu conta da limitação da linguagem para produzir enunciados mais que arbitrários pode entender quão precária é a posição de Parrochia neste artigo (pequeno e ‘grosso’ até para a mídia artigo, em minha opinião).

Não entendo como o autor pode pensar que a matemática do século XXI refuta o texto da Fenomenologia do Espírito (que de qualquer modo já se tornou superável filosoficamente no séc. XIX). Ao pesquisar sobre as relações hegelianas com Pitágoras e as exatas, imaginei que encontraria um artigo de um matemático filósofo (ou vice-versa), mas este artigo resulta inútil aos meus propósitos de entender melhor a “fixação hegeliana” pelo triplete, ou pela tríade, pitagórico e pelo Timeu de Platão, além de seu fetiche pelo número 3 que é cerne em sua dialética (vide afirmação sobre Kant contida no início).

No fim do artigo, quase esquecido do tema da introdução, Parrochia diz que Lakatos “se tornou um hegeliano através da matemática, porém sem os erros de Hegel”. Ora, Lakatos não é um metafísico (filósofo na verdadeira acepção da palavra), apenas um epistemólogos de sua disciplina, i.e., um filósofo da matemática. Ainda assim, outra figura conhecida criticou o alcance de Lakatos, o “ídolo de Parrochia”, com propriedade:


Paul Feyerabend argued that Lakatos’s methodology was not a methodology at all, but merely ‘words that sound like the elements of a methodology’. He argued that Lakatos’s methodology was no different in practice from epistemological anarchism, Feyerabend’s own position. He wrote in Science in a Free Society (after Lakatos’s death) that:

Lakatos realized and admitted that the existing standards of rationality, standards of logic included, were too restrictive and would have hindered science had they been applied with determination. He therefore permitted the scientist to violate them (he admits that science is not <rational> in the sense of these standards). However, he demanded that research programmes show certain features in the long run — they must be progressive… I have argued that this demand no longer restricts scientific practice. Any development agrees with it.’

Lakatos and Feyerabend planned to produce a joint work in which Lakatos would develop a rationalist description of science, and Feyerabend would attack it. The correspondence between Lakatos and Feyerabend, where the two discussed the project, has since been reproduced, with commentary, by Matteo Motterlini.”

Em suma, Parrochia usou um “alvo fácil” ou um simples outsider e figura histórica como trampolim para “brilhar em seu campo”, provavelmente para conseguir mais repercussão em seu artigo; mas é sem fundamento que o faz ao buscar (ou apenas ‘maquiar’) uma pseudo-interdisciplinaridade com a filosofia. De todo modo, a sinopse do livro de Parrochia que encontrei não parece de todo mal. Aos interessados, vale a pena se arriscar:


Benson, The Moment of Proof : Mathematical Epiphanies, Oxford University Press, Oxford, 1999.

Cartan, E., Leçons sur la théorie des spineurs vol. 1 and 2, Hermann, Paris, 1938.

Kocik, J., ‘Clifford Algebras and Euclid’s Parameterization of Pythagorean Triples’, Advances in Applied Clifford Algebras 17 (2007), 71-93.

Larvor, B., ‘Lakatos’s Mathematical Hegelianism’, The Owl of Minerva Vol 31, Issue 1, 23-44, 1999.

Loomis, E.S., The Pythagorean Proposition: Its Demonstration Analyzed and Classified and Bibliography of Sources for Data of the Four Kinds of ‘Proofs’ (1940), National Council of Teachers of Mathematics, Washington, DC, 1968.

Schroeder, M.R., Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise, Courier Corporation, New York, 2012.

Thom, A., Megalithic Sites in Britain, Oxford University Press, Oxford, 1967.

Trautman, A., ‘L’échiquier spinoriel’, Bulletin de la Classe des Sciences, Académie Royale de Belgique, 6e série, tome 1, 6-9, 187-194, 1990.