Neo-Pythagoreanism

Here is the exchange between my friend who is interested in Duhem:

this comes from a recent article my "Duhem" Google alert found for me:

Babette Babich, “Heidegger’s Jews: Inclusion/Exclusion and Heidegger’s Anti-Semitism,” Journal of the British Society for Phenomenology 47, no. 2 (April 2, 2016): 133–56, doi:10.1080/00071773.2016.1139927.

Babich, as fn. #62 says, is a student of Lonergan.

Here's where he discusses Duhem, ending with an interesting observation. I haven't read Duhem's La science allemande in its entirety; just the part I sent you regarding Einstein and the "geometric/'mathematical' mind."

By contrast, especially for those of us up on our Laruelle or our Stiegler or Meillassoux or our Simondon, and even in conjunction with the present theme, thinking of identities and differences, thinking of Heidegger and his Jews in a French context, who reads Pierre Duhem, author of the posthumous German Science?⁵⁰ Have we today – even those of us interested, as I am interested, in the history and philosophy and sociology of science – any political geography of theory in science in connection to or with hermeneutic reflection? In connection to Heidegger, ah, yes, but not with respect to questioning science much less technology? We leave to those in power their game plan, and we do so without remainder. Critical theory has thus managed not to be critical for years.

Where Duhem in 1916 criticizes the German turn of mind as it finds expression in Gustav Kirchoff, a theorist of mathematical physics, we could find Heidegger's words along with Duhem's critique: “We can and will posit [poser] …  Wir können und wollen setzen … ”⁵¹ Note that this stipulative posing is by no means limited to a dogmatic and axiomatic controversy. The mathematician David Hilbert made this the watchword of the so-called Göttingen programme, which project included Husserl.⁵² As Duhem continues to refute Heinrich Hertz's explicitly deductive construction of mechanics,⁵³ the problem is not that the postulate is arbitrary but rather that it is, out of context and history, thereby articulated, “imperiously”: “Sic volo, sic jubeo, sit pro ratione voluntas. [I will it thus, I order it thus; let my will stand in the place of reason.]”⁵⁴ What Duhem ultimately sought, fierce as his gainsaying was, was only the inclusion of specifically French science—this would be the torch later taken up by Bachelard and Canguilhelm and today, if less and less, Serres—finally admitted to the table along with German science: “Scientia germanica ancilla scientiae gallicae.”⁵⁵

In the published version we can read Duhem citing Nietzsche's contemporary and fellow philologist, Hermann Diels: “the German is, here and now, on this earth, the sanctuary in which the principle of order takes refuge.”⁵⁶ Yet it is Duhem's extended citation of Wilhelm Ostwald that is arguably the most disturbing, even using the language of a “great secret” with respect to the German:

Germany wants to organize Europe which, until now, has not been organized. I shall now explain to you the great secret of Germany. We, or perhaps rather the German race, have discovered the factor of organization. Other people still live under the regimes of individualism, when we are under that of organization.⁵⁷

When Duhem asks “[w]as Scholasticism not essentially, as German science is, a work of the mathematical mind”⁵⁸ he can seem to approximate Heidegger's standpoint in his Beiträge with respect to what Heidegger names Machenshaft, where Heidegger writes in GA 95 (Überlegungen VIII, 5) of the Black Notebooks currently under discussion. “One of the most secret forms of the gigantic, and perhaps the oldest, is the persistent skillfulness in calculating, pushing, and intermingling through which the worldlessness of Jewry is grounded.”⁵⁹ Or else and still more troublingly, when Heidegger writes:

the temporary increase in the power of Jewry has its ground in the fact that the metaphysics of the West, especially in its modern development, served as the point of attachment for the diffusion of an otherwise empty rationality and calculative skill, which in this way lodged itself in the “spirit” without ever being able to grasp the concealed domains of decision on its own. The more original and inceptive the coming decisions and questions become, the more inaccessible will they remain to this “race.”⁶⁰

This last is only a prelude to the most infamous of these quotes:

That in the age of machination, race is elevated to the explicit and specially erected “principle” of history (or just of historiology) is not the arbitrary stipulation of “doctrinaires” but a consequence of the power of machination, which must cast down beings, in all their regions, into planned calculation.⁶¹

It can be argued that what Duhem calls “German science” corresponds to what would come to be called “Jewish science”.⁶²

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52 I discuss this with attention to the time that was the first few decades of the twentieth century in the philosophy of science (and mathematics) in Babich, “Early Continental Philosophy of Science.

53 “Let us agree that this point – which is itself nothing but an algebraic expression, only a world of geometric consonance take to designate an ensemble of n numbers – changes, from one instant to another, by an algebraic formula. From this convention, so perfectly algebraic in nature, so completely arbitrary in appearance, we deduce, with perfect rigor, the consequences that calculation can draw from it, and we say that we are setting forth mechanics.” Duhem, “Some Reflections on German Science”, p. 93

54 Ibid.

55 Ibid., p. 112.

56 Duhem, “German Science and German Virtues”, here p. 122.

57 Ibid.

58 Ibid., p. 123

59 Heidegger, Überlegungen VIII, 5, GA 95, p. 97.

60 Heidegger, GA 96, pp. 46 (from Überlegungen XII, 24).

61 Heidegger, Ibid., p. 38.

62 That argument can be made, but for his part, Duhem is talking about “scholasticism”, that is what my old Jesuit teacher, the Canadian Thomist, Bernard Lonergan, author of the conspicuously named Method in Theology (1972) and Insight: A Study of Human Understanding (1957), with its famous listings of points to the seemingly nth degree, no mere sic et non, took the mid-twentieth century to an extraordinary pitch well beyond the tradition of generalized empirical method of “transcendental Thomism” inaugurated by the Belgian Jesuit philosopher, Joseph Maréchal. Indeed, Maréchal was probably one of the reasons Lonergan was able to answer my questions regarding the intersection of mysticism and empiricism as well as he did. Maréchal's initial main works included: Le point de départ de la métaphysique: leçons sur le développement historique et théorique du problème de la connaissance, 5 vols (Bruges-Louvain, 1922–47) and Études sur le psychologie des mystiques, 2 vols (1926, 1937).

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German/Jewish science (my attempt at a compact definition): a fabricated (magical) new “physical” science derived from hypothetical equations—e.g. relativistic dynamics and its geometric implications dictating the replacement of a planar space with a gravitational field by a curved spacetime without gravitational field. The transition from the former to the latter is essentially magical, thanks to the concoction of highly elegant algebraic tools. It would therefore be worth investigating how the actual magic, in the truly occult sense of the term, has impacted “Jewish science” and its use of mathematics (as the intellectual Talmudic culture is riddled with magic practices related to gematric ideas).

Also, as I read a little bit of Duhem’s writings and get a sense of his Ampèrian affinities, I’m wondering whether he was aware of (and, if so, what he thought of) the tensor-based reformulation of electrodynamics (relativistically rewriting Maxwell’s equations to account for the electromagnetic potential in terms of R⁴ using an electromagnetic field tensor and a current tensor). Electrodynamics assuming GR has always bothered me a great deal because it essentially unifies Einsteinian gravitation (its spacetime continuum) and electromagnetism by way of geometry, not physics (as though the physics of E&M was ultimately dependent upon and constrained by the non-physical relativistic dogma of gravitation). 

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Yes, that was fascinating. And the author quoted Duhem's German. Duhem published in French, English, and German, and knew Latin and Greek as well as reading comprehension in Italian! He was a true polymath and polyglot.

"Ampèrian affinities":
Duhem certainly admired Ampère's experimental and theoretical genius, but he disagreed with Ampère's Newtonian inductivism, which held that "phénomènes électrodynamiques" are "uniquement déduite de l’expérience," as Ampère subtitled his famous work. Duhem discusses this in his Théorie physique ch. 6,

• §4 (PDF p. 154): a critique of Newton himself ("Critique de la méthode newtonienne. - Premier exemple : La Mécanique céleste")
• §5 (PDF p. 158): a critique of Ampère ("Critique de la méthode newtonienne (suite ). – Second exemple : L’Électrodynamique").

(Duhem was very good at making everyone on all sides of a debate very uncomfortable. ☺)

Duhem was familiar with Riemann's mathematics. We know this based on a citation he made to a paper by the Italian mathematician Enrico Betti. Duhem's 2nd doctoral dissertation (the accepted one) was, after all, in the mathematics department; his 1st (the rejected one) was in the physics department.

By "tensor-based reformulation of electrodynamics," are you referring to Maxwell's quaternion way of writing things? For example, in his Treatise on Electricity & Magnetism (vol. 2), p. 232-233, where he introduces the displacement current, he wrote the Maxwell-Ampère Law
$$\nabla \times \vec{B} = \mu_{0} \vec{ J} + \mu_{0} \varepsilon_{0} \frac{{\partial \vec{E}}}{\partial t$$as
$$4\pi \mathfrak{C} = V \nabla \mathfrak{H$$
where
$${\mathfrak{C}} = \Re + \dot{\mathfrak{D}}$$i.e., (true current) = (conduction current) + (displacement current);
$\mathfrak{H}$ is the magnetic force;
$V\nabla$ is the vector part of $\nabla$ (i.e., curl).

Regarding "occult" and "gematric [geometric?] ideas" in physics:
    I like Steiner (2009)'s term "Pythagorean analogies," i.e., "analogies inexpressible in any other language but that of pure mathematics" (Karam 2014); he cites Maxwell's displacement current as the first example of a "Pythagorean analogy" in physics.

Here are Duhem's views of quaternions and vector analysis (Théorie physique ch. 4, §6 "L’École anglaise et la Physique mathématique," PDF p. 62-63):

Mais chez les Anglais seuls l’amplitude d’esprit se trouve d’une manière fréquente, habituelle, endémique ; aussi est-ce seulement parmi les hommes de science anglais que les Algèbres symboliques, le calcul des quaternions, la vector-analysis, sont usuels ; la plupart des traités anglais se servent de ces langages complexes et abrégés. Ces langages, les mathématiciens français ou allemands ne les apprennent pas volontiers ; ils n’arrivent jamais à les parler couramment ni surtout à penser directement sous les formes qui les composent ; pour suivre un calcul mené selon la méthode des quaternions ou de la vector-analysis, il leur en faut faire la version en Algèbre classique. Un des mathématiciens français qui avaient le plus profondément étudié les diverses espèces de calculs symboliques, Paul Morin, me disait un jour : « Je ne suis jamais sûr d’un résultat obtenu par la méthode des quaternions avant de l’avoir retrouvé par notre vieille Algèbre cartésienne. »

Also, you would be very interested in the

Notice sur les Titres et Travaux scientifiques de Pierre Duhem rédigée par lui-même lors de sa candidature à l'Académie des sciences (mai 1913)

While Duhem wrote most of it—summarizing all his scientific, philosophical, and historical researches—, Jordan wrote the biography section, Hadamard wrote the section on the mathematical aspects of Duhem's works, and Darbon (whom you may not have heard of) wrote the section on Duhem's history of physics.

By the way, some considered Duhem "anti-Semitic" because of his stance on the Dreyfus affair, yet he was close friends with the Jew Hadamard, who held a very high opinion of Duhem.

Also, Duhem's influence has been vast, across many fields. For example, the economist Schumpeter, in his preface to Fr. Dempsey, S.J.'s erudite defense of the moderns' understanding of interest and the medievals' arguments against usury, Interest & Usury, mentions how Fr. Dempsey did for economics what Duhem did for physics; they both showed the medievals' contributions to their respective modern disciplines.

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Thanks for this rich piece of Duhemian insights and many references! Quite dense, owing to the vast extent (“across many fields”) of Duhem’s multi-layer thought and writings.

Yes Maxwell’s equations can be reformulated using matrix operators to represent quaternions. But, in field theories, quaternions are broader algebraic tools than tensors and need not include any “relativistic” alteration. Thus “Maxwell’s quaternion way of writing things” is not exactly the same as a tensor-based reformulation of electrodynamics. The tensor version of Maxwell’s equations (deriving E&M from the deformation of R⁴ geometry) describes the relationship between the electromagnetic potential A^µ (which, from the perspective of quaternion operators, would be defined as a quadri-vector potential), the electromagnetic field strength tensor F^µv (when F^µv = ω^µv), and the current tensor j^µ, yielding the “homogenous” form of Maxwell’s equations:

∂_kF_µv + ∂_µF_vk + ∂_vF_kµ = 0.

∂²A^µ = j^µ ,

where ∂² = ∂₀² - ∂₁²- ∂₂² - ∂₃² = c^-2∂_t² - ∇_x²

“Gematric”, referring (adjectively) to the Jewish gematria and its many occult misuses of mathematics and numbers.

Fascinating quote pertaining to “Duhem’s views of quaternions and vector analysis”! Besides Cauchy and his stress tensor, I cannot think of many French mathematicians and natural philosophers with a taste for the kind of algebraic operations used in quaternion, vector, and tensor analyses.

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Oh, I see. You were referring to relativity theory's tensorial "simplification" of Maxwell's equations. To my knowledge, Duhem never wrote about that (at least not directly, by writing relativity's "'homogenous' form of Maxwell’s equations").

Re: "Duhem’s multi-layer thought and writings":
    Duhem even spoke about Loti, Corneille, Shakespeare, and Dickens in his Théorie physique. Duhem contrasts Corneille with Shakespeare (∵ he considers both as not strictly having an esprit de géométrie) and Loti with Dickens (∵ both are prime examples of an esprit de géométrie). I read Tale of Two Cities because I was curious if Dickens indeed has an "English mind" (esprit de géométrie), and he certainly does; I wasn't that impressed with Tale of Two Cities because it didn't have much of an "Ariadne's thread" (coherent idea/theme) running through it. It was a disconnected smorgasbord of events and myriads of characters. As Hertz said, Maxwell's theory is nothing more than Maxwell's equations. Maxwell did not even derive these equations from a single principle, like an energy law, which was customary to do in E&M in the era between Ampére and Maxwell; thus, Maxwell's theory is an example par excellence of the English/German/geometrical mind, juggling many disconnected ideas around—which Poincaré said, in that quote I sent you awhile back [translated on p. 8 of this], makes French minds ill-at-ease when reading Maxwell for the first time.
    Have you read any Corneille? I know he wrote Le Cid; have you seen/read that? Does it exemplify the French esprit de finesse?

Duhem certainly is not opposed to analogies in physics. Classification is impossible without the ability to form analogies, and Duhem defines physical theory as a classification of experimental laws (not a classification of equations!). Duhem explicitly mentions "analogie" in Physique du croyant p. 146 ff. (on the analogy between cosmology [natural philosophy] and physical theory; Duhem essentially proves Aristotle Physica 191a7-8: "The underlying nature is known by analogy."), which you may have already read. He describes very well what Fr. Wallace, O.P., says is the "teaching that is distinctive of Thomism," i.e., that "analogical middle terms are sufficient for a valid demonstration" (cf. this). This is vital for there to be "mixed sciences" or scientia media, where minor and major premises are taken from distinct fields, like mathematics and physics, with distinct principles of their own. Fr. William A. Wallace, O.P., who pioneered research into Galileo's logical treatises, describes this very well in the best logic-of-science work I've ever read: The Modeling of Nature (if Duhem wrote a logical work, which I wish he did!, it would probably be similar to Fr. Wallace's).

Thus, what I think best describes "German" or "Jewish science" is not that it uses analogy, which all physical theory does, but that it's Neo-Pythagorean, inverting the 1st (physical) and 2nd (mathematical) degrees of abstraction. Duhem, where he mentions Einstein in the La science allemande, makes it clear that one cannot define time from a mathematical equation as Einstein does. The inversion of the first two degrees of abstraction has become so extreme that Max Tegmark, who is a hardcore Pythagorean, even wrote a paper on the "Mathematical Universe Hypothesis," i.e., that the universe is mathematics! Pythagoreans appear to be the first gematrists.

I was pleased to see Steiner (2009) quote Peirce regarding analogies between physical theories (p. 52fn9):

These universal super-laws were, to Peirce's thinking, the key to the formal mathematical analogies we see between laws—such as the inverse square laws in gravity and electricity—analogies that demand explanation (7.509-7.511). But Peirce looked to these super-laws also to explain, not only the mathematical form of laws, but even the specific values of the constants (like the gravitational constant) appearing in them.

This reminds me of what St. Thomas said is impossible in that Super Iob quote I sent you, where he says some things cannot have a natural explanation but are up to the will of God. A "physical" theory being the analogy between two mathematical laws is not a physical theory, but a mathematical one, at best, and a confusing of mathematics with the will of God, at worst.

This reminds me: I need to read the article ["De Analogia secundum Doctrinam Aristotelico-Thomisticam"] by Fr. Ramírez, O.P., that formed the foundation of his multi-volume De Analogia. Fr. Rimírez is the master of analogy, and I'm curious what he has to say about Duhem claim that (p. 147):

…si l'on prononce à cet endroit les mots de preuve par analogie, il convient d'en fixer exactement le sens et de ne point confondre une telle preuve avec une véritable démonstration logique. Une analogie se sent ; elle ne se conclut pas ; elle ne s'impose pas à l'esprit de tout le poids du principe de contradiction. Là où un penseur voit une analogie, un autre, plus vivement frappé par les contrastes des termes à comparer que par leurs ressemblances, peut fort bien voir une opposition ; pour amener celui-ci à changer sa négation en affirmation, celui-là ne saurait user de la force irrésistible du syllogisme…

How is "preuve par analogie" not "une véritable démonstration logique" thet "ne se conclut pas" and "ne s'impose pas à l'esprit de tout le poids du principe de contradiction"?

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The formalism of quaternions was very much Maxwellian in spirit and has actually been extended in Germano-Anglo-American electromagnetic field theory on the basis of the tensor field equations of Einstein. But it is the introduction of the electromagnetic tensor field F_µv, combined with the spin connection vector ω^a_b (the electrodynamics simplification of this amounts to equating F and ω by retranslating the latter into a rotation tensor, ω_µv) that makes up for “relativity theory’s tensorial "simplification" of Maxwell’s equations”, which consists in the merging of electrodynamics with GR I was referring to (and wondering whether Duhem had had any thought on this “tensorization” of E&M, which is all the rage today among pan-relativists).

About tragedians and the difference between “esprit de finesses” and “esprit de géométrie”, Duhem was probably keen on his contrasting assessment of Corneille and Shakespeare on the one hand and Loti Dickens on the other. Corneille, as Molière, also was a comedy writer and therefore, to my view, does exemplify something of “un esprit de finesse”, since finesse was rather characteristic of the kind of spirit 17th century French political/social satires would famously convey through penetrating comedies (hardly the under-the-belt level of our contemporary late wisecracking shows on T.V.).     

Regarding the centrality of analogies to the life of the created intellect, you correctly write: “Classification is impossible without the ability to form analogies”, and logically justify the validity of Duhem’s definition of “physical theory as a classification of experimental laws”, namely by way of analogous abstraction (which is what modeling physical data in the logical formal of a physical theory really amounts to). In a broader (cross-field) sense, I would define analogicity (ἀνα-λογία, meaning quite literally the logic of comparative relation between that which is lower to that which is higher) as a critical way of both intuitively and intellectually seeing the similitude of the invisible (the higher) mirrored in the visible (the lower). Both the Hebrew word for intellectus (בִינָה, “understanding” as the attitude of the intellect seeing in between two things, meaning re-cognizing the like features by which two essentially distinct things can coherently be related) and the Aramaic word for “comparison” (ܒ݁ܡܰܬ݂ܠܶܐ ,מתלא) clearly suggest that an analogy is actually more than simply a ratio-nal proportion (in the arithmetic, Aristotelian sense). In revealed anthropology, it appears (my theory) that the created intellect (whose life is intellectus) is constitutionally analogic (together in inner structure and cognitive motion).

I did notice Duhem’s mention of analogy in Physique du croyant.  The distinction he makes (referring to your final question) between “preuve par analogie” and “une véritable démonstration logique” does not imply that the first is less intellectually powerful and epistemologically meaningful than the second. In fact, a mere logical demonstration, however valid, may not have the same epistemological value as a proof by analogy, even though the latter’s proof value is technically more limited than a syllogistic demonstration. What this means is that analogical knowledge is not reducible to logical validity. Thus a little like he did in La science allemande—first lesson (Les Sciences de Raisonnement)—when distinguishing between axioms and theorems (which the following sentence on p. 6 summarizes like an aphorism: “Les principes se sentent, les propositions se concluent…”), Duhem is essentially right to say on p. 147 of Physique du croyant: “Une analogie se sent ; elle ne se conclut pas…”      

The confusion of degrees of abstraction you talk about is indeed critical! My sense is that it is typically indulged in because there is actually more to numbers and their multifaceted relations and properties than their simply being abstract entities bereft of positive existence outside the mind (entia rationis).¹ However, no one really knows how much more, and what the true nature of this “more” is. That is the reason why mathematical physics can really lead to ontological problems (as the nature of the 2nd degree of abstraction is so evasive), but without possibly providing a solution. If you remember, it was the sense of my comparative (analogical) “syllogism” used here to extend Gödel’s incompleteness results from mathematics to all-encompassing Neo-Pythagorean physical theories whose ultimate Galilean assumption is that “the universe is mathematics.”

“Duhem, where he mentions Einstein in the La science allemande, makes it clear that one cannot define time from a mathematical equation as Einstein does.”

Yes, and it is very significant that Newton did not include “time” in his descriptive “universal super-law” of gravitation, while Einstein did. The Neo-Pythagorean thinking undergirding GR and its inclusion of the time dimension times the square root of - 1 was never meant to account for empirical results (contrary to the regular claims appealing to the “countless corroborations” of the curved geometry of Einsteinian gravitational field). GR was intended to provide a mathematical way out of the contradiction between the instantaneous Newtonian gravitational field and the new principles couched in the Einsteinian formulas pertaining to spacetime in SR (implying action at a distance).

I read Fr. Ramírez’s The authority of St. Thomas, not his De Analogia.