Abstract
In the Crisis of the European Sciences Husserl raised a fascinating question, namely (broadly paraphrasing), why is it that the axioms of mathematical physics are not self-evident despite the evidence and clarity that is gained through the deductive processes that flow from them? In this chapter, I hope to illuminate Husserl’s foundational question by pursuing the idea that physics is a form of life. This idea should not be taken in a naive metaphorical sense but quite literally. The meaning of life must not be restricted to a biological definition but should be construed broadly as a manifold phenomenon appearing in historical contexts and linguistic frameworks. I will argue that nature manifests certain of her aspects to us, but that in her totality (including ourselves as observers of nature), crucially, she resists our insight. This being hidden of the totality nature, or her desire or necessity to hide herself, explains why the axioms of mathematical physics must appear to our intuition as obscure, according as Husserl noted. It is because they point us to nature as a totality, or put in another way, because nature cannot know herself in her totality. A phenomenologically oriented physics is grounded in diverse mathematical styles that evolve in history and are ultimately rooted in natural languages and in the life-worlds of the physicists. From this phenomenological viewpoint physics is not concerned with truth in the sense of a psychophysical parallelism (the conformity of mind and reality). Indeed, axioms cannot be true in this psychophysical sense given their unintelligibility and unobservability. Rather physics is a form of life coming to be in history and language.
“We are products of the past and we live immersed in the past which encompasses us. How can we move towards the new life, create new activities without getting out of the past ― without placing ourselves above it? (And how can we place ourselves above the past if we are in it and it is in us?) There is no other way out except through thought which does not break off relations with the past but rises ideally above it and converts it into knowledge… Only historical judgment liberates the spirit from the pressure of the past; (it is pure and extraneous to conflicting parties, and guarding itself against their fury, their lures, and their insidiousness,) it maintains its neutrality and seeks only to furnish light ― it alone makes possible the fixing of a practical purpose; opens a way to the development of action (and, in the process of action, to the struggle of good against bad, useful against harmful, beautiful against ugly, true against false, in a word, value against non-value).” Benedetto Croce.
“Noi siamo prodotto del passato, e viviamo immersi nel passato, che tutt’intorno ci preme. Come muovere a nuova vita, come creare la nostra nuova azione senza uscire dal passato, senza metterci di sopra di esso? E come metterci disopra del passato, se vi siamo dentro, ed esso è in noi? Non v’ha che una sola via d’uscita, quella del pensiero, che non rompe il rapporto col passato ma sovr’esso s’innalza idealmente e lo converte in conoscenza.
[…] Solo il giudizio storico, che libera lo spirito dalla stretta del passato e, puro qual è ed estraneo alle parti in contrasto, guardingo contro i loro impeti ed i loro allettamenti e le loro insidie, mantiene la sua neutralità, ed attende unicamente a fornire al luce che gli si chiede, sol esso rende possibile il formarsi del pratico proposito e apre la via allo svolgersi dell’azione e, col processo dell’azione, alle opposizioni, tra le quali questa si deve travagliare, di bene contro male, di utile contro dannoso, di bello contro brutto, di vero contro falso, del valore insomma, contro il disvalore.” —Benedetto Croce. (Croce, 1937, pp. 21, 24).
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Notes
- 1.
- 2.
I have paraphrased and interpreted Husserl’s text from the Crisis. The original is as follows. “Die Natur ist in ihrem „wahren Sein an sich” mathematisch. Von diesem An-sich bringt die Reine Mathematik der Raumzeitlichkeit eine Gesetzesschicht in apodiktischer Evidenz als unbedingt allgemein gültige, zur Erkenntnis: unmittelbar die axiomatischen Elementargesetze der apriorischen Konstruktionen, in unendlichen Mittelbarkeiten die übrigen Gesetze. Hinsichtlich der Raumzeitform der Natur besitzen wir eben das uns (wie es später heißt) „eingeborene” Vermögen, wahres Ansichsein als Sein in mathematischer Idealität (vor aller wirklichen Erfahrung) bestimmt zu erkennen. Implizite ist sie selbst uns also eingeboren. Anders steht es mit der konkreteren universalen Naturgesetzlichkeit, obwohl auch sie durch und durch mathematisch ist. Sie ist,a posteriori”, von den faktischen Erfahrungsgegebenheiten aus induktiv zugänglich. Vermeintlich voll verständlich stehen sich scharf unterschieden gegenüber: apriorische Mathematik der raumzeitlichen Gestalten und induktive — obschon reine Mathematik anwendende — Naturwissenschaft. Oder auch: Scharf unterscheidet sich das rein mathematische Verhältnis von Grund und Folge von dem des realen Grundes und der realen Folge, also dem der Naturkausalität. Und doch macht sich allmählich ein unbehagliches Gefühl der Unklarheit über das Verhältnis zwischen der Naturmathematik und der ihr doch zugehörigen Mathematik der Raumzeitform, zwischen dieser „eingeborenen” und jener nicht eingeborenen Mathematik geltend.“(Husserl, 1954, pp. 54–55).
- 3.
Many thanks to Harald Wiltsche for alerting me to the nuances of Husserl’s views on Platonism in mathematics.
- 4.
“Aber ist nicht die Natur an sich durchaus mathematisch, muß nicht auch sie als einheitliches mathematisches System gedacht werden, also wirklich darstellbar sein in einer einheitlichen Naturmathematik: eben jener, die die Naturwissenschaft immer nur sucht, sucht als umgriffen von einem der Form nach „axiomatischen” Geesetzessystem, dessen Axiomatik immer nur Hypothese ist, also nie wirklich erreichbar? Warum eigentlich nicht, warum haben wir keine Aussicht, das der Natur eigene Axiomensystem als ein solches echter apodiktisch evidenter Axiome zu entdecken? Weil uns hier faktisch das eingeborene Vermögen fehlt?” (Husserl, 1954, pp. 55–56).
- 5.
For a more technical exposition, I take the liberty of referring the reader to Palmieri, 2001.
- 6.
As is well known, the correct principle which Galileo eventually adopted is the proportionality speed and time. In uniformly accelerated fall from rest, both along a vertical path and an inclined plane, a body’s degree of speed is proportional to the time elapsed from the beginning of the fall. Galileo would not have accepted a proportionality between non-homogeneous quantities, and would have said more precisely that the ratios of the speeds are the same as the ratios of times. In effect, the document in which Galileo derived the times-squared law from the ‘erroneous’ principle communicated to Sarpi, though traditionally associated with the 1604 letter, cannot be dated with certainty. It was first published in Le opere di Galileo Galilei, Edizione Nazionale, edited by Antonio Favaro (1890–1909). I have quoted this edition as: Galilei, 1890–1909, followed by the Roman numeral of the volume and the page numbers in Arabic numerals. Cf. Galilei, 1890–1909, VIII, pp. 373–74. See Koyré, 1966, pp. 86ff., quotation from p. 96.
- 7.
By equimultiple, or equal multiple, Euclid means multiples of magnitudes according to the same multiplication factor. Cf. Heath’s comments on the long-winded philosophical debates among mathematicians concerning the obscure meaning of this definition, in Euclid, 1956, II, pp. 120–129.
- 8.
Galilei, 1974, p. 149.
- 9.
‘During the same equable motion, the space completed in a longer time is greater than the space completed in shorter time’. Galilei, 1974, p. 148.
- 10.
Drake, 1995, p. 424.
- 11.
Drake, 1995, p. 426.
- 12.
Drake, 1995, pp. 427–428.
- 13.
Cf. Galilei, 1890–1909, XI, pp. 340–343, a famous letter where Galileo expands on art, esthetics and the imitation of nature.
- 14.
Panofsky, 1954, pp. 35–36, my emphases.
- 15.
Helmholtz, 1885, p. 5.
- 16.
Helmholtz, 1877, pp. 251 ff.
- 17.
Corti, 1851.
- 18.
Helmholtz, 1885, p. 145.
- 19.
A ‘general’ solution to a special case of the anharmonic oscillator that Helmholtz had investigated was found recently. See Rand, 1990, where a discussion of the special case is presented which, however, does not clarify what is meant by ‘general’.
- 20.
Perturbation methods came of age in the nineteenth century, rising to prominence after the publication of Lagrange’s second edition of his Mėcanique Aanlitique (1814), and particularly in an effort to determine the moon’s exact orbit. “Lagrange was imagining the planet or satellite as moving at each instant in an ellipse characterized by its six orbital elements, with the elements changing from instant to instant due to perturbation. […] Two simultaneous processes had to be taken into account: the continuous change in shape and orientation of the instantaneous elliptical orbit in which the perturbed body was conceived to be traveling, and the body’s motion along this protean orbit” (Wilson, 2010, p. 17).
- 21.
Cardano, 1570, p. 131.
- 22.
“Es gibt eine Nachreife auch der festgelegten Worte. Was zur Zeit eines Autors Tendenz seiner dichterischen Sprache gewesen sein mag, kann später erledigt sein, immanente Tendenzen vermögen neu aus dem Geformten sich zu erheben. Was damals jung, kann später abgebraucht, was damals gebräuchlich, später archaisch klingen. Das Wesentliche solcher Wandlungen wie auch der ebenso ständigen des Sinnes in der Subjektivität der Nachgeborenen statt im eigensten Leben der Sprache und ihrer Werke zu suchen, hieße — zugestanden selbst den krudesten Psychologismus — Grund und Wesen einer Sache verwechseln, strenger gesagt aber, einen der gewaltigsten und fruchtbarsten historischen Prozesse aus Unkraft des Denkens leugnen. Und wollte man auch des Autors letzten Federstrich zum Gnadenstoß des Werkes machen, es würde jene tote Theorie der Übersetzung doch nicht retten. Denn wie Ton und Bedeutung der großen Dichtungen mit den Jahrhunderten sich völlig wandeln, so wandelt sich auch die Muttersprache des Übersetzers. Ja, während das Dichterwort in der seinigen überdauert, ist auch die größte Übersetzung bestimmt in das Wachstum ihrer Sprache ein-, in der erneuten unterzugehen. So weit ist sie entfernt, von zwei erstorbenen Sprachen die taube Gleichung zu sein, daß gerade unter allen Formen ihr als Eigenstes es zufällt, auf jene Nachreife des fremden Wortes, auf die Wehen des eigenen zu merken.” (Benjamin, 1972, pp. 12–13.)
- 23.
- 24.
I will loosely follow the exposition in Heisenberg, 1925.
- 25.
Aitchison, MacManus, Snyder 2004. The word ‘magical’ emphasized by Aitchison, MacManus, and Snyder is quite appropriate here, I take it literally.
- 26.
Gracia, 1984.
- 27.
“He [Heisenberg] proposed to replace the kinematical framework LN of classical mechanics by a new quantum theoretic framework, let us call it LQ, which would fulfill the five conditions of the relativistic model. Condition Hi is satisfied by that part of ordinary and scientific language LP which is neutral to the transposition from LN to LQ and includes, therefore, the language of electromagnetic theory as well as the language of the manifest image of the world. Conditions Hii and Hiii represent different aspects of Bohr’s Correspondence Principle. Condition Hiv states that LQ will contain ‘only relations between quantities which are observable in principle’ [Heisenberg 1925, p. 879]. Condition Hv implies that a semantical re-interpretation of the variables — a hermeneutical transformation — accompanies the transposition from LN to LQ.” (Heelan, 2016, p. 30).
- 28.
Aitchison, MacManus, Snyder 2004. Puzzled by the mystery of the recursive formulae for a quantum correlate of (4.8), which had been put forward by Heisenberg without any hint of the processes by which they became present to his consciousness, the authors reinvented the calculation by which, according to them, Heisenberg must have discovered them.
- 29.
Aitchison, MacManus, Snyder 2004, pp. 1372–1372. Already Patrick Heelan, in 1970, had written the same form for the quantum anharmonic oscillator by plugging the array X(n, n − α, t) into (4.8), but he did not pursue the question of how Heisenberg had arrived at the recursive formulas derived from the application of perturbation theory to the quantum case. See Heelan, 2016, pp. 15, 29–32.
- 30.
Bell & Nauenberg, 1987, p. 27.
- 31.
Bell & Nauenberg, 1987, p. 27.
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Acknowledgments
I am grateful to my colleague John Norton for many casual conversations, failed jokes, and insightful platitudes on the parasitic nature of quantum mechanics.
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Palmieri, P. (2020). Physics as a Form of Life. In: Wiltsche, H.A., Berghofer, P. (eds) Phenomenological Approaches to Physics. Synthese Library, vol 429. Springer, Cham. https://doi.org/10.1007/978-3-030-46973-3_4
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