Clearly, science, whose aim is to comprehend nature, must assume the comprehensibility of nature and must reason and investigate according to this presupposition until incontrovertible facts may force it to recognize its limits.
(Helmholtz 1847, p. 4)
Abstract
The relativistic revolution led to varieties of neo-Kantianism in which constitutive principles define the object of scientific knowledge in a domain-dependent and historically mutable manner. These principles are a priori insofar as they are necessary premises for the formulation of empirical laws in a given domain, but they lack the self-evidence of Kant’s a priori and they cannot be identified without prior knowledge of the theory they purport to frame. In contrast, the rationalist endeavors of a few masters of theoretical physics have led to comprehensibility conditions that are easily admitted in a given domain and yet suffice to generate the theory of this domain. The purpose of this essay is to compare these two kinds of relativized a priori, to discuss the nature of the comprehensibility conditions, and to demonstrate their effectiveness in a modular conception of physical theories.
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Notes
On the second approach, see Darrigol (2014, 2015).
Darrigol (2014).
Kant (1781).
This is indeed the judgment expressed in Cassirer (1910).
Cassirer (1910, pp. 402–405).
Cassirer (1910, pp. 186–191, 216 (ether), 241 (space and time)).
Cassirer (1910, p. 352; also p. 368): "Jedes spätere Glied der Reihe [der Gesichtspunkte] hängt mit den früheren an deren Stelle es sich setzt, notwendig zusammen, sofern es die Antwort auf eine Frage geben will, die in ihnen latent ist. Wir stehen hier vor einem ständig sich erneuernden Prozeß, der nur relative Haltpunkte kennt: und diese Haltpunkte sind es, die uns den jeweiligen Begriff der ‘Objektivät' definieren."
Cassirer (1910, p. 357): "Apriorisch können nur jene letzten logischen Invarianten heißen, die jeder Bestimmung naturgesetzlicher Zusammenhänge überhaupt zugrunde liegen"; p. 427 (projektierte Einheit) pp. 248, 351–355, 362–363 (on successive forms). The regulative character of the convergence of the forms of knowledge toward a well-defined limit is asserted in Cassirer (1921, p. 24).
Cassirer (1910, p. 355).
Cassirer (1910, pp. 356–358, 411 (logische Invariante), p. 357 (citation)).
All page numbers are from Cassirer (1921). Cassirer attempts to give finer distinctions between various principles were not too felicitous: see, e.g., his distinction between the "material" principle of the constancy of the velocity of light and the "formal" principle of relativity, and his assertion that the former principle becomes invalid in general relativity (Cassirer 1921, p. 38).
Cassirer (1921, p. 42).
Schlick (1918).
Reichenbach (1920, p. 40): "Wir konstatieren die Merkwürdigkeit, dass die definierte Seite die Einzeldinge der undefinierten Seite erst bestimmt, und dass umgekehrt die undefinierte Seite die Ordnung der definierten Seite vorschreibt"; p. 43: "Eindeutigkeit heißt für die Erkenntiszuordnung, daß eine physikalische Zustandsgröße bei ihrer Bestimmung aus verschiedenen Erfahrungsdaten durch dieselbe Messungszahl wiedergegeben ist."
Reichenbach (1920, pp. 51–52).
Reichenbach (1920, p. 86): "So ist es offenbar nicht in dem Charakter der Wirklichkeit begründet, daß wir sie durch Koordinaten beschreiben, sondern dies ist die subjektive Form, die es unserer Vernunft erst möglich macht, die Beschreibung zu vollziehen."
Reichenbach, ref. 22, p. 88: "Das Verfahren, durch Transformationsformeln den objektiven Sinn einer physikalischen Aussage von der Subjektiven Form der Beschreibung zu eliminieren, ist, indem es indirekt diese subjektive Form charakterisiert, an Stelle der Kantischen Analyse der Vernunft getreten. Es ist allerdings ein sehr viel komplizierteres Verfahren als Kants Versuch einer direkten Formulierung, und die Kantische Kategorientafel muß neben dem modernen invarianten-theoretischen Verfahren primitiv Erscheinen."
Cassirer to Reichenbach, 7 July 1820, doc. 285, DVD in Cassirer (2009): "Unsere Gesichtspunkte sind verwandt – decken sich aber, so viel ich bis jetzt ersehen kann, gerade nicht mit Bezug auf die Bestimmung des Begriffs der Apriorität und mit Bezug auf die Interpretation der Kantischen Lehre, die Sie meiner Ansicht nach zu psychologisch sehen u. daher in einen zu scharfen Gegensatz zu ihrer ,wissenschaftsanalytischen’ Betrachtung rücken. Der streng ,transzendental’ verstandene Kant steht dieser Auffassung glaube ich viel näher, als es bei Ihnen erscheint."
Friedman (2001, pp. 64–65 (on Cassirer), 32 (on Carnap), 19–21 (on Kuhn), 40–41 (anti-Quine); pp. 20, 37–40, 76–80 (constitutive principles)).
Friedman (2001, pp. 46, 63–64, 66).
This understanding (Friedman 2001, p. 79) is contrary to common usage, according to which the equivalence principle and the geodetic principle are two separate principles.
Friedman (2001, pp. 77 (mechanics), 79–80 (relativity)). Samaroo (2015) refuses to regard the mathematical background as constitutive and retains only Friedman's "coordinating principles." In his view, a properly Kantian constitutive principle constitutes or interprets a theoretical concept by expressing a criterion for its application. In Friedman's view, the mathematical background is constitutive insofar as it is also needed to express the properly empirical laws.
This statement contradicts the now common view that the three laws of Newtonian mechanics together define the way we can apply mechanical concepts and therefore cannot be tested independently of a given law of force (such as the law of universal gravitation). As will appear in Sect. 2 of this article, this view is not necessary because empirically established modules (geometry, kinematics, statics, etc.) may permit the testing of individual laws of a theory without implying its entire structure.
Friedman (2009).
Ryckman (2010).
As was mentioned above, in most circumstances Friedman's "equivalence principle" truly is the geodetic principle.
Friedman (2012, p. 48): "I now think … that this notion of distinctively constitutive principles is too thin, in so far as it does not attribute to what is given in sensibility a sufficiently rich and sufficiently independent a priori structure."
Friedman (2012, pp. 48–49).
Friedman (2010, p. 698).
On the status of the second law, cf. Darrigol (2014, pp. 23, 43–44). Again (see note 41), the second law of mechanics becomes an empirical law in a proper modular framework. Newton himself states that the three laws of motion (as the law of gravitation) are inductive generalizations (cf. Darrigol 2014, pp. 5–6). This view is not incompatible with the laws of motion being a precondition for formulating the law of universal gravitation.
Goethe to Zelter, cited in Cassirer (1921, pp. 30–31): "Die größte Kunst in Lehr- und Weltleben besteht darin, das Problem in ein Postulat zu verwandeln, damit kommt man durch."
Friedman (2001, p. 23) for the light postulate and the needed meta-framework. A similar strategy could be used to distinguish between Einstein's general relativity and the equivalent flat-spacetime version by Suraj Gupta and Richard Feynman.
According to this geodetic principle, point-like particles follow the geodesics of space–time in the absence of non-gravitational interactions.
Cf. Darrigol (2008).
Jürgen Renn's notion of "integration of knowledge" through "mental models" implicitly involves such anticipation of modular structure: see Büttner et al. (2003).
On Boltzmann's pluralism, cf. de Courtenay (1999).
Cf. Smith and Wise (1989).
Galison (1997, Chap. 9).
Darrigol (2008, pp. 215–222).
Typically, comprehensibility arguments only determine theoretical frameworks in the sense given at the beginning of this essay. Some additional empirical laws are needed to apply these frameworks to specific concrete systems. In conformity with the physicists' practice, in the following I use "theory" instead of "theoretical framework."
The two following subsections are taken from Darrigol (2015).
Helmholtz (1868). The following is a very free reconstruction of Helmholtz's argument.
I use "locally" and "local" to indicate approximate validity in small domains of space, not in the modern mathematical sense.
See Darrigol (2014, Chap. 4).
Lagrange (1798). In a concrete connected system on earth, the \( {\mathbf{F}}_{\alpha } \)'s would include the weight of the various components of the system, so that Lagrange's construction can only be an imaginary one (the more so because W itself is subjected to gravitation).
See Darrigol (2014).
Cf. Mittelstaedt (2011, p. x): "stepwise reduction of prejudices."
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Acknowledgements
I thank João Príncipe and Tom Ryckman for instructive conversations, and two anonymous reviewers for useful comments.
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Darrigol, O. Constitutive principles versus comprehensibility conditions in post-Kantian physics. Synthese 197, 4571–4616 (2020). https://doi.org/10.1007/s11229-018-01948-2
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DOI: https://doi.org/10.1007/s11229-018-01948-2