Abstract
It is common knowledge that, next to experimentation, mathematics is the most important pillar of modern natural science. Less well known is how strongly the mathematical ideal of knowledge shaped modern science, especially so-called ‘rational mechanics’, which was regarded by most scientists and philosophers as the foundation and backbone of all natural sciences.
The following chapter examines how this ideal of a ‘mechanical Euclideanism’, as I call it, shaped different programmes of mathematical natural philosophy in the late seventeenth and eighteenth centuries. It shows that this ideal was in opposition to the modern, hypothetico-decuctive understanding of science, and reveals how it was supported by epistemological and methodological arguments from both traditional rationalism and empiricism. The analysis of these processes is directed against an understanding of the development in question as one that was shaped primarily by Newton’s mechanics (as Ernst Mach, Thomas S. Kuhn and others claimed). Rather, it attempts to reconstruct this development as a dispute and competition between different programmes, guided by different scientific metaphysics and striving for different conceptual foundations of rational mechanics. It also tries to reveal how the attempts to integrate the achievements of these programmes into a unified formal framework of analytical mechanics alter and ultimately undermine the ideal of mechanical Euclideanism.
The first edition of ‘Between Leibniz, Newton and Kant’ already contained a slightly different version of this paper as a chapter. As the former one was quite widely received and positively evaluated by the readership, this new version could make do with some updates and a few corrections. Readers interested in a more detailed account than can be given in this outline, perhaps also in continuing the story beyond Lagrange and Kant into the nineteenth century up to Neumann and Einstein, are referred to my book Axiomatik und Empirie.
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Notes
- 1.
“[…] rational mechanics [mechanica rationalis] will be the science of motions resulting from any forces whatsoever, and of the forces required to produce any motions, accurately proposed and demonstrated. This part of mechanics, as far as it extended to the five powers which relate to manual arts, was cultivated by the ancients, who considered gravity (it not being a manual power) no otherwise than in moving weights by those powers. But I consider philosophy rather than arts and write not concerning manual but natural powers, and consider chiefly those things which relate to gravity, levity, elastic force, the resistance of fluids, and the like forces, whether attractive or impulsive; and therefore I offer this work as the mathematical principles of philosophy, for the whole burden of philosophy seems to consist in this—from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena […]” (Newton, Mathematical Principles, XVII–XVIII). In one of his early papers on the history of rational mechanics, Clifford Truesdell asked for any precursor of Newton using the term ‘rational mechanics’, and I. Bernard Cohen later put forward the same question. Alan Gabbey has shown that it was used in Goclenius’s Lexicon philosophicum Graecum and therefore “was in (probably common) use during the first decade of the seventeenth century, at the latest” (Gabbey, 309, n. 13). His argument, that Newton’s Principia “was and was not a treatise on mechanics” (ibid., 308) seems to be in line with my understanding of ‘mathematical philosophy of nature’; see Pulte, Axiomatik und Empirie.
- 2.
Concerning the foundations of mechanics, Newton’s principles (axiomata sive leges motus) and his law of gravitation should be distinguished. What was soon understood as ‘revolutionary’ (in the sense of an obvious and irreversible break with the past) was his celestial mechanics, i.e., the application of his three laws and the law of gravitation to the motion of the moon and the planets. In the last decades, however, more and more publications have shown that Newton’s three laws of motion were neither entirely new, nor understood as new by his contemporaries and his immediate successors; for an overview see Bos, Mathematics and Rational Mechanics.
- 3.
“Die Newtonschen Prinzipien sind genügend, um ohne Hinzuziehung eines neuen Prinzips jeden praktisch vorkommenden mechanischen Fall […] zu durchschauen. Wenn sich hierbei Schwierigkeiten ergeben, so sind dieselben immer nur mathematischer (formeller) und keineswegs mehr prinzipieller Natur” (Mach, Mechanik, 272). Mechanics after Newton is characterised by Mach as a deductive, formal and mathematical development on the basis of his principles (ibid., 179).
- 4.
“The Principia […] did not always prove an easy work to apply, partly because it retained some of the clumsiness inevitable in a first venture and partly because so much of its meaning was only implicit in its applications. For many terrestrial applications, in any case, an apparently unrelated set of Continental techniques seemed vastly more powerful. Therefore, from Euler and Lagrange in the eighteenth century to Hamilton, Jacobi and Hertz in the nineteenth, many of Europe’s most brilliant mathematical physicists repeatedly endeavoured to reformulate mechanical theory in an equivalent but logically and aesthetically more satisfying form. They wished, that is, to exhibit the explicit and implicit lessons of the Principia and of Continental mechanics in a logically more coherent version, one that would be at once more uniform and less equivocal in its application to the newly elaborated problems of mechanics. Similar reformulations of a paradigm have occurred repeatedly in all of the sciences, but most of them have produced more substantial changes in the paradigm than the reformulations of the Principia cited above.” (Kuhn, Structure, 33). Kuhn’s marginal note (“Principia and of Continental mechanics”) reveals the main problem which he failed to address because of the ‘Machian shaping’ of his history of mechanics.
- 5.
In this paper, I will use the term ‘scientific metaphysics’ for all assumptions which define the ‘hard core’ of a scientific research program in the sense of Lakatos. They belong to metaphysics, in so far as they are immune from empirical falsification, and they are scientific, in so far as they determine the problems, basic concepts and acceptable explanations of the science in question. Elkana, inspired by Lakatos, defines scientific metaphysics as “those untestable hypotheses which deal with the structure of the physical world and which direct scientists in their research” (Elkana, Euler and Kant, 278). The scientific metaphysics of mechanics shapes the understanding of matter and motion. It has genuinely to do with the concepts of space, time, mass and (eventually) with the concept of force and (or) energy and their mutual relations.
- 6.
See Lagrange’s letter to d’Alembert of January 27, 1778 (Lagrange, Oeuvres, XIII 336).
- 7.
See Mittelstraß, Neuzeit, esp. 302. ‘Euclidean’ and ‘synthetical’ are obviously used as synonyms; see Mittelstraß, Möglichkeit, 119 and 236 note 19. The case study ‘analytical mechanics’ is also picked up in Mittelstraß, Rationale Rekonstruktionen.
- 8.
A remarkable, though not very influential exception is Lazare Carnot; see his Principes. A detailed analysis of Carnot’s work can be found in Gillispie, Lazare Carnot Savant.
- 9.
Lakatos makes clear that the dichotomy ‘Euclidian/Empiricist’ (or later: ‘Euclidian/Quasi-empirical’) applies for whole theories, while single propositions are traditionally qualified as ‘a priori/a posteriori’ or ‘analytic/synthetic’: “[…] epistemologists were slow to notice the emergence of highly organized knowledge, and the decisive role played by the specific patterns of this organization” (ibid., 6) This holds true especially for mechanics. The traditional empirical/rationalistic dichotomy conceals the common basis of infallibility and is not very useful historiographically (ibid., 70–103).
- 10.
In case of Lagrange, the term “Rubber Euclideanism” (ibid., 7, 9) would be more appropriate; see Sect. 5.4.3 of this paper.
- 11.
For a detailed discussion see Hanson, Newton’s First Law.
- 12.
Ronald N. Giere exemplifies in ch. 3 of his Explaining Science that this classical demand of ‘metatheoretical invariance’ is not accepted by modern philosophy of science.
- 13.
The concept of force in Leibniz’s physics is analysed in some detail by Stammel, Kraftbegriff.
- 14.
“Force was an entity ontologically existent in the universe” (Westfall, Force, 87).
- 15.
It has to be kept in mind that mathematical realism in my sense only implies the ontological relevance of all concepts which are actually used in mathematical principles. This does not mean, however, that all ontologically relevant concepts enter these principles. The concept of impenetrability, for example, is ontologically relevant for all important programs of the time in question but does not (and cannot) play a role in its mathematical formulation, because it has no quantitative meaning. It is a concept which later disappears from the textbooks of mechanics, though it is still present in some textbooks of general physics in the first decades of the nineteenth century.
- 16.
This is probably the reason why it was frequently presented as a model of the ‘hypothetical-deductive’ concept of science (see, for example, Blake, Isaac Newton).
- 17.
Therefore, I can and will restrict my attention to Newton in this context: I take it for granted that mechanics in the tradition of Cartesian or Leibnizian rationalism is accepted as ‘Euclideanistic’ in the sense described above.
- 18.
Remember, for example, the instances given as ‘empirical’ support of his first law, according to which “every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it”: We find projectiles, “so as far as they are not retarded by the resistance of the air,” a rotating top which “does not cease its rotation,” and “even the greater bodies of the planets and comets” (Newton, Mathematical Principles, 13). There is obviously no observation which shows the uniformity and rectilinearity of ‘natural’ motion.
- 19.
Newton to Cotes on March 28, 1713 (Newton, Correspondence, V 396–397). Newton’s statement was provoked by an example which was used by Cotes in order to explicate his foundations of mechanics: “[…] ‘till this Objection be cleared I would not undertake to answer any one who should assert You do Hypothesim fingere […]” (ibid., 392). Though Cotes’ thought experiment is untenable (and therefore is not discussed here), it should be noted that Newton’s rejection relies on the undubitable truth and generality of his axiomata.
- 20.
See, for example, his famous letter to Bentley of December 10, 1692 (Newton, Correspondence, III 233). The law of gravitation does not, of course, belong to his axioms in a strict sense. But its certainty is vital for Newton in order to show that his celestial mechanics (presented in Book III of the Principia) can be based on a set of certain principles, i.e., his three laws of motion and the law of gravitation.
- 21.
Newton, Principia, 64. Wolfer’s German translation (“von den Mathematikern angenommen”; Newton, Mathematische Prinzipien der Naturlehre, 39) promotes ‘conventionalistic’ misinterpretations, while the brand-new translation by Volkmar Schüller (“von den Mathematikern allgemein anerkannt.”; Newton, Mathematische Prinzipien der Physik; 40) does justice to the original meaning.
- 22.
“[…] for the description of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us how to draw these lines, but requires them to be drawn, for it requires that the learner should first be taught to describe these accurately before he enters upon geometry, then it shows how by these operations problems may be solved. To describe right lines and circles are problems, but not geometrical problems. The solution of these problems is required from mechanics, and by geometry the use of them, when so solved, is shown; and it is the glory of geometry that from those few principles, brought from without, it is able to produce so many things” (Newton, Mathematical Principles, XVII).
- 23.
A label already applied to him by Jammer, Problem des Raumes, 110; see Burtt, Metaphysical Foundations and Strong, Newton’s Mathematical Way, for similar judgements.
- 24.
See Sect. 5.1.3 for the ‘non-Kuhnian’ implications of this change.
- 25.
Cassirer, Erkenntnisproblem, 472. According to Cassirer the second description was given by a historian of mathematics of his time, but he agrees with this judgement, especially “with respect to the methodological manner of the interpretation and treatment” of scientific problems.
- 26.
See Hankins, Jean d’Alembert, for a detailed discussion of d’Alembert’s Cartesian leanings.
- 27.
Lagrange, Mécanique Analytique (2nd ed.), I 243. This passage is not included in the first edition.
- 28.
To be more precise: the ‘explicit’ use. It is well known that Newton made use of calculus, but later ‘translated’ his results in a geometric language in order to facilitate the reception of his Principia.
- 29.
See Euler, Mechanik, I 49. I cannot discuss his various ‘demonstrations’ in this paper.
- 30.
See Mittelstraß, Neuzeit, esp. 301–302; see endnote 7 above, and Pulte, Axiomatik und Empirie, ch. IV.6.
- 31.
The relevance of Euler’s Découverte is underlined by Truesdell, Program and Pulte, Prinzip, esp. 151.
- 32.
Euler, Découverte, 88–89. Later he discovered that the principle of moment of momentum has to be added as a separate ‘axiom’.
- 33.
‘Basic concepts’ in this context always means ‘concepts which are used in the actual axiomatisation of mechanics’.
- 34.
For all details see Pulte, Prinzip, 150–181, esp. 176.
- 35.
As d’Alembert put it in the title of his great textbook: “Traité de Dynamique, dans lequel les loix de l’equilibre & du Mouvement des Corps sont réduites au plus petit nombre possible, & démontrèes d’une manière nouvelle, & où l’on donne un Principe général pour trouver le Mouvement de plusieurs Corps qui agissent les uns sur les autres, d’une maniére quelconque.”
- 36.
See, for example, Euler’s De la force and his Anleitung.
- 37.
“Wenn von der geistigen oder logischen Evidenz als einem Wissenschaftskriterium [klassischer Wissenschaft] gesprochen wird, so soll ein relativ neutraler und umfassender Begriff verwendet werden, da damit ein Komplex mit vielerlei Nuancen gemeint ist. Genau genommen gilt das Wort nur für die frühe Wissenschaft, später tritt mehr und mehr die Idee der logischen Struktur, schließlich der logischen Ordnung als System an seine Stelle. Im einzelnen ist dazu folgendes zu sagen: Daß Wissen im Sinne des später so verstandenen wissenschaftlichen Wissens keine unmittelbare Evidenz in sich trägt, keine unmittelbare Wahrheit in sich birgt, ist eine implizite Voraussetzung der Theorie—eigentlich bis zur Gegenwart. So stand wie über der episteme der nous, so über [der] scientia als der ‘mittelbaren’ der intellectus bzw. Die intelligentia als die eigentliche unmittelbare Einsicht der letzten Wahrheiten, der Axiome. Durch die Evidenz der Ableitung, der ‘De-duktion’—(‘Apagoge’)—wird dann die Sicherheit und Gewißheit der anderen Sätze garantiert. Die wissenschaftliche Gewißheit und insofern die Wissenschaftlichkeit liegt also nicht so sehr in der ursprünglichen Schau als der gesicherten, d.h. systematischen Ableitung. Dies wird zunächst unmittelbar gesagt und sinngemäß versucht, die entsprechenden Syllogismusstrukturen als die entsprechenden Wege zu entwickeln. In zunehmendem Maße verlagert sich dann später der Schwerpunkt; er rutscht gewissermaßen ‘abwärts’ […].” (Diemer, Begründung, 30–31).
- 38.
Note my general use of this technical term: Not all mechanics which uses calculus is called ‘analytical’ (Euler’s Mechanica, for example, is not ‘analytical’ in this sense), but only mechanics in so far as it makes use of principles, which are based on analytical principles, i.e., integral variational principles (like the principle of least action) or differential variational principles (like the principle of virtual velocities).
- 39.
See Pulte, Prinzip, 230–261, for a more detailled discussion of this development.
- 40.
See endnote 4 above; see also Sect. 5.4 for more details.
- 41.
Here I do not distinguish between Maupertuis’ and Euler’s formulations, though they do differ in various details. It is noteworthy that Euler and Maupertuis always stressed that they discovered and elaborated the same principle.
- 42.
This emerges even from the titles of some of Maupertuis’ and Euler’s essays on the principle of least action. See, for example, Maupertuis, Accord de différentes Loix de la Nature qui avoient jusqu’ici paru incompatibles and Les Lois du Mouvement et du Repos déduites d’un Principe Metaphysique and Euler, Harmonie entre les principes générales de repos et de mouvement de M. de Maupertuis.
- 43.
Letter to Euler of May 19, 1756 (Lagrange, Oeuvres, Vol. 14, 391–392).
- 44.
Euler, for example, frequently makes use of geometrical figures, even if he deals with ‘analytical mechanics’ (in the narrow sense defined in endnote 38).
- 45.
In order to avoid misunderstandings, I must emphasise that ‘instrumentalism’ here refers to philosophy of nature (I1) rather than to philosophy of science (I2): Lagrange did not base his mathematical formulation of mechanics on an analysis of the fundamental concepts of philosophy of nature like matter, force, space and time, as did Descartes, Newton, Leibniz, d’Alembert, or Euler. Instead, he chose the basic concepts and laws of his theory in a mathematically convenient manner. This is what I call instrumentalism (I1) and which is best illustrated by Lagrange’s switch described in Sect. 5.4.1. In contrast, instrumentalism with respect to philosophy of science (I2) is characterised by the view that the whole theory of mechanics or at least one of its principles is only a tool to describe and predict phenomena without having any real content itself. Lagrange certainly would not have accepted being called an instrumentalist in this second sense (see Sect. 5.4.3, but also Sect. 5.4.1). He could not, however, have refuted such an ascription: A consistent instrumentalism (I1), which is not supported by an adequate theory of representation inevitably leads to (I2). Therefore the distinction is generally unnecessary, but as Lagrange’s view is not consistent in this respect, it has to be kept in mind.
- 46.
That is the main reason why Lagrange’s mechanics was sterile in some respects: It contains no truly new principles, nor new concepts of mechanics, as Truesdell and others have justly remarked.
- 47.
The Méchanique Analitique “réunira & présentera sous un même point de vue, le différens Principes trouvés jusqu’ici pour faciliter la solution des questions de Méchanique, en montrera la liaison & la dépendance mutuelle, & mettra à portée de juger de leur justesse & de leur étendue.” (Lagrange, Méchanique Analitique, v)
- 48.
Regardless of its philosophical shortcomings, the Méchanique Analitique became for some time a model of how mathematics should be used in physics. It is its advanced mathematical and anti-metaphysical style which made his textbook attractive for working mathematicians like Fourier as well as for positivistic philosophers like Comte (see Fraser, Lagrange’s Analytical Mechanics). It was widely accepted as the best realisation of a ‘purely mathematical’ Euclideanism in physics.
- 49.
ibid., 8–12. Lagrange uses the history of mechanics partly as a substitute for missing philosophical justification.
- 50.
It was probably brought to his attention by Fourier’s Mémoire from 1798.
- 51.
From Fourier (1798), de Prony (1798), Laplace (1799), L. Carnot (1803) and Ampère (1806) to Cournot (1829), Gauss (1829), Poisson (1833), Ostrogradsky (1835, 1838), and Poinsot (1806, 1838, 1846). They aimed at an extension of Lagrange’s principle, taking into account conditions of constraints given by inequalities (Fourier, Cournot, Gauss, Ostrogradsky) and/or at its better foundation.
- 52.
In the passage above I reported upon the interpretation of Pulte, Jacobi’s Criticism, 158–160; see 160–181 for the subsequent development of analytical mechanics, especially with respect to C.G.J. Jacobi.
- 53.
See Diemer, Begründung, as well as Diemer and König, Was ist Wissenschaft?.
- 54.
Kant, Metaphysische Anfangsgründe, A XIII; see also the mottos of this paper.
- 55.
See Pulte, Axiomatik und Empirie, Ch. IV.7 for Kant’s contribution in the context of the general decline of ‘mechanical Euclidianism’.
- 56.
This gap was—curiously enough—often ignored (see, for example, Gloy, Die Kantische Theorie, Schäfer, Kants Metaphysik) or belittled. Eric Watkins shows in his The Laws of Motion in some detail that Kant’s omission was by no means unique in eighteenth-century German attempts to justify the laws of motion. Euler and others, however, tried to give a justification of the second law, and Kant’s strong orientation towards the introductory part of Newton’s Principia also urges toward further explanation. Some notes in his Opus postumum suggest that Kant (later) might have regarded the relation between force and motion as a matter of empirical investigation. This would mean, however, a serious drawback for his claim to provide a foundation of mathematical philosophy of nature including dynamics, mechanics and phenomenology (in his terms).
- 57.
See for example Gloy, Die Kantische Theorie, Plaass, Kants Theorie and Schäfer, Kants Metaphysik.
- 58.
In a certain sense this was, restricted to the area of mechanics, Lagrange’s problem, too: His attempt to organise mechanics by means of analytical principles starts with the fact that there are a number of accepted laws, but that no order and unity can be found among these laws. This may serve as a second reason for calling this projection ‘analytical’.
- 59.
This kind of teleology bears some similarities to the ‘architectural’ function of mathematics in Leibniz’ programme; see Pulte, Mathesis pura und Mathesis mixta, 233–238.
- 60.
For a criticism of Kant’s approach see Pulte, Physikotheologie, esp. 320–327. I have tried to show that Kant’s adherent J. F. Fries gave a more satisfying methodological solution of this problem (ibid., 327–341 and Pulte, Kant, Fries and the Expanding Universe of Science).
- 61.
As Lakatos aptly states: “An Euclidean never has to admit defeat: his programme is irrefutable. One can never refute the pure existential statement that there exists a set of trivial first principles from which all truth follows. Thus science may be haunted for ever by the Euclidean programme as a regulative principle, ‘influential metaphysics’” (Lakatos, Philosophical Papers, II 6).
- 62.
Ibid, 5.
- 63.
References to publications in academic periodicals usually bear two dates (1748/1750, for example). The first refers to the year when the contribution was read (in the Berlin Academy, for example), the second to the year when the volume in question (of the Berlin Histoires, for example) was published. Abbreviations to collected works etc. are added in square brackets (as [AA] in case of Kant’s Gesammelte Schriften, for example). Quotations and references taken from these editions are indicated by square brackets at the end of the contribution in question (as [AA 2, 63–163] for Kant’s Der einzig mögliche Beweisgrund).
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References to publications in academic periodicals usually bear two dates (1748/1750, for example). The first refers to the year when the contribution was read (in the Berlin Academy, for example), the second to the year when the volume in question (of the Berlin Histoires, for example) was published. Abbreviations to collected works etc. are added in square brackets (as [AA] in case of Kant’s Gesammelte Schriften, for example). Quotations and references taken from these editions are indicated by square brackets at the end of the contribution in question (as [AA 2, 63–163] for Kant’s Der einzig mögliche Beweisgrund).
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Pulte, H. (2023). Order of Nature and Orders of Science. In: Lefèvre, W. (eds) Between Leibniz, Newton, and Kant. Boston Studies in the Philosophy and History of Science, vol 341. Springer, Cham. https://doi.org/10.1007/978-3-031-34340-7_5
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