Abstract
This paper discusses Ernst Mach’s interpretation of the principle of energy conservation (EC) in the context of the development of energy concepts and ideas about causality in nineteenth-century physics and theory of science. In doing this, it focuses on the close relationship between causality, energy conservation and space in Mach’s antireductionist view of science. Mach expounds his thesis about EC in his first historical-epistemological essay, Die Geschichte und die Wurzel des Satzes von der Erhaltung der Arbeit (1872): far from being a new principle, it is used from the early beginnings of mechanics independently from other principles; in fact, EC is a pre-mechanical principle which is generally applied in investigating nature: it is, indeed, nothing but a form of the principle of causality. The paper focuses on the scientific-historical premises and philosophical underpinnings of Mach’s thesis, beginning with the classic debate on the validity and limits of the notion of cause by Hume, Kant, and Helmholtz. Such reference also implies a discussion of the relationship between causality on the one hand and space and time on the other. This connection plays a major role for Mach, and in the final paragraphs its importance is argued in order to understand his antireductionist perspective, i.e. the rejection of any attempt to give an ultimate explanation of the world via reduction of nature to one fundamental set of phenomena.
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Notes
On Helmholtz’s attitude towards the terminological shift Kraft/Energie and for a discussion of Tyndall’s translation, see Elkana (1974, pp. 122–137).
About the relationship between Mach and Carus see Holton (1993, pp. 4–7).
It is also remarkable that Mach (1895) does not mention the view expressed by Planck (1887, p. 139), that EC actually consists of two parts: the impossibility to create work from nothing and the impossibility to destroy work without any other effect. Planck argues that the two assumptions are logically independent. This could obviously imply a potential criticism of Mach’s position, but Mach does not seem to have taken it into account in the subsequent presentations of his argument. In the Preface to the second edition of the Erhaltung der Arbeit (1909) he only complains that Planck has paid scanty attention to his book (Mach 1872, p. V; Engl. transl., p. 10).
According to Mach (1872, p. 39; Engl. transl., p. 65), an application of the principle of sufficient reason can be exemplified in this way: “Let us take a straight horizontal bar, which we support in its middle and at both ends of each we hang equal weights. Then we perceive at once that equilibrium must subsist, because there is no reason why the bar should turn in one direction rather than in the other”.
See on this Guzzardi (2005, pp. 17–35).
In the referred Sect. 8 of the “Antimetaphysical Introductory Remarks” (Mach 1886, chapt. 1) Mach specifies the “monistic point of view” he had expounded in the previous paragraphs: both the Self and the outer World are made of the same elements which only differ in their mutual connection. They are sensations when they present themselves in a physio-psychological connection; but “in another functional relation they are at the same time physical objects”. So the Self and the World are not conflicting terms, for “all elements […] form a single coherent mass [zusammenhängende Masse]” (Mach 1886, pp. 11–14; see also Mach 1905, pp. 8–9, Engl. transl., pp. 6–7: “The mental and physical have common elements and are not in stark opposition as commonly supposed”).
On the relationship between Mach and Hume see also Hamilton 1990; Guzzardi 2010, pp. 63–67. Banks (2003, pp. 42–44) finds in Mach’s function concept some important theoretical differences between Hume’s and Mach’s empiricism; nevertheless, these don’t seem to affect the notion of causality as a regulative principle.
See also other similar expressions implying temporality, such as “were the power or energy of any cause discoverable by the mind, we could foresee the effect” (Hume 1748, p. 51; emphasis added), “one event follows another” and “the only immediate utility of all sciences, is to teach us, how to control and regulate future events by their causes” (Hume 1748, p. 60; emphasis added), etc.
On Kant’s relationship with Hume regarding the Second Analogy, see Ward (1986).
There is of course a connection between both principle. See Ward (2001), pp. 405–406.
On Kant’s influence over Helmholtz’s On the Conservation of Force, with particular reference to the notion of regulative principles, see Hyder (2009, pp. 76–88).
“The first product of the thoughtful comprehension of the phenomena is lawfulness […]. We acknowledge it as an existence enduring independent of the way in which we form representations, and call it the cause, i.e. that which primarily remains and endures behind what alternates […]. Inasmuch as we then acknowledge the law to be something compelling our perception and the course of natural processes […] we call it ‘force’”. See Helmholtz (1977, p. 139).
On Kant’s view of force as a cause see the sections “Dynamics” and “Mechanics” in Kant (1786), in particular A31–A37 and A119, “Lehrsatz 3”). See also Kant (1787, p. 593: A649/B677): note that in the English translation the crucial passage: “[die] Kausalität einer Substanz, welche Kraft [force] genannt wird” is rendered as: “the causality of a substance, which is called ‘power’”. On the development of Helmholtz’s views on causality, see Schiemann (1997, pp. 259–264, 369–374).
Taking account of Helmholtz’s dislike for mere mathematical expedients in physical equations, it is important here to recognize that in the subsequent equation the q a ’s are Lagrangian multipliers, i.e. they are endowed with a clear physical meaning. On their physical significance see Lanczos (1970), pp. 83–86. On Helmholtz’s inclination to introduce “physically sound concepts” (logically) before their mathematical counterparts, see Helmholtz (1854, in particular p. 82); for useful comments on this tendency, see Bevilacqua (1993, p. 328), and Hyder (2009, pp. 78–79).
For a presentation of the derivation of EC from Hamilton’s principle in modern terms, which happens to be close to Helmholtz’s, see Lanczos (1970, pp. 111–122).
I cannot agree with Michael Heidelberger, that Helmholtz found in the principle of the least action “a more general law into which he could embed the conservation principle” (Heidelberger 1993, p. 496). In fact, this seems to be literally excluded by Helmholtz himself in stating that the principle of the least action “does not applies in all cases where the constance of energy takes place [gewahrt ist]. So the latter tells us [sagt… aus] more than the former.” (Helmholtz 1886, p. 211).
On technical aspects of this analogy see Banks (2003, pp. 169–171), who also briefly accounts for a number of differences between Mach and the energeticists (pp. 222–224). Mach was fully aware of possible analogies and differences at least with Ostwald. For instance, he declared in a letter on January 27th, 1902: “You may have already received the third edition of Analyse der Empfindungen […] you will find many points in common between us, and ideed, perhaps also some differences between our views. […] As a scientist, wherever we agree makes me happy; and let the philosophers live with differences” (Blackmore 1992, p. 79, emphasis in original text). For a comprehensive study on Mach and the energeticists see Neuber (2005).
Jourdain’s original translation reads “is to be deduced” instead of “results”; however, ergibt sich aus is not meant here in strictly logical terms; it rather means ‘it historically results’.
Note that Ostwald considered his Vorlesungen as philosophically inspired by Mach, as he stated in a letter on May 31st, 1901: “This semester under the designation Naturphilosophie I am giving a series of lectures which are essentially on the theory of knowledge, and of which my foundations for the most part I owe to you” (Blackmore 1992, p. 76).
On Ostwald’s peculiar Kantian background see Moiso (1998, pp. 270–275).
So Thomas McCormack, the English translator of Mach’s paper, correctly rendered Mach’s phrase as “Black’s material conception of heat” (Mach 1895, p. 53).
It is worthwhile to notice that this paper, barely taken into consideration by historians and philosophers of science, shows a remarkable convergence between Mach and Boltzmann, all too often regarded as antagonists by influential books like Popper (1974, pp. 181–182) and Janik and Toulmin (1973, in particular chapter 5).
Note that, like Helmholtz (1847, p. 68), Mach uses here the verb voraussetzen (presuppose) in order to qualify the law of causality. So the whole phrase, which is emphasized in the original text, reads in German: “Das Causalgesetz ist also hinreichend charakterisiert, wenn man sagt, es setzte eine Abhängigkeit der Erscheinungen von einander voraus”.
Holton (1993, pp. 60–65) has questioned Mach’s correct understanding of Minkowski’s conception of space–time as a geometrical four-dimensional variety, because of Mach’s lack of the appropriate mathematical tools. It is not my aim to discuss this thesis in the present paper. It might be true that Mach’s education in mathematics was not sufficient in order to penetrate all the technical aspects for example in the case of differential geometry and topology, or even in the case of Hertz’s geometric image, as it is based upon “a differential geometry of configuration space” (Lützen 2005, p. v). But he surely could realize the significance of Bernhard Riemann’s or Felix Klein’s works and recognize the meaning and the coherence of Hertz’s efforts from the theory of electromagnetism to mechanics. Against Holton’s thesis, this is also substantiated by the fact that Mach discusses the importance of Riemann’s and Klein’s results (not to say of Hertz’s) on several occasions (e.g. Mach 1905, pp. 415–421; Engl. transl., pp. 321–327).
For a critical reading of Mach on space and time see DiSalle (2002).
It is to be noted that Mach developed his space conception quite independentely from Riemann, whose writing Über die Hypothesen, welche der Geometrie zu Grunde liegen (1868) he probably knew in 1871–1872. However, Herbart’s philosophy, particularly because of his insistence on the concept of serie (Rehie) in order to construct space, was an obvious reference shared both by Mach and Riemann. A detailed account of Mach’s relationship with Riemann is given by Banks (2003, pp. 77–90).
Mach first attempt to regard the sound scale (or better, sound serie) as a one-dimensional space traces back to his ‘Herbartian’ physiological essay Zur Theorie des Gehörorgans, whose reader will find this meaningful passage: “We order sounds in a serie [Rehie]. How do we get to this? […] There are completely analogous phenomena in other sensory apparatus which have been already explained. We also order our space sensations in series, namely spatial series” (Mach 1863, p. 298). The example of the one-dimensional sound serie reappears in other works with more emphasis on space (while the concept of serie remains in the background): see Mach (1872, pp. 27, 55; Engl. transl., pp. 50–51, 87), Mach (1886, pp. 225, 227, where the serial order of sounds and the space analogy are treated separately), and Mach (1905, p. 393 and footnote 1: “My attention was drawn to this analogy [with space] while studying the organ of hearing; since then I have further developed this subject [i.e. the analogy itself]”; see also Engl. transl., pp. 302, 328).
For this description of the concept of intensive magnitudes and a related discussion, see Hyder (2009, pp. 129–130). Hyder also shows that the originally Kantian distinction between intensive and extensive magnitudes came to a turning point with the emergence of the colour-space theories in the second half of the nineteenth century—a debate of which Mach was fully aware (see Mach 1905, pp. 392–395).
It is remarkable that Mach ascribes the beginnings of this conception to Lagrange, who actually poses the bases of the concept of generalized co-ordinates—but to say that with Lagrange time is to be considered a fourth dimension is at least very questionable from a historical point of view. This is much more Mach’s own view than Lagrange’s.
On Mach’s economic view of science in a broader context, also discussing the role of Richard Avenarius and Joseph Petzoldt, see Moiso 1998.
Mach has treated this subject in various works and under many aspects. In Erkenntnis und Irrtum he describes the difference between physiological and physical space and time as follows: “As regards physiology, time and space are systems of sensations of orientation that, together with the sensations of the sensory apparatus [Sinnesempfindungen], determine the release of biologically appropriate reactions of adaptation. As regards physics, time and space are special dependences of physical elements on each other” (Mach 1905, p. 434, emphasis in original text; Engl. transl., p. 339: my translation slightly differs from the usual English translation).
Leaving aside the question of Mach’s understanding of the essence and role of topology, a philosophical speculation is that the most appropriate concept of space to express Mach’s notion of causality and EC could be that of a topological space, of which the physical space, energetic relations and the causality principle are mere forms. From a historical point of view it would be of course strained to seek a translation of Mach’s insights into topology. Moreover, this would add nothing to the understanding of Mach’s science and philosophy themselves. Nevertheless, such an attempt can be philosophically interesting in the light of further developments of the very “fundamental transformation” which affected the space conception in the second half of the nineteenth century and the subsequent geometrization of physics. See e.g. Boi (1995, in particular pp. 481–484; Boi refers to Misner and Wheeler’s idea that “physics is geometry” as a development of Clifford’s “new theory of matter”).
Mach would regard as mysticism a view like this: “All we agree that it makes no great difference in which of the three forms an equation is written, f (α β γ…) = 0, α = F (α β γ…), F (α β γ…) = const., and that in the last of these forms there lies no specially higher wisdom than in the others. But it is merely by this form that the law of the conservation of work differs from other laws of nature. We can easily give a similar form to any other law of nature […]. However beautiful, simple, and perspicuous much in the form of the principle [Satz] of the conservation of work looks, I cannot feel any enthusiasm for the mysticism which some people love to push forwards by means of this principle” (Mach 1872, p. 46; Engl. transl., p. 73. See also Mach 1895, p. 54, Germ. version, p. 214, where the same term Mystik/mysticism occurs).
A pithy formulation of Mach’s principle is accorded to Stephen Hawking: “Matter possesses inertia only relatively to other matter in the Universe” (Hawking 1990, p. 189). For a discussion of Mach’s principle see Barbour and Pfister (1995). About the substantial equivalence of various formulation of Mach’s principle, see Bartocci and Guzzardi (2007). For Einstein’s expression of Mach’s principle, see Einstein (1916, p. 771) and Einstein (1917, p. 50). Both papers have been reprinted in chronological order in Einstein (1996).
In the Preface to Hertz’s Prinzipien there was a second image which Hertz undermined: the energetic image. Hertz was well aware that the theory of energy could provide an alternative foundation for mechanics, which could solve some problems of the standard force-based mechanics. But after having enthusiastically looked at energetic concepts for some time, he became convinced that they could even lead to paradoxes, if one was willing to assume a substantial conception of energy at the basis of physics (Lützen 2005, pp. 78–79). Mach’s suspicious attitude towards energetism and the substantialist conception of energy went in the same direction.
The meaning and translation of the term Fallkraft (fallforce) in this passage may be problematic. Jourdain’s original English translation as “tendency to fall” is not exactly Mayer’s original idea. In fact, in Mayer’s Bemerkungen über die Kräfte der unbelebten Natur (1842) Fallkraft is described as follows: “A cause that brings about the raising of a weight [die Hebung einer Last] is a force; its effect, the raised weight, is thus likewise a force; expressed more generally this means that the spatial difference of ponderable objects is a force; since this force brings about the fall of the body, we thus call it fallforce [Fallkraft]”. After few lines Mayer makes clear that Fallkraft is not Schwerkraft (i.e. gravitational force): “As far as one regards gravity [Schwere] as the cause of fall, one speaks of a force of gravity [Schwerkraft] and thereby confuses the concepts of force and property […]. If one calls gravity a force, one thereby imagines a cause which, without itself diminishing, produces effect, and one thereby thus entertains incorrect ideas [Vorstellungen] about the causal connection of things” (Mayer 1867, pp. 235–236; a slightly different translation of these passages is given in Caneva 1993, p. 24).
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Acknowledgments
I owe a debt of gratitude to Claudio Bartocci, Fabio Bevilacqua, Giulio Giorello, and Riccardo Pozzo, who read an early version of this paper and made many useful suggestions. Finally I wish to thank three anonymous referees for their constructive comments, which helped in significantly improving the text.
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Guzzardi, L. Energy, Metaphysics, and Space: Ernst Mach’s Interpretation of Energy Conservation as the Principle of Causality. Sci & Educ 23, 1269–1291 (2014). https://doi.org/10.1007/s11191-012-9542-9
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DOI: https://doi.org/10.1007/s11191-012-9542-9