Abstract
The development of rigorous foundations of differential calculus in the course of the nineteenth century led to concerns among physicists about its applicability in physics. Through this development, differential calculus was made independent of empirical and intuitive notions of continuity, and based instead on strictly mathematical conditions of continuity. However, for Boltzmann and Poincaré, the applicability of mathematics in physics depended on whether there is a basis in physics, intuition or experience for the fundamental axioms of mathematics—and this meant that to determine the status of differential equations in physics, they had to consider whether there was a justification for these mathematical continuity conditions in physics. For this reason, their ideas about continuity and discreteness in nature were entangled with epistemology and philosophy of mathematics. They reached opposite conclusions: Poincaré argued that physicists must work with a continuous representation of nature, and thus with differential equations, while Boltzmann argued that physicists must ultimately take nature to be discrete.
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Notes
“Il est très difficile, pour les mathématiciens contemporains, de comprendre les contradictions que nos devanciers croyaient découvrir dans les principes du calcul infinitésimal.”
“...il semble qu’en s’arithmétisant, en s’idéalisant pour ainsi dire, la mathématique s’éloignait de la nature et le philosophe peut toujours se demander si les procédés du calcul différentiel et intégral, aujourd’hui complètement justifiés au point de vue logique, peuvent être légitimement appliqués à la nature.”
The reason why the rational numbers do not suffice as a basis for differential calculus is that a sequence of rational numbers may converge to an irrational limit, and it is essential for the foundations of the calculus that for any converging sequence, the limit point exists (this is needed among others in order to define continuous functions in terms of limits).
“Gegen die Fiktion oder die willkürliche begriffliche Construktion eines solchen Systems ist nichts einzuwenden.
Der Naturforscher, der nicht bloss reine Mathematik treibt, hat sich aber die Frage vorzulegen, ob einer solchen Fiktion auch in der Natur etwas entspricht?”
Mach writes that it is at least thinkable, and compatible with experience, that in reality, there are discrete elements rather than a continuum: “Wherever we believe to find a continuum, this only means that we can make observations of the smallest observable parts of the system in question that are analogous to observations of larger parts, and observe an analogous behaviour. How far this continues can only be decided through experience. As long as experience has raised no objections, we can maintain the in no way harmful but merely convenient fiction of the continuum.” (Mach 1896, p. 77).
This is reminiscent of the famous remark, attributed to Kronecker, that “God made the integers; everything else is the work of man”. Kronecker argued that positive integers were the only numbers that could be accepted in mathematics, and mathematics should be rewritten in terms of only these numbers (Kline 1972, p. 1197).
De Courtenay (2002, 2010) argues that Boltzmann embraced the arithmetization of analysis, and that according to Boltzmann, mathematics is an autonomous domain, of which the applicability to nature is not intrinsic, but dependent on practical context. This autonomy of mathematics, however, is hard to reconcile with the central role that Boltzmann gives to experience in his philosophy of mathematics in his Lectures on natural philosophy (Fasol-Boltzmann 1990). According to the views that Boltzmann here presents, the basic concepts and theorems in mathematics must ultimately be grounded in experience and the justification for the introduction of mathematical entities depends on whether they can be made empirically relevant, which undermines the autonomous development of mathematics.
“Man kann eigentlich nicht sagen, dass diese Widersprüche vollkommen gelöst worden sind, aber sie wurden, wie wir sagen, durch diese Mengenlehre wenigstens mit Erfolg umgangen. Wir lernten da eine Methode zu operieren, ohne dass wir irgendwie genötigt sind, an diesen Widersprüchen Anstoss zu nehmen.”
“Wenn wir den Begriff des streng Unendlichen festhalten, kommen wir immer zu solchen Fällen, wo wir keine Entscheidung treffen können”.
“Wenn wir sie [matter] aber wirklich kontinuierlich denken, kommen wir in die Mengenlehre hinein; wir kommen alle Augenblicke an Stellen, wo wir nicht eindeutig schliessen können, und der Zweck des Denkens ist ja, überall eindeutig schliessen zu können; daher müssen wir unsere Sprach-, Schrift- und Denkzeichen so zu bilden suchen, dass wir uns selbst eindeutig ausdrücken und uns selbst eindeutig verstehen.”
“Man verzeihe den etwas banalen Ausdruck, wenn ich sage, dass derjenige, welcher die Atomistik durch Differentialgleichungen losgeworden zu sein glaubt, den Wald vor Bäumen nicht sieht.”
“Während früher die Annahme einer bestimmten Grösse der Atome als eine rohe, willkürlich über die Tatsachen hinausgehende Vorstellung galt, so erscheint sie jetzt gerade als die natürlichere, und die Behauptung, dass niemals Unterschiede zwischen den Tatsachen und den Limitenwerten entdeckt werden könnten, weil solche bis heute (vielleicht nicht einmal in allen Fällen) noch nicht entdeckt wurden, fügt dem Bilde etwas Neues, Unerwiesenes bei.”
Wilholt (2008) has shown that the debate about atomism in the late nineteenth century was to an important degree a conceptual debate: it concerned not merely the degree of ontological commitment that we should make to unobservable entities in our theories, but the very legitimacy of the use of atomistic versus continuous conceptions of matter. He shows that whereas among others Du Bois-Reymond pointed out inconsistencies in the atomic conception of matter, Boltzmann pointed out conceptual problems that can arise in continuous representations of nature.
“C’est là le postulat que nous admettons implicitement quand nous appliquons à la nature les lois de l’analyse mathématique et en particulier celles du calcul infinitésimal”.
This is not entirely true: in 1876, Lipschitz had shown that a differential equation of the form only has a unique solution if F(r) satisfies a condition that is now known as Lipschitz continuity (see Van Strien 2014).
According to Poincaré, arithmetic is based on “the affirmation of the power of the mind which knows itself capable of conceiving the indefinite repetition of the same act when once this act is possible” (Poincaré 1902, p. 39); “We have the faculty of conceiving that a unit can be added to a collection of units; thanks to experience, we have occasion to exercise this faculty and we become conscious of it; but from this moment we feel that our power has no limit and that we can count indefinitely, though we have never had to count more than a finite number of objects” (Poincaré 1902, p. 47; and see Folina (1992), p. 94). In (1905a), however, he leaves open the question of whether the intuition involved is inner or sensible intuition, as an issue that should be left to psychologists and metaphysicians (Poincaré 1905a, p. 221). For Helmholtz’ account of the foundations of arithmetic and its influence, see Helmholtz (1887), Darrigol (2003).
“sorte de fusion des éléments voisins”.
The example comes from Fechner, it is related to the (Weber-) Fechner law of perception (Poincaré refers to Fechner in (1917[1913], p. 68)). Poincaré argues that measurement in general shows us a physical continuum rather than a mathematical continuum. It is of course possible to make more precise measurements of weight, for example by using a scale which can distinguish between 10 and 11 g, but no matter how precise we make our measurements, in the end we always have to appeal to our senses to read off the measurement apparatus, “which will bring along the characteristics of the physical continuum and its essential imprecision” (Poincaré 1905b, p. 295).
In (1917[1913], p. 71), Poincaré refines his view by saying that the physical continuum is not directly derived from the senses, in the sense that it can only be constructed when certain sensations are isolated, through abstraction, e.g. by only paying attention to weight.
“We might conceive the stopping of this operation [of further division] if we could imagine some instrument sufficiently powerful to decompose the physical continuum into discrete elements, as the telescope resolves the milky way into stars. But this we can not imagine; in fact, it is with the eye we observe the image magnified by the microscope, and consequently this image must always retain the characteristics of visual sensation and consequently those of the physical continuum” (Poincaré 1902, p. 47).
“le monde donné est un continu physique, et les savants supposent que le monde réel est un continu mathématique, mais quelques métaphysiciens ont préféré admettre que le monde est discontinu”.
In (1902), Poincaré treats continuity of matter as a matter of convenience: “In most questions the analyst assumes at the beginning of his calculations either that matter is continuous or, on the contrary, that it is formed of atoms. He might have made the opposite assumption without changing his results. He would only have had more trouble to obtain them; that is all.” (Poincaré 1902, p. 135). His point here is that atomism is a ’neutral hypothesis’ in the sense that it can be neither empirically confirmed nor falsified; by implication, the same holds for the assumption that matter is continuous (which is, according to Poincaré, the more convenient option). On Poincaré’s atomism, see Ivanova (2013).
“cela ne serait pas aussi simple qu’avec le système de M. Evellin, et il serait difficile sans doute de donner à cette idée une forme mathématique et de la rendre compatible avec le déterminisme absolu”. For Poincaré, the possibility to formulate laws of nature in terms of differential equations implies determinism; he seems to take for granted that these equations will always have unique solutions for given initial conditions and will thus be deterministic (Poincaré 1917[1913], p. 8). Therefore, to give up continuity in physics threatens the possibility to have a deterministic physics. However, the connection between determinism and differential equations was in fact not so straightforward: while Poincaré refers to Bertrand’s (1878) paper in which Bertrand argues for a discontinuous conception of reality, he does not remark on the fact that the motivation for Bertrand’s argument was to save determinism. Bertrand’s paper was a reaction to an argument by Boussinesq, who had shown that there can be mechanical systems for which the (differential) equations of motion fail to have a unique solution for given initial conditions and thus allow for indeterminism. Bertrand argues that the indeterminism that Boussinesq describes is an artefact of the use of differential equations, and that it can be avoided through the assumption that physical reality is fundamentally discrete and that differential equations offer mere approximations (Van Strien 2014).
“Bien entendu, on ne peut non plus fixer de tangente en aucun point de la trajectoire, même de la façon la plus grossière. C’est un des cas où l’on ne peut s’empêcher de penser à ces fonctions continues qui n’admettent pas de dérivée, qu’on regarderait à tort comme de simples curiosités mathématiques, puisque la nature peut les suggérer aussi bien que les fonctions à dérivées.”
Cassirer later remarked that this proposal by Perrin “shatters one of the essential bases on which the edifice of classical analysis as well as that of classical physics rests”. He adds: “It is now shown that ’macrostates’ do not permit immediate inference to ’microstates.’ Leibniz, at times, formulated his continuity principle in such a manner as to demand exactly this analogy.” (Cassirer 1956 [1936], pp. 164–165).
“hinlänglich sicher gestellte Erfahrungstatsache”.
“...wir kennen auch Beispiele sehr rascher Oszillationen und können nicht exakt beweisen, ob nicht in gewissen Fällen Bewegungen vorhanden sind, wie z. B. die Wärmebewegungen der Moleküle, welche durch eine der Weierstrassschen Funktion ähnliche besser als durch eine differenzierbare dargestellt werden.”
“Wir können ja dann unser Bild formen, wie wir wollen und einfach die Differenzierarbeit von vornherein in dasselbe aufnehmen, es damit rechtfertigend, dass das Bild hinterher mit der Erfahrung stimmt.”
The French original: “on vit surgir toute une foule de fonctions bizarres qui semblaient s’efforcer de ressembler aussi peu que possible aux honnètes fonctions qui servent à quelque chose. Plus de continuité, ou bien de la continuité, mais pas de dérivées, etc., etc. (...) Autrefois, quand on inventait une fonction nouvelle, c’était en vue de quelque but pratique; aujourd’hui, on les invente tout exprès pour mettre en défaut les raisonnements de nos pères, et on n’en tirera jamais que cela.” (Poincaré 1899)
Poincaré writes on simplicity assumptions: “No doubt, if our means of investigation should become more and more penetrating, we should discover the simple under the complex, then the complex under the simple, then again the simple under the complex, and so on, without our being able to foresee what will be the last term.
We must stop somewhere, and that science may be possible, we must stop when we have found simplicity.” (Poincaré 1902, p. 133).
And in (1905b): “Il y aura donc toujours moyen de représenter les observations, quelles qu’elles soient, par des fonctions qui s’écarteront moins que ne le comporte l’incertitude des mesures et qui jouiront de la continuité, de la propriété d’avoir une dérivée, de toutes les propriétés des fonctions analytiques. Une fonction quelconque étant donnée, on peut toujours trouver une fonction analytique qui en diffère aussi peu que l’on veut.”
“Ainsi le physicien peut toujours appliquer les règles du calcul infinitésimal sans craindre un démenti de l’expérience”.
For a function to be approximable as closely as one wishes by a continuous function, this function must be the (pointwise) limit of a sequence of continuous functions. Functions for which this holds are called Baire class I functions. Not all discontinuous functions are Baire class I functions which means that not all functions can be approximated as closely as one wishes by a continuous function.
Poincaré’s claim did not go completely uncontested. In (1905b), Poincaré argues that any function can be approximated as closely as one wishes by an analytic function, where analyticity is a stronger property than differentiability. But this claim was criticized by Hadamard in (1923): “I have often maintained, against different geometers, the importance of this distinction. Some of them indeed argued that you may always consider any functions as analytic, as in the contrary case, they could be approximated with any required precision by analytic ones. But, in my opinion, this objection would not apply, the question not being whether such an approximation would alter the data very little, but whether it would alter the solution very little.” (Hadamard 1923, p. 33). See also Wilson (2006, pp. 308–309) on the problems with Poincaré’s claim that functions in physics can always be taken to be analytic. Wilson emphasizes that analytc functions have a specific character that one cannot expect all functions in physics to have; in particular, if you know how the function behaves within a certain finite interval, you can derive how it behaves elsewhere.
“La discontinuité va-t-elle régner sur l’univers physique et son triomphe est-il définitif? Ou bien reconnaîtra-t-on que cette discontinuité n’est qu’apparente et dissimule une série de processus continus. Le premier qui a vu un choc a cru observer un phénomène discontinu, et nous savons aujourd’hui qu’il n’a vu que l’effet de changements de vitesse très rapides, mais continus. Chercher dès aujourd’hui à donner un avis sur ces questions, ce serait perdre son encre.”
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Acknowledgments
I would like to thank two anonymous referees for their helpful comments. Furthermore, I would like to thank Eric Schliesser and Boris Demarest for commenting on earlier drafts of this paper. I am also indebted to audiences at the workshop on Scientific Metaphysics in Ghent (17–18 February, 2014), the “Mathematizing science: limits and perspectives II” conference in Norwich (1–3 June, 2014), and the HOPOS conference in Ghent (3–5 July, 2014) for helpful feedback. The research for this paper was largely done at Ghent University, with funding from the Research Foundation Flanders (FWO). I would also like to thank the Max Planck Institute for the History of Science, where I finished the paper.
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van Strien, M. Continuity in nature and in mathematics: Boltzmann and Poincaré. Synthese 192, 3275–3295 (2015). https://doi.org/10.1007/s11229-015-0701-9
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DOI: https://doi.org/10.1007/s11229-015-0701-9