Boltzmann's H-theorem, its discontents, and the birth of statistical mechanics

A comparison is made of the traditional Loschmidt (reversibility) and Zermelo (recurrence) objections to Boltzmann's H-theorem, and its simplified variant in the Ehrenfests’ 1912 wind-tree model. The little-cited 1896 (measure-theoretic) objection of Zermelo (similar to an 1889 argument due to Poincaré) is also analysed. Significant differences between the objections are highlighted, and several old and modern misconceptions concerning both them and the H-theorem are clarified. We give particular emphasis to the radical nature of Poincaré's and Zermelo's attack, and the importance of the shift in Boltzmann's thinking in response to the objections taken together.     

History of Science through Koyré's Lenses

Alexandre Koyré was one of the most prominent historians of science of the twentieth century. The standard interpretation of Koyré is that he falls squarely within the internalist camp of historians of science—that he focuses on the history of the ideas themselves, eschewing cultural and sociological interpretations regarding the influence of ideologies and institutions on the development of science. When we read what Koyré has to say about his historical studies (and most of what others have said about them), we find him embracing and championing this Platonic view of his work. Ultimately I think this interpretation of Koyré's history of science is lopsided and in need of correction. I claim, rather, that a careful reading of Koyré's work suggests that a tension exists between internal and external methodological considerations. The external considerations stem from Koyré's commitment to the unity of human thought and the influence he admits that the ‘transscientifiques’ (philosophy, metaphysics, religion) have on the development of science. I suggest in conclusion then, that if we are to put a philosophical label on his work, rather than ‘Platonist’, as has been the custom, ‘Hegelian’ makes a better fit.     

Concepts, anomalies and reality: a response to Bloor and Fehér

In this article I respond to the defences of the Strong Programme put forward by David Bloor and Márta Fehér in this issue. I dispute the claim that it is attention to only limited parts of the Strong Programme framework that allows me to argue that this approach: (i) leads to weak idealism, (ii) undermines the idea that theories have varying levels of instrumental success, and (iii) challenges the theoretical claims of scientific actors. Rather, I argue that these problematic positions are entailed by the constructionist tenets at the core of the Strong Programme.     

Hélène Metzger: the history of science between the study of mentalities and total history

In this article, I examine the historiographical ideas of the historian of chemistry Hélène Metzger (1886–1944) against the background of the ideas of the members of the groups and institutions in which she worked, including Alexandre Koyré, Gaston Bachelard, Abel Rey, Henri Berr and Lucien Febrve. This article is on two interdependent levels: that of particular institutions and groups in which she worked (the Centre de Synthèse, the International Committee for History of Science, the Institut d'Histoire des Sciences et Techniques (Sorbonne) and the École Pratique des Hautes Études) and that of historiographical ideas. I individuate two particular theoretical aspirations pursued by the historians in Metzger's milieu: the ideal of total history and the study of the human mind. These two objectives were seen by Metzger and many others as implicating each other. Moreover, Metzger and other historians wanted to integrate the practice of commentary of texts in the realisations of those ideals. I argue, however, that these objectives proved very difficult to realise at the same time. One tradition which stemmed out of these discussions, exemplified by Bachelard, Canguilhem and Foucault, focused on the mind and knowledge, and renounced commentary of texts and total history as it was understood by the historians of the Centre de Synthèse. The latter, however, did not really pursue the study of the mind. Moreover, historians like Metzger and Koyré who practised an attentive analysis of texts could not realise total history.     

The conceptual import of Carnot's theorem to the discovery of the entropy

Summary Among many other things, Carnot stated a principle and proved a theorem. In 1850, Clausius corrected Carnot's theory, modifying it according to Joule's principle. He might have considered a corollary of the theorem as the mathematical formulation of Carnot's principle. We challenge the corollary: it is based on hidden assumptions, nor is it true for all cycles. Clausius realized the corollary's lack of generality, but on different grounds. In 1854, he generalized the theorem, and gave an (other) expression to Carnot's principle. We analyze Clapeyron's account of Carnot's theory, Thomson's account of 1849 and some of Clausius belated comments on his 1850 paper, as well Clausius' paper of 1854. We hope that they shed light on the corollary's tacit hypotheses and on the meaning of Carnot's principle. It is our contention: Clausius took seriously a contemporary meaning of the principle, and looked for a condition of integrability that could express recovery of the initial conditions of the reservoirs. Furthermore, he seems to have had some prior knowledge of the form the expression of the principle should take. Actually, this was the theory's natural candidate.     

An empirical approach to symmetry and probability

We often rely on symmetries to infer outcomes’ probabilities, as when we infer that each side of a fair coin is equally likely to come up on a given toss. Why are these inferences successful? I argue against answering this question with an a priori indifference principle. Reasons to reject such a principle are familiar, yet instructive. They point to a new, empirical explanation for the success of our probabilistic predictions. This has implications for indifference reasoning generally. I argue that a priori symmetries need never constrain our probability attributions, even for initial credences.     

Subjective probability and quantum certainty

In the Bayesian approach to quantum mechanics, probabilities—and thus quantum states—represent an agent's degrees of belief, rather than corresponding to objective properties of physical systems. In this paper we investigate the concept of certainty in quantum mechanics. Particularly, we show how the probability-1 predictions derived from pure quantum states highlight a fundamental difference between our Bayesian approach, on the one hand, and Copenhagen and similar interpretations on the other. We first review the main arguments for the general claim that probabilities always represent degrees of belief. We then argue that a quantum state prepared by some physical device always depends on an agent's prior beliefs, implying that the probability-1 predictions derived from that state also depend on the agent's prior beliefs. Quantum certainty is therefore always some agent's certainty. Conversely, if facts about an experimental setup could imply agent-independent certainty for a measurement outcome, as in many Copenhagen-like interpretations, that outcome would effectively correspond to a preexisting system property. The idea that measurement outcomes occurring with certainty correspond to preexisting system properties is, however, in conflict with locality. We emphasize this by giving a version of an argument of Stairs [(1983). Quantum logic, realism, and value-definiteness. Philosophy of Science, 50, 578], which applies the Kochen–Specker theorem to an entangled bipartite system.     

Poincaréan intuition revisited: what can we learn from Kant and Parsons?

This paper provides a comprehensive critique of Poincaré’s usage of the term intuition in his defence of the foundations of pure mathematics and science. Kant’s notions of sensibility and a priori form and Parsons’s theory of quasi-concrete objects are used to impute rigour into Poincaré’s interpretation of intuition. In turn, Poincaré’s portrayal of sensible intuition as a special kind of intuition that tolerates the senses and imagination is rejected. In its place, a more harmonized account of how we perceive concrete objects is offered whereby intuitive knowledge is consistently a priori whatever the domain of application.     

Quantum sidelights on The Material Theory of Induction

John Norton's The Material Theory of Induction bristles with fresh insights and provocative ideas that provide a much needed stimulus to a stodgy if not moribund field. I use quantum mechanics (QM) as a medium for exploring some of these ideas. First, I note that QM offers more predictability than Newtonian mechanics for the Norton dome and other cases where classical determinism falters. But this ability of QM to partially cure the ills of classical determinism depends on facts about the quantum Hamiltonian operator that vary from case to case, providing an illustration of Norton's theme of the importance of contingent facts for inductive reasoning. Second, I agree with Norton that Bayesianism as developed for classical probability theory does not constitute a universal inference machine, and I use QM to explain the sense in which this is so. But at the same time I defend a brand of quantum Bayesianism as providing an illuminating account of how physicists' reasoning about quantum events. Third, I argue that if the probabilities induced by quantum states are regarded as objective chances then there are strong reasons to think that fair infinite lotteries are impossible in a quantum world.     

Becquerel and the choice of uranium compounds

Conclusion The common assumption that Becquerel had no special reason to study uranium compounds in his search for substances emitting penetrating radiation cannot explain (a) Becquerel's own accounts, which refer to his choice as due to the peculiar harmonic series of bands; (b) Becquerel's systematic test of all uranium compounds (and metallic uranium), in contrast to his neglect of other substances; and (c) Becquerel's belief in invisible phosphorescence as an explanation of the radiation emitted by uranium compounds, even after his discovery that non-luminescent and metallic uranium also emit penetrating radiation.By comparing Becquerel's older studies of uranium to his radioactivity research, this paper has presented a reconstruction that can explain all of these points above. According to the historical evidence presented here, it is likely that Becquerel concentrated his attention on uranium and its compounds because the mechanical theory of luminescence opened up the possibility that, precisely in the case of uranium and its compounds, a violation of Stokes's law could occur, and penetrating short-wavelength radiation could be emitted through a special type of phosphorescence.     

‘The happy thought of a single man’: On the legendary beginnings of a style of reasoning

In this paper I direct attention to one feature of Hacking’s recent work on styles of reasoning and argue that this feature is of far greater philosophical significance than Hacking’s limited discussion of this suggests. The feature in question is his use of ‘legendary beginnings’ in setting out a given style, viz. the method of introducing a style of reasoning by recounting a popular and quasi-mythical narrative that ties the crystallisation of that style to a particular person in a particular place and at a particular time. Whilst Hacking both deploys and discusses this method, his comments suggest that this is primarily a stylistic device employed for reasons of expedience. In contrast, it is argued here that recounting the legendary origins of a style of reasoning affords a distinctive way of vindicating that style, a vindication from within the style itself.     

Everettian quantum mechanics and physical probability: Against the principle of “State Supervenience”

Everettian quantum mechanics faces the challenge of how to make sense of probability and probabilistic reasoning in a setting where there is typically no unique outcome of measurements. Wallace has built on a proof by Deutsch to argue that a notion of probability can be recovered in the many worlds setting. In particular, Wallace argues that a rational agent has to assign probabilities in accordance with the Born rule. This argument relies on a rationality constraint that Wallace calls state supervenience. I argue that state supervenience is not defensible as a rationality constraint for Everettian agents unless we already invoke probabilistic notions.     

The development of the Neurath principle: unearthing the Romantic link

Otto Neurath’s thoroughgoing anti-foundationalism is connected to the recognition that protocol sentences are not inviolable, that is they are fallible and their choice cannot be determined: ‘Poincaré, Duhem and others have adequately shown that even if we have agreed on the protocol statements, there is a not limited number of equally applicable, possible systems of hypotheses. We have extended this tenet of the uncertainty of systems of hypotheses to all statements, including protocol statements that are alterable in principle’ (Neurath, 1983, p. 105). Later historiography has called Neurath’s extension of Duhemian holism the Neurath principle. Based on a study of Neurath’s early works on the history of optics, the paper investigates a previously unnoticed influence on the development of this principle, Neurath’s reading of Goethe’s Theory of colours. The historical and polemical parts of Goethe’s tripartite book provided Neurath with ideal examples for the vertical extension of Duhem’s thesis to observation statements. Moreover, Goethe’s critique of the language of science and his views on the theory-ladenness of observation, as well as on the history of science show strong parallels to many of Neurath’s ideas. These demonstrate the existence of surprisingly direct textual links between Romantic views on science and the development of twentieth-century philosophy of science. Neurath’s usage of Goethe’s examples also indicates that the birth of the Neurath principle is more tightly connected to actual scientific practice than to theory-testing, and that by admitting the theory-ladenness of observation reports and fallibility of protocol statements Neurath does not throw empiricism overboard.     

Interactions between mechanics and differential geometry in the 19th century

Conclusion 79. This study of the interaction between mechanics and differential geometry does not pretend to be exhaustive. In particular, there is probably more to be said about the mathematical side of the history from Darboux to Ricci and Levi Civita and beyond. Statistical mechanics may also be of interest and there is definitely more to be said about Hertz (I plan to continue in this direction) and about Poincaré's geometric and topological reasonings for example about the three body problem [Poincaré 1890] (cf. also [Poincaré 1993], [Andersson 1994] and [Barrow-Green 1994]). Moreover, it would be interesting to find out how the 19^th century ideas discussed here influenced the developments in the 20^th century. Einstein himself is a hotly debated case.Yet, despite these shortcommings, I hope that this paper has shown that the interactions between mechanics and differential geometry is not a 20^th century invention. Klein's view (see my Introduction) that Riemannian geometry grew out of mechanics, more specifically the principle of least action, cannot be maintained. On the other hand, when Riemannian geometry became known around 1870 it was immediately used in mechanics by Lipschitz. He began a continued tradition in this field, which had several elements in common with the new view of mechanics conceived by the physicists and explicitly carried out by Hertz.Before 1870 we found only scattered interactions between differential geometry and mechanics and only direct ones for systems of two or three degrees of freedom. For more degrees of freedom the geometrical ideas were in some interesting cases taken over by analogy, but these analogies did not lead to formal introduction of geometries of more than three dimensions.     

Styles of reasoning: A pluralist view

Styles of reasoning are important devices to understand scientific practice. As I use the concept, a style of reasoning is a pattern of inferential relations that are used to select, interpret, and support evidence for scientific results. In this paper, I defend the view that there is a plurality of styles of reasoning: different domains of science often invoke different styles. I argue that this plurality is an important source of disunity in scientific practice, and it provides additional arguments in support of the disunity claim. I also contrast Ian Hacking’s broad characterization of styles of reasoning with a narrow understanding that I favor. Drawing on examples from molecular biology, chemistry and mathematics, I argue that differences in style of reasoning lead to differences in the way the relevant results are obtained and interpreted. The result is a pluralist view about styles of reasoning that is sensitive to nuances of inferential relations in scientific activity.     

The literature and the science of ‘two cultures’ historiography

This paper discusses the historiography of the ‘two cultures’ controversy. C. P. Snow’s lament about the ‘two cultures’, literary and scientific, has inspired a wide range of comment—much of which begins by citing Snow and his thesis, before going on to discuss very different things. This paper focuses upon one strand of this commentary, the historical analysis of the controversy itself. A ‘historical’ analysis is defined here as one that resists the impulse to enter the argument on behalf of Snow or Leavis, to conceive of their argument in the terms that Snow defined, or to invoke their argument as a precursor to some contemporary issue. Instead, a historical interpretation registers distance between that day and this, takes the controversy itself as its object of study, and explores the tensions and associations that came to be packed into those now familiar terms. As the fiftieth anniversary of Snow’s Rede Lecture nears, this approach—rather than the repetition of clichés about the bridging of cultures—offers both analytical perspective on the controversy and interpretive possibilities for its examination.     

Francis Hauksbee's theory of electricity

Conclusion Historians of science have usually assumed that the science of electricity developed in the period prior to Franklin, or at least prior to Nollet, in what amounted to a theoretical vacuum. It has been my aim in this paper to demonstrate the falsity of that assumption. I have shown, I hope, that Hauksbee's important researches were guided throughout by strong theoretical considerations, and I have indicated that Dufay's even more important studies were guided by exactly the same considerations. Nor was their theory in any sense a stagnant one. As it was developed by Hauksbee, it could give a fairly adequate explanation of almost all the known electrical phenomena; it even enabled him to predict the outcome of experiments such as the one involving the rubbing of a globe while it was positioned near a second, exhausted, globe. With the discovery of so many new phenomena in the 1730's, the theory turned out to be no longer adequate, but it is not at all surprising that it was a few years before the full extent of its inadequacies was appreciated, nor is it surprising that a strong continuity is evident between it and the theory which eventually replaced it. In the meantime, the theory continued to serve a useful function by suggesting new lines of research to its adherents. The theory functioned, then, in the same way as any other scientific theory, and it deserves a more serious treatment than it has usually received. This paper, I hope, can serve as a beginning.     

François Viète: between analysis and cryptanalysis

François Viète is considered the father both of modern algebra and of modern cryptanalysis. The paper outlines Viète’s major contributions in these two mathematical fields and argues that, despite an obvious parallel between them, there is an essential difference. Viète’s ‘new algebra’ relies on his reform of the classical method of analysis and synthesis, in particular on a new conception of analysis and the introduction of a new formalism. The procedures he suggests to decrypt coded messages are particular forms of analysis based on the use of formal methods. However, Viète’s algebraic analysis is not an analysis in the same sense as his cryptanalysis is. In Aristotelian terms, the first is a form of ‘’, while the second is a form of . While the first is a top-down argument from the point of view of the human subject, since it is an argument going from what is not actual to what is actual for such a subject, the second one is a bottom-up argument from this same point of view, since it starts from what is first for us and proceed towards what is first by nature.     

Prelude to dimension theory: The geometrical investigations of Bernard Bolzano

This paper treats Bernard Bolzano's (1781–1848) investigations into a fundamental problem of geometry: the problem of adequately defining the concepts of line (or curve), surface, solid, and continuum. Bolzano's interest in this problem spanned most of his creative lifetime. In this paper a full discussion is given of the philosophical and mathematical motivation of Bolzano's problem as well as his two solutions to the problem. Bolzano's work on this part of geometry is relevant to the history of modern mathematics, because it forms a prelude to the more recent development of topological dimension theory.