Mathematics is an essential tool for today’s physicists. Only with mathematics can physics be made precise, and accurate forecasts be made, to be verified through observations of nature and experiments in laboratories.
In the next few posts, I will be writing about the origins of the infinitesimal calculus, without which Newtonian physics could not have been developed.
In this post, however, I will focus on some critiques of the use of mathematics in thermodynamics as an example of how not to use mathematics. I am grateful to Johan Hoffman, Claes Johnson and Murtazo Nazarov, who collected a number of wry or amusing quotations in Chapter 4, entitled “Telling the Truth”, of their book Computational Thermodynamics1.
In addition to being my first step in discussing mathematics, this post is also my first step in discussing thermodynamics. Already, I presented in a previous post, The Old Productive Research Programs Refuse to Die, that thermodynamics might be problematic:
Did the work of Lazare and Sadi Carnot, another father-and-son pair, on thermodynamics suffer the same treatment as did the work of Fresnel and Ampère? Because Sadi Carnot was accused by no less than William Thompson of having violated the principle of conservation of energy. Oh, that sounds familiar, that is exactly what they accused Weber of having done.
So let us look at some of the aforementioned quotations. We begin with the great Soviet-Russian mathematician V.I. Arnol’d2 (1937-2010):
Every mathematician knows it is impossible to understand an elementary course in thermodynamics. The reason is that thermodynamics is based—as Gibbs has explicitly proclaimed—on a rather complicated mathematical theory, on the contact geometry. Contact geometry is one of the few ‘simple geometries’ of the so-called Cartan’s list, but it is still mostly unknown to the physicist—unlike the Riemannian geometry and the symplectic or Poisson geometries, whose fundamental role in physics is today generally accepted. [my emphasis]
If one of the world’s foremost mathematicians of the twentieth century writes that “thermodynamics is based … on a rather complicated mathematical theory”, and that “Every mathematician knows it is impossible to understand an elementary course in thermodynamics”, how is thermodynamics to be understood by mere mortals who are not professional mathematicians?
The next quotation comes from Stephen G. Brush3 (1935-), writing an explanatory note with respect to Rudolf Clausius’s (1822-1888) Mechanical Theory of Heat.
As anyone who has taken a course in thermodynamics is well aware, the mathematics used in proving Clausius’ theorem (the 2nd Law) is of a very special kind, having only the most tenuous relation to that known to mathematicians. [p.581, n.39]
Brush’s biting sarcasm is perhaps the sharpest criticism I have ever seen of any scientific work! He continues that note by quoting from The Tragicomical Story of Thermodynamics: 1822-18544, written by the very colourful Clifford Truesdell (1919-2000). The remainder of the critiques of thermodynamics in this post will come from that work.
Truesdell’s book focuses on the development of thermodynamics from Sadi Carnot (1796-1832) to Rudolf Clausius. He praises the intuition of Carnot, but criticizes him intensely for not carefully developing his concepts and expressing them mathematically. Here is how Truesdell begins his chapter on Carnot:
Despite his schooling at the Ecole Polytechnique and despite his early training by his father, who was a thoughtful mathematician, Carnot does not follow the tradition of eighteenth-century rational mechanics he has just praised for its generality and extent. Instead, the sardonic muse directs him to write in a medium that anybody can understand. An obvious necessary condition is that no mathematics be used in the main text. This condition did not turn out to be sufficient. Among all writers on natural philosophy only Carnot equals the pre-Socratics in ability to provoke an infinite sequence of cyclic quandary, acute and painful ponderation, conjecture, gloss, controversy, and quandary again, which bears witness that the outcome is comprehensible by nobody. [p.80, my emphasis]
And this is how Truesdell begins his conclusion on Carnot:
In Carnot’s work there is no sign of the scrupulous analysis of concepts, the inexorable rejection of every postulate or axiom not necessary to the end desired, that is the essence of mathematics in general, of the rational mechanics of the seventeenth and eighteenth centuries in particular.
It was not experiment that was wanting; it was mathematics. [p.134, my emphasis]
Truesdell accepts that Carnot made many conceptual advances in thermodynamics, but failed in making them clear. But Truesdell’s harshest critique is for Carnot’s successors, who made no attempt to clarify the problematic aspects of the latter’s work.
Few men have done so much to found any science as has Carnot for thermodynamics. Our analysis of his work has revealed magnificent success in part, though failure in the end. Carnot was by no means alone in failing much while doing much. The curse of thermodynamics has been, not that, as happened in every other branch of physics, the great creators occasionally erred or failed, but that their successors have treasured the errors and the deficiencies while neglecting to seize, purify, and exploit the successes. [p.135, my emphasis]
According to Truesdell, not only had Carnot not properly “stated his assumptions clearly in mathematical terms and given explicit mathematical proofs of his deductions from them”, but these problematic aspects of his work “were cherished, enshrined, and magnified by his successors”.
[H]is preference for undisciplined, unmathematical arguments; his primitive use of the infinitesimal calculus; his tendency to sweeping claims about maxima without specifying what is held fixed and what is allowed to vary, or even the variables upon which the thing maximized depends; his predilection for steam and coal; his appeal to irrelevant or at best merely ancillary experimental details; his reluctance to face the test of comparing his own results with the successful theories of others; and, finally and above all, his confusion of the constitutive properties of special substances with the general relations between heat, work, and change of temperature—all these were cherished, enshrined, and magnified by his successors. Such became the tradition of the subject. An eruption of paper covered with symbols and the data of experiments could have [been] spared, had Carnot stated his assumptions clearly in mathematical terms and given explicit mathematical proofs of his deductions from them.
Among physicists of the first rank, Carnot is the first who was not in at least equal measure a mathematician. Thermodynamics is the first mathematical science to have been invented without the control afforded by patient, merciless, mathematical criticism. It has suffered from this congenital defect from 1824 until now. [p.136, my emphasis]
Truesdell does not just take to task Carnot’s mathematics, but also his physics. In particular, he focusses on the fact that time is not a factor in his thermodynamics, nor that of his successors!
On the physical side, most unfortunate is Carnot’s failure to give any position to irreversible processes, despite their everyday familiarity and despite Fourier’s work, already famous though only recently published. Here, too, Carnot’s limitation is to become characteristic of thermodynamics: Rather than extend the existing successful theories so as to embrace new ranges of phenomena, thermodynamics will rule them out from the start. Having put on the stage as protagonist a pygmy, the ideal gas, Carnot appoints as director a Mephistopheles who tells him it makes no difference which way he goes. The “reversible” process, a prototype of Liberal Philosophy, is to keep thermodynamics turning in ineluctable circles for over a century. For such processes the time makes no difference. Thus the letter t is free, and Carnot, unlike Fourier and Laplace, uses it for temperature. As all thermodynamicists were to follow Carnot, it came to seem impossible that thermodynamics could ever mention the time. The very letter for it was already used up! Even Kelvin, a virtuoso in heat conduction and mechanics, in his papers on thermodynamics refrained from using t for anything but temperature and from introducing any letter at all to denote the time. Thus begins that quality of classical thermodynamics that to the modern student is most striking: its timelessness. [pp.136-137, my emphasis]
Truesdell’s criticism of Carnot is clearly harsh, but this is nothing compared to the words used to address Rudolf Clausius.
Clausius’ verbal statement of the “Second Law” makes no sense, for “some other change, connected therewith” introduces two new and unexplained concepts: “other change” and “connection” of changes. Neither of these finds any place in Clausius’ formal structure. All that remains is a Mosaic prohibition. A century of philosophers and journalists have acclaimed this commandment; a century of mathematicians have shuddered and averted their eyes from the unclean.
The brief remarks on irreversible processes make no sense either, since a “process” has not been defined or illustrated except within a structure that provides only bodies susceptible of no processes but reversible ones. Clausius tells us that he could easily calculate “the equivalence-values of the uncompensated transformations” but gives no illustration and no idea what we could do with such a quantity if we had it. [p.333, my emphasis]
Truesdell’s difficulty in understanding Clausius is not for lack of trying.
More attention was to be paid to the oracles of Clausius, which have been repeated, embroidered, and glossed in all the textbooks. What they are to mean is another matter.
Seven times in the past thirty years have I tried to follow the argument Clausius offers to conclude that the integrating factor T exists in general; is a function of θ alone, and is the same for all bodies, and seven times has it blanked and gravelled me…. I cannot explain what I cannot understand. [p.335, my emphasis]
Truesdell states that Clausius transforms thermodynamics from “largely a model for the way things are” to “a model for the way things are not”.
Clausius’ first paper, while entangled and slack, was in aim and result constructive. From his second paper, on the contrary, through the murk and gloom emerges a growing aura of retreat and impending failure. While in all work analysed up to now there was no hint that conditions were any more specific than the equations themselves suggested, in this paper Clausius assumes that “the pressure always changes very gradually,” though he specifies no time scale sufficient to give meaning to the term “gradual”. Here the tergiverse “quasistatic process”, hinted at by Reech, first slithers onto the scene. It joins the “state” as a principal engine of the mystic double-talk that makes thermodynamics different in kind from all the rest of classical physics.
But that is far from the worst. Hitherto thermodynamics had been, like any other theory in mathematical physics, pretty largely a model for the way things are. In Clausius’ hands it now begins to change into a model for the way things are not. The old theory, based on the Doctrine of Latent and Specific Heats, makes all processes “reversible”. Clausius seems suddenly to see that in nature we cannot run engines backward. To the great geometers of the previous century this idea (which, if I may be permitted an unhistorical conjecture, would not have seemed the least bit startling or new) would have been a challenge to construct a general and inclusive theory. To Clausius it was sufficient reason to confine the circumstances so as to fit the existing theory. [pp.337-338, my emphasis]
Truesdell concludes with yet another biting remark. With thermodynamics, physicists decided that if their knowledge of mathematics was not sufficient, they would simply “cut down the problem to its size”!
The thermodynamics of the nineteenth century began a new style, in which the physicist applied what mathematics he happened to know. If that mathematics did not suffice, he cut down the problem to its size. Such a physicist did not think it necessary for his students to learn any mathematics beyond what he himself had been taught when he was a student. Mathematical research meanwhile advanced swiftly, but little of it was learnt by physicists. It became purer and purer. Physicists finally began to think that all this new mathematics was useless; that mathematicians neglected their duty to teach the good old mathematics already used in physics and hence (obviously!) destined to suffice it forever; and that they themselves should teach “physical” mathematics to their students: mathematical tools for the physicist! [p.339, my emphasis]
Reading the above quotations, it is clear that from a mathematical perspective, there is something wrong with thermodynamics as a discipline. But, unlike my original intuition, it appears that the problems did not begin with people like Clausius. Rather, the very fact that Carnot did not put thermodynamics on a sound foundation right from the beginning has plagued the discipline with serious problems ever since.
I would like to conclude with Truesdell’s hope, expressed in the introduction to The Tragicomical History of Mathematics, entitled “The Producer’s Apology to the Spectators”:
Finally, I confess to a heartfelt hope—very slender but tough—that even some thermodynamicists of the old tribe will study this book, master the contents, and so share in my discovery: Thermodynamics need never have been the Dismal Swamp of Obscurity that from the first it was and that today in common instruction it is; in consequence, it need not so remain. [p.6, my emphasis]
Physics and mathematics are hard. If parts of these from previous generations are unclear, it is the responsibility of following generations to clarify or replace these parts to improve the situation. In future posts, I hope to look at these issues in more detail.
Johan Hoffman, Claes Johnson and Murtazo Nazarov. Computational Thermodynamics. 2012.
V.I. Arnold. Contact geometry: the geometrical method of Gibbs’ thermodynamics. In D.G. Caldi, G.D. Mostow, editors, Proceedings of the Gibbs Symposium: Yale University, May 15-17, 1989. American Mathematical Society, 1990, pp.163-80.
Stephen G. Brush. The Kind of Motion We Call Heat: A History of the Kinetic Theory of Gases in the 19th Century. Book 2: Statistical Physics and Irreversible Processes. North-Holland, 1976.
C. Truesdell. The Tragicomical Story of Thermodynamics: 1822-1854. Springer, 1980.
Thanks for this post. Truesdell continues to be an inspiration for physicists aiming to make sense of reality.
Very interesting! A law of physics is not usually provable. It is curious that the fathers of thermodynamics felt the need to prove it.