Epistemic status: exploratory

Introduction: Starting point

One of the aspects of deconfusion that consistently trip people up is “where to start”. You have your phenomenon or cluster of intuitions, and then you… make it less confusing? There’s a bunch of heuristics (find as many examples as possible, focus on applications, try a naive synthesis…), but nothing that satisfying.

So one of my goals during my epistemology training is to look at a bunch of examples of deconfusion, and find out patterns about how they got started and how important/problematic were their starting points.

Let’s start with Sadi Carnot’s “Reflections on the Motive Power of Fire”, which founded thermodynamics as a science. And the fascinating part is that he seems to have drawn heavily on an analogy with his father Lazare Carnot’s work on a general theory of machines. At least that’s the thesis of philosopher of science John D. Norton in “How Analogy Helped Create the New Science of Thermodynamics”, the paper which was my starting point here.

Following Norton, I’ll use the names “Sadi” and “Lazare” to disambiguate the two Carnot, even though as a fellow frenchman I feel uncomfortable going on a first name basis that quickly.

What did Sadi discover?

My thermodynamics is quite rusty (never liked the subject in college), and I never studied its history. So I at least needed a reminder on what Sadi’s work was all about, and how it fit into the knowledge of its day and the narrative of thermodynamics.

From the Norton paper and Wikipedia, I get that Sadi searched for a general theory of heat engines. Actually, it looks like he introduced the more general concept of heat engine, or at least made the explicit distinction between steam and heat engine, with the former being only a specific form of the latter. What Sadi sought and found was a general enough theory of heat engines to answer questions like the best possible design of a heat engines, or the relevance of using steam vs other gases.

His answer, in a fundamental move establishing a shared frame for the next 200 years of thermodynamics, was that the efficiency of such an engine only depended on two points:

The difference of temperature between the hot source and the cold sink
How close to reversibility the transfer of heat was in the engine.

That’s a pretty cool example of a shared frame simplifying a whole mess of engineering problems into two fundamental points. And the abstractions and proof methods Sadi introduced (thermodynamically reversible processes, Carnot’s theorem, thinking in terms of cycles) proved fundamental for establishing the more secure basis of thermodynamics 50 years later.

Normally with that sort of incredible insight, I expect the path taken to be lost to time, with maybe a couple of pointers here and there. Yet it seems well accepted by historians of science that Sadi’s work was heavily inspired by an analogy to his father (Lazare Carnot) work on a general theory of machine. And Norton makes quite a convincing case of it.

Lazare’s theory of machines and the analogy to heat engines

Norton dedicates a good 16 pages of his paper to explaining Lazare Carnot’s general theory of machines. I don’t plan on going that far, but the gist is important for the analogy.

Basically, Lazare published a theory about what mattered the most in designing efficient machines, where machines are broadly speaking constructions that transmit work or “motive power” as Lazare wrote. This includes levers, winches, pulleys, but also hydraulic and pneumatic machines.

This is the first part of the analogy Norton builds: Lazare showed his son that you could unify a whole class of systems within one theory. He thus showed him that going to the abstract was a possible fruitful move.

What was this theory? Well, Lazare showed in his (apparently quite badly written) essay that the most efficient machines are those for which the percussive shocks between parts are minimized.

That doesn’t look that much like Sadi’s later thermodynamical work, until we look at Lazare characterization of motions that minimize these shocks, the so-called “geometric motions”. Honestly, I had trouble following Norton’s analysis of these geometric motions, and he implies that Lazare’s writing is even worse. The important point seems to be that Lazare characterizes them in terms of reversibility: geometric motions can be reversed, like rotation around a center of two weights linked by a taut wire.

So in this characterization of most efficient movements of machine parts through reversibility, Norton sees the seeds of thermodynamically reversible processes, which analogously characterize the design of optimal heat engines in Sadi’s work.

Another aspect of Lazare’s work that Norton highlights is its fundamental dissipative ontology: Lazare works within a mechanics of hard bodies with inelastic shocks and loss of “energy” (technically the concept wasn’t really formalized, but that’s what it amounts to). Such an ontology was ill fated in mechanics, as I gather from the fundamental nature of basically all conservation laws I’ve heard of.

Yet this mistake might have served his son and Science incredibly well; Norton argues that

Whatever may have directed Lazare’s choice of this ill-fated dissipative ontology, it was most fortuitous for Sadi. For Lazare mapped out ways of understanding systems that are inherently dissipative. When Sadi turned to analyze just such a system, heat engines, he had available to him the model of Lazare’s work. He could copy its ways and methods and, using them, devise the basis of what becomes the modern theory of thermodynamics.

To summarize, Norton draws an analogy in three points between Lazare and Sadi’s work:

The goal of a general theory for a whole class of systems
A dissipative ontology
An approach to characterizing the most efficient systems in such an ontology, based on reversibility

Surprising benefits of this analogy

Why is this exciting? Because that highlights a probable direct path Sadi took in proposing one of the most important concepts in thermodynamics: thermodynamically reversible processes. Now, there is no direct quote of Sadi referencing his father’s work by name, but Norton has a bunch of quotes where Sadi basically reexplains and restates Lazare’s work as his example and starting point.

(From Sadi’s Reflexions)

Machines which do not receive their motion from heat, those which have for a motor the force of men or of animals, a waterfall, an air-current, etc., can be studied even to their smallest details by the mechanical theory. All cases are foreseen, all imaginable movements are referred to these general principles, firmly established,and applicable under all circumstances. This is the character of a complete theory.
[...]
A similar theory is evidently needed for heat-engines. We shall have it only when the laws of Physics shall be extended enough, generalized enough, to make known beforehand all the effects of heat acting in a determined manner on any body.

I for one find this analogy quite convincing. And it helps explain why Sadi thought about such a counterintuitive idea as thermodynamically reversible processes — they are the thermodynamical analogous to Lazare’s geometric motions.

Norton definitely stresses out the weirdness of reversible processes a lot. He’s not content with just calling them ideal processes or thought experiments, because they rely on logical contradiction: they must constantly be at equilibrium (or infinitesimally close to equilibrium) while changing. He has a whole paper on it called “The Impossible Process”!

I’m not sure if I agree with Norton’s analysis, but the idealized nature of reversible processes, their importance (for example in proving Carnot’s theorem) and the surprising aspect of their invention seems widely recognized. And it all apparently follows naturally from Lazare’s theory!

Even more fascinating, Sadi drew from Lazare’s work a concept (reversibility) that should by all means break when going from the mechanical to the thermodynamical. Lazare’s reversible processes can be in a state of equilibrium (from percussive shocks) while moving because of inertia; but that doesn’t work for thermal processes. This is a part where the analogy should break, as Norton writes:

This, then, is a significant disanalogy between Lazare’s machines and Sadi’s engines. Realization of the most efficient processes for Lazare’s machines does not contradict the laws of mechanics. Realization of the most efficient processes of Sadi’s engines, however, does contradict basic thermal laws.

And yet, that was the right move to make! Pushing the analogy “too far” led to a fundamental building block of thermodynamics, itself one of the main foundations of modern physics.

Other epistemic curiosities

A bunch of other aspects of Norton’s paper titillated my curiosity, even if I haven’t yet gone deeply into any. I’m not promising I will, but they’re definitely on my mind and in my reading list.

Epistemic Analysis of Thermodynamically Reversible Processes

Already mentioned that one above, but Norton and other philosophers of science have a whole pan of the literature on analyzing the weirdness of reversible processes as a concept in thermodynamics, and how to make sense of them in light of both their logical contradiction and their fruitfulness.

The starting point would be two of Norton’s papers on the subject, and the reviews/responses in the philosophy of science literature. Plus some digging into the actual thermodynamics to get a better grasp of the use of these processes.

Impossibility of perpetual motion

Sadi’s main result, Carnot’s theorem, relies on proof by contradiction leading to perpetual motion (and heat) machines, which are considered impossible. That surprised me, because I was expecting the theorem to prove or give grounding for the impossibility of perpetual motion machines. So I became more curious of where that comes from. It sounds like there’s a path through the Second Law of Thermodynamics and Noether’s theorem, but I haven’t followed it yet.

And with the relevance of this impossibility result to physics and to Yudkwoskian’s analogies about the security mindset, that sounds like a great thing to clarify.

Caloric fluid, an interesting mistake?

One aspect of Sadi’s work that I haven’t discussed here is that he subscribed to the caloric theory of heat, which sees heat as a self-repellent fluid. This has been superseded by the kinetic theory later in the history of physics, in part because the caloric theory couldn’t deal with conservation of energy and the second law (at least that’s what Norton and Wikipedia says).

Where Norton and Wikipedia disagree is on the fruitfulness of this ontological mistake. Wikipedia describes Sadi’s accomplishment as “despite” the caloric theory, whereas Norton argues that seeing heat as a fluid pointed the way to the analogy with Lazare’s work, and also made obvious the important of the entry and exit points of heat, which end up being the only two part that matter for the efficiency limit of a heat engine.

That looks like a fascinating example of an interesting or fruitful mistake, which paved the way to getting things actually right.

Sadi’s abstraction carried over to the next paradigms

More generally, Sadi introduced a bunch of concepts and intuitions (reversible processes, his results, and his proof method for Carnot’s theorem) which resisted multiple paradigm changes and reformalization of the underlying physics. That’s a great example of principles that carry over to the next paradigm, as John writes.

Who wouldn’t want to understand better how to be so right with such imperfect foundations?

Conclusion

If we follow Norton (and I'm quite convinced by his case), Sadi got a clear inspiration and analogy from his father's work, which helped tremendously in making his theory of heat engines so powerful and eventually right.

Yet that doesn't solve our initial problem (how did Sadi discover his result?); it only refines the questions. Sure, he might have leveraged Lazare's analogy, but why did that work? Why is Lazare work so productive when applied to thermodynamics, when it's a dead end in its original field (mechanics)? And how come that the analogy leads to the right insight when it actually breaks?

Eventually I aim for epistemic tools that refine, correct and distill the underlying mental moves so that we can more easily emulate this type of intellectual progress. The first step is to realize that there's a confusion here: something that doesn't fit with our nice and clean narratives of scientific progress. And then investigate and make sense of it.