I wanted to talk about some interesting observations made by Stephen Brush, an important commentator of the history of the molecular interpretation of thermodynamics .
As noted by Brush , Friedrich Nietzsche wrote:
If the universe may be conceived as a definite quantity of energy, as a definite number of centres of energy -and every other concept remains indefinite and therefore useless- it follows therefrom that the universe must go through a calculable number of combinations in the great game of chance which constitutes its existence. In infinity, at some moment or other, every possible combination must once have been realized; not only this, but it must have been realized an infinite number of times. And inasmuch as between every one of these combinations and its next recurrence every other possible combination would necessarily have been undergone, and since every one of these combinations would determine the whole series in the same order, a circular movement of absolutely identical series is thus demonstrated: the universe is thus shown to be a circular movement which has already repeated itself an infinite number of times, and which plays its game for all eternity.
This short text amounts to a confutation of the second law of thermodynamics.
The second law of thermodynamics says: when you take a cake out of the oven, it cools down. By imagining stars to be hot cakes in the process of cooling down, and applying the second law to the whole universe, some thinkers had come to the conclusion, before the time of Nietzsche, that the unavoidable fate of the universe might be to cool down, in its entirety, and to become a shapeless pudding.
However, Nietzsche’s argument suggests that the second law, along with its depressing consequences, may be at odds with the idea of a universe that is made up of a huge number of molecules moving about in frenetic way (an idea which we now take for granted).
Because, if a cake cools down due to the frenetic motion of its molecules, the same frenesy will necessarily drive the cake molecules to try all possible arrangements available to them, thus eventually causing any cake to heat itself up again and again an infinite number of times without need of an oven.
Nietzsche’s argument was arrived at independently by the French mathematician Henri Poincaré: it is in fact Poincaré’s argument that is (often unwillingly) incorporated into the general knowledge of the average physicist.
However, as stressed by Brush, Nietzsche’s aim was to find an argument, supported by the molecular interpretation of matter, to use against those that he considered unacceptable consequences of the second law. On the contrary, the aim of Poincaré and his supporters was to confute the existence of molecules, which back then was a very controversial idea.
In other words, even though physics in our day does not attribute any scientific importance to Nietzsche's argument, Nietzsche's interpretation of the conflict between molecules and thermodynamics -which is an undeniable mathematical difficulty- is equivalent to that of modern physics. However, since Nietzsche is usually seen as a crazy-philosopher-guy, and since the respectable mathematician Poincaré was using his same argument to confute the existence of molecules (which is something that no one today would dare to do), the whole matter nowadays tends to be downplayed as one of little relevance.
And maybe it is of little relevance. No one I know has ever heard of a cake heating itself up.
On the other hand, though, maybe we are living in one of those cakes. We are usually told that the formation of our planetary system can be modelled as a big cold pudding-like cloud of gas, which collapsed on itself to form the big-hot-cake which we call the Sun.
Those stubborn enough to question whether this is in line with the second law of thermodynamics are usually derided for not understanding thermodynamics properly, and are given the simple homework of figuring out why thermodynamics is still safe and sound.
I'll just say that, so far, I have found only this and this actual explanations of how it is possible, according to the second law, that the Sun-cake heated itself up. The two explanations don't seem to be very in line with each other. In particular, the first explanation crucially invokes a concept called "coherence", which does not appear in the second explanation (nor in the statement of the second law of thermodynamics, for that matter). The second explanation is discussed here by people who don't seem to be able to make heads or tails out of it.
As they say in tv shows, all this is very circumstantial. But it also suggests that something is rotten in the state of Denmark.
In general, I have the feeling that the gross overlooking of the parallel between Poincaré's recurrence paradox and Nietzsche's idea of eternal recurrence will come increasingly under scrutiny in the near future. In particular, the unresolved conflict between the second law and the existence of molecules resonates quite deeply with the apparent dissonance between the second law (which seems to transform things into a pudding) and the process of evolution of life (since life is clearly not a pudding... or is it.). So far, the only academic response I’ve heard to the latter problem is this: “if you think there’s a contradiction, then you don’t understand thermodynamics properly, because the Sun provides the Earth with plenty of ordered energy, and this is enough to justify the existence of life”. But, like I was saying, it seems to me that our Sun-cake itself has some explaining to do regarding its own self-baking.
In any case, I think that there is a more immediate way in which Stephen Brush's observations may be useful.
Nietzsche is often pictured as a crazy philosopher who might have provided Hitler with the inspiration for the best ideas on how to kill Jews. Physicists are more or less fine with this, because, due in particular to the influence of people like Richard Feynman, during the last century philosophers have come to be considered as rambling time-wasters that have mainly hindered the development of so-called “modern science”.
Regarding this point, it’s worth noticing that the framework of “modern physics” has been laid by Ludwig Boltzmann, who was a philosopher as much as a physicist, and that he developed this framework to answer the question “why does the second law of thermodynamics exist”, which is an inherently philosophical question (because of the why). In other words, Feynman owed, and his community of followers still owe, their comfortable jobs and social positions to a philosopher. I find it impossible not to have a visceral respect for people who dedicate their whole lives to tackle difficult mathematics. But that doesn't make it ok for them to spit on the plate where they eat.
The response of the philosophical community, concerning Nietzsche’s image as a mad forerunner of the nazis, seems to be that, since the things that Nietzsche was saying are too difficult for ordinary people to understand anyway, it's ok to just let the poor uneducated fools think that he was crazy.
However, in the argument which I quoted above, Nietzsche describes his eternal recurrence idea in very concrete terms. In fact, he describes a very concrete mathematical result with a clarity that a mathematician could hardly have been able to afford.
Brush here anticipates a criticism, by saying that many physicists and mathematicians will say that Poincaré's argument was a rigorous proof, whereas Nietzsche's was just another one of his ramblings. However, Brush says, Poincaré's proof would not be considered rigorous by today’s standards, whereas it is undeniable that Nietzsche’s text contains the main idea of the proof.
Nietzsche wasn’t a mathematician: he purposely decided to educate himself in the physical sciences in order to tackle a difficulty he was experiencing in his understanding of the world. By doing so, he identified one of the most widely known results of one of the last great mathematicians in history. To me this hints strongly to the fact that Nietzsche was up to something more than just rhetorics.
Just in passing, I think it's worth remarking that this argument of Nietzsche's is included in The Will to Power, a work which was published after his death, and which many consider to have been manipulated by his sister in order to associate Nietzsche and his reputation with ideas akin to those which later gave birth to nazism. This might be at the root of the fact that this text is not usually used to elucidate the concept of eternal recurrence, and the idea is usually introduced through other works, such as Zarathustra, where the philosopher expresses it in far more cryptic terms.
Be it as it may, the overlap between Nietzsche’s and Poincaré’s arguments vindicates the possibility of tackling knowledge as a unitary entity from which specific instances ramify, rather than as a fragmented collection of disparate notions. This may offer a foothold on which to step in order to rise out of the swamp of dogmas in which we somehow have come to find ourselves.
1. The importance of Brush's work in putting order in this field is demonstrated by the fact that his books are one of the few places where English translations of the original works of Ludwig Boltzmann can be found. Accordingly, the difficulty in finding these translations is a vivid manifestation of the lack of commitment that physicist have shown during the last century to understanding the foundations of their own subject.
2. Stephen G Brush, Nancy S Hall (2003). The kinetic theory of gases: an anthology of classic papers with historical commentary History of Modern Physical Sciences DOI: 10.1142/9781848161337, available from here.
3. F. Nietzsche, Der Wille zur Macht, in Gesammelte Werke, Musarion Verlag, Munich, 1926, Vol. 19, Book 4, Part 3; English translation by O. Manthey-Zorn in Nietzsche, An Anthology of his works, Washington Square Press, New York, 1964, p. 90. Another translation available from here.