[FOM] In response to M. Detlefsen's post "From theorems of infinity to axioms of infinity" of March 9, 2013

Gregory Taylor Gregory.Taylor at baruch.cuny.edu
Sun Mar 31 17:48:12 EDT 2013

Regarding Zermelo's adoption of the Axiom of Infinity in his 1908 paper, a chapter from the history of science seems relevant.  I have in the mind the so-called ``Boltzmann Controversy,'' wherein Zermelo, Max Planck's assistant in Berlin, was himself a central player, albeit on the losing side.

Everything I am about to write is very well known, and I draw heavily on H.-D.~Ebbinghaus' biography of Zermelo (see pp.~15--26 including ample bibliographic citations).  However, the story may be worth repeating since no FOM member has mentioned it thus far, I believe.  In the end, I see a rather clear analogy between Zermelo's position with respect to the Second Law in and around 1896 and his adoption of the Axiom of Infinity in 1908.  To jump ahead somewhat in our exposition, he saw both principles as amply justified on empirical grounds and, accordingly, in no need of additional, questionable, theoretical explanation.  Therefore, it may be unfair to suggest that Zermelo's motivation was ``expedience,'' as does M. Detlefsen in his initial posting.

During the last decade of the nineteenth century a controversy arose among German physicists concerning the correct interpretation of the Second Law of Thermodynamics, which states that a closed physical system undertakes a spontaneous transition to a state of equilibrium.  In the context of the Kinetic Theory of Gases, a closed volume containing gas molecules will, according to the Second Law, migrate toward a state of homogeneity or maximum entropy whereby a total absence of differentiation-reducing processes would be observed.

Ludwig Boltzmann, already well before this period, had argued for a mechanistic reduction of the Second Law to a probabilistic account of the manner in which closed systems like the one just mentioned tend toward states of highest probability, where the probability of any macrostate $s$ of the system is the ratio of the number of system configurations realizing $s$ to the number of possible system configurations overall (if my understanding is accurate).  ``Mechanistic'' here refers to the view that the heat energy of a system is reducible to the motions (kinetic energy) of the individual atomic particles contained within that system.

In contrast, those Ebbinghaus calls ``energeticists''---he mentions Mach and Ostwald in addition to Planck---regarded energy (heat) as a fundamental, irreducible concept in terms of which ultimate physical principles like the Second Law are to be expressed.  The young Zermelo sided with the energeticists and, in 1896, published a paper in which he presents an argument to the effect that Boltzmann's reduction of the Second Law to probability-theoretic principles contains an internal inconsistency.  The details of his argument seem not so relevant.  Suffice it to mention that Zermelo's argument rests on Poincare's so-called ``Recurrence Theorem,'' of which Zermelo gives his own proof.

Most relevant here is what Zermelo writes in a rejoinder, also published in 1896, to Boltzmann's response to Zermelo's first paper, and I quote.

As for me (and probably I am not alone in this opinion), I believe that a single universally valid principle summarizing an abundance of established experimental facts according to the rules of induction is more reliable than a \emph{theory} that by its nature can never be directly verified; so I prefer to give up the \emph{theory} rather than the \emph{principle}, if the two are incompatible.  (Ebbinghaus, p.~22)

The ``single universally valid principle'' mentioned is of course the Second Law of Thermodynamics.  The reference to the lack of direct verification would concern the statistical nature of the mechanistic reduction of the Second Law and of mechanistic principles generally.

Turning now to the Axiom of Infinity and 1908, Zermelo would surely have seen that principle as ``universally valid'' based on ``an abundance of experimental facts.'' Those facts would surely have included, on his view, the reductions, due to Dedekind and Cantor, of the concept of a real number to that of natural number (and an as-yet-unexplicated concept of order).
Zermelo's response, from the same period, to objections to his Axiom of Choice seems relevant.  He writes that a decision to adopt a mathematical principle is justified, in part, by its theoretical fruitfulness.
Zermelo was never called upon to justify the Axiom of Infinity in a similar manner.  We are convinced that, if the need had arisen, he would have done just that.

Finally, I will mention an apparent, temporal discrepancy between the situation with respect to the Second Law of Thermodynamics and that of the Axiom of Infinity.  Namely, in the case of physics, the Second Law came first and the, by Zermelo's lights, problematic mechanistic reduction (``theory'') only afterward.  In contrast, in the case of mathematics, the problematic justifications, in the guise of psychologistic proofs of the existence of infinite collections, were given before the Axiom of Infinity itself.  This seems not so important in the end, however.  My guess is that Zermelo believed himself to see something like his Axiom of Infinity in the writings of his predecessors.

One final thing.  Although I have not seen the German of Zermelo's second paper from 1896, ``universally valid'' in the English translation is probably a rendering of the same German phrase (likely ``allgemein gueltig'') used in his 1908 paper on the Axiom of Choice.

I wrote a paper published in the Notre Dame Journal that addresses the justification issue in connection with Zermelo, and S. Shapiro followed up with a paper in the Review of Symbolic Logic in 2009.  Neither of those two papers mentions the Second Law of Thermodyamics.  However, it seems quite relevant to me now.

Gregory Taylor (gregory.taylor at baruch.cuny.edu)
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