Question. Has anything other than what can be guessed from this obituary written by Max Noether survived of the 'defense' of infinitesimals that Paul Gordan gave in his doctoral disputation on March 1, 1862 in Berlin?


  • This is an interesting historical morsel that I accidentally stumbled over and that Mikhail Katz suggested to be made an MO question.

  • It should be noted that the philosophical substance of this 'discovery' is not surprising at all, rather expected and boring in fact (the only interesting result of this question would be a mathematical argument): that Noether says that Gordan thought that infinitesimals are good and that Gordan defended 'implicit solutions better than explicit solutions' only because he had to, is squarely in line with the typology found in the usual 'narrative', i.e.,

Gordan : $\mathsf{explicit}$ , Hilbert : $\mathsf{implicit}$

  • Someone having less reservations against such titles than the writer of these lines might have entitled the question "The King of Invariants and his Defense of the Infinitesimals: a lost Latin swan song?", or something like that.

  • For readers' convenience, here are relevant passages from the obituary. Source is Mathematische Annalen 75, 1914:

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My translation:

In Breslau Gordan got the topic of his dissertation; it arose from an article entitled "De linea geodetica", with which in August 1861 Gordan won a competition started by a prize question asked by the faculty. This doctoral dissertation that Gordan defended on March 1, 1862 in Berlin treated the geodetic line on the spheroid of the earth. Of the questions which Gordan defended at this disputation---in a Latin, of course, which was deemed not exactly classical by Kronecker---the following two may be mentioned; the first, because it is in line with Gordan's views both then and later in life; the second, for the opposite reason:

"The method of the Infinitely-Small is, I claim, no less precise than the method of limits."

"It is of greater interest to investigate the implicit properties of a function defined by a differential equation than to investigate in terms of which known functions it can be expressed."

Complex analysis attracted Gordan in autumn 1862 to go to Riemann [Riemann was employed at the university of Göttingen at the time]. He [...]

  • Max Noether writes as if he knew more about Gordan's thoughts on infinitesimals than what he tells readers in the obituary.

  • Noether insinuates that Kronecker was present at the disputation.

  • I don't know what the regulations for disputations were in the 1860s at University of Berlin. (This is the 'ancestor' of what is by now one of at least three universities in Berlin.) If notes were taken, then they may have survived, though I think this unlikely. (And even if they have survived: anyone who knows what minutes of exams usually look like will not expect there to be much information in them. The best one could hope for is that Gordan later himself wrote carefully about infinitesimals.)

  • One could even hope that this question has (some sort of) mathematical answer; 155 years are not much by historians' standards. Of course, one should not expect there to be much in exam minutes, even conditioning on the, by itself, unlikely event that such were taken and have survived.

Update. I now took the time to read the whole of the obituary written by Noether ('op. cit.'). I here summarize all that seems not irrelevant to the present question (to cut a long story short: nothing strictly relevant to the question will follow; the last item will border on the off-topic; Noether is basically saying that Gordan did not use limits simply because he didn't like them, or just didn't care, and avoided them, not because he had a thought-out positive theory about infinitesimals; it seems more of a rather boring avoidance of second-order definitions and a preference for first-order definitions, than any positive theory about infinitesimals, making the chances of any appreciable answer to this question seem even slimmer than they already seem just from thinking about the question; I don't expect there to ever be much of an answer):

  • The most relevant item in this 'update' is the following passage in op. cit., in which Noether describes Gordan as a teacher:

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My attempt at a translation:

In his own field [Noether has written in the immediately preceding paragraph that Gordan liked to read German literature] it was not so much the perusal of the work of others than a 'big picture' of the inner connections and an instinctive feeling for the paths and goals of mathematical endeavours which enabled him to distinguish the valuable from the inferior, and small hints were sufficient for this. However, Gordan never did justice to foundational conceptual developments: also, in his lectures he entirely avoided any basic definitions of a conceptual kind, even the definition of 'limit'. [emphasis added] His lectures extended only to mathematics of the common kind, occasionally also to the theory of binary forms; the exercises were preferably taken from algebra. He liked to lecture on the work of Jacobi [Gordan's doctoral advisor], e.g. on functional determinants, yet never about complex analysis, higher geometry or mechanics1; he also did not have anyone give seminar presentations. The lectures [of Gordan], essentially, had an effect rather due to his vivacious way of speaking, and due to a vigour which encouraged further independent study, than due to systematizing and rigor.

  • Noether on the first page of op. cit. names his 'informants', which explains the pretense Noether's of being informed about so much in Gordan's life:

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Part of the relevant [i.e.: relevant for the biographical parts of the obituary] data was taken from [Gordan's] dissertation (I of the bibliography at the end). For other bits of information I am indebted to R. Sturm, a fellow student of Gordan at Breslau, C.F. Geiser, one of the opponents at the Berlin disputation in 1862, J. Thomae, a fellow student at Göttingen, and A. Brill, a colleague at Gießen; furthermore, I am indebted to L. Schlesinger for information from files at Gießen.

  • Noether also describes what he thinks is the only non-algebraic work Gordan ever did after his dissertation:2

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My translation:

Only once did Gordan ever work on a non-algebraic topic: in 1893 Hilbert had, based on a new idea, given a simple proof of the transcendence of the numbers $e$ and $\pi$; and immediately afterwards Hurwitz had given a modification of the proof, by way of avoiding the integrals used by Hilbert, essentially by using integration-by-parts, and by the use of the simplest case of the intermediate-value-theorem. In an article (54), which by the way only had the aim of giving an exposition of Hurwitz's deduction, 'played backwards', Gordan took one more step in towards an elementary proof, by replacing also the still remaining differential quotients by use of the Taylor series of $e^x$. [emphasis added; by the way: what is nowadays the 'most' elementary proof that $\pi$ and $e$ are transcendental; the only proof I ever studied in detail used integrals, and, as such, limits] Regarding the $r!=h^r$ symbolism, which has been emphasized by various others after Gordan's paper, it must however be said that it does not play any further role in the proof than being a mere abbreviating notation, and that it moreover has simply been lifted from Hurwitz's note. The estimation of the vanishing remainder term, an estimate which is essential for the proof and was left out of the note (54), has been carried out by Gordan in 1900 in an unpublished notebook in Gordan's Nachlass. 3

  • The 'MathSci-Net' and 'Zentralblatt' of the 19th century was the 'Jahrbuch über die Fortschritte der Mathematik'; therein, Heinrich Burkhardt reviewed Gordan's 'transcendent note'; for convenience, since it is not entirely off-topic (it is 'Gordan on the non-algebraic'), I reproduce the review here (it is also something of a self-contained summary of Hilbert's proof, and as such, maybe of interest in and of itself):

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The blue link in the yellow lead to a review which is (displayed to me as) empty.

1To my way of thinking, this is all rather expected and unsurprising and in line with the usual reputation of Gordan: all the subjects Noether says Gordan avoided are second-order theories; in particular, in the mechanics of the time variational principles had an important role, and variational principles involve quantifier over sets of functions, i.e., second-order quantifiers; 19th century mechanics/variational principles/principles of least action, etc. to me seem recognizably non-algebraic. Also, one could now very much overinterpret and say something about whether and how much infinitesimals help in complex analysis, and that there, perhaps, limits are more necessary than in real analysis, and Gordan avoided complex analysis because of this. This, however, is evidently an overinterpretation.

2 And to me, it seems evident that if there is any hope of a mathematical answer, then this hope comes from the possibility that Gordan might later have written about infinitesimals/used them for something in published printed work. However, unfortunately, I think this is unlikely.

3 It might be an interesting separate historical problem to locate this "unpublished" notebook. Please note, though, that I don't think this relevant for the question: what Noether describes sounds like a rather ordinary estimate of a remainder term, presumably by inequalities; it does not sound like a use that Gordan made of infinitesimals, which don't yield effective estimates anyway, as far as I know. One should be clear that there is no sign that Gordan ever recognizably used infinitesimals, except for his thesis defense.

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    $\begingroup$ The last paragraph of your answer can probably be deleted. First, the kind of argument you propose involves a distinction between theory and model that was simply not available at the time, so Gordan could not have formulated even a remote approximation of such an argument (perhaps Frege conceivably could, but Gordan was no Frege). Second, we have no way of knowing whether the archives for the disputatio exist and/or have possibly been preserved, so there is no reason to be overly pessimistic about it. $\endgroup$ Commented Sep 15, 2017 at 7:38
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    $\begingroup$ Perhaps we should note that this (1861) is before Dedekind, Weierstrass, Cantor et. al. made limits more rigorous. So in fact we could argue that in 1861 infinitesimals and limits were equally bad in terms of rigor. By around 1900, limits had reached a modern sort of rigor. And by around 1960 (Robinson), so did infinitesimals. $\endgroup$ Commented Sep 15, 2017 at 13:28
  • $\begingroup$ @GeraldEdgar, I thought Cauchy was the one who rigorized analysis in your view. Cauchy died in 1857. Thus Cauchy had been dead for 4 years in 1861. Barring any posthumous contributions by Cauchy (that I am not aware of), analysis would have been rigorized at least 4 years before 1861 then. Why do you say then that "limits were bad in terms of rigor" in 1861? $\endgroup$ Commented Sep 17, 2017 at 12:58
  • $\begingroup$ @MikhailKatz: if one takes the conclusions of [Gordon M. Fisher: Cauchy and the infinitely small. Historia Mathematica 5 (1978), 313-331] at face value, then Gerald Edgar is right with "in 1861 infinitesimals and limits were equally bad in terms of rigor". Roughly, Fisher demonstrates that Cauchy was not committed to the contemporary notion of 'limit'. E.g., op. cit. p. 330: "In his textbooks, Cauchy sometimes argued as if he were dealing with actual infinitesimals. The variables with limit zero that he often used were not fully analyzed, but they had "values" [...]" I recommend op.cit. $\endgroup$ Commented Sep 18, 2017 at 5:41
  • $\begingroup$ Hi Peter, in his fine answer Carlo seems rather pessimistic about finding any trace of this but it may still be worth contacting the experts you mentioned in case other universities had other practices. As far as Cauchy is concerned I would prefer communication by private mail if you are interested. $\endgroup$ Commented Sep 18, 2017 at 15:31

1 Answer 1


Paul Gordan's theses were published in De linea geodetica and digitised by Google, from which I reproduce the relevant page:


  • I. The method of functional division, proposed by the respectable Cournot as an analytical and empirical method, appears unsuitable, since one does not have fully empirical functions.
  • II. The method of the infinitely-small is, I claim, no less precise than the method of limits.
  • III. It is of greater interest to investigate the implicit properties of a function defined by a differential equation than to investigate in terms of which known functions it can be expressed.
  • IV. The principles of Democritus remain to the present day as the foundation of the theory of atoms.

One can also learn from this publication that the opponents of Gordan at the March 1, 1862 Disputatio were Dr. Kretschmer, Mr. Geiser, and Mr. Rathke, presumably E.E. Kretschmer and C.F. Geiser. Gordan acknowledges as his teachers in mathematics the professors Kummer, Joachimstal, Richelot, Rosenhain, Schroeter, and Galle --- but it is unclear whether any of these were present at the defense ceremony.

Concerning the question "what was actually said at the defense"; at least in my University (Leiden) the practice through the centuries has been that the examination itself is not recorded, the secretary records only the outcome (pass/fail/cum laude etc.).


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