On the Hypotheses Which Lie at the Bases of Geometry

by Bernhard Riemann, Jürgen Jost (Editor)

Vengono qui raccolti, oltre al testo della famosa prolusione del 1854 sui fondamenti della geometria, alcuni scritti del grande matematico tedesco Bernhard Riemann: a saggi di carattere tecnico-scientifico, i cui compaiono alcuni dei temi più discussi del periodo (la natura dell'etere, la fisiologia, la psicologia della percezione, ecc.), si affiancano scritti metodologici e anche filosofici in senso stretto. Questi ultimi mostrano il grande interesse di Riemann per le idee e i personaggi del dibattito filosofico dell'epoca, e gettano nuova luce sulla sua stessa produzione scientifica. La concezione della scienza offerta da questi scritti è assai significativa del personaggio e della sua epoca. Per Riemann la matematica non è un mero strumento esteriore da applicare, appunto dall' "esterno", ai fenomeni. Al contrario, essa consente di spingersi con rigore necessario oltre la superficie delle cose e di penetrare sempre più a fondo nella realtà, nell'ottica di una concezione unitaria del sapere scientifico.

Genres: Science, Mathematics

182 pages, Paperback

First published February 22, 2010

About the author

Georg Friedrich Bernhard Riemann [ˈʁiːman] ( listen) (September 17, 1826 – July 20, 1866) was an influential German mathematician who made lasting contributions to analysis, number theory, and differential geometry, some of them enabling the later development of general relativity.

Riemann was born in Breselenz, a village near Dannenberg in the Kingdom of Hanover in what is the Federal Republic of Germany today. His mother, Charlotte Ebell, died before her children had reached adulthood. Riemann was the second of six children, shy and suffering from numerous nervous breakdowns. Riemann exhibited exceptional mathematical skills, such as calculation abilities, from an early age but suffered from timidity and a fear of speaking in public.

Riemann held his first lectures in 1854, which founded the field of Riemannian geometry and thereby set the stage for Einstein's general theory of relativity. In 1857, there was an attempt to promote Riemann to extraordinary professor status at the University of Göttingen. Although this attempt failed, it did result in Riemann finally being granted a regular salary. In 1859, following Lejeune Dirichlet's death, he was promoted to head the mathematics department at Göttingen. He was also the first to suggest using dimensions higher than merely three or four in order to describe physical reality—an idea that was ultimately vindicated with Albert Einstein's contribution in the early 20th century. In 1862 he married Elise Koch and had a daughter.

Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866. He died of tuberculosis during his third journey to Italy in Selasca (now a hamlet of Verbania on Lake Maggiore) where he was buried in the cemetery in Biganzolo (Verbania). Meanwhile, in Göttingen his housekeeper discarded some of the papers in his office, including much unpublished work. Riemann refused to publish incomplete work and some deep insights may have been lost forever.

Riemann's published works opened up research areas combining analysis with geometry. These would subsequently become major parts of the theories of Riemannian geometry, algebraic geometry, and complex manifold theory. The theory of Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz. This area of mathematics is part of the foundation of topology, and is still being applied in novel ways to mathematical physics.

Riemann made major contributions to real analysis. He defined the Riemann integral by means of Riemann sums, developed a theory of trigonometric series that are not Fourier series—a first step in generalized function theory—and studied the Riemann–Liouville differintegral.

He made some famous contributions to modern analytic number theory. In a single short paper (the only one he published on the subject of number theory), he investigated the Riemann zeta function and established its importance for understanding the distribution of prime numbers. He made a series of conjectures about properties of the zeta function, one of which is the well-known Riemann hypothesis.

He applied the Dirichlet principle from variational calculus to great effect; this was later seen to be a powerful heuristic rather than a rigorous method. Its justification took at least a generation. His work on monodromy and the hypergeometric function in the complex domain made a great impression, and established a basic way of working with functions by consideration only of their singularities.

Reviews

[William Bies · Rating 5 out of 5]

Bernhard Riemann’s brilliant inaugural lecture of 1854 – a prime instance of the originality which (in contrast to ours) the German university system of the nineteenth century was designed to foster – does not merely found the field of differential geometry that has come to bear his name but outlines a blueprint for how, in principle, we are to think about mathematical space in general and its relation to the world of experience, as encompassed by the natural sciences [Über die Hypothesen, welche der Geometrie zu Grunde liegen (Aus dem Nachlass des Verfassers mitgetheilt durch R. Dedekind), Habilitationsschrift, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, mathem. Classe 13, 133-152 (1868)].

About the present edition released in 2016 by Springer Verlag: it is rather unfortunate that the German version of Riemann’s text is not included at all, despite the fact that it easily could have been (an index of the low level of scholarship expected). The editor Jürgen Jost’s somewhat extended commentaries contain nothing memorable, more than about what anyone could draw from any modern textbook on differential geometry, thus fail altogether to rise to the level one has a right to expect from an intellectual historian. Indeed, he scarcely engages Riemann’s ipsissima verba at all, so as history of mathematics it remains even less distinguished than Bourbaki’s (our review here). In particular, Jost eschews any discussion of the philosophical grounds behind Riemann’s ground-breaking work. Let us stress that our five-star rating reflects the supreme merit of Riemann’s innovative ideas as described in his text itself, and not the indifferent quality of Jost’s editorial interventions.

The most effectual jumping-off point here will be for us to review how Riemann himself thinks of manifolds in the habilitation lecture. The first item that strikes one’s notice is that Riemann views space as something that has to be constructed in spatial intuition [den Begriff einer mehrfach ausgedehnten Größe zu construiren]. Thus, space will consist in a collection of locations [Orte] which are to be determined quantitatively. At this point, Riemann introduces what will be for him a programmatically important distinction between relations of extension versus relations of measure [Ausdehnungs- bzw. Maßverhältnisse]. Before entering into the significance of what Riemann wants to convey with this terminological difference, let us first discuss the quantitative determination of place by means of coordinate functions. From the etymology (coordinate = backformation from Latin, coordination = co- + ordination-, ordinatio from ordinatus, past participle of ordinare = to put in order, arrange), it is implicit that one has gathered a collection of sensible markers disposed at various places in space which are susceptible to quantitative comparision, by estimation of sight or the visual image on our retina (failing this, by estimation and comparison of tactile sensations). When taken to the limit of greatest possible refinement, one obtains the coordinate function as an idealization. Riemann applies to the resulting variables the term, extension [Ausdehnung].

Given a suitable set of variables to serve as coordinate functions, one can differentiate places in space quantitatively by means of the values assumed there by the corresponding variables (Bestimmung des Orts durch Veränderlichen). The space of experience represents only the special case of dimension 𝑛 = 3. In particular, one can answer questions of greater, lesser or how much (Das Mehr, das Minder, das Wieviel) by resort to comparison of numerical values of these variables (Vergleichung der Quanta durch Messung bzw. Zählung). Thereby, according to Riemann, one has generated the concept of a manifold (Erzeugung des Begriffs der mehrfach ausgedehnten Mannigfaltigkeit) by resolving the determination of place into questions of quantity (Zurückführung der Ortsbestimmungen auf Quantitätsbestimmungen). For instance, in the simplest case of one dimension, one can move forwards or backwards by marginally increasing respectively decreasing the numerical value of the corresponding variable (einfach ausgedehnte Mannigfaltigkeit: vorwärts oder rückwärts). The case of manifolds of higher dimension (mehrfach ausgedehnte Mannigfaltigkeit) is obtained by appending additional coordinate functions. Conversely, one can go down in dimension by fixing the numerical value of one of the coordinate functions (Zerlegung) (cf. Theorem 5.51, local slice condition for embedded submanifolds, in John M. Lee, Introduction to Smooth Manifolds (second edition), Springer Verlag (2013), our review here).

So far, we have been discussing solely relations of extension [Ausdehnungsverhältnisse]. Riemann understands quite well that, owing to the possibility of change of coordinates, these do not suffice to tell us everything we could wish to know about space. For instance, we have an intuitive concept of distance (which, as Fichte states, is connected with the amount of force it takes to move ourselves from one place to another [J.G. Fichte, Wissenschaftslehre nova methodo, Philosophische Bibliothek 336, second edition, Felix Meiner Verlag (1994), at 134-135]), which ought to be independent of any arbitrary conventions entering into the definition of the coordinate functions in terms of which the space is to be constructed. Thus Riemann, we posit in addition relations of measure [Maßverhältnisse] in order to obtain an invariant sense of distance (Länge unabhängig von Lage). Of course, in principle, a given space could admit diverse relations of measure (verschiedene Maßverhältnisse fahig bzw. denkbar). Riemann, in fact, invents his metric tensor as a device by which to capture these relations of measure. He supposes that the line element 𝑑𝑠 should be related to the quantities 𝑥 and 𝑑𝑥 by making 𝑑𝑠 linearly proportional to 𝑑𝑥 up to corrections of second order, which will be negligible.

At this point, we can commence a critique of the concept of coordinate function from everyday experience, to be expanded upon below. How does it extend to physics, under what conditions? First of all, as Riemann teaches us, the theorems of geometry cannot in general be deduced from mere relations of extension (Sätze der Geometrie sich nicht aus allgemeinen Größenbegriffen ableiten lassen). Moreover, our initial assumption as to the form of the metric tensor is by no means necessary; ultimately, it is justified by the extent to which the geometrical results to which it leads enable us to render intelligible the manifold of experience (diese Thatsachen nicht notwendig sondern nur von empirischer Gewissheit); thus, the pertinent assumptions go under the heading of hypotheses (which constitute the hypotheses referred to in the very title of the inaugural lecture). Therefore, a scientific investigation into the experience of space demands that the hypotheses of Riemannian geometry be criticized and judged as to their empirical adequacy when we press beyond everyday experience into the realm of the infinitely large respectively infintely small (ihre Zulässigkeit [sc. die Hypothesen] nach der Seite des Unmeßbargrossen und nach der Seite des Unmeßbarkleinen urtheilen). As befits his truly profound power of spatial intuition, Riemann himself provides us with the clue as to how to proceed:

Im Laufe der bisherigen Betrachtungen wurden zunächst die Ausdehnungs- order Gebietsverhältnisse von den Maßverhältnissen gesondert, und gefunden, daß bei denselben Ausdehnungsverhältnissen verschiedene Maß- verhältnisse denkbar sind; es wurden dann die Systeme einfacher Maßbestimmungen aufgesucht, durch welche die Maßverhältnisse des Raumes völlig bestimmt sind und von welchen alle Sätze über dieselben eine nothwendige Folge sind; es bleibt nun die Frage zu erörtern, wie, inwelchem Grade und in welchem Umfange diese Voraussetzungen durch die Erfahrung verbürgt werden. In dieser Beziehung findet zwischen den bloßen Ausdehnungs- verhältnissen und den Maßverhältnissen eine wesentliche Verschiedenheit statt, insofern bei erstern, wo die möglichen Fälle eine discrete Mannigfaltigkeit bilden, die Aussagen der Erfahrung zwar nie völlig gewiß, aber nicht ungenau sind, während bei letzeren, wo die möglichen Fälle eine stetige Mannigfaltigkeit bilden, jede Bestimmung aus der Erfahrung immer ungenau bleibt—es mag die Wahrscheinlichkeit, daß sie nahe richtig ist, noch so groß sein. Dieser Umstand wird wichtig bei der Ausdehnung dieser empirischen Bestimmungen über die Grenzen der Beobachtung in’s Unmeßbargroße und Unmeßbarkleine; denn die letztern können offenbar jenseits der Grenzen der Beobachtung immer ungenauer werden, die ersteren aber nicht. (pp. 11-12)

As is well known to historians of science, Riemann himself follows Herbart in epistemology but not in natural philosophy [J.F. Herbart, Lehrbuch zur Einleitung in die Philosophie, ed. W. Henckmann, Philosophische Bibliothek 453, Felix Meiner Verlag (1993); esp. §§116-164 and annotations thereon. See also E. Scholtz, Herbart’s influence on Bernhard Riemann, Historia Mathematica 9, 413-440 (1982)]. Riemann makes two elementary points in this connection: first, our picture of the world is true when the connection among our representations [Zusammenhang der Vorstellungen] corresponds to the connection among things; and second, the connection among things is to be discovered from that among the appearances [Zusammenhang der Erscheinungen]. Natural science seeks to frame nature through its concepts, which apply not only to the present but to the future and will be either necessary [nothwendig] or contingent [möglich], to the extent to which the system of concepts remains incomplete. If, however, something unexpected happens (which is to say, either impossible or improbable according to existing concepts), then one must extend or revise the existing concepts in order to account for the unexplained observation. Thus, the pursuit of natural science consists in a continual development of concepts [Bildung der Begriffe]:

Die Geschichte der erklärenden Naturwissenschaften, soweit wir sie rückwarts verfolgen können, zeigt, dass dieses in der That der Weg ist, auf welchem unsere Naturerkenntniss fortschreitet. Die Begriffssysteme, welche ihnen jetzt zu Grunde liegen, sind durch allmählige Umwandlung älterer Begriffssysteme entstanden, und die Gründe, welche zu neuen Erklärungsweisen trieben, lassen sich stets auf Widersprüche oder Unwahrscheinlichkeiten, die sich in den älteren Erklärungsweisen herausstellten, zurückführen. [B. Riemann, Erkenntnistheoretisches, pp. 489-492 in Gesammelte mathematische Werke und wissenschaftlicher Nachlass, hrsg. R. Dedekind and H. Weber, Teubner, Leipzig (1876), at p. 489]

With Kant, Riemann supposes that the concepts of being and of causality are not analytic, but conditions of possible experience. Yet, with Herbart against Kant, in order to reduce the world of experience to intelligibility, we require concepts whose origin and development we cannot follow because they are given to us without being in themselves thrust upon our notice [unvermerkt] through language. But, in principle, they could be derived from pure forms of connection of simple sensible representations, rather than from Kant’s synthetic a priori. For instance, the concepts of continuity and of causality arise from the effort [Aufgabe] to preserve the economy of existing concepts as much as possible when seeking to embrace a wider range of phenomena. The derivation of new concepts in so far as they are accessible to observation occurs via the process just described, and

Dieser Nachweis ihres Ursprungs in der Auffassung des durch die sinnliche Wahrnehmung Gegebenen ist für uns desshalb wichtig, weil nur dadurch ihre Bedeutung in einer für die Naturwissenschaft genügenden Weise festgestellt werden kann....[op. cit., p. 490]

Perhaps, the philosopher of science will argue that, in Riemann’s schema, the language we inherit from our forebears plays the role of the synthetic a priori in Kant but that we must have recourse to the latter if we are to explain the origin of the former. If this were so, then Riemann’s epistemology would amount to a genealogy of the abstract synthetic a priori in Kant (who adopts a perspective outside time). To forestall such a move, though, and to stay consistent with our remarks above, let us appeal to Herder’s metacritique of the critique of pure reason of 1799. J.G. Herder, building upon J.G. Hamann, goes beyond the latter’s rejoinder to the second so-called purism of reason in Kant, to the effect that any critique of pure reason has to be enunciated in language, but that language itself takes shape from empirical origins, both in the case of the individual and in the case of society at large; hence, how can it be as pure—prior to all experience—as it puts itself out to be? [J.G. Herder, Eine Metakritik zur Kritik der reinen Vernunft, in Sprachphilosophie: Ausgewählte Schriften, Philosophische Bibliothek 574, Felix Meiner Verlag (2005), pp. 181-226, see our review here; J.G. Hamann, Writings on Philosophy and Language, trans. and ed. K. Haynes, Cambridge Texts in the History of Philosophy, Cambridge University Press (2007)]. Herder shores up Hamann’s point with a brilliant investigation into the formation of language itself. In a transcendental aesthetic, Herder shows how experience is constituted in the very process of defining a language. Thus, the distinction between time and space and our notions of being, perdurance and force themselves originate in the derivation of the spoken word from the sign [Merkmal]. The concepts of the understanding and the categories of all possible experience are the products of its work, or functioning [handeln].

To tie everything together, then, the material upon which poetic reflection operates is our interaction with things in the world, the most primitive stratum of which forms the subject of Jean Piaget’s developmental psychology of the child. But the salient point to raise in the present connection is just this: Piaget assumes that the child’s learning terminates in the adult concepts of number and space familiar to us [The Child’s Conception of Number, our review here, and The Child’s Conception of Space, our review here]. To a degree, this will be true. Nevertheless, these concepts themselves have a history, a non-trivial observation into which Herder enters in his metacritique. But, once we acknowledge their historical character, we have no grounds for supposing that the development of the concepts of number and space must come to a halt at the guise in which mathematicians of the present day formulate them. Rather, we ought to expect that renewed poetic reflection will, by seeking to reorder the manifold of experience, bring about refinements of these concepts of number, space and time.