Abstract
This paper explores two respects in which a study of the history of mathematics can enrich the philosophy of mathematics. First, central concepts in the informal methodology of mathematical research—understanding, explanation, the “proper context”, the correct or natural definition of a concept, etc.—can in many cases only be identified, refined and adjudicated as the practice evolves over time. Second, the distinction between mathematics and philosophy in many cases is not sharply delineated. Many paradigmatically “philosophical” goals—identifying central concepts and providing rationales for such choices, analysing concepts, establishing criteria of “rigour” etc.—arise organically within mathematical practice itself. I illustrate these observations by exploring the historical sequence beginning with the rationales of Gauss and Riemann for studying real functions in the complex numbers, with special attention to the double periodicity of elliptic functions. I illustrate the profundity of some of the methodological issues that can arise via a contrast between the Riemann and Weierstrass approaches to elliptic functions and their generalisations.
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Notes
- 1.
Indeed, you could see much of the present essay as an extended reflection on the discussion of Fourier series in Gray (2009, §3).
- 2.
Lawrence Sklar’s Locke Lectures (Sklar, 2000) make a similar observation for physics: in many cases foundational philosophy arises organically within physics. Of course, there are also important cases where philosophy and mathematics interact at more of a distance, so to speak, such as Riemann’s immersion in Naturphilosophie (cf. Bottazzini and Tazzioli, 1995) and Herbart’s writings in particular (Scholz, 1982), (Ferreirós, 2006), and (Laugwitz, 1999, §3.3). Gray (2008) explores a wealth of other interactions between mathematics and philosophy.
- 3.
“…the ring and the corresponding affine scheme are equivalent objects. The scheme is, however, a more natural setting for many geometric arguments” (Eisenbud and Harris, 1992, p. 5).
- 4.
- 5.
“[Many mathematicians may use expressions like “explanatory”] to mean little more than “of the kind I like”. And different kinds of mathematicians like different kinds of proofs” (Burgess, 2015, p. 96 fn. 22).
- 6.
- 7.
- 8.
- 9.
The point is not just that complex numbers should be used, but that they are collectively treated as an object of study in their own right, and as supporting rigorous proof, in contrast to (say) Poisson, who saw them as a tool for discovery but not rigorous proof (Bottazzini and Gray, 2013, p. 127.).
- 10.
- 11.
- 12.
- 13.
- 14.
- 15.
- 16.
For example: “I have tried to avoid Kummer’s elaborate computational machinery so that here too Riemann’s principle may be realized and the proofs compelled not by calculations but by thought alone.” Hilbert (1998, p. X).
- 17.
This “conceptual” dimension of Riemann’s approach is a theme running throughout Laugwitz (1999). I discuss Riemann’s reorientation in a philosophical context in Tappenden (2006) and more briefly in Tappenden (2013, §9.3). A useful brief historical overview of the rivalry is Rowe (2000). For details and philosophical discussion of Dedekind’s Riemann-inspired methodology see Avigad (2006) and Tappenden (2008). Some of the antecedents of the “conceptual” interpretation in Gauss’s work are explored in Ferreirós (2007).
- 18.
This does not mean that Riemann didn’t carry out virtuoso calculations when needed; cf. Siegel (1932, esp. p. 771).
- 19.
- 20.
- 21.
Riemann’s definition and the Weierstrass series-based definition I’ll discuss in Sect. 24.5 are essentially equivalent, though it would be some decades before that was proven. The qualifier “essentially” is necessary here; note for example the Weierstrass objection in footnote 50.
- 22.
Riemann (1851, p. 41). The quoted words are from the description of §20 in the table of contents.
- 23.
Jones and Singerman (1987, p. 76).
- 24.
I discuss the “inner characteristic properties” versus “external forms of representation” theme, with other quotes from Dedekind and others, in Tappenden (2006, esp §II). A more systematic examination of this and other aspects of Dedekind’s methodology is Avigad (2006). A more recent paper, which connects Dedekind’s interpretation of Riemann’s methods with contemporary “structuralism” in mathematics is Ferreirós and Reck (2020).
- 25.
Note Bottazzini and Gray (2013, p. 70).
- 26.
- 27.
- 28.
An aside: This is already a point where the choice of the complex numbers as the domain is theoretically crucial. The inverted function makes sense if restricted to the real numbers, but the fundamental property of double periodicity only appears if the function ranges over all the complex numbers. Except in trivial cases, one of the periods must be a complex, non-real number and the other a real number. This illustrates a point that is often neglected in connection with the subject of ontology in mathematics: extending a domain can change which classifications of objects in the original domain count as “natural”.
- 29.
This “length” is more formally the net effect of a complex velocity along the whole curve, where complex velocities combine differently than real velocities do (so that visually very different paths can all have the same “length” in this sense of same net effect). I’m grateful to Colin McClarty here.
- 30.
On the construction of the Riemann surface for \(\frac {1}{p(z)}\), with \(p(z)\) a polynomial, see Jones and Singerman (1987, pp. 157–67).
- 31.
Wilson (2006, pp. 312–19 ) discusses some of the relevant complexities in a philosophical context.
- 32.
- 33.
The main reference point for Riemann’s remarks were a specific cluster of results he had derived on hypergeometric series. I’m taking his words to be intended more broadly.
- 34.
No doubt some of the psychological force arises from the fact that this Riemann surface can be visualised. It’s an issue worth study, and visualisation in mathematics has been the topic of excellent research in recent decades, but I’ll set it aside in this paper. Giaquinto (2020) is an excellent survey of this work. The methodological significance of visualisation is complicated for reasons I discuss in Tappenden (2005b) (see especially Sect. 2.3).
- 35.
Speaking on the 100th anniversary of Riemann’s thesis, Ahlfors wrote: “Very few mathematical papers have exercised an influence on the later development of mathematics which is comparable to the stimulus received from Riemann’s dissertation. It contains the germ to a major part of the modern theory of analytic functions, it initiated the systematic study of topology, it revolutionized algebraic geometry, and it paved the way for Riemann’s own approach to differential geometry.” (Ahlfors, 1953, p. 3). The list could easily be extended: Lie groups, algebraic number theory, …(Farkas and Kra, 1992, p. 1).
- 36.
See for example Fillion (2019). Jeremy Gray, Stephen Menn and Philip Kitcher were especially helpful in coaxing me to a more balanced point of view.
- 37.
This conjures up a visual analogy with the periodicity of the sine function, different from the “go around the circle” one. Why is sine periodic with period \(2\pi \)? Because adding \(2\pi \) just shifts the sine curve to the right in such a way that the curve remains unchanged.
- 38.
cf. McKean and Moll (1997, section 2.8 (pp. 84–87)) or Stein and Shakarchi (2003, pp. 266–73) for clear explanations of why the series is the simplest and most reasonable one to pick, given what work the \(\wp \) function is meant to do. For a variety of illustrations of the ubiquity of the \(\wp \)-function, see McKean and Moll (1997, pp. 84–104).
- 39.
For example, avoiding integration proved useful for an extensive range of number types, such as p-adics (Roy, 2017, p. 183).
- 40.
- 41.
- 42.
- 43.
This point is also made in Remmert (1991, pp. 351–2 and 431).
- 44.
- 45.
Siegel (1973, p. 1); the German original appeared in 1966.
- 46.
On the \(\bar {\partial }\)-equation, I’m indebted to correspondence with Jon-Erik Fornæss. The emergence of one style of n-dimensional generalization of Riemann’s approach is charted in Remmert (1998).
- 47.
“…function theory …must be built on the foundation of algebraic truths, and that it is therefore not the right path when the “trancendant” …is taken as the basis of simple and fundamental algebraic propositions. [Appeal to the “transcendant”] seems so attractive at first sight, in that through it Riemann was able to discover so many of the important properties of algebraic functions. (It is self-evident that, as long as he is working, the researcher must be allowed to follow every path he wishes; it is only a matter of systematic foundations.)” (Werke II p. 235).
- 48.
For one contemporary opinion, in connection with elliptic functions in particular, note Remmert’s remark, concerning Abelian functions, which generalise elliptic functions: Though Weierstrass regarded the topic as crucial, “here Riemann’s ideas were more fruitful” (Weierstraß, 1988, p. ix).
- 49.
Weierstrass himself noted several Riemannian lapses of rigour, for example Riemann’s overly general formulation of the Dirichlet principle, as noted in Sect. 24.6.2.2. Another of Weierstrass’s objections directly addressed §20 of Riemann’s thesis, indicating a limitation on the generality of Riemann’s approach. (Weierstrass says §19, but §20 is clearly intended.) Riemann states at the end of §20 that he is not proving that the class of (“monogenic”) functions he is defining coincide with those that can be “expressed by operations on quantities”, but that such a proof would be needed to view his approach as foundational to “a general theory of operations on quantities”, and he certainly appears to believe that such a proof can be found. But though the equivalence holds for the most part, Weierstrass produced a class of counterexamples displaying that “the concept of a monogenic function of one complex variable does not coincide with the concept of a dependence that can be expressed by means of (arithmetical) operations on magnitudes” (Weierstrass, 1880, p. 79). Weierstrass was no doubt pleased to note, “the contrary has been stated by Riemann”. (ibid p. 79 footnote) I am here indebted to Bottazzini and Gray (2013, p. 465), and to a communication from Bottazzini.
- 50.
For present purposes I’m presupposing that we can make sense of the idea that a precise set of concepts and techniques can to some degree or other accurately represent concepts/techniques that are only vaguely implicit in earlier work. There has been philosophical scrutiny of this, tracing back to Burge (1979) on Frege’s view of partial grasp of concepts and Peacocke (1998) on the relation between the Leibniz/Newton treatment of calculus and later ones. Smith (2015) on the derivative is a good recent discussion.
- 51.
See, for example, Clebsch and Gordan (1866). A rich exploration of (inter alia) Clebsch’s approach to Riemann is Gray (1989). An extensive, clear presentation of the Brill-Noether approach is Casas-Alvero (2019). Lê (2017) is a revealing immersion in Clebsch’s computational conception of geometry, including the genus/deficiency relation (see esp. §4.1, §4.2) touched on in footnote 58 below.
- 52.
There were other objections, as noted in footnote 50. Another turned on “purity of method” considerations, as Mittag-Leffler (presumably channeling Weierstrass) wrote that even if Riemann’s approach could be developed rigorously, it would “[introduce] elements into function theory that are in principle altogether foreign”. Frostman (1966, pp. 54–5) cf. Tappenden (2006, p. 113) and Bottazzini and Gray (2013, p. 424).
- 53.
- 54.
Indeed, Weyl seems to suggest that Riemann was well aware of this possibility but chose to hold back from conveying “too strange ideas” to his contemporaries. Weyl (1997, p. VII) (Though perhaps this was among the things that prompted Weyl’s mature 3rd edition reflection that his “enthusiastic preface betrayed the youth of the author” (Weyl, 1955, p. VII)).
- 55.
- 56.
Mittag-Leffler notes this in an 1875 letter, for example Frostman (1966, p. 54).
- 57.
Clebsch called this the Geschlecht, usually translated “genus”. (Cayley’s term “deficiency” is sometimes used for genus defined in Clebsch’s way, especially in older textbooks (for example Hilton, 1920, p. 113).) Popescu-Pampu (2016) is illuminating on the history of the variations on the genus concept.
- 58.
This is not to endorse the negative side of Klein’s evaluation. Clebsch’s non-topological approach to genus also marks out a central property as a birational invariant—in this case a central property of algebraic curves, which interacts systematically with the degree of the curve. For reasons of space I won’t explore Weierstrass’ important non-topological definition of genus via his “Lückensatz” (gap theorem). For the history, see Bottazzini and Gray (2013, §6.8.6) and Del Centina (2008). Edwards (2005, ch. 4) (“The Genus of an Algebraic Curve”) explores the value of a further non-topological definition.
- 59.
- 60.
The addition theorem for \(\wp \) directly entails an algebraic addition theorem for \(\wp \), since \(\wp '(z_1)\) and \(\wp '(z_2)\) are algebraic functions of \(\wp (z_1)\) and \(\wp (z_2)\) cf. Akhiezer (1990, p. 45).
- 61.
Bottazzini and Gray (2013, §6.6.3) and Del Centina (2019). A clear textbook discussion of the mathematical details is Prasolov and Solovyev (1997, §2.9). In this section I’m also indebted to an unpublished manuscript by Mark Villarino, which cites the proof of Weierstrass’s characterisation as indicating that “the ‘cause’ or ‘explanation’ of the existence of a period of the meromorphic function” is the combination of the AAT and the dispersion of points around essential singularities.” (Villarino, 2022, p. 7).
- 62.
For the use of the \(\wp \)-function to demonstrate the group law for elliptic curves see Koblitz (1993, §7 ) or Lang (1978, §§2–3). Griffiths and Harris (1994, p. 240) write of the group structure of cubic curves and the addition theorem for elliptic functions as arising from different “interpretations” of a basic equation containing the \(\wp \)-function. For connections to cryptography and number theory, see Washington (2008). Pastras (2020), as the name suggests, is an illustration of the long reach of the Weierstrass conception, the \(\wp \)-function in particular. Another illustration of the addition-theorem centred conception of elliptic functions in an applied context is the textbook (Hietarinta et al., 2016). See especially Appendix B. See also Nijhoff (2022) Appendix A for more explicit framing remarks.
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It is a great pleasure to dedicate this essay, with gratitude, to Jeremy Gray, as he has been a friend and intellectual mentor almost from the beginning. My happy encounter, as a graduate student, with Gray (1989) transformed my understanding of the potential interactions of history and philosophy of mathematics. It will be clear to anyone reading this paper what a debt it owes to Jeremy’s writings, especially Gray (2015) and Bottazzini and Gray (2013). I’ve accumulated debts to so many people during the long evolution of this paper that a full list would be longer than any editor would allow, so this list is quite compressed. My apologies to anyone I have failed to mention. Early versions of the material in Sect. 24.4 were read at the University of Toronto, CUNY graduate center, Columbia University, UC Irvine, McGill and the University of Paris VII. I am grateful to Howard Stein for instructive written comments on that material. A later evolution, augmented with early versions of Sect. 24.5 was presented to the PoSe seminar at the University of Michigan, the 2015 APMP meetings in Paris, the 2017 MPMW at Notre Dame, and as a plenary address to the 2015 Canadian Mathematical Association meetings in Montréal. I am grateful to those audiences for questions and commentary, not just during the talks themselves but in subsequent discussions and correspondence over the years, especially the late Andrew Arana, Margaret Morrison, Philip Kitcher, Stephen Menn, Lydia Patton, Michael Hallett, Colin McClarty, Paolo Mancosu, Umberto Bottazzini, Marco Panza, Ivahn Smadja, Hourya Sinaceur, Karine Chemla, Bruno Belhoste, Renaud Chorlay and Emmylou Haffner. I’ve learned much about Riemann’s philosophical dimensions from José Ferreirós. I am grateful to the referees of this paper for helpful comments. Thanks to Yu Zhongmiao for helpful feedback. I would have been lost without my Michigan mathematical colleagues, beginning with fondly remembered conversations with the late Juha Heinonen and Mario Bonk when these ideas were just taking form. On these topics I learned from James Milne, Mel Hochster, Al Taylor, Dan Burns and Jens-Eric Fornæss. Most recently I’ve learned much from email and conversation with Lizhen Ji. Finally I’d like to thank the editors of this volume collectively for conceiving and bringing this volume to completion.
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Tappenden, J. (2023). History of Mathematics Illuminates Philosophy of Mathematics: Riemann, Weierstrass and Mathematical Understanding. In: Chemla, K., Ferreirós, J., Ji, L., Scholz, E., Wang, C. (eds) The Richness of the History of Mathematics. Archimedes, vol 66. Springer, Cham. https://doi.org/10.1007/978-3-031-40855-7_24
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