Abstract
“Classical” set theory in the style of Cantor, Dedekind and Zermelo enjoyed dominance during the early twentieth century, playing a prominent role in many of the “modern” mathematical developments, in analysis, algebra, topology, and so on. Yet the theory was polemical, and one finds a characteristic pattern of second thoughts about set theory, after an initial enthusiasm; examples we briefly discuss are Borel, Weyl, and Rey Pastor, which contrast with Hilbert’s optimism. As a result, one can speak of a proliferation of set theories – in the plural – during the period 1910–1950. Classic examples are the (more or less coherent) predicative proposals of Russell and Weyl, the “intuitionistic set theory” developed by Brouwer, and several other systems which we cannot discuss here. We consider in this paper whether such criticism of set theory was a symptom of traditionalism, which leads to an analysis of the notion of modernism, paying especial attention to the case of L.E.J. Brouwer. I shall argue that modernism is a somewhat ambiguous notion, and that Brouwer (like Weyl) can indeed be regarded as prototypically “modern” in a sense that was characteristic of the Inter War period 1918–1939.
Thanks are due to Moritz Epple, Erhard Scholz, Leo Corry, Javier Ordóñez, and very especially to Mark van Atten, who greatly improved my discussion of Brouwer, and Jeremy Gray for the many conversations that have helped shape my views on the topic. No doubt other historians have also influenced me; may they forgive my lack of memory. This work had a complicated story: it is partly based on a chapter meant for a collective book, which was ready in 2008, but I have eliminated some sections and combined it with new material for the present volume. Work on it was supported financially by research grants FFI2009-10224, P07-HUM-02594 and FFI2017-84524-P.
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Notes
- 1.
Hilbert 1932, 466, from the obituary of his also famous friend Minkowski, originally published in 1910. Zermelo said that it was only “the influence of D. Hilbert” which made him realise the importance and deep significance of the fundamental problems of set theory.
- 2.
First Cantor’s continuum problem, second the consistency of the set-theoretic definition of the real numbers.
- 3.
For a masterful exposition of these novelties, insisting on matters of ontology, epistemology and methodology, the reader is referred to (Gray 2008). The paradigmatic definition of a structure around 1950 was as a kind of set-theoretic construct (Bourbaki), but more recently a lively debate has been developing about other possible approaches, notably the one based on category theory.
- 4.
The expression is literal from (Hilbert 1897, iii) and it became a ready-made phrase later on.
- 5.
B. L. Van der Waedern, Moderne Algebra (Berlin: Julius Springer, 1930). See (Corry 1996).
- 6.
- 7.
Mehrtens (1990), 108. The colourful expression was employed by Hilbert’s friend and colleague Hermann Minkowski in a letter commenting on Hilbert’s celebrated lecture of 1900 on ‘Mathematical problems.’
- 8.
Mehrtens 1990, 9–10. A not very careful reader of Mehrtens will thus extract the idea that there were merely two opposite sides – the modernists or progressives, and the conservatives calling for reactionary reform. Such a simplistic scheme would be untenable, but Mehrtens is more sophisticated and faithful to the events. His essay is about “the emergence of mathematical modernism around the turn of the century and about its nemesis [Abwehr], the «countermodernism,» that set itself as an opposition, as a shadow that would not lose it despite all the successes” (Mehrtens 1990, 7). He intimates that there is an internal link between both, so that the modernists could not exist without their shadow – some kind of inner dialectics.
- 9.
In fact this category seems to be reserved for those who developed political orientations akin to National-socialism (Mehrtens 1990, 14–15, 308ff).
- 10.
Judging from the reactions of some colleagues and referees, I come to think that perhaps there is a cultural element in this perspective of mine: perhaps there is a reflection of my cultural background in Spain and its complex cultural and political history in the fact that, from the beginning, it seemed obvious to me that modernism and modernity are to be distinguished.
- 11.
- 12.
- 13.
On this topic see Corry (2004). With R. Courant, Hilbert authored a key book for mathematical physics, Methoden der Mathematischen Physik (Berlin: Julius Springer, 1931), in the tradition of Riemann and Klein.
- 14.
Even though his public presentation was limited to 10 of the 23 problems in his list, those two were among the chosen ones. On this topic, see Ferreirós (2007), chap. IX.
- 15.
- 16.
In the 1920s, Arthur Schoenflies did historical work that established Kronecker’s reputation as a malevolent enemy of Cantor, and even a major cause of his mental illness. Much earlier, Hilbert had obtained an impression of Kronecker’s ways of promoting his enmity to the “new mathematics” from his friends Minkowski and Hurwitz, who knew well the Berlin master.
- 17.
Interestingly, Brouwer thought that mental proofs are, in general, infinite objects (see his Collected Works, vol. 1, p. 394); I thank van Atten for referring me to this passage.
- 18.
This is an aspect of the figures under review that we shall take into account. Considering modern as an actor’s category, we avoid any need to define it more properly, but it might be worthwhile to ask whether there is a prototypic “modern persona” (in the sense of scientific personae, see Daston and Sibum 2003).
- 19.
A detailed analysis of his foundational views would show this in full clarity. But here it may suffice to indicate a simple symptom: in the 1910 quotation given above, first page, Hilbert wrote Mengentheorie – and not Mengenlehre. In doing so he was avoiding the traditionalistic overtones so frequent in Cantor’s work.
- 20.
See (Rowe 1989) on the “intellectual alliance” between Klein and Hilbert.
- 21.
Perhaps Betrand Russell and Alfred N. Whitehead might be other good candidates. See on Hausdorff (Mehrtens 1990) or (Epple 2006), on Brouwer see van Dalen’s biography (1999) and (van Stigt 1990) (1996), on Weyl the book edited by E. Scholz (2001) and (Scholz 2006), and about Tarski (Feferman and Feferman 2004).
- 22.
Although Weyl’s allegiance to Brouwer’s ideas only lasted for 4–5 years, from his 1918 book Das Kontinuum to the end of his career he remained a critic of set theory and preferred alternative restricted methods; it has been said that, throughout his career, he consciously avoided employing the most significant and polemical axiom of set theory, the Axiom of Choice (Ferreirós 2007, 339, quoting Dieudonné).
- 23.
In papers such as ‘Auswahlaxiom und Kontinuumshypothese’ (1938), Sierpiński offered a most competent exposition of the uses of and paradoxical consequences of the axiom of choice (AC) and the continuum hypothesis (CH) in mathematics, but he declared himself to be neither for nor against AC.
- 24.
Fields medal W. P. Thurston e.g. calls the axioms of set theory ‘polite fictions’ (Thurston 1994). Many first-rate mathematicians would opt for minimalism as regards set theory and the higher infinite: the Bourbaki themselves, Quine or Wang proposed restrictions on set theory around 1950; but also much later we find influential examples such as Martin-Löf or Feferman from the 1970s, Voevodsky in the 2000s.
- 25.
The fourth in 1950. First edn was a mere 134 pages, the fourth featured 150 more of appendixes.
- 26.
- 27.
Van Atten calls my attention to a 1909 letter to Van Scheltema: “This summer the first mathematician of the world was in Scheveningen; I was already in contact with him through my work, but now I have repeatedly made walks with him, and talked as a young apostle with a prophet. He was 46 years old, but with a young soul and body; he swam vigorously and climbed walls and barbed wired gates with pleasure. It was a beautiful new ray of light through my life.”
- 28.
- 29.
- 30.
This is a feature not only of intuitionism, but more generally of constructivism; the development of analysis is severely affected by having to circumvent that simplifying principle.
- 31.
See also Ferreirós (2008). The point is discussed in any good textbook on philosophy of mathematics.
- 32.
That’s something Scheltema sought to remedy in their student years, endeavouring “tirelessly … to make him come closer to the material world.”
- 33.
I take these sentences verbatim from M. van Atten, “Luitzen Egbertus Jan Brouwer”, The Stanford Encyclopedia of Philosophy (Summer 2011 Edition), Edward N. Zalta (ed.), URL = http://plato.stanford.edu/archives/sum2011/entries/brouwer/. Gerrit Mannoury, a polymath who among other things published mathematics and philosophy, was one of Brouwer’s most important and influential teachers.
- 34.
I thank Mark van Atten for calling this passage to my attention, and also for the translation. From E. Husserl’s Briefwechsel Vol. IV, (1994), p. 156.
- 35.
- 36.
Nihilism was a natural consequence of the philosophical solipsism of young Brouwer. As early as 1898, being only a boy of 17, he wrote: “the only truth is my own ego of this moment, surrounded by a wealth of representations in which the ego believes, and that makes it live” (van Dalen 1999, 18).
- 37.
As he described it himself around the turn of the century: people of both sexes in vest with bare black feet and blue nails, the sunbathing of bare backs, the gnawing of raw turnips and carrots (see van Dalen 1999, 28).
- 38.
As regards his wife, Life, Art and Mysticism makes clear the expectations of submission that Brouwer had at the time of choosing her (see van Dalen 1999, 73).
- 39.
- 40.
Proofs abound: the Göttingen Association for the Promotion of Applied Physics (1898), created by Klein in association with industrialists; his efforts to hire Ludwig Prandtl, who became head of the Institute for Technical Physics in 1905 and did pioneering work on aerodynamics; the professorship of applied mathematics created in 1904 for Carl Runge; the important contributions that Hilbert and Minkowski made to physics, their deep involvement with the subject in the 1900s and 1910s; the close links with the physicists, which Max Born for instance describes. See Rowe (1989).
- 41.
Kramer (1995), p. 3. How that relationship was to be redefined was explored by Kandinsky not only in his paintings but in his influential treatise Concerning the Spiritual in Art (1911).
- 42.
Notice that Herf is not defining reactionary modernism here, but modernism in general. In this respect, van Atten comments that intuitionism was hailed by figures from both left and right, such as A. Khinchin in 1926 Soviet Russia (Verburgt and Hoppe-Kondrikova 2016) and O. Becker in 1933 Nazi Germany (van Atten 2003).
- 43.
I thank the organisers of a Frankfurt meeting that took place in 2006, in particular M. Epple, for bringing the relevant quote to my attention.
- 44.
I have made this same point for Romanticism in Ferreirós (2003).
- 45.
- 46.
See Ferreirós 2017, 68. One can find attributed to Hilbert this sentence: “Mathematics is a game played according to certain simple rules with meaningless marks on paper.” But even if the phrase is repeated in a thousand web pages, it cannot be found anywhere in his work – it is just made up by putting together several things that, jointly, amount to a severe misrepresentation of Hilbert’s thought. In effect, he often emphasised the meaningfulness of mathematical statements and the depth of conceptual content expressed in them.
- 47.
In questions like this it becomes obvious that Hilbert was indeed Klein’s follower in his pragmatic and well-thought disciplinary politics. In my view, this is fully modern (in the sense of modernisation) but not modernist at all.
- 48.
Hilbert’s views on this topic evolved, until he (together with Bernays) embraced a kind of as-if position in the context of his metamathematics of the 1920s; see Ferreirós (2009). Hans Vaihinger, who like Hilbert presented himself as a follower of Kant, published his book The Philosophy of As If in 1911.
- 49.
In a public lecture, 1930, he emphasised that mathematics is “the instrument that mediates between theory and practice, between thought and observation,” without which “today’s astronomy and physics would be impossible.” This was recorded and is available on the internet: see http://math.sfsu.edu/smith/Documents/HilbertRadio/HilbertRadio.mp3 (accessed Sept. 2012).
- 50.
See van Dalen (1990) or the short presentation in Hesseling (2003), 81–86, but notice that there were political issues at stake, tensions within the German mathematical community that led to a dramatic power struggle for control of the journal Mathematische Annalen. For this wider context, see Rowe and Felsch (2019).
- 51.
Perhaps he should have gone one step further, as modern mathematics has been so successful that anxiety is not one of its distinguishing traits – unlike the case of the arts.
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Ferreirós, J. (2023). On Set Theories and Modernism. 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_18
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