Abstract
As Weyl was interested in infinitesimal analysis and for some years embraced Brouwer’s intuitionism, which he continued to see as an ideal even after he had convinced himself that it is a practical necessity for science to go beyond intuitionistic mathematics, this note presents some remarks on infinitesimals from a Brouwerian perspective. After an introduction and a look at Robinson’s and Nelson’s approaches to classical nonstandard analysis, three desiderata for an intuitionistic construction of infinitesimals are extracted from Brouwer’s writings. These cannot be met, but in explicitly Brouwerian settings what might in different ways be called approximations to infinitesimals have been developed by early Brouwer, Vesley, and Reeb. I conclude that perhaps Reeb’s approach, with its Brouwerian motivation for accepting Nelson’s classical formalism, would have suited Weyl best.
dedicated to the memory of Richard Tieszen, 1951–2017
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Notes
- 1.
- 2.
See, e.g., Palmgren 1995; van den Berg et al. 2012; Ferreira and Gaspar 2015; Sanders 2018; van den Berg and Sanders 2017 and Dinis and Gaspar 2018. Such approaches are of particular interest for ‘proof mining’, which is the extraction of explicit information from proofs (e.g., explicit bounds for existential quantifiers, or rates of convergence).
- 3.
[Note MvA: An example of Leibniz’ saying this is found in his well-known letter to Varignon of February 2, 1702 (Leibniz 1859, pp. 93–94).]
- 4.
- 5.
The culmination of this is HST in Kanovei and Reeken 2004; but it does not contain full ZFC as a proper part. See also footnote 45, below.
- 6.
It is, in fact, possible to look at Robinson’s nonstandard analysis in an entirely formalistic way, and take it not to introduce new objects, but new ways of deducing theorems. Robinson points this out at the very end of his book:
Returning now to the theory of this book, we observe that it is presented, naturally, within the framework of contemporary Mathematics, and thus appears to affirm the existence of all sorts of infinitary entities. However, from a formalist point of view we may look at our theory syntactically and may consider that what we have done is to introduce new deductive procedures rather than new mathematical entities. (Robinson 1966, p. 282)
I have not highlighted this in the main text, so as to be able to show the contrast between the model-theoretical and the syntactical approaches.
- 7.
The predicates ‘finite’ and ‘infinite’ are defined as usual, in terms of the presence or absence of a bijection between x and the set {m | m < n} for some n ∈ ℕ.
- 8.
Nelson himself later provided a purely syntactical proof that proofs in IST can be reduced to proofs in a standard system ZFC[V] which is itself conservative over ZFC (Nelson 1988). Kanovei and Reeken have shown that actually the presentation of IST there is stronger than that in Nelson 1977, and that not all properties of the later version are shared by the earlier one. However, they add that this is the case if only bounded sets are considered, and that in practice these are the ones that matter (Kanovei and Reeken 2004, p. 128).
- 9.
- 10.
In IST it is also possible to prove of ℝ directly that it contains infinitesimals. But the approach through ℕ fits Reeb’s motivation better.
- 11.
But (S) does not guarantee that it does not also contain nonstandard elements.
- 12.
See on this also chapter 32, ‘A modified Hilbert Program’, in Nelson 1986.
- 13.
In two dimensions a simplex is a triangle with all its interior points; in three dimensions a pyramid with a triangle as its base.
- 14.
The primary reference for this paragraph and the next is Brouwer’s dissertation, Brouwer 1907, pp. 8–11, but the argument is general.
- 15.
An extensive historical treatment is Ehrlich 2006.
- 16.
See on this point Ehrlich 2006, 69–71.
- 17.
A recent English translation and introduction is van Atten and Sundholm 2017.
- 18.
Gödel 1906–1978, Arbeitsheft 14, pp. 21–23; see its page 14 for the year.
- 19.
Brouwer’s Vienna lectures were invited by a committee of which Hahn was a member (van Dalen 2005, p. 561).
- 20.
- 21.
The same construction is also in Brouwer 1948, but there Brouwer does not add the comment quoted above.
- 22.
In the next section, we will see that Brouwer also had a proof of the actual negation, ¬ ∀ α(α > 0 → α ⋗ 0) (Theorem 28).
- 23.
Brouwer could have given this argument in terms of an undecidable proposition instead of an untestable one. The reason he uses an untestable one is that in his paper he exploits almost the same construction to prove that ≠ cannot be defined as a disjunction of < and >, as that would lead to the contradiction that an untestable proposition is testable. For further discussion of Brouwer’s weak and strong counterexamples, see van Atten 2018.
- 24.
Let ϵ be given, and determine an n such that 2−n < ϵ. Construct the sequence α up to α(n), which can be done as each choice is decidable. If α(n) = 0, all further choices will be in the interval [0, 2−(n + 1)] and hence within ϵ from one another. If α(n) ≠ 0, then the choices in α have already been fixed, and hence within ϵ from one another.
- 25.
Before, it was a growing construction for a real number that had yet acquired neither the property of being rational, nor that of being irrational.
- 26.
I don’t think Vesley knew of that particular passage in Brouwer, which was published only in 1981, but he was of course very familiar with this kind of reasoning, e.g. Kleene and Vesley 1965.
- 27.
See Brouwer 1954, p. 4 for Brouwer’s demonstration, and Myhill 1966, p. 295 for the observation that this is KS. Brouwer does not literally state KS; he constructs, from an arbitrary proposition P that as yet cannot be tested, an infinite sequence C(γ, P), and shows that truth of P and rationality of C(γ, P) are equivalent. However, the construction of a witness for KS from C (γ, P) is immediate; and Brouwer’s reasoning towards the existence of C(γ, P) goes through for any P, not only untestable ones.
- 28.
One might think the permanent existence of an α-infinitesimal can be assured by starting a sequence starting with 0’s and stipulating that one will never make the decision between (a) restricting the remaining choices to 0 and (b) making a choice that is not 0. This however will not do, because by choosing 0 until that decision is made, and at the same time resolving always to postpone that decision, the result is that the sequence will be constant 0.
- 29.
Strictly speaking, Brouwer does not consider the question of rationality of each α ∈ [0, 1], but of each α ∈ J, where J is a fan that coincides with [0, 1]. Note also that our notational use of α is different from that in Brouwer 1949a.
- 30.
There, ‘nilpotents’, which are δ such that δ ≠ 0 but δ 2 = 0, may be cancelled when universally quantified.
- 31.
Compare Brouwer’s objection to Veronese’s postulate above, p. 11.
- 32.
For Reeb’s (philosophy of) nonstandard analysis, see, in French, Reeb 1979, 1981; Barreau and Harthong 1989; Diener and Reeb 1989; Lobry 1989; Reeb and Harthong 1989; L’Ouvert 1994, and Salanskis 1999. There is not much about Reeb’s (philosophy of) nonstandard analysis in English; see Fletcher et al. 2017 for a few recent remarks.
- 33.
Nelson has written:
One of the most treasured experiences of my life is my friendship with Georges Reeb. We had many strong discussions together, intuitionist versus formalist. What he created was unique in my experience. His rare spirit, gentle but fiercely demanding of the highest standards, inspired a group of younger mathematicians with an unmatched ethos of collegiality. And their discoveries are extraordinary.
Reeb found, and led others to find, not only knowledge and beauty in mathematics, but also virtue. His insights into the nature of mathematics will point the way towards the mathematics of the future. (Nelson 1996, p. 8)
- 34.
In Reeb 1981, p. 153, he had stated that his notion of naïve numbers leads to ideas that ‘show some analogy with IST’, and this is what one expects of a motivation in constructive reality of a distinction in an idealised, classical formal theory. Note that Reeb in his writings does not much discuss his philosophical differences with Nelson. On Nelson’s philosophy of mathematics, see, besides his own papers, also Buss 2006.
- 35.
1984 is the year in which a first version was written and began to circulate.
- 36.
‘la conception intuitionniste (aussi bien la nôtre que celle de Brouwer) …’
- 37.
This attitude was later described by Sundholm and myself as the intuitionists’ ‘ontological descriptivism’, an attitude they share with Platonists, the disagreement being over the nature of that reality (Sundholm and van Atten 2008, p. 71). If we had known the paper by Reeb and Harthong then, we would surely have taken it into account.
- 38.
Besides the main influence Brouwer, in Reeb one finds quotations or echos from for example Hilbert, Poincaré, Löwenheim, Skolem, and Von Neumann.
- 39.
Brouwer makes the point in Brouwer 1928.
- 40.
[The role of Brouwer’s ‘contentual mathematics’ corresponds to that of Reeb’s ‘mathematical reality’; but the former is richer than the latter.]
- 41.
- 42.
Briefly, the point is that a recursive function is defined by a set of equations, and if the function is to be considered as computable, there must be an effective method to determine that set; but now to understand ‘effective’ as’ recursive’ would be circular. A detailed presentation is given in Heyting 1958, pp. 340–342. For discussion and further references, see Coquand 2014 and Sundholm 2014.
- 43.
Personal communication from Jean-Michel Salanskis, who was a member of Reeb’s group.
- 44.
For Reeb, a constructive proof can exist without having been found: ‘ou bien il y a une démon- stration constructive, déjà connue ou non …’ (Reeb and Harthong 1989, section 16). For Brouwer, on the contrary, the only sense in which a proof can be said to exist is that it has been constructed. However, for the matter at hand this makes no difference.
- 45.
This idealisation need not lead to IST; it was the theory Reeb knew and liked, but closely related axiomatic nonstandard theories have been developed in the meantime (Kanovei and Reeken 2004). Just as in the natural sciences, different theories of the same phenomena in reality may be developed, and have different theoretical virtues.
- 46.
As Salanskis emphasises (Salanskis 1999, p. 140), Reeb writes ‘properties’, not ‘theorems’, so as to distinguish assertions about reality from provable formulas in a formal system.
- 47.
This last consideration is one among several that leads to the question of constructive analogues to IST (which was not a particular concern to Reeb, to whom, on the contrary, the idea of using a classical formal theory was attractive). For this, I refer to the papers mentioned in footnote 2.
- 48.
- 49.
But, as we have seen (the four ‘insights’, p. 21), Brouwerian intuitionism does not exclude a formalist foundation of classical mathematics; it includes it as a proper part. However, it is not the part of intuitionistic mathematics that is concerned with the development of contentual mathematics; and the contentual mathematics that Brouwer sought to develop is far richer than the minimum required to get the formalist foundation going.
- 50.
Given the properties of human vision, even at an everyday scale infinitesimal analysis can be useful, as shown by the analysis of the moiré effect in Harthong 1981. Further applications are presented in, e.g., Cutland 1988; Arkeryd et al. 1997; and Lobry and Sari 2008. A recent view from a philosopher of science is Wenmackers 2016.
- 51.
- 52.
In a retrospective remark of 1920 Brouwer wrote that, when he had just begun to develop intuitionism, ‘in my contemporary philosophy-free mathematical papers I have frequently also used the old [i.e., non-intuitionistic] methods, trying however to derive only such results as could be hoped to find, after the completion of a systematic construction of intuitionistic set theory, a place in the new system and claim a value, perhaps in modified form.’ (Brouwer 1920, p. 204, trl. mine)
- 53.
See in particular Bernard 2013, pp. 246–248, and Bernard’s instructive, unpublished manuscript Bernard.
- 54.
Palmgren indicates that Nelson’s nonstandard analysis, which corresponds to part but not all of Robinson’s, may well lend itself to constructivisation (Palmgren 1998, p. 234). Weyl, on the other hand, would presumably have been interested in the classical formalism.
- 55.
Scans are available online at https://library.ias.edu/godelpapers, and a full transcription of one notebook (vol. X) at https://halshs.archives-ouvertes.fr/hal-01459188
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Acknowledgements
Earlier versions of this paper were presented at the conference ‘Weyl and the Problem of Space: From Science to Philosophy’, University of Konstanz, May 2015, and at the workshop ‘Workshop on the Continuum in the Foundations of Mathematics and Physics’, University of Amsterdam, April 2017. I am grateful to the organisers for their invitations, and to the audiences for their questions and comments. I have also benefited from exchanges with Julien Bernard (who also shared his instructive, unpublished manuscript ‘New insights on Weyl’s Problem of Space, from the correspondence with Becker’ with me), Dirk van Dalen, Bruno Dinis, Mikhail Katz, Carlos Lobo, David Rabouin, Jean-Michel Salanskis, Sam Sanders, Wim Veldman, and Freek Wiedijk. Gödel’s shorthand notes on the non-Archimedean number system in Brouwer’s dissertation, mentioned in footnote 1.5, were kindly transcribed by Eva-Maria Engelen. These notes are owned by the Institute for Advanced Study and kept in the Department of Rare Books and Special Collections at the Firestone Library, Princeton University.Footnote 55
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van Atten, M. (2019). Weyl and Intuitionistic Infinitesimals. In: Bernard, J., Lobo, C. (eds) Weyl and the Problem of Space. Studies in History and Philosophy of Science, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-030-11527-2_5
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