## 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

## Access this chapter

Tax calculation will be finalised at checkout

Purchases are for personal use only

### Similar content being viewed by others

## 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

## References

Ardourel, V. 2012. La physique dans la recherche en mathématiques constructives.

*Philosophia Scientiae*16 (1): 183–208.Arkeryd, L.O., N.J. Cutland, and C.W. Henson, eds. 1997.

*Nonstandard analysis. Theory and applications*. Dordrecht: Springer.Baron, M. 1969.

*The origins of infinitesimal calculus*. Oxford: Pergamon Press.Barreau, H., and J. Harthong, eds. 1989.

*La Mathématique non standard*. Paris: Éditions du CNRS.Bernard, J. 2013.

*L’idéalisme dans l’infinitésimal. Weyl et l’espace à l’époque de la relativité*. Presses universitaires de Paris Nanterre, Nanterre. Available online at http://books.openedition.org/pupo/3917.Brouwer, L.E.J. Notebooks, 1904–1907.

*Brouwer Papers*. Haarlem: Noord-Hollands Archief. Available at http://www.cs.ru.nl/F.Wiedijk/brouwer/index.html.———. 1907.

*Over de grondslagen der wiskunde*. PhD thesis, Universiteit van Amsterdam.———. 1908. De onbetrouwbaarheid der logische principes.

*Tijdschrift voor Wijsbegeerte*2: 152–158.———. 1909.

*Het wezen der meetkunde*. Amsterdam: Clausen.———. 1917. Addenda en corrigenda over de grondslagen der wiskunde.

*Nieuw Archief voor Wiskunde*12: 439–445.———. 1919.

*Wiskunde, waarheid, werkelijkheid*. Groningen: Noordhoff.———. 1920. Intuitionistische Mengenlehre.

*Jahresbericht der deutschen Mathematiker-Vereinigung*28: 203–208.———. 1924. Über die Bedeutung des Satzes vom ausgeschlossenen Dritten in der Mathematik, insbesondere in der Funktionentheorie.

*Journal für die reine und angewandte Mathematik*154: 1–7. 1923B2 in Brouwer (1975).———. 1927. Über Definitionsbereiche von Funktionen.

*Mathematische Annalen*97: 60–75.———. 1928. Intuitionistische Betrachtungen über den Formalismus.

*KNAW Proceedings*31: 374–379.———. 1929. Mathematik, Wissenschaft und Sprache.

*Monatshefte für Mathematik und Physik*36: 153–164.———. 1930.

*Die Struktur des Kontinuums*. Komitee zur Veranstaltung von Gastvorträgen ausländischer Gelehrter der exakten Wissenschaften, Wien.———. 1948. Essentieel negatieve eigenschappen.

*Indagationes Mathematicae*10: 322–323.———. 1949a. De non-aequivalentie van de constructieve en de negatieve orderelatie in het continuum.

*Indagationes Mathematicae*11: 37–39.———. 1949b. Contradictoriteit der elementaire meetkunde.

*KNAW Proceedings*52: 315–316.———. 1949c. Consciousness, philosophy and mathematics. In

*Proceedings of the 10th international congress of philosophy, Amsterdam 1948*, ed. E. Beth, H. Pos, and J. Hollak, vol. 2, 1235–1249. Amsterdam: North-Holland.———. 1951. On order in the continuum, and the relation of truth to non-contradictority.

*KNAW Proceedings*54: 357–358.———. 1954. Points and spaces.

*Canadian Journal of Mathematics*6: 1–17.———. 1975.

*Collected works. Vol. 1: Philosophy and foundations of mathematics*, ed. A. Heyting. Amsterdam: North-Holland.———. 1981.

*Brouwer’s Cambridge lectures on intuitionism*, ed. D. van Dalen. Cambridge: Cambridge University Press.Buss, S. 2006. Nelson’s work on logic and foundations and other reflections on foundations of mthematics. In

*Diffusion, quantum theory, and radically elementary mathematics*, ed. W. Faris, 183–208. Princeton: Princeton University Press.Coquand, T. 2014. Recursive functions and constructive mathematics. In

*Constructivity and calculability in historical and philosophical perspective*, ed. Dubucs, J., and M. Bourdeau, 159–167. Dordrecht: Springer.Cutland, N.J., ed. 1988.

*Nonstandard analysis and its applications*. Cambridge: Cambridge University Press.Diener, F., and G. Reeb. 1989.

*Analyse non standard*. Paris: Hermann.Dinis, B., and J. Gaspar. 2018. Intuitionistic nonstandard bounded modified realisability and functional interpretation.

*Annals of Pure and Applied Logic*169 (5): 392–412.Dubucs, J., and M. Bourdeau, eds. 2014.

*Constructivity and calculability in historical and philosophical perspective*. Dordrecht: Springer.Dummett, M. 1975. Wang’s Paradox.

*Synthese*30: 301–324.Eckes, C. 2011.

*Groupes, invariants et géométries dans l’œuvre de Weyl: Une étude des écrits de Hermann Weyl en mathématiques, physique mathématique et philosophie, 1910–1931*. PhD thesis, Université Jean Moulin Lyon 3.Ehrlich, P. 2006. The rise of non-Archimedean mathematics and the roots of a misconception I: The emergence of non-Archimedean Systems of Magnitudes.

*Archive for the History of the Exact Sciences*60: 1–121.Ferreira, F., and J. Gaspar. 2015. Nonstandardness and the bounded functional interpretation.

*Annals of Pure and Applied Logic*166 (6): 701–712.Fletcher, P., K. Hrbaček, V. Kanovei, M. Katz, C. Lobry, and S. Sanders. 2017. Approaches to analysis with infinitesimals following Robinson, Nelson, and others.

*Real Analysis Exchange*42 (2): 193–252.Fraenkel, A. 1929.

*Einleitung in die Mengenlehre. Eine elementare Einführung in das Reich der unendlichen Grössen. 3te umgearbeitete und stark erweiterte Auflage*. Berlin: Springer.Freudenthal, H. 1955. Hermann Weyl. Der Dolmetscher zwischen Mathematikern und Physikern um die moderne Interpretation von Raum, Zeit und Materie. In

*Forscher und Wissenschaftler im heutigen Europa. Weltall und Erde: Physiker, Chemiker, Erforscher des Weltalls, Erforscher der Erde, Mathematiker*. Oldenburg: Gerhard Stalling.Gödel, K.

*Papers*, 1906–1978. Department of rare books and special collections. Princeton: Firestone Library.Hahn, H. 1907. Über die nichtarchimedischen Größensysteme.

*Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien*116: 601–655.Harthong, J. 1981. Le moiré.

*Advances in Applied Mathematics*2: 24–75.———. 1989. Commentaires sur Intuitionnisme 84. In

*Barreau and Harthong*, 253–273.Heyting, A. 1934.

*Mathematische Grundlagenforschung, Intuitionismus, Beweistheorie*. Berlin: Springer.———. 1958. Blick von der intuitionistischen Warte.

*Dialectica*12: 332–345.Hilbert, D. 1899.

*Grundlagen der Geometrie. Festschrift zur Feier der Enthüllung des Gauss-Weber Denkmals in Göttingen*. Leipzig: Teubner.Kanovei, V., and M. Reeken. 2004.

*Nonstandard analysis, axiomatically*. Berlin: Springer.Kleene, S., and R. Vesley. 1965.

*The foundations of intuitionistic mathematics, especially in relation to recursive functions*. Amsterdam: North-Holland.Kock, A. 2006.

*Synthetic differential geometry*. 2nd ed. Cambridge: Cambridge University Press.Kreisel, G. 1967. Informal rigour and completeness proofs. In

*Problems in the philosophy of mathematics*, ed. I. Lakatos, 138–186. Amsterdam: North-Holland.L’Ouvert. 1994.

*Numéro spécial Georges Reeb*. Institut de recherche sur l’enseignement des mathématiques (IREM) de Strasbourg, Strasbourg, septembre 1994.Laugwitz, D. 1986.

*Zahlen und Kontinuum. Eine Einführung in die Infinitesimalmathematik*. Mannheim: BI Wissenschaftsverlag.Leibniz, G.W. 1859. In

*Leibnizens mathematische Schriften*, ed. C. Gerhardt, vol. 4. Halle: Schmidt.Lobry, C. 1989.

*Et pourtant… ils ne remplissent pas N*. Lyon: Aléas.Lobry, C., and T. Sari. 2008. Non-standard analysis and representation of reality.

*International Journal of Control*81 (3): 517–534.Mancosu, P. 1999.

*Philosophy of mathematics and mathematical practice in the seventeenth century*. Oxford: Oxford University Press.Mancosu, P., and T. Ryckman. 2002. Mathematics and phenomenology: The correspondence between O. Becker and H. Weyl.

*Philosophia Mathematica, New Series*10: 130–202.Myhill, J. 1966. Notes towards an axiomatization of intuitionistic analysis.

*Logique et Analyse*9: 280–297.Nelson, E. 1977. Internal set theory: A new approach to nonstandard analysis.

*Bulletin American Mathematical Society*83: 1165–1198.———. 1986.

*Predicative arithmetic*. Princeton: Princeton University Press. Available at https://web.math.princeton.edu/~nelson/books/pa.pdf.———. 1988. The syntax of nonstandard analysis.

*Annals of Pure and Applied Logic*38 (2): 123–134.———. 1996

*. Ramified recursion and intuitionism*. Available at https://web.math.princeton.edu/~nelson/papers/ramrec.pdf. The year 1996 is that in the date of the TeX file on the same server; the original talk was presented to the Colloque Trajectorien, Strasbourg/Obernai, June 12–16, 1995.———. 2002.

*Internal set theory*. First chapter of an unfinished book on nonstandard analysis, available at https://web.math.princeton.edu/~nelson/books/1.pdf. The year 2002 is that of the pdf file on the server.Palmgren, E. 1993. A note on mathematics of infinity.

*The Journal of Symbolic Logic*58 (4): 1195–1200.———. 1995. A constructive approach to nonstandard analysis.

*Annals of Pure and Applied Logic*73 (3): 297–325.———. 1998. Developments in constructive nonstandard analysis.

*Bulletin of Symbolic Logic*4 (3): 233–272.Reeb, G. 1979.

*La mathématique non standard vieille de soixante ans ?*. References are to the reprint in Appendix A to Salanskis 1999.———. 1981. La mathématique non standard vieille de soixante ans ?

*Cahiers de Topologie et Géométrie Différentielle Catégoriques*22 (2): 149–154.———. 1989.

*0, 1, 2, etc … ne remplissent pas (du tout) N, 1989*. Included as chapter 9 in*Analyse non standard*, ed. Diener, F., and G. Reeb. Paris: Hermann.Reeb, G., and J. Harthong. 1989. Intuitionnisme 84. In

*La Mathématique non standard*, ed. Barreau, H., and J. Harthong, 213–252. Reprinted in L’Ouvert (1994, pp. 42–77).Robert, A. 1988.

*Nonstandard analysis*. New York: Wiley.Robinson, A. 1965. Formalism 64. In

*Logic, methodology, and philosophy of science*, ed. Y. Bar-Hillel, 228–246. Amsterdam: North Holland.———. 1966.

*Non-standard analysis*. Amsterdam: North-Holland.Salanskis, J.-M. 1994. Un Maître. In

*Numéro spécial Georges Reeb*, ed. L’Ouvert, 25–32. Institut de recherche sur l’enseignement des mathématiques (IREM) de Strasbourg, Strasbourg.———. 1999.

*Le Constructivisme non standard*. Villeneuve d’Ascq: Presses Universitaires du Septentrion.Sanders, S. 2017.

*Nonstandard analysis and constructivism!*. https://arxiv.org/abs/1704.00281.———. 2018. To be or not to be constructive, that is not the question.

*Indagationes Mathematicae*29 (1): 313–381.Schmieden, C., and D. Laugwitz. 1958. Eine Erweiterung der Infinitesimalrechnung.

*Mathematische Zeitschrift*69: 1–39.Scholz, E. 2001. Weyls Infinitesimalgeometrie (1917–1925). In

*Hermann Weyl’s Raum-Zeit-Materie and a general introduction to his scientific work*, ed. E. Scholz, 48–104. Basel: Birkhäuser.Schubring, G. 2005.

*Conflicts between generalization, rigor, and intuition. Number concepts underlying the development of analysis in 17–19th century France and Germany*. New York: Springer.Skolem, T. 1929. Über die Grundlagendiskussionen in der Mathematik. In

*Den syvende skandinaviske matematikerkongress i Oslo 19–22 August 1929*. Oslo: Broegger.———. 1934. Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen.

*Fundamenta Mathematicae*23: 150–161.Sundholm, G. 2014. Constructive recursive functions, Church’s thesis, and Brouwer’s theory of the creating subject: afterthoughts on a Parisian Joint Session. In

*Constructivity and calculability in historical and philosophical perspective*, ed. Dubucs, J., and M. Bourdeau. Dordrecht: Springer. 1–35.Sundholm, G., and M. van Atten. 2008. The proper interpretation of intuitionistic logic. On Brouwer’s demonstration of the Bar Theorem. In

*One hundred years of intuitionism (1907–2007). The Cerisy conference*, ed. M. van Atten, P. Boldini, M. Bourdeau, and G. Heinzmann, 60–77. Basel: Birkhäuser.Tieszen, R. 2000. The philosophical background of Weyl’s mathematical constructivism.

*Philosophia Mathematica*3: 274–301.Troelstra, A. 1982. On the origin and development of Brouwer’s concept of choice sequence. In

*The L. E. J. Brouwer centenary symposium*, ed. A. Troelstra and D. van Dalen, 465–486. Amsterdam: North-Holland.Troelstra, A., and D. van Dalen. 1988.

*Constructivism in mathematics*. Amsterdam: North-Holland.van Atten, M. 2004.

*On Brouwer*. Belmont: Wadsworth.———. 2015.

*Essays on Gödel’s reception of Leibniz, Husserl, and Brouwer*. Cham: Springer.———. 2018. The creating subject, the Brouwer-Kripke Schema, and infinite proofs.

*Indagationes Mathematicae*29: 1565–1636.van Atten, M., and G. Sundholm. 2017. L. E. J. Brouwer’s “Unreliability of the logical principles”. A new translation, with an introduction.

*History and Philosophy of Logic*38 (1): 24–47.van Atten, M., D. van Dalen, and R. Tieszen. 2002. Brouwer and Weyl: The phenomenology and mathematics of the intuitive continuum.

*Philosophia Mathematica*10 (3): 203–226.van Dalen, D. 1988. Infinitesimals and the continuity of all functions.

*Nieuw Archief voor Wiskunde*6 (3): 191–202.———. 1999.

*Mystic, geometer, and intuitionist. The life of L. E. J. Brouwer. Volume 1: The dawning revolution*. Oxford: Oxford University Press.———. 2001.

*L.E.J. Brouwer en de grondslagen van de wiskunde*. Utrecht: Epsilon.———. 2005.

*Mystic, geometer, and intuitionist. The life of L. E. J. Brouwer. Volume 2: Hope and disillusion*. Oxford: Clarendon Press.van den Berg, B., and S. Sanders. 2017. Reverse mathematics and parameter-free transfer. https://arxiv.org/abs/1409.6881.

van den Berg, B., E. Briseid, and P. Safarik. 2012. A functional interpretation for nonstandard arithmetic.

*Annals of Pure and Applied Logic*163 (12): 1962–1994.van Heijenoort, J., ed. 1967.

*From Frege to Gödel: A sourcebook in mathematical logic, 1879–1931*. Cambridge, MA: Harvard University Press.Veronese, G. 1891.

*Fondamenti di Geometria a più dimensioni e a più specie di unità rettilinee, esposti in forma elementare*. Padova: Tipografia del Seminario.Vesley, R. 1981. An intuitionistic infinitesimal calculus. In

*Constructive Mathematics (Lecture Notes in Mathematics, 873)*, ed. F. Richman, 208–212. Berlin: SpringerWavre, R. 1926. Logique formelle et logique empirique.

*Revue de Métaphysique et de Morale*33: 65–75.Wenmackers, S. 2016. Children of the cosmos. Presenting a toy model of science with a supporting cast of infinitesimals. In

*Trick or truth?*, ed. A. Aguirre, B. Foster, and Z. Merali, 5–20. Dordrecht: Springer.Weyl, H. 1922. Die Einzigartigkeit der Pythagoreischen Maßbestimmung.

*Mathematische Zeitschrift*12: 114–146.———. 1925. Die heutige Erkenntnislage in der Mathematik.

*Symposion*1: 1–32.———. 1926.

*Philosophie der Mathematik und Naturwissenschaft*. München: Leibniz Verlag. Weyl 1949 (*Philosophy of mathematics and natural science*. Princeton: Princeton University Press.) is an expanded English version.———. 1949.

*Philosophy of mathematics and natural science*. Princeton: Princeton University Press.———. 1988. In

*Riemanns geometrische Ideen, ihre Auswirkung und ihre Verknüpfung mit der Gruppentheorie*, ed. K. Chandrasekharan. Berlin: Springer.Wright, C. 1975. On the coherence of vague predicates.

*Synthese*30: 325–365.

## 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}

## Author information

### Authors and Affiliations

## Editor information

### Editors and Affiliations

## Rights and permissions

## Copyright information

© 2019 Springer Nature Switzerland AG

## About this chapter

### Cite this chapter

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

### Download citation

DOI: https://doi.org/10.1007/978-3-030-11527-2_5

Published:

Publisher Name: Springer, Cham

Print ISBN: 978-3-030-11526-5

Online ISBN: 978-3-030-11527-2

eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)