Brouwer meets Husserl. On the phenomenology of choice sequences
Philosophers in the English-speaking tradition have shown great interest in an intuitionism cleansed of Brouwer's philosophy, as the latter is seen as, to put it mildly, out of place. In their view, Brouwer stumbled upon a remarkable discovery, for which he provided the wrong underlying philosophy -- an outgrowth of personal idiosyncrasies -- which can be safely ignored. This view calls for an entirely different justification for intuitionism, which is what philosophers in that tradition (first and foremost Dummett) have done. For `continental' philosophers the problem has been not that of discarding Brouwer from intuitionism, but that of placing Brouwer's intuitionism into the mainstream of `continental' philosophy by finding points of contact with established philosophies or ways to justify intuitionistic practice from tenets closely resembling Brouwer's but put forward by a different philosopher in a different context. As its title suggests, van Atten's book belongs to this category, which includes the reviewer's [``Mathematik, Intuition und die Existenzweise des Seins'', in: Wissenschaft vom Menschen: Jahrbuch der Internationalen Erich-Fromm-Gesellschaft, Münster, Vol. 3, 87--119 (1992; Zbl 0763.00003)], \textit{M. Detlefsen} [``Constructive existence claims'', in: M. Schirn (ed.), The philosophy of mathematics today. Oxford: Clarendon Press, 307--335 (1998; Zbl 0922.03006)], \textit{J. Largeault} [Intuition et intuitionisme. Paris: Librairie Philosophique J. Vrin (1993; Zbl 0866.03002)], \textit{T. Koetsier} [``Arthur Schopenhauer and L. E. J. Brouwer: a comparison'', in: T. Koetsier et al. (eds.), Mathematics and the divine. Amsterdam: Elsevier/North-Holland, 569--593 (2005; Zbl 1078.01016)], and \textit{Z. Fraser} [``The law of the subject: Alain Badiou, Luitzen Brouwer and the Kripkean analysis of forcing and Heyting calculus'', Cosm. Hist. 2, 94--133 (2006; Zbl 1117.03010)]. Its purpose is to show that Brouwer's argument for choice sequences, which are not static mathematical objects, can be reconstructed in Husserl's phenomenology. The first attempt to bring intuitionism and phenomenology together goes back to \textit{O. Becker} [``Beiträge zur phänomenologischen Begründung der Geometrie und ihrer physikalischen Anwendungen'', Jahrb.\ Philos.\ Phänomenolog.\ Forsch.\ 6, 385--560 (1923; JFM 49.0391.01)], followed by his [``Mathematische Existenz. Untersuchungen zur Logik und Ontologie mathematischer Phänomene'', ibid. 8, 439--809 (1927; JFM 53.0038.04)], where choice sequences do show up, but Brouwer's and Weyl's view are so hopelessly mixed up that non-lawlike choice sequences are dismissed as genuine mathematical objects. They suffer a similar fate at the next attempt to render them phenomenologically respectable in \textit{F. Kaufmann}'s [Das Unendliche in der Mathematik und seine Ausschaltung. Eine Untersuchung über die Grundlagen der Mathematik. Wien: F. Deuticke (1930; JFM 56.0039.01, Zbl 0402.03001)]. Given that Husserl relied primarily on Becker for the mathematics of choice sequences (with some additional input from Weyl and Kaufmann), it is not surprising that they found no place in his philosophy, for he saw mathematical objects as omnitemporal (they exist in time, but are not subject to change), after having seen them in an earlier part of his life as atemporal (existing outside of time). The author shows that Husserl could have, had he followed the essential ideas of his transcendental idealism and had he been better informed about choice sequences, provided a phenomenological justification for choice sequences by admitting that `some mathematical objects are omnitemporal, some are not.' The key that allows for the softening of the widely held position that mathematical objects are either omni- or a-temporal is the following quote from \textit{E. Husserl}'s [Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie. Erstes Buch. K. Schuhmann (ed.), Vol.\ III/1 of \textit{Husserliana}. Den Haag: Martinus Nijhoff (1976)]: ``Prinzipiell entspricht (im Apriori der unbedingten Wesensallgemeinheit) jedem ``wahrhaft seienden'' Gegenstand die Idee eines möglichen Bewußtseins, in welchem der Gegenstand selbst originär und dabei voll\-kommen adäquat erfaßbar ist. Umgekehrt, wenn diese Möglichkeit gewährleistet ist, ist eo ipso der Gegenstand wahrhaft seiend.'' This rejection of the Kantian dichotomy between phenomenon and noumenon is at the heart of Brouwer's philosophy as well, making it seem, literally `out of place', for it is, at heart, \textit{Eastern}, if we use this term in the sense of \textit{William S. Haas} [The destiny of the mind. London: Faber and Faber; New York: Macmillan (1956)], for whom ``Eastern cognition is interested in consciousness itself. Western cognition is interested in the object of consciousness'' (p.\ 167) and ``In the mind of the East the subject holds the predominant place occupied by the object in the West'' (p.\ 180). And it is precisely the insistence on the predominance and ontological primodialness of the object that has led philosophers (such as F. Kaufmann -- ``Denn Brouwers Auffassung, daß\, sich die mathematischen Tatsachen mit der mathematischen Erkenntnis wandeln, schließt ein, daß\, es ein zu Erkennendes gebe, welches erst durch die darauf bezogene Erkenntnis geschaffen wurde, was dem Wesen der Erkenntnis zuwiderläuft. Denn jede Erkenntnis setzt \(\ldots\) einen unabhängig von seinem Erkanntwerden als bestehend zu denkenden Gegenstand voraus.'' (op.\ cit., p.\ 65) -- and \textit{N. Hartmann} [Zur Grundlegung der Ontologie. Berlin: Walter de Gruyter (1948), p.\ 236] -- ``Die Wahrheit der Intuition ist, daß\, sie nicht gebender, sondern aufnehmender (rezeptiver) Akt ist, und daß\, die gebende Instanz hinter ihr beim Gegenstande zu suchen ist. Dieser bestimmt die Anschauung, insoweit er sich ihr `darbietet' (erscheint), und zwar als ein gegen den Anschauungsakt selbst indifferenter. Er ist also schon als ansichseiender vorausgesetzt. Bietet sich ein solcher Gegenstand nicht dar, liegt also kein Seiendes vor, das sein bestimmtes Sosein schon an sich hätte, so ist auch der Akt kein schauendes Erfassen.'') to reject intuitionism. Van Atten's book is very well written and argued, with all premises and conclusions clearly stated and labeled, containing, besides a presentation of the original positions of Brouwer and Husserl, an argument for the phenomenological incorrectness of the original arguments of both Husserl and Brouwer (containing a very rare instance of criticism of Brouwer's philosophy, as presented in [\textit{L. E. J. Brouwer}, Consciousness, philosophy, and mathematics. Library of the Tenth International Congress of Philosophy, Amsterdam, August 11--18, 1948, Vol. I, Proceedings of the Congress, 1235--1249 (1949; MR 10,422a)], on textual grounds, namely that a philosophy that asserts the existence of three (or four) kinds of consciousness (stillness, sensational, mathematical, and (perhaps) wisdom) cannot reflect on itself, since the authorship of the assertion that there are these three (or four) kinds of consciousness cannot be attributed to any of these states of consciousness), an in-depth analysis of choice sequences as mathematical objects, and a phenomenological argument for a weak continuity principle. There is also an appendix on \textit{E. Husserl}'s analysis of finite number in his [Philosophie der Arithmetik. Halle a.\ S.: C. E. M. Pfeffer (R. Stricker) (1891; JFM 23.0058.01, Zbl 1073.01026)], in which the author finds that ``Brouwer's account of ordinal number is descriptively more accurate than Husserl's'', and that, although all the elements needed for an understanding in the sense of descriptive psychology of the concept of finite number in the manner Brouwer understood it (based on the perception of time, making ordinal numbers prior to cardinal ones) were present in Husserl's writings, Husserl never advanced beyond his original cardinal number based understanding of 1887, as presented in his Habilitationsschrift, which makes the set and not the sequence the psychologically basic concept.
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