Brouwer's Intuition of Twoity and Constructions in Separable Mathematics

ABSTRACT

My first aim in this paper is to use time diagrams in the style of Brentano to analyze constructions in Brouwer's separable mathematics more precisely. I argue that constructions must involve not only pairing and projecting as basic operations guaranteed by the intuition of twoity, as sometimes assumed in the literature, but also a recalling operation. My second aim is to argue that Brouwer's views on the intuition of twoity and arithmetic lead to an ontological explosion. Redeveloping the constructions of natural numbers and systems sketched in an appendix to Brouwer's Cambridge lectures, I observe that the only plausible way he can make some elementary arithmetic in his separable mathematics is by allowing for the same canonical number to be determined by multiple separable entities, resulting in an overabundant mathematical ontology.

KEYWORDS:

• Brouwer
• intuition of twoity
• intuitionism
• time diagrams
• ontological explosion

Acknowledgments

I would like to thank Ansten Klev, Vincent Peluce, Will Stafford, Göran Sundholm, Colin Zwanziger, and an anonymous reviewer for comments on an early version of this paper. I am especially grateful to Carl Posy for his continuous support and encouragement throughout my studies of Brouwer's philosophy.

Notes

1 Systems in separable mathematics are roughly non-empty discrete sets.

2 Due to space limitations, I must set aside the question of whether my diagrams represent constructions in a solipsistic mind or the idealized mind of a transcendental subject which allows for the possibility of communication to other minds equally equipped with all the mental inventory assumed by Brouwer. I refer the reader to Placek 1999, pp. 5–9, 22–27, ch. 2, § 4, for an excellent discussion of the problem of other minds in Brouwer and a defense of a non-solipsistic reading of his philosophy.

3 Before Brentano, we find a short outline of a retentional model in Kant (1998, A102). It is hard to tell to what extent Brentano's description of time consciousness was influenced by Kant's.

4 See also van Atten (2018, p. 1594). For an in-depth study of protentions in a Husserlian account of the constructions of natural numbers and sets, see Tieszen (1989, pp. 107–108, 137, 147–148).

5 Another natural approach to the study of constructions is through Kripke schema (see van Atten 2018). Due to its formulation in the intuitionistic theory of sequences of natural numbers, it has the advantage that it can go beyond separable mathematics. But, to my mind, this alternative is more directly amenable to the analysis of the construction of true propositions (e.g. the study of weak counterexamples in van Atten (2018, § 4)) rather than that of objects like separable entities, our focus in this paper. This distinction corresponds to that between the intuition that a propositions is true and intuition of an object (see Tieszen 1989, ch. 1, § 3).

6 See alternative (A) in Benacerraf  (1965, p. 56).

Additional information

Funding

This research was funded in part by the Zhejiang Federation of Humanities and Social Sciences project ‘Philosophy of the Future’ (23YJRC04ZD).