Dirk van Dalen

How Connected is the Intuitionistic Continuum

In the twenties Brouwer established the well-known continuity theorem “every real function is locally uniformly continuous”, [Brouwer 1924, Brouwer 1924a, Brouwer 1927]. From this theorem one immediately concludes that the continuum is indecomposable (unzerlegbar), i.e. if R = A∪B and A∩B = ∅ (denoted by R = A+B), then R = A or R = B. Brouwer deduced the indecomposability directly from the fan theorem (cf. the 1927 Berline Lectures, [Brouwer 1992, p. 49]). The theorem was published for the first time in [Brouwer 1928], it was used to refute the principle of the excluded middle: ¬∀x ∈ R(x ∈ Q ∨ ¬x ∈ Q).

Arguments for the Continuity Principle

There are two principles that lend Brouwers mathematics the extra power beyond arithmetic. Both are presented in Brouwers writings with little or no argument. One, the principle of bar induction, will not concern us here. The other, the continuity principle for numbers, occurs for the first time in print in [4]. It is formulated and immediately applied to show that the set of numerical choice sequences is not enumerable. In fact, the idea of the continuity property can be dated fairly precisely, it is to be found in the margin of Brouwers notes for his course on Pointset Theory of 1915/16. The course was repeated in 1916/17 and he must have inserted his first formulation of the continuity principle in the fall of 1916 as new material right at the beginning of the course. In modern language, the principle reads where α and β range over choice sequences of natural numbers, m and x over natural numbers, and stands for ⟨α(0), α(1), …, α( m − 1)⟩, the initial segment of α of length m . An immediate consequence of WC-N is that all full functions are continuous, and, as a corollary, that the continuum is unsplittable [28]. Note that WC-N is incompatible with Churchs thesis, [22], section 4.6. After Brouwer asserted WC-N, Troelstra was the first to ask in print for a conceptual motivation, but he remained an exception; most authors followed Brouwer by simply asserting it, cf. [18]. Let us note first that in one particular case the principle is obvious indeed, namely in the case of the lawless sequences. The notion of lawless sequence surfaced fairly late in the history of intuitionism. Kreisel introduced it in [17] for metamathematical purposes. There is a letter from Brouwer to Heyting in which the phenomenon also occurs [7]. This is an important and interesting fact since it is (probably) the only time that Brouwer made use of a possibility expressly stipulated in, e.g., [5], see below.

From Brouwerian Counter Examples to the Creating Subject

The original Brouwerian counter examples were algorithmic in nature; after the introduction of choice sequences, Brouwer devised a version which did not depend on algorithms. This is the origin of the ‘creating subject’ technique. The method allowed stronger refutations of classical principles. Here it is used to show that ‘negative dense’ subsets of the continuum are indecomposable.The original Brouwerian counter examples were algorithmic in nature; after the introduction of choice sequences, Brouwer devised a version which did not depend on algorithms. This is the origin of the ‘creating subject’ technique. The method allowed stronger refutations of classical principles. Here it is used to show that ‘negative dense’ subsets of the continuum are indecomposable.

‘Outside’ as a primitive notion in constructive projective geometry

In intuitionistic (or constructive) geometry there are positive counterparts, ‘apart’ and ‘outside’, of the relations ‘=’ and ‘incident’. In this paper it is shown that the relation ‘outside’ suffices to define ‘incident’, ‘apart’ and ‘equality’. The equivalence of the new system with Heytings system is shown and as a simple corollary one obtains duality for intuitionistic projective geometry.

Ishihara's proof technique in constructive analysis

Abstract Two surprising constructive lemmas of Ishihara, with extremely useful proof techniques, are placed in a general setting. This both clarifies the ideas underlying those lemmas and raises the possibility that some other applications of their proof techniques in constructive analysis are, in fact, corollaries of our general results.

Brouwer and Fraenkel on Intuitionism

In the present paper the story is told of the brief and far from tranquil encounter of L.E.J. Brouwer and A. Fraenkel. The relationship which started in perfect harmony, ended in irritation and reproaches.1 The mutual appreciation at the outset is beyond question. All the more deplorable is the sudden outbreak of an emotional disagreement in 1927. Looking at the Brouwer–Fraenkel episode, one should keep in mind that at that time the so-called Grundlagenstreit2 was in full swing. An emotional man like Brouwer, who easily suffered under stress, was already on edge when Fraenkel’s book Zehn Vorlesungen uber die Grundlegung der Mengenlehre, [Fraenkel 1927] was about to appear. With the Grundlagenstreit reaching (in print!) a level of personal abuse unusual in the quiet circles of pure mathematics, Brouwer was rather sensitive, where the expositions of his ideas were concerned. So when he thought that he detected instances of misconception and misrepresentation in the case of his intuitionism, he felt slighted. We will mainly look at Brouwer’s reactions. since the Fraenkel letters have not been preserved. The late Mrs. Fraenkel kindly put the Brouwer letters that were in her possession at my disposal. I am grateful to the Fraenkel family for the permission to use the material. I am indebted to Andreas Blass for his valuable suggestions and corrections.

Brouwer’s ϵ-fixed point and Sperner’s lemma

Abstract It is by now common knowledge that in 1911 Brouwer gave mathematics a miraculous tool, the fixed point theorem, and that later in life, he disavowed it. It usually came as a shock when he replied to the question “is the fixed point theorem correct?” with a point blank “no”. This rhetoric exchange deserves some elucidation. At the time that Brouwer did his revolutionary topological work, he had suspended his constructive convictions for the time being. He was well aware that he was using the principle of the excluded middle, indeed in Brouwer (1919) [1] , p. 950, he remarked that “In my philosophy-free mathematical papers I have regularly used the old methods, while at the same time attempting to deduce only those results, of which I could hope that they would find a place and be of value, if necessary in a modified form, in the new doctrine after the carrying out of a systematic construction of intuitionistic set theory”. And in the case of the fixed point theorem we are presented with exactly such a result. From the intuitionistic point of view the theorem is not correct because the fixed point that is promised can in general not be found, that is to say, approximated.

Gentzens problem. mathematische logik im nationalsozialistischen Deutschland.

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The New Topology

The next step in Brouwer’s topological research was the study of continuous maps on manifolds. The program opened with a bang: in a brief note Brouwer proved the invariance of dimension under homeorphisms. This publication led to an unpleasant altercation with Lebesgue, who claimed to have already found a proof. In fact he had deduced the invariance from the paving principle, but failed to prove the paving principle. In the end Brouwer’s priority and superior insight was fully vindicated. In subsequent papers Brouwer enriched the arsenal of basic notions of topology with simplicial approximation and the mapping degree. The contacts with Baire, Hadamard, Blumenthal, and Hilbert, are described. Brouwer’s name became lastingly attached to his fixed point theorem. Brouwer also proved the invariance of domain theorem, which he subsequently used to salvage Klein’s continuity method for proving uniformisation. This brought him into a conflict with Paul Koebe, who was the uncrowned king of uniformisation and complex function theory. This first topological period closed with a significant feat: Brouwer defined, following Poincare’s first approach, the general notion of dimension, and proved its ‘correctness’, i.e. showed that ℝ n is n-dimensional.

Another look at Brouwer’s dissertation

Brouwer’s dissertation marked the beginning of two separate research activities that played an important role in the mathematics of the twentieth century. The first of these were his first steps in topology and Lie group theory, the second one opened up new directions in the foundations of mathematics. It is with the second one that we are concerned here; for the first one see Freudenthal’s comments in (Brouwer 1976).