Chapter 3 The continuum, intuitive and measurable 3.1 Introduction This chapter covers Brouwer’s analysis of the result of the ur-intuition, viz. the intuitive continuum, its character and its properties.1 We ended the preceding chapter with the remark that the intuitive continuum as the immediate corol- lary of the ur-intuition was emphasized repeatedly in the dissertation, as well as in other publications of later date. Also after the introduction and the develop- ment of the ‘perfect spread’, which gives us the ‘full continuum’ of all unfinished choice sequences on the unit interval, the continuum still remains the immedi- ate result of the ur-intuition. We will see that, in order to prevent confusion, the different concepts about the continuum (like the ‘intuitive’ continuum, the ‘reduced’ continuum, the ‘full’ continuum) have to be distinguished sharply. We find the origin of Brouwer’s continuum concept, as we saw, in the flow of time as the medium of cohesion between two well-separated events. Apart from the construction of the integers and of the rational scale, more properties of the continuum and of these scales can be deduced, still on the basis of the mathematical intuition and of the concept of mathematics as a free creation of the individual human mind. Several properties of the continuum were discussed in the previous chapter, and new ones will come up in the present one, again requiring interpretation about what exactly Brouwer had in mind, like the idea of a ‘freely constructed scale’, or the property of being ‘partlially unknown’ of an infinite sequence approximating a point. We will argue that ‘partially unknown’ can be interpreted in two ways: 1. Partially unknown on principle. This is the case with lawless choice se- quences; the respective elements are not individualized. 1See Brouwer’s dissertation, page 9 ff. 63 64 CHAPTER 3. THE CONTINUUM, INTUITIVE AND MEASURABLE 2. Partially unknown but individualized. The elements are not yet known since they are not yet computed, but√ they are ‘intensionally known’; this is the case with e.g. the sequences defining 2 or π. Another important item to be discussed is the notion of the ‘everywhere dense scale of the rationals’. Brouwer claimed that in the process of construct- ing the rational scale, there might be ‘points not reached’; we can even leave (or find out afterwards that we have left) a complete segment of the continuum unpenetrated by inserted points. Brouwer declared the everywhere dense char- acteristic of the rational scale η to be the result of an ‘agreement’ (which we will argue to be the result of a free act of the mind); but then the following two questions have to be examined: 1. How can we exclude a segment of the continuum from the insertion of new elements, given the fact that the awareness of the non-coincidence of two ex- periences is the inserted element? In other words, how do we create an empty segment under this interpretation? 2. Once we succeed in the construction of such a segment or we become aware of it, how do we eliminate it again? Another point of discussion is the Bolzano-Weierstrass theorem. Brouwer stated that this theorem is a direct corollary of the measurability of the contin- uum, but closer inspection reveals that he employed in its proof the principle of ‘reductio ad absurdum’, as well as the ‘tertium exclusum’. Of special interest for this subject are the notebooks, which show a de- velopment from a possible constructibility of the continuum (albeit that this possibility was only expressed once, at the level of a thought experiment), via ‘something mysterious but nevertheless intuitively known’ to the continuum ‘as the result of the ur-intuition’. An evolution in Brouwer’s thoughts about the exact nature of the continuum can be observed in the nine notebooks. In the first section of this chapter a concise overview will be presented of just a few different (and sometimes mutually opposing) opinions about the contin- uum, as well as the names of the philosophers and mathematicians attached to them. This overview serves as an introductory remark, to make a comparison with Brouwer’s ideas possible, which ideas were often deviating from those of the mainstream of mathematics. Moreover, this overview shows the importance that, through the ages, mathematicians and philosophers have attached to the continuum. 3.1.1 Aristotle Euclid held the position that a line is composed of points, as can be concluded form definition 4 in Book 1 of The Elements: 4. A straight line is a line which lies evenly with the points on itself.2 2A not too lucid definition; see [Heath 1956], page 153. 3.1. INTRODUCTION 65 Aristotle, who lived some fifty years earlier, took up a different position. For him, a line (the continuum) is not composed of points; all one can say is that a line is indefinitely divisible. In his Physics, in Book III, under B–6, a–14, this is explained as follows: Now, ‘to be’ means either ‘to be potentially’ or ‘to be actually’, and a thing may be infinite either by addition or by division. I have argued that no actual magnitude can be infinite, but it can still be infinitely divisible (it is not hard to disprove the idea that there are indivisible lines), and we are left with things being infinite potentially.3 and the first part of Book VI contains the following sections: Proof that no continuum is made up of indivisible parts. and Proof that distance, time and movement are all continua. In the first section of book VI Aristotle argued as follows: For instance, a line, which is continuous, cannot consist of points, which are indivisible, first because in the case of points there are no limits to form a unity (since nothing indivisible has a limit which is distinct from any other part of it), and second because in their case there are no limits to be together (since anything which lacks parts lacks limits too, because a limit is distinct from that of which it is a limit).4 A similarity between Aristoteles’ conclusions and Brouwer’s ideas will be (or will become) clear, and the contrast with e.g. Cantor is obvious. 3.1.2 Georg Cantor In chapter 1 of this dissertation (see page 8 and page 14) Cantor’s views on the continuum were given: the continuum is composed of points. Cantor’s continuum is the arithmetical continuum of the real numbers. The concepts of time and space presuppose the continuum concept. 3.1.3 Hermann Weyl In 1918 Weyl published Das Kontinuum. In chapter II of this booklet, Zahlbegriff und Kontinuum, the system of the real numbers is defined in a way similar to Dedekind. In § 6 the ‘anschauliches Kontinuum’ (intuitive continuum) and the ‘mathematisches Kontinuum’ are discussed. In this section it is stated: 3[Aristotle 1999], page 71. 4Op. cit. page 138. 66 CHAPTER 3. THE CONTINUUM, INTUITIVE AND MEASURABLE Bleiben wir, um das Verh¨altnis zwischen einem anschaulich gegebe- nen Kontinuum und dem Zahlbegriff besser zu verstehen (..), bei der Zeit als dem fundamentalsten Kontinuum; (...) Um zun¨achst einmal ¨uberhaupt die Beziehung zur mathematischen Begriffswelt herstellen zu k¨onnen, sei die ideelle M¨oglichkeit, in dieser Zeit ein streng punktuelles ‘jetzt’ zu setzen, sei die Aufweisbarkeit von Zeitpunkten zuzgegeben. (...) Zwei Zeitpunkte A, B, von denen A der fr¨uhere ist, begrenzen eine Zeitstrecke AB; in sie hinein f¨alltjeder Zeitpunkt, der sp¨ater als A, aber fr¨uher als B ist. Der Erlebnisgehalt, welcher die Zeitstrecke AB erf¨ullt, k¨onnte ‘an sich’, ohne irgendwie ein andrer zu sein als er ist, in irgend eine andere Zeit fallen; die Zeitstrecke, die er dort erf¨ullen w¨urde, ist der Strecke AB gleich.5 Note the similarity with Brouwer’s concept of the combination discrete and continuous which we observed in the previous chapter. From available evidence we know that Weyl was in the possession of the English translation of Brou- wer’s inaugural address from 1912, in which this inseparable combination is stressed. He may also have read Die m¨oglichen M¨achtigkeiten (1908) in which also the ‘Zeitstrecke’ between two events as a matrix for ‘Zeitpunkte’ is under- lined.6 Weyl then posed the question whether we can use this concept of ‘points in a time continuum’ and their mutual relation of ‘earlier’ and ‘later’ as the foundation of the real number concept, and thus, whether this time continuum can serve as a model for the mathematical continuum of real numbers, and he claimed: Gewiß: das anschauliche und das mathematische Kontinuum decken sich nicht; zwischen ihnen ist eine tiefe Kluft befestigt. Aber doch sind es vern¨unftige Motive, die uns in unserm Bestreben, die Welt zu begreifen, aus dem einen ins andere hin¨ubertreiben; (...) Beschr¨anken wir uns hinsichtlich des Raumes auf die Geometrie der Geraden! Will man nun doch versuchen, eine Zeit- und Raumlehre als selbst¨andige mathematisch-axiomatische Wissenschaft aufzurich- ten, so muß man immerhin folgendes beachten. 1. Die Aufweisung eines einzelnen Punktes ist unm¨oglich. Auch sind Punkte keine Individuen und k¨onnen daher nicht durch ihre Eigenschaften charakterisiert werden. (W¨ahrend das ‘Kontinuum’ 5[Weyl 1918], page 67. 6Note that Brouwer’s thesis On the Foundations of Mathematics was not yet available in any other language except Dutch. 3.1. INTRODUCTION 67 der reellen Zahlen aus lauter Individuen besteht, ist das der Zeit- oder Raumpunkte homogen.)7 Note the terminology: the continuum of time- or spacepoints is ‘homoge- neous’, which has to be understood in the sense of ‘having no gaps’, i.e.