Can the straight line be analysed mathematically such that it does not fall apart into a set of discrete points, as is usually done but through which its fundamental continuity is lost? And are there objects of pure mathematics that can change through time? Mathematician and philosopher L.E.J. Brouwer argued that the two questions are closely related and that the answer to both is "yes''. To this end he introduced a new kind of object into mathematics, the choice sequence. But (...) other mathematicians and philosophers have been voicing objections to choice sequences from the start. This book aims to provide a sound philosophical basis for Brouwer's choice sequences by subjecting them to a phenomenological critique in the style of the later Husserl. (shrink)

ON BROUWER, like other titles in the Wadsworth Philosopher's Series, offers a concise, yet comprehensive, introduction to this philosopher's most important ideas. Presenting the most important insights of well over a hundred seminal philosophers in both the Eastern and Western traditions, the Wadsworth Philosophers Series contains volumes written by scholars noted for their excellence in teaching and for their well-versed comprehension of each featured philosopher's major works and contributions. These titles have proven valuable in a number of ways. Serving as (...) standalone texts when tackling a philosophers' original sources or as helpful resources for focusing philosophy students' engagements with these philosopher's often conceptually daunting works, these titles have also gained extraordinary popularity with a lay readership and quite often serve as "refreshers" for philosophy instructors. (shrink)

On Friday, November 15, 1940, Kurt Gödel gave a talk on set theory at Brown University. The topic was his recent proof of the consistency of Cantor’s Continuum Hypothesis with the axiomatic system ZFC for set theory. His friend from their days in Vienna, Rudolf Carnap, was in the audience, and afterward wrote a note to himself in which he raised a number of questions on incompleteness.

This book summarizes the intense research that the author performed for his Ph.D. thesis , revised and with the addition of an intuitionistic critique of Husserl's concept of number. His starting point consisted of a double conviction: 1) Brouwerian intuitionism is a valid way of doing mathematics but is grounded on a weak philosophy; 2) Husserlian phenomenology can provide a suitable philosophical ground for intuitionism. In order to let intuitionism and phenomenology match, he had to solve in general two problems: (...) 1) the question of the reciprocal indifference that the authors had toward each other's theorizing which indeed they knew; 2) Husserl's general attitude of accepting classical mathematics, which contrasts with the critical attitude of the intuitionists.Moreover, in the specific case focused on by this book—concerning choice sequences, that is, sequences that can be completely lawless and that were admitted as mathematical entities by Brouwer—van Atten had to handle a further problem: such sequences do not fit the characteristic of omnitemporality that the late Husserl seemed to consider a necessary attribute of entities if they were to be considered mathematical. In as far as they are lawless we cannot predict at any moment how they will develop in the future, hence they are not determined at any particular time—they will be determined only in the future.Let us begin with van Atten's first conviction, i.e., that Brouwerian intuitionism is grounded on a weak philosophy. We have to recall that Brouwer provided a mystical attitude for proposing his intuitionism: man can obtain happiness only by keeping himself closed inside the inner Self. Any attempted movement towards the exterior world is a source of pain: in particular language and the sciences. Mathematics, in order to be morally acceptable …. (shrink)

Gödel always claimed that he did not know Skolem's Helsinki lecture when writing his dissertation. Some questions and doubts have been raised about this claim, in particular on the basis of a library slip showing that he had requested Skolem's paper in 1928. It is shown that this library slip does not constitute evidence against Gödel's claim, and that, on the contrary, the library slip and other archive material actually corroborate what Gödel said.

In his text 'The modern development of the foundations of mathematics in the light of philosophy' from around 1961, Go¨del announces a turn to Husserl's phenomenology to find the foundations of mathematics. In Go¨del's archive there are two draft letters that shed some further light on the exact strategy that he formulated for himself in the early 1960s. Transcriptions of these letters are presented, together with some comments.

Can the straight line be analysed mathematically such that it does not fall apart into a set of discrete points, as is usually done but through which its fundamental continuity is lost? And are there objects of pure mathematics that can change through time? The mathematician and philosopher L.E.J. Brouwer argued that the two questions are closely related and that the answer to both is "yes''. To this end he introduced a new kind of object into mathematics, the choice sequence. (...) But other mathematicians and philosophers have been voicing objections to choice sequences from the start. This book aims to provide a sound philosophical basis for Brouwer's choice sequences by subjecting them to a phenomenological critique in the style of the later Husserl. "It is almost as if one could hear the two rebels arguing their case in a European café or on a terrace, and coming to a common understanding, with both men taking their hat off to the other, in admiration and gratitude. Dr. van Atten has convincingly applied Husserl's method to Brouwer's program, and has equally convincingly applied Brouwer's intuition to Husserl's program. Both programs have come out the better." Piet Hut, professor of Interdisciplinary Studies, Institute for Advanced Study, Princeton, U.S.A. (shrink)

In his text ‘The modern development of the foundations of mathematics in the light of philosophy’ from around 1961, Gödel announces a turn to Husserl's phenomenology to find the foundations of mathematics. In Gödel's archive there are two draft letters that shed some further light on the exact strategy that he formulated for himself in the early 1960s. Transcriptions of these letters are presented, together with some comments.

Gödel once offered an argument for the general reflection principle in set theory that took the form of an analogy with Leibniz' Monadology. I discuss the mathematical and philosophical background to Gödel's argument, reconstruct the proposed analogy in detail, and argue that it has no justificatory force.

Joint Session of the two Divisions of the International Union for History and Philosophy of Science.

Leibniz had a well-known argument against the existence of infinite wholes that is based on the part-whole axiom: the whole is greater than the part. The refutation of this argument by Russell and others is equally well known. In this note, I argue (against positions recently defended by Arthur, Breger, and Brown) for the following three claims: (1) Leibniz himself had all the means to devise and accept this refutation; (2) This refutation does not presuppose the consistency of Cantorian set (...) theory; (3) This refutation does not cast doubt on the part-whole axiom. Hence, should there be an obstacle to Gödel's wish to integrate Cantorian set theory within Leibniz' philosophy, it will not be this famous argument of Leibniz'. (shrink)

The Dutch mathematician and philosopher L. E. J. Brouwer (1881-1966) developed a foundation for mathematics called 'intuitionism'. Intuitionism considers mathematics to consist in acts of mental construction based on internal time awareness. According to Brouwer, that awareness provides the fundamental structure to all exact thinking. In this note, it will be shown how this strand of thought leads to an intuitionistic function that is effectively computable yet non-recursive.

As is well-known Edmund Husserl's Logical Investigations (1900-1901) had their first, broader impact not on those who attended Husserl's lectures and seminars in Göttingen but on the students of Theodor Lipps, the psychologist and philosopher at the University of Munich.

On Friday, November 15, 1940, Kurt Gödel gave a talk on set theory at Brown University. The topic was his recent proof of the consistency of Cantor’s Continuum Hypothesis with the axiomatic system ZFC for set theory. His friend from their days in Vienna, Rudolf Carnap, was in the audience, and afterward wrote a note to himself in which he raised a number of questions on incompleteness.

The article comments on the foregoing article by Christophe Sigwart on concepts of number and compares Sigwart's text to the "intuitionistic" mathematical work developed by Dutch logician and philosopher L. E. J. Brouwer. The two men's positions are compared on such concepts as the intersubjective validity of mathematics, inner time consciousness, and infinite numbers.

We compare Gödel’s and Brouwer’s explorations of mysticism and its relation to mathematics.

Encyclopedia of Philosophy. - Detroit : Macmillan Reference, 2006, 2nd edition.