Kant held that under the concept of √2 falls a geometrical magnitude, but not a number. In particular, he explicitly distinguished this root from potentially infinite converging sequences of rationals. Like Kant, Brouwer based his foundations of mathematics on the a priori intuition of time, but unlike Kant, Brouwer did identify this root with a potentially infinite sequence. In this paper I discuss the systematical reasons why in Kant's philosophy this identification is impossible.
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.
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.
Brouwer's demonstration of his Bar Theorem gives rise to provocative questions regarding the proper explanation of the logical connectives within intuitionistic and constructivist frameworks, respectively, and, more generally, regarding the role of logic within intuitionism. It is the purpose of the present note to discuss a number of these issues, both from an historical, as well as a systematic point of view.
On the intended interpretation of intuitionistic logic, Heyting's Proof Interpretation, a proof of a proposition of the form p -> q consists in a construction method that transforms any possible proof of p into a proof of q. This involves the notion of the totality of all proofs in an essential way, and this interpretation has therefore been objected to on grounds of impredicativity (e.g. Gödel 1933). In fact this hardly ever leads to problems as in proofs of implications usually (...) nothing more is assumed about a proof of the antecedent than that it indeed is one, and this assumption does not require a further grasp of the totality of proofs. The prime example of an intuitionistic theorem that goes beyond that assumption is Brouwer's proof of the 'bar theorem': For every tree x, if x contains a decidable subset of nodes such that every path through the tree meets it (a 'bar'), then there is a well-ordered subtree of x that contains a bar for the whole of x. Instantiated with an arbitrary tree t, this proposition takes the form P(t) -> Q(t). Brouwer's proof of the bar theorem mainly consists in an analysis of the inner structure that a proof of P(t) must have, where proofs are taken to be primarily mental objects. So here Brouwer engages in phenomenological reflection by considering the acts in which we think about bars. From that analysis he obtains the information from which to construct a proof of Q(t). In this talk I will argue that Brouwer circumvents the problem of impredicativity by resorting to a transcendental argument based on phenomenological description, and defend this application by showing how common objections to transcendental arguments do not apply here. Finally, I will indulge in some historical speculation by relating the foregoing considerations to the remarkable change that Gödel's view on the Proof Interpretation underwent between his Yale Lecture (1941) and the Dialectica paper (1958). (shrink)
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)
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 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.
"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)
Even though Husserl and Brouwer have never discussed each other's work, ideas from Husserl have been used to justify Brouwer's intuitionistic logic. I claim that a Husserlian reading of Brouwer can also serve to justify the existence of choice sequences as objects of pure mathematics. An outline of such a reading is given, and some objections are discussed.
Husserl repeatedly has claimed that (1) mathematics without a philosophical foundation is not a science but a mere technique; (2) philosophical considerations may lead to the rejection of parts of mathematical practice; but (3) they cannot lead to mathematical innovations. My thesis is that Husserl's third claim is wrong, by his own standards.
I argue that Brouwer''s general philosophy cannot accountfor itself, and, a fortiori, cannot lend justification tomathematical principles derived from it. Thus it cannot groundintuitionism, the jobBrouwer had intended it to do. The strategy is to ask whetherthat philosophy actually allows for the kind of knowledge thatsuch an account of itself would amount to.
Brouwer and Weyl recognized that the intuitive continuum requires a mathematical analysis of a kind that set theory is not able to provide. As an alternative, Brouwer introduced choice sequences. We first describe the features of the intuitive continuum that prompted this development, focusing in particular on the flow of internal time as described in Husserl's phenomenology. Then we look at choice sequences and their logic. Finally, we investigate the differences between Brouwer and Weyl, and argue that Weyl's conception of (...) choice sequences is defective on several counts. (shrink)