Completeness of modal logic S4 for real line R, real interval (0,1) etc. (with modality interpreted as interior) was proved by McKinsey and Tarski. Several attempts to simplify this proof contain gaps. A new proof presented here combines the ideas in a book by Mints and paper by Aiello, van Benthem and Bezhanishvili to construct an open and continuous map of an arbitrary rooted Kripke S4-model onto the interval (0,1). Existence of such map implies completeness.
Gödel started out almost dismissive of a platonist philosophy of set theory, and only over many years moved to the fully platonist point of view with which he is usually identified. By suggesting that his final position was the one he had always held, Gödel has made it difficult to trace his evolving views. In this talk I will point out various markers on his intellectual odyssey. If time permits, I will say something about my own views.
I will present a generalization of the binomial function to infinite cardinals and discuss some of its properties. In particular, I will try to explain Shelah's philosophy about infinite cardinal arithmetic, by explaining his version of the GCH inside ZFC.
I will also describe two old problems of Erdos and Hechler on maximal almost disjoint families over singular cardinals, and present their recent solutions with pcf techniques by Kojman, Kubis and Shelah.
Quine famously holds that philosophy is "continuous with natural science". In order to find out what exactly the point of this claim is, I shall take up one of his preferred phrases and trace it through his writings, i.e., the phrase "Science itself teaches that ...". Unlike Wittgenstein, Quine did not take much interest in determining what might be distinctive of philosophical investigations, or of the philosophical part of scientific investigations. I find this limitation regrettable, and I shall take a fresh look at his metaphilosophy, trying to defuse his avowed naturalism by illustrating how little influence his naturalistic rhetoric has on the way he actually does philosophy.
A proof is methodologically pure, roughly, if it uses methods `close' or `akin' to the statement being proved. Advocates of purity in one form or another include Aristotle, Descartes, Newton, Bolzano, and Hilbert. In this talk, we distinguish between two main breeds of purity, and explain their distinctive epistemic benefits. To highlight the importance of purity concerns, we discuss a particular case study in mathematics, concerning the so-called `casus irreducibilis' for cubic polynomial equations. This case study follows the development of algebra from the Italian Renaissance through the late nineteenth century, and highlights the role of the introduction of complex numbers in algebra. We close by considering briefly what impact purity concerns might have on a search for natural kinds in mathematics.
The picture of mathematics as being about constructing objects of various sorts and proving the constructed objects equal or unequal is an attractive one, going back at least to Euclid. On this picture, what counts as a mathematical object is specified once and for all by effective rules of construction.
In the last century, this picture arose in a richer form with Brouwer's intuitionism. In his hands (for example, in his proof of the Bar Theorem), proofs themselves became constructed mathematical objects, the objects of mathematical study, and with Heyting's development of intuitionistic logic, this conception of proof became quite explicit. Today it finds its most elegant expression in the Curry-Howard theory of types, in which a proposition may be regarded, at least in principle, as simply a type of object, namely the type of its proofs. When we speak of 'proof-theoretic semantics' for mathematics, it is of course this point of view that we have in mind.
Much of my discussion applies equally to constructive mathematics. But the type-theoretic point of view remains, for many people, restricted to the domain of constructive mathematics. The term "classical" is included in the title to indicate that, on the contrary, classical mathematics can also be understood in this way and does not need to be founded on an inchoate picture of truth-functional semantics in the big-model-in-the-sky, a picture that can in any case never be coherently realized.
Three commonly acknowledged desiderata in an account of the logic of vague discourse are the following: (1) to assign the correct meanings to ``or,'' ``and,'' etc., which probably means to give a truth-functional account of these connectives; (2) to allow for so-called penumbral connections, precise logical relations holding between vague statements; (3) to avoid introducing sharp boundaries where there should be none. Several known accounts accomplish (1) or (2), but it is doubtful that any of them accomplish both. Perhaps no known account accomplishes (3), lest it be an account that is itself couched in vague language. The degree theories constitute one important class of accounts of the logic of vagueness. Degree theories hold that sentences containing vague terms can take on a large number of truth-values intermediate between truth and falsity. Usually truth-values are taken to range over the unit interval [0,1]. Some degree theories treat sentential connectives truth-functionally; some do not. The former are subject to what seem intuitively to be counterexamples. The latter do not seem to give any real account of the logic of vagueness. All of them have problems with regard to desideratum (3). Several people have proposed that the unit interval is not the right range for truth-values. I would like to put forward the suggestion that the range of possible truth-values of a vague sentence is a complete Boolean algebra, with different vocabularies usually leading to different algebras. This suggestion, even though accompanied by little or no attempt to characterize the algebras associated with particular vocabularies, does more work than one might think. It accomplishes both (1) and (2), for example. Whether it is compatible with (3) is, however, more problematic.