- Ingo Blechschmidt (Augsburg) "Exploring hypercomputation with the effective topos"
Wednesday 8 February 17:00-18:00 D.107 (Mathematics Institute)
To any model of computation, such as lambda calculus or Turing machines, there is an associated "effective topos". This is an alternate mathematical universe in which the usual laws of logic don't hold.
For instance, for some models, anti-classical dream axioms like "any function is continuous" and "any function is computable" hold. The law of excluded middle however, "any proposition is either true or not true", is falsified in those toposes.
The effective topos has especially wondrous properties in case we employ models of hypercomputation, where computers can perform
infinitely many calculational steps in finite time: Andrej Bauer proved that then the effective topos contains an injection .
We can also employ physical models about the real world. In this case, statements which are in classical mathematics simply true become non-trivial statements about the nature of the physical world when interpreted in the effective topos.
Topos theory therefore provides an apt vehicle to study computation and alternative axioms of logic. Other applications include mechanical extraction of programs from proofs and a reconciliation of platonism with formalism.
- Andrew Arana (Paris 1) "Depth in mathematics"
Mathematicians judge some theorems, definitions, and proofs to be deep, ordinarily using this as an evaluation of merit. We articulate several different criteria for measuring depth, indicating pros and cons of each. We use Szemerédi's theorem as an example throughout, making the various proposals more concrete.
- Marcus Giaquinto (UCL) "Numbers: what they are and how we know them" (Warwick Philosophy Colloquium)
- Daniel Wolf (Leeds) "Model theory: Your new favourite area of maths"
Model theory is a thriving area of mathematics with appeal to mathematicians and philosophers alike. In this talk I will act as a salesman for model theory, doing my best to convince the audience of both its utility and its beauty. I will attempt to mix some practical, hands-on technical aspects of the subject with its more philosophical features. I will start with some history and then move on to a key notion of model theory, namely that of a type. I will then go over the very useful technique of quantifier elimination before discussing some of model theory's great (and ongoing) successes, for example nonstandard analysis, Hrushovski's proof of the Manin–Mumford conjecture, and neo-stability. I will remark on philosophical implications throughout the talk. Note that I will assume knowledge of first-order logic, such as the compactness and Löwenheim–Skolem theorems.
- Juliette Kennedy (Helsinki/Newton Institute) "Turing, Gödel and some new Inner Models of Set Theory" (Warwick Mathematics Colloquium)
Friday 4 March 16:00-17:00, B3.02 (Mathematics Institute)
Abstract: In his 1946 Princeton Bicentennial Lecture Gödel suggested the problem of finding a notion of definability for set theory which is "formalism free" in a sense similar to the notion of computable function --- a notion which is very robust with respect to its various associated formalisms. One way to interpret this suggestion is to consider standard notions of definability in set theory, which are usually built over first order logic, and change the underlying logic. We show that constructibility is not very sensitive to the underlying logic, and the same goes for hereditary ordinal definability (or HOD). This is joint work with Menachem Magidor and Jouko Väänänen.
- Richard Kaye (Birmingham) "Toogle machines"
- Walter Dean (Warwick) Mathematical existence and the arithmetized completeness theorem (Warwick Mathematics Colloquium)
- Andrew Arana (University of Paris 1 Panthéon-Sorbonne) "Non-euclidean geometry and geometrical content"
Wednesday 28 October 17:00-18:00, B3.02 (Mathematics Institute)
Abstract: In 1868, Beltrami described himself as having identified a "real substrate” for hyperbolic geometry, "rather than admit the necessity for a new order of entities and concepts." He sought to show that hyperbolic geometry described the geometry on curved surfaces in "real", i.e. Euclidean, space. In 1873, Klein wrote that such investigations are "by no means intended to decide the validity of the parallel axiom, but only whether the parallel axiom is a mathematical consequence of the remaining axioms of Euclid". But relative consistency is a logical relationship between statements, whereas Beltrami’s reduction of hyperbolic geometry to the geometry of the surfaces of Euclidean objects is a geometric relationship between objects. The logical relationship identified by Klein, and soon Poincaré and Hilbert, is the heart of what we now consider at issue in the legitimization of non-euclidean geometry. Yet the move from Beltrami's view to this "modern" one involves a considerable conceptual leap, in which geometers came to accept that they could "reinterpret" the geometrical terms of geometric statements. In this talk I want to consider the nature of this conceptual leap, toward better understanding this "modern" understanding of the content of geometrical statements.
- Sam Sanders (Ghent/Munich) "Reverse Mathematics: The playground of logic"
Wednesday 18 February 17:00-18:00, B3.03 (Mathematics Institute)
Abstract: Reverse Mathematics (RM) is a program in the foundations of mathematics. The aim of RM is to find the minimal axioms needed to prove theorems of ordinary mathematics. What emerges from RM is a classification of mathematical theorems based on Turing computability. In particular, one of the main results of RM is that most mathematical theorems fall into only five logical categories. We review the (pre-)history of RM and relate it to earlier foundational/philosophical programs by Hilbert, Brouwer, Weyl, and Turing.
- Andrew Brooke-Taylor (Bristol) "Computing from oracles and uncountable cardinals"
Abstract: Given an infinite string of 0s and 1s - an "oracle" - one can ask what can be computed from it, when you give your computer the ability to query what the nth bit is for any n. There are many results in the literature relating different properties of such oracles, mostly arrived at in an ad hoc manner, but I will show how many of them can be organised in a systematic way. This is done by analogy with so-call cardinal characteristics of the continuum, which are very well-studied (definitions for) uncountable cardinals that may be strictly less than the cardinality of the reals. Inequalities between these cardinals correspond to implications between properties of oracles, and whilst imperfect the analogy is instructive, highlighting some natural open questions.
- Colin McLarty (Case Western) "What does it take to prove X?"
Wednesday 12 March, 17:00-18:00 MS.02 (Mathematics Institute)
Abstract: David Hilbert made a mistake about the relation between finite and infinite mathematics -- a great mistake only a giant could have made. It was revealed by a great achievement: Gödel's incompleteness theorems. Incompleteness is a reality in arithmetic and set theory, and central to a great deal in logic, but not because it arises often in specific mathematics. In fact one productive tactic in model theory is to focus on complete theories. But in using that tactic one is constantly aware of incompleteness, so as to guard against it. The talk will introduce and survey aspects of incompleteness.
Colin will taking part in several other events on 18-21 March as described here.
- Zachiri McKenzie (Cambridge) "Ultraproducts in model theory"
Abstract: The ultraproduct construction is an algebraic technique that allows one to build models whose existence would normally be proved using compactness or omitting types arguments. Since its invention in early twentieth century this technique has found a variety of applications in model theory and set theory. I will describe the ultraproduct construction. I will prove the Łoś Theorem, which characterises the theory satisfied by an ultraproduct, and sketch a proof of the fact that every filter can be extended to an ultrafilter. I will then show how ultraproducts can be used to construct non-standard models of set theory and arithmetic.
- Christopher Porter (University of Paris 7) "Defining Randomness"
Abstract:In the 1960s, there were a number of major developments in the study of random sequences. Two are particularly noteworthy: First, in 1965, Kolmogorov published a definition of random finite sequence based on a measure of complexity known as Kolmogorov complexity. Second, in 1966, Martin-Löf published a definition of random infinite sequence given in terms of a certain collection of effective statistical tests which are nowadays referred to as Martin-Löf tests. Although Martin-Löf had also shown that there is some relationship between his own definition of randomness for infinite sequences and Kolmogorov's for finite sequences, the exact relationship between these two definitions was not resolved until the early 1970s. The goal of my talk is to present the details of these two definitions of randomness and to explain precisely how they relate to one another.