Readings or seminar hosted by authors in the field; ususally presenting their own work.
Format may vary, depending on preferences of the guest.
Professor Joel David Hamkins will present to Φ-Math his upcoming book Lectures on the Philosophy of Mathematics. The presentation will contain an overview of the book’s contents and motivation with a focus on selected philosophical problems tackled in it, followed by a discussion/questions from attendants.
Anna Bellomo presented her ongoing PhD research on Bolzano’s conceptions of infinity, as well as her involvement on the e-Ideas framework.
‘In the philosophy of mathematics circles, Bernard Bolzano (1871-1848) is mostly known for two things: his contributions to the so-called rigorisation of analysis, and his proto-Cantorian theory of size for infinite sets. In this talk, I will focus on the latter and summarise some recent findings suggesting that, contrary to what has been so far the default interpretation of Bolzano’s treatment of the countable infinite, his focus was not a theory of size for infinite sets, but solving some problems relating to the treatment of (non-convergent) infinite sequences.’
The P vs. NP problem, one of the seven Millenium Problems, is one of the most relevant unsolved question in theoretical computer science. The progress in the last decade, however, has been little. Can information theory and philosophy of information provide new insights as to why these classes should be distinct (or the same)? Pieter Adriaans will be offering a talk on the subject, followed by a discussion.
‘In this talk, I will present a perspective on the P vs. NP problem in the context of philosophy of information. P is the class of decision problems that can be solved in time polynomial to the length of their input by a deterministic computer. NP is the class of problems that can be solved in polynomial time by a non-deterministic computer. This description gives a direct relation with the philosophical question of determinism versus non-determinism and the problem of the interaction between information and computation. This suggests that a careful analysis of the flow of information through computational processes might help us to understand the P vs. NP problem better. Unfortunately, the existing theories of information are not very adequate for this purpose. Shannon information is based on a statistical notion of entropy that is not directly applicable. Kolmogorov complexity defines a structural concept of entropy that is asymptotic and not computable. Apart from that the classes of problems known to be in NP do not have a structure that easily facilitates the application of information theory. There is, for example, no univocal theory of information measurement for finite sets of numbers. This complicates an information theoretical analysis of the subset sum problem and related problems in NP considerably. I therefore propose to look at a new class of problems that I call Multiple Mutual Key (MMK) decision problems. The case for MMK not being in P is at least prima facie stronger than for other problems in NP since the checking function is a repeated application of one time pad code ciphers, which is a provably safe encryption technique. Apart from that problems in MMK are constraint free: any binary string of an adequate length defines an MMK decision problem. This implies that the average case complexity is high and makes it easy to apply maximum entropy techniques.’
On the last Friday of January, we will have the pleasure to host Dean McHugh presenting Newcomb’s Paradox, a game-theoretical problem in decision theory with many diverse philosophical implications.
Luca Incurvati will be presenting his recently published book Conceptions of Set and the Foundations of Mathematics. The book is accessible through the UvA Library.
Book summary:
Sets are central to mathematics and its foundations, but what are they? In this book Luca Incurvati provides a detailed examination of all the major conceptions of set and discusses their virtues and shortcomings, as well as introducing the fundamentals of the alternative set theories with which these conceptions are associated. He shows that the conceptual landscape includes not only the naïve and iterative conceptions but also the limitation of size conception, the definite conception, the stratified conception and the graph conception. In addition, he presents a novel, minimalist account of the iterative conception which does not require the existence of a relation of metaphysical dependence between a set and its members. His book will be of interest to researchers and advanced students in logic and the philosophy of mathematics.