# Logic and Foundations

## Geometry, Logic, and Philosophy: The Case of the Parallels Postulate

One of the most important techniques provided by modern logic is the use of models to show the consistency of theories. The technique burst onto the scene in the late 19th century, and had its most important early instance in demonstrating the consistency of non-Euclidean geometries. This talk investigates the development of that technique as it transitions from a geometric tool to an all-purpose tool of logic. I’ll argue that the standard narrative, according to which our modern technique provides answers to centuries-old questions, is mistaken. Once we understand how modern models work, I’ll argue, we see important differences between the kinds of consistencyclaims that would have made sense e.g. to Kant and the kinds of consistency-claims that we can demonstrate today. We’ll also see some philosophically-interesting shifts, over this time period, in the kinds of things that we take proofs to demonstrate.

### Speaker

Patricia Blanchette is Professor of Philosophy and Glynn Family Honors Collegiate Chair in the Department of Philosophy at the University of Notre Dame. Prior to coming to Notre Dame, Blanchette taught in the Department of Philosophy at Yale University. Blanchette works in the history and philosophy of logic, philosophy of mathematics, history of analytic philosophy, and philosophy of language. She is an editor of the Bulletin of Symbolic Logic, and serves on the editorial boards of the Notre Dame Journal of Formal Logic and of Philosophia Mathematica. She is the author of Frege’s Conception of Logic (Oxford University Press 2012).

## Univalence as a New Principle of Logic

It is often convenient or useful in mathematics to treat isomorphic structures as the same. The Univalence Axiom for the foundations of mathematics elevates this idea to a foundational principle in the setting of Homotopy Type Theory. It states, roughly, that isomorphic structures can be identified. In his talk, Prof. Awodey will explain this principle and how it can be taken as an axiom, and explore the motivations and consequences, both mathematical and philosophical, of making such an assumption.

## Alan Turing and Enigma

Central to Alan Turing's posthumous reputation is his work with British codebreaking during the Second World War. This relationship is not well understood, largely because it stands on the intersection of two technical fields, mathematics and cryptology, the second of which also has been shrouded by secrecy. This lecture will assess this relationship from an historical cryptological perspective. It treats the mathematization and mechanization of cryptology between 1920-50 as international phenomena. It assesses Turing's role in one important phase of this process, British work at Bletchley Park in developing cryptanalytical machines for use against Enigma in 1940-41. It focuses on also his interest in and work with cryptographic machines between 1942-46, and concludes that work with them served as a seed bed for the development of his thinking about computers.

### Turing 2012 - Calgary

This talk is part of a series celebrating the Alan Turing Centenary in Calgary. The following mathtube videos are part of this series

- Alan Turing and the Decision Problem,
*Richard Zach*. - Turing's Real Machine,
*Michael R. Williams*. - Alan Turing and Enigma, John R. Ferris.

- Read more about Alan Turing and Enigma
- 8878 reads

## Turing's Real Machines

While Turing is best known for his abstract concept of a "Turing Machine," he did design (but not build) several other machines - particularly ones involved with code breaking and early computers. While Turing was a fine mathematician, he could not be trusted to actually try and construct the machines he designed - he would almost always break some delicate piece of equipment if he tried to do anything practical.

The early code-breaking machines (known as "bombes" - the Polish word for bomb, because of their loud ticking noise) were not designed by Turing but he had a hand in several later machines known as "Robinsons" and eventually the Colossus machines.

After the War he worked on an electronic computer design for the National Physical Laboratory - an innovative design unlike the other computing machines being considered at the time. He left the NPL before the machine was operational but made other contributions to early computers such as those being constructed at Manchester University.

This talk will describe some of his ideas behind these machines.

### Turing 2012 - Calgary

This talk is part of a series celebrating The Alan Turing Centenary in Calgary. The following mathtube videos are also part of this series

- Alan Turing and the Decision Problem,
*Richard Zach*. - Turing's Real Machine,
*Michael R. Williams*. - Alan Turing and Enigma,
*John R. Ferris*.

- Read more about Turing's Real Machines
- 9517 reads

## Alan Turing and the Decision Problem

Many scientific questions are considered solved to the best possible degree when we have a method for computing a solution. This is especially true in mathematics and those areas of science in which phenomena can be described mathematically: one only has to think of the methods of symbolic algebra in order to solve equations, or laws of physics which allow one to calculate unknown quantities from known measurements. The crowning achievement of mathematics would thus be a systematic way to compute the solution to any mathematical problem. The hope that this was possible was perhaps first articulated by the 18th century mathematician-philosopher G. W. Leibniz. Advances in the foundations of mathematics in the early 20th century made it possible in the 1920s to first formulate the question of whether there is such a systematic way to find a solution to every mathematical problem. This became known as the decision problem, and it was considered a major open problem in the 1920s and 1930s. Alan Turing solved it in his first, groundbreaking paper "On computable numbers" (1936). In order to show that there cannot be a systematic computational procedure that solves every mathematical question, Turing had to provide a convincing analysis of what a computational procedure is. His abstract, mathematical model of computability is that of a Turing Machine. He showed that no Turing machine, and hence no computational procedure at all, could solve the Entscheidungsproblem.

### Turing 2012 - Calgary

This talk is part of a series celebrating the Alan Turing Centenary in Calgary. The following mathtube videos are also part of this series

- Alan Turing and the Decision Problem,
*Richard Zach*. - Turing's Real Machine,
*Michael R. Williams*. - Alan Turing and Enigma,
*John R. Ferris*.

## On the Chromatic Number of Graphs and Set Systems

During this series of lectures, we are talking about infinite graphs and set systems, so this will be infinite combinatorics. This subject was initiated by Paul Erdös in the late 1940’s.

I will try to show in these lectures how it becomes an important part of modern set theory, first serving as a test case for modern tools, but also influencing their developments.

In the first few of the lectures, I will pretend that I am talking about a joint work of István Juhász, Saharon Shelah and myself [23].

The actual highly technical result of this paper that appeared in the **Fundamenta** in 2000 will only be stated in the second or the third part of these lectures. Meanwhile I will introduce the main concepts and state—--and sometimes prove—--simple results about them.