Noson S. Yanofsky

A classification of hidden-variable properties

Hidden variables are extra components added to try to banish counterintuitive features of quantum mechanics. We start with a quantum-mechanical model and describe various properties that can be asked of a hidden-variable model. We present six such properties and a Venn diagram of how they are related. With two existence theorems and three no-go theorems (EPR, Bell and Kochen?Specker), we show which properties of empirically equivalent hidden-variable models are possible and which are not. Formally, our treatment relies only on classical probability models, and physical phenomena are used only to motivate which models to choose.

A universal approach to self-referential paradoxes, incompleteness and fixed points

Following F. William Lawvere, we show that many self-referential paradoxes, incompleteness theorems and fixed point theorems fall out of the same simple scheme. We demonstrate these similarities by showing how this simple scheme encompasses the semantic paradoxes, and how they arise as diagonal arguments and fixed point theorems in logic, computability theory, complexity theory and formal language theory.

Towards a Definition of an Algorithm

We define an algorithm to be the set of programs that implement or express that algorithm. The set of all programs is partitioned into equivalence classes. Two programs are equivalent if they are essentially the same program. The set of equivalence classes forms the category of algorithms. Although the set of programs does not even form a category, the set of algorithms form a category with extra structure. The conditions we give that describe when two programs are essentially the same turn out to be coherence relations that enrich the category of algorithms with extra structure. Universal properties of the category of algorithms are proved.

Galois Theory of Algorithms

Many different programs are the implementation of the same algorithm. The collection of programs can be partitioned into different classes corresponding to the algorithms they implement. This makes the collection of algorithms a quotient of the collection of programs. Similarly, there are many different algorithms that implement the same computable function. The collection of algorithms can be partitioned into different classes corresponding to what computable function they implement. This makes the collection of computable functions into a quotient of the collection of algorithms. Algorithms are intermediate between programs and functions:

Why Mathematics Works so Well

\begin{aligned} \hbox {Programs}\twoheadrightarrow \hbox {Algorithms} \twoheadrightarrow \hbox {Functions}. \end{aligned}

An Introduction to Quantum Computing

Galois theory investigates the way that a subobject sits inside an object. We investigate how a quotient object sits inside an object. By looking at the Galois group of programs, we study the intermediate types of algorithms possible and the types of structures these algorithms can have.

Finding Structure in Science and Mathematics

Theories are a canonical way of describing categories with extra struc- ture. 2-theory-morphisms are used when discussing how one structure can be replaced with another structure. This is central to categorical coher- ence theory. We place a Quillen model category structure on the category of 2-theories and 2-theory-morphisms where the weak equivalences are biequivalences of 2-theories. A biequivalence of 2-theories (Morita equiv- alence) induces and is induced by a biequivalence of 2-categories of alge- bras. This model category structure allows one to talk of the homotopy of 2-theories and discuss the universal properties of coherence.

The Role of Symmetry in Mathematics

A major question in philosophy of science involves the unreasonable effectiveness of mathematics in physics. Why should mathematics, created or discovered, with nothing empirical in mind be so perfectly suited to describe the laws of the physical universe? To answer this we review the well-known fact that the defining properties of the laws of physics are their symmetries. We then show that there are similar symmetries of mathematical facts and that these symmetries are the defining properties of mathematics. By examining the symmetries of physics and mathematics, we show that the effectiveness is actually quite reasonable. In essence, we show that the regularities of physics are a subset of the regularities of mathematics.