The Axiom of Choice Isn’t Always Non-Constructive

The Axiom of Choice is usually introduced as a non-constructive axiom that mathematicians used to care about but don’t really pay much attention to anymore. It’s true that mainstream mathematicians often don’t pay much attention to it, but it turns out that AC isn’t inherently non-constructive: it depends on what the base system it’s being […]

A Nice Definition of “Field Theory”

Like most people, I don’t really know anything about quantum field theory. But the other day I stumbled across this paper by Stefano Gogioso, Maria E. Stasinou, and Bob Coecke that provides a very nice framework for what sort of thing a “quantum field theory” (or really, any “field theory”) is. It certainly doesn’t mean […]

What does it mean to extend the manipulability of differentials?

In an interesting paper called Extending the Manipulability of Differentials, the authors Jonathan Bartlett and Asatur Zh. Khurshudyan describe an interesting proposal for representing higher-order derivatives. The argument is basically this: As is well-known, the chain rule for first derivatives seems to follow algebraically if you use Leibniz notation for the derivatives: However, there’s a […]

Hoping for Lane Closures

Two lanes of a four lane highway are closed and you’re stuck in a traffic jam. If there are no on- or off-ramps, what should you hope to see on the road ahead of you: the other two lanes re-open or one of the two open lanes close? I think you should hope to see […]

The CGP Grey Sheaf of Continents

CGP Grey is a youtuber with a variety of interesting videos, often about the quirks of geography and political boundaries.  In this video, he asks the question “How many continents are there?”, discusses a variety of subtleties in the notion of “continent”, and concludes that it is not well-defined enough to provide an answer. Let’s […]

Thermodynamics is Easier Than I Thought

Actually, thermodynamics is hard and I don’t understand it.  But even without totally understanding thermodynamics, it turns out its possible to do a surprising number of useful calculations with just a couple of simple rules about entropy. The setup is as follows: Imagine that there is some set of states of the world, called the […]

Gravity is Stronger Than I Thought

I’m not a physicist, and I’d always supposed that, while the Earth has a significant gravitational pull because it’s so massive, the gravitational pull between everyday objects must be completely undetectable, or maybe only detectable with modern laboratory equipment. But I only thought that because I never bothered to actually plug in any numbers.  Using […]

The Arithmetic Hierarchy Meets the Real World

Mathematical logic has a categorization of sentences in terms of increasing complexity called the Arithmetic Hierarchy.  This hierarchy defines sets of sentences and for all nonnegative integers .  The definition is as follows: and are both equal to the set of sentences such that a computer can determine the truth or falsity of in finite […]

Two Constants: Khinchin and Chaitin

Take a real number, .  Write out its continued fraction: It’s an intriguing fact that if you look at the sequence of geometric means this approaches a single constant, called Khinchin’s constant, which is approximately , for almost every .  This means that if you were to pick (for convenience, say it’s between 0 and 1) […]

A Good Definition of Randomness

Most mathy people have a pretty good mental model of what a random process is (for example, generating a sequence of 20 independent bits). I think most mathy people also have the intuition that there’s a sense in which an individual string like 10101001110000100101 is more “random” than 0000000000000000000 even though both strings are equally […]

Mathematica and Quantifier Elimination

In 1931, Alfred Tarski proved that the real ordered field allows quantifier elimination: i.e., every first-order formula is equivalent to one with no quantifiers.  This is implemented in Mathematica’s “Resolve” function. The Resolve function is called like Resolve[formula,domain] where domain gives the domain for the quantifiers in formula.  Since we’ll always be working over in […]

A Logical Interpretation of Some Bits of Topology

Edit: These ideas are also discussed here and here (thanks to Qiaochu Yuan: I found out about those links by him linking back to this post). Although topology is usually motivated as a study of spatial structures, you can interpret topological spaces as being a particular type of logic, and give a purely logical, non-spatial […]

Topology and First-Order Modal Logic

The normal square root function can be considered to be multi-valued. Let’s momentarily accept the heresy of saying that the square root of a negative number is , so that our function will be total. How can we represent the situation of this branching “function” topologically?

Two Interesting Observations about Voting I Hadn’t Seen Until Recently

By “voting”, I mean the following general problem:  Suppose there are candidates and voters.  Each voter produces a total ordering of all candidates.  A voting procedure is a function which takes as input all orderings, and produces an output ranking of all candidates.  Arrow’s impossibility theorem states that there is really no satisfactory voting procedure […]

A Suite of Cool Logic Programs

You may have heard about the Tarski-Seidenberg theorem, which says that the first-order theory of the reals is decidable, that the first-order theory of the complex numbers is similarly decidable, or that the first order theory of the integers without multiplication is decidable. In the course of John Harrison‘s logic textbook Handbook of Practical Logic […]

An Interesting Puzzle in Propositional Logic

Suppose that you’re translating an ancient text, and in this text you come across three words whose meaning you are unsure of: , , and .  So, you head down to the ancient language department of your local university. The first professor you come across, , knows what means, the second professor you come across, […]

What Happens When You Iterate Gödel’s Theorem?

Let be Peano Arithmetic.  Gödel’s Second Incompleteness Theorem says that no consistent theory extending can prove its own consistency. (I’ll write for the statement asserting ‘s consistency; more on this later.) In particular, is stronger than .  But certainly, given that we believe that everything proves is true, we believe that does not prove a […]

A Simple Introduction to Quantum Groups

In the course of reading some background material for an article by James Worthington on using bialgebraic structures in automata theory, I was led to finally reading up on what a Hopf algebra (sometimes called a “quantum group“) is. Although it is not strictly related to logic, I’ll write up what I learned here.

Games Which are Impossible to Analyze

In the last post, I mentioned the computational complexity of various games. To be explicit, we consider each “game” to actually be a sequence of games for . For example, would be checkers played on an board. The problem was then to analyze the computational complexity of the function which takes and tells you which […]

How to Show that Games are Hard

Peg Solitaire is a pretty popular game, often found in restaurants (including Cracker Barrel, if I remember correctly). It’s also NP-complete (by which I mean determining a winning strategy given the initial set-up is an NP-complete problem). You may have also heard of computational complexity results for Minesweeper (see here, for example). There are a […]

When are the Real Numbers Necessary?

The natural numbers can all be finitely represented, as can the rational numbers. The real numbers, however, cannot be so represented and require some notion of “infinity” to define. This makes it both computationally and philosophically interesting to determine for what purposes you need the real numbers, and for what purposes you need only the […]

A language which does term inference

Many strongly typed languages like OCaml do type inference. That is, even though they’re strongly typed, you don’t have to explicitly say what the type of everything is since a lot of the time the compiler can figure it out by itself. For example, if you define a function which takes an x and adds […]

Integrability Conditions (Guest Post!)

Please enjoy the following guest post on differential geometry by Tim Goldberg. A symplectic structure on a manifold is a differential -form satisfying two conditions: is non-degenerate, i.e. for each and tangent vector based at , if for all tangent vectors based at , then is the zero vector; is closed, i.e. the exterior derivative […]

Lots of Fun Math Papers

In the course of looking up a link for my last blog entry, I discovered the MAA Writing Awards site, which collects many pdfs of articles that have won MAA writing awards.  From browsing it a bit, it seems to be a goldmine of fun math articles.

Non-Rigorous Arguments 1: Two Formulas For e

I’m a big fan of non-rigorous arguments, especially in calculus and analysis. I think there should be a book cataloging all the beautiful, morally-true-but-not-actually-true proofs that mathematicians have advanced, but until that time I’ll try to at least catalog a few of them on my blog. This first one is Euler’s original argument for the […]

Almost a Number-Theoretic Miracle

An arithmetic statement is one made up of quantifiers “,” “,” the logical connectives “and,” “or,” “not”, function symbols , , constants , , and variables which are bound by the aforementioned quantifiers. It is known that there is no algorithm which will decide whether or not an arithmetic statement is true or not. This […]

Set Theory and Weather Prediction

Here’s a puzzle: You and Bob are going to play a game which has the following steps. Bob thinks of some function (it’s arbitrary: it doesn’t have to be continuous or anything). You pick an . Bob reveals to you the table of values of his function on every input except the one you specified […]