XOR’s Hammer

My four-year-old has recently started asking me questions about infinity (“is there a biggest number?”

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.

Like most people, I don’t really know anything about quantum field theory.

We think of a proof as being non-constructive if it proves “There exists an such that without ever actually exhibiting such an .

In an interesting paper called Extending the Manipulability of Differentials, the authors Jonathan Bartlett and Asatur Zh.

Two lanes of a four lane highway are closed and you’re stuck in a traffic jam.

If you do a google search for how to derive the facts that and , most of the derivations you’ll find rely on knowing something like the double angle formula and go through a direct, non-trivial evaluation of .

You can often get the gist of a mathematical subject via an informal explanation involving infinitesimals.

Until this morning, I didn’t really understand how it was possible for a sailboat to sail into the wind: popular descriptions like Wikipedia’s talk about keels and lift and the Bernoulli effect and so forth, but this feels like a leap beyond my understand…

A while ago, I wrote a blog post called Making Money Disappear Through Infinite Iteration, and I just put out a video version of this post on youtube.

CGP Grey is a youtuber with a variety of interesting videos, often about the quirks of geography and political boundaries.

Actually, thermodynamics is hard and I don’t understand it.

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 laborato…

Mathematical logic has a categorization of sentences in terms of increasing complexity called the Arithmetic Hierarchy.

Popular explanations of physical phenomena like gyroscopes or magnetic fields often end up having to explain the right-hand rule to explain how rotational quantities add (say, by using the right-hand rule to convert angular momenta into vectors, then addi…

The Strong Law of Small Numbers (see also Wikipedia) says that “There aren’t enough small numbers to meet the many demands made of them.”

Take a real number, .

Most mathy people have a pretty good mental model of what a random process is (for example, generating a sequence of 20 independent bits).

A generating function is a formal power series where the sequence of coefficients is the object of interest.

Generalizing a problem can make the solution simpler or more complicated, and it’s often hard to predict which beforehand.

There is a class of all cardinalities , and it has elements , and operations , , and so forth defined on it.

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.

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).

Let be the set of propositions considered by some rational logician (call her Sue).

The normal square root function can be considered to be multi-valued.

By “voting”, I mean the following general problem: Suppose there are candidates and voters.

In the book Good and Real, author Gary Drescher, who received his PhD from MIT’s AI lab, defends the view that determinism is a consistent and coherent view of the world.

A function from to is called even if for all , .

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 mul…

Suppose that you’re translating an ancient text, and in this text you come across three words whose meaning you are unsure of: , , and .

Let be Peano Arithmetic.

Let be a -periodic function.

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.

Nonstandard Analysis is usually used to introduce infinitesimals into the real numbers in an attempt to make arguments in analysis more intuitive.

In the last post, I mentioned the computational complexity of various games.

Peg Solitaire is a pretty popular game, often found in restaurants (including Cracker Barrel, if I remember correctly).

We can think of a computably enumerable (or c.e.) set as a bag which some computer program puts more and more numbers into over time.

Let be the set of all such that the th Turing machine halts.

The natural numbers can all be finitely represented, as can the rational numbers.

Suppose that we wanted to construct a mathematical universe where all objects were computable in some sense.

Many strongly typed languages like OCaml do type inference.

Chomp is a two-player game which is played as follows: The two players, A and B, start with a “board” which is a chocolate bar divided into small squares.

Intuitively, for any property of sets, there should be a set which has as its members all and only those sets such that holds.

In the book A=B, the authors point out that while the identity is provable (by a very simple proof!)

There are many functions from to that cannot be computed by any algorithm or computer program.

Please enjoy the following guest post on differential geometry by Tim Goldberg.

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.

I’m a big fan of non-rigorous arguments, especially in calculus and analysis.

Why did the chicken cross the island on Lost?

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.