XOR’s Hammer

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?

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 [...]

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.   In doing so, he enters many different arenas: ethics, decision theory, and physics. In his chapter on quantum mechanics, he defends the “many-worlds” interpretation [...]

A function from to is called even if for all , .  We might call it even about the point if, for all , . Conversely, we can call a function strongly non-even if for all , , . Finding strongly non-even functions is easy, as any injective function provides a trivial example.  We can [...]

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 [...]

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, [...]

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 [...]

Let be a -periodic function. It may or may not have a representation as a trigonometric series A natural question to ask is whether or not the representation of as a trigonometric series is unique, if it has one. It was the consideration of this question that led Cantor to the invention of set theory. [...]

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.

Nonstandard Analysis is usually used to introduce infinitesimals into the real numbers in an attempt to make arguments in analysis more intuitive. The idea is that you construct a superset which contains the reals and also some infinitesimals, prove that some statement holds of , and then use a general “transfer principle” to conclude that [...]

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 [...]

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 [...]

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 [...]

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. With Player A starting, they take turns choosing a square and eating it together with all squares above and to the right. The catch is that [...]

Intuitively, for any property of sets, there should be a set which has as its members all and only those sets such that holds. But this can’t actually work, due to Russell’s Paradox: Let , and then you can derive a contradiction from both and . The standard solution to this is essentially to forbid [...]

In the book A=B, the authors point out that while the identity is provable (by a very simple proof!), it’s not possible to prove the truth or falsity of all such identities. This is because Daniel Richardson proved the following: Let denote the class of expressions generated by The rational numbers, and . The variable [...]

There are many functions from to that cannot be computed by any algorithm or computer program. For example, a famous one is the halting problem, defined by if the th Turing machine halts and if the th Turing machine does not halt. Another one in the same spirit is the busy beaver function. We also [...]

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 [...]

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.

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 [...]

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. It is known that there is no algorithm which will decide whether or not an arithmetic statement is true or not. This [...]