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.
Entries from August 2008
August 29, 2008
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 [...]
August 25, 2008
A Curious Application of Ambiguity with Respect to the Possessive Form
Why did the chicken cross the island on Lost?
August 25, 2008
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 [...]
August 23, 2008
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 [...]
August 22, 2008
Making Money Disappear Through Infinite Iteration
In Joel David Hamkin’s paper Supertasks and Computation, he relates the following puzzle: Suppose that you have a countable infinity of dollar bills, and one day you meet the devil, who offers you the following bargain: In the first half minute from now, the devil will give you two dollar bills, and take one from [...]
August 21, 2008
What do We Have to Know About a Function in Order to Compute its Definite Integral?
Suppose that is a continuous function from to and that we have a program which computes it. (Ignore for now exactly what it means to “compute” a real-valued function of the reals. Suffice it to say that almost every natural continuous function you come across is computable). If we want to compute , say to [...]
August 19, 2008
Rosser’s Clever Improvement to Gödel’s Original First Incompleteness Theorem
Sometimes it’s the case that in a first-order system T in which you can do number theory there is a property P(n) of natural numbers such that the following two seemingly contradictory statements hold: For every n, T proves P(n) and T does not prove that for all n, P(n) To see that these statements [...]
August 19, 2008
Great Intro to Intuitionistic Logic
At Mathematics and Computation, there’s a really good accessible introduction to intuitionistic logic called Intuitionistic Logic for Physics. It also includes some nice accessible remarks on smooth infinitesimal analysis.
August 18, 2008
Is the “Hardest Logic Puzzle Ever” too Easy?
In 1992, the philosopher George Boolos gave what he called the “Hardest Logic Puzzle Ever”, which he attributed to Raymond Smullyan. In 2008, a clever paper by two graduate students, Brian Rabern and Landon Rabern, appeared in the philosophical journal “Analysis” which gave a simpler solution to the puzzle than Boolos gave—and furthermore claimed that [...]
August 16, 2008
One Puzzle with Two Totally Different Solutions
Peter Winkler‘s excellent book Mathematical Puzzles: A Connoisseur’s Collection has in it the problem of finding a partition of into disjoint non-trivial circles. (Here “non-trivial” means “not a point.”) Winkler gives a very clever solution which is purely geometric. Later, I read the same problem in Krzysztof Ciesielski‘s excellent book Set Theory for the Working [...]
August 16, 2008
Multivariable Calculus with Nilpotent Infinitesimals: More Smooth Infinitesimal Analysis
This is a continuation of my earlier post on smooth infinitesimal analysis. In this installment, I’ll show how the definition of a “stationary point” in Smooth Infinitesimal Analysis leads directly to a nice substitute for the Lagrange multipliers method. Then I’ll show how you can define differential forms as objects which assign a “signed volume” [...]
August 15, 2008
Ax’s Theorem: An Application of Logic to Ordinary Mathematics
There are a number of applications of logic to ordinary mathematics, with the most coming from (I believe) model theory. One of the easiest and most striking that I know is called Ax’s Theorem. Ax’s Theorem: For all polynomial functions , if is injective, then is surjective. Very rough proof sketch: The field has characteristic [...]
August 13, 2008
Did we learn all there is to know about exponentiation in sixth grade?
By sixth grade (I think), you’ve learned some basic facts about addition, multiplication, and exponentiation over the natural numbers: you’ve learned that addition and multiplication are commutative and associative, that multiplication distributes over addition, that 0 is an identity for addition and 1 is an identity for multiplication, and the following simplification rules for exponents: [...]
August 13, 2008
Löb’s Theorem: Santa Claus and Provability
Consider the following argument for the existence of Santa Claus (which is called Curry’s paradox): Let S be the sentence If S is true, then Santa Claus exists. Lemma: S is true. Proof. S is of the form “If P, then Q.” so to show S we just have to assume P and show Q. [...]
August 13, 2008
Kreisel’s No-Counterexample Interpretation: How can we turn proofs in elementary number theory into programs?
For every true number-theoretic statement of the form “For all n there is an m such that P(n, m)” there is a witnessing function f such that for all n, P(n, f(n)) is true. For example, a witnessing function for the statement “For all n there is an m such that m > n and [...]
August 11, 2008
How can one do calculus with (nilpotent) infinitesimals?: An Introduction to Smooth Infinitesimal Analysis
Many mathematicians, from Archimedes to Leibniz to Euler and beyond, made use of infinitesimals in their arguments. These were later replaced rigorously with limits, but many people still find it useful to think and derive with infinitesimals. Unfortunately, in most informal setups the existence of infinitesimals is technically contradictory, so it can be difficult to [...]
