Monthly Archives: August 2008
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 … Continue reading
Filed under Uncategorized
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 … Continue reading
Filed under Uncategorized
A Curious Application of Ambiguity with Respect to the Possessive Form
Why did the chicken cross the island on Lost?
Filed under Uncategorized
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 … Continue reading
Filed under Uncategorized
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 … Continue reading
Filed under Puzzles, Set Theory
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 … Continue reading
Filed under Ordinals, Puzzles, Set Theory
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 … Continue reading
Filed under Programming
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) … Continue reading
Filed under Provability Logic, Self-Reference
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.
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 … Continue reading
Filed under Puzzles, Self-Reference
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, … Continue reading
Filed under Ordinals, Puzzles, Set Theory
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 … Continue reading
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 … Continue reading
Filed under Model Theory
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 … Continue reading
Filed under Universal Algebra
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, … Continue reading
Filed under Provability Logic
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 … Continue reading
Filed under Proof Theory
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 … Continue reading
Filed under Intuitionistic Logic, Smooth Infinitesimal Analysis, Toposes
