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 […]
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 […]
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 […]
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.
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 […]
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 […]
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” […]
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 […]
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: […]
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. […]
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 […]
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 […]
XOR's Hammer 