Programming languages as theorem verifiers
(This post is also available as a PDF). One of the most interesting ideas in modern logic is the Curry-Howard correspondence. This is the informal observation that, from a certain perspective,...
View ArticleNotes on proving the completeness theorem for propositional logic
NB: I’ve opted to just get straight to the point with this post rather than attempting to introduce the subject first, so it may be of little interest to readers who aren’t already interested in...
View ArticleA proof of the boundedness theorem by induction
As usual, this post can be viewed as a PDF. Theorem (Boundedness Theorem). A continuous real-valued function on a closed interval is bounded. Proof. Suppose f is such a function and [a, b] is its...
View ArticleNotes on periodic functions
A period of a function f : ℝ → X is a T ∈ ℝ such that for every t ∈ ℝ, we have f(t + T) = f(t). Note that X may be any set; the definition does not refer to any structure on X other than its equality...
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A simple motivation for the notion of a natural transformation
A natural transformation is an operation on a category, or more precisely a family of operations, one for each object in the category, which is preserved by morphisms in the category. Each operation in...
View ArticleDualities between depth-first search and breadth-first search
Something which I think is fairly well-known among programmers (I first learned it from reading Higher Order Perl) is that depth-first search and breadth-first search can be implemented in such a way...
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Enumerating the ordered k-partitions of an integer
Given two integers and , where is non-negative, an ordered -partition of is a -tuple of positive integers such that For example, the ordered 3-partitions of 5 are , , , , , and . The problem I'm...
View ArticleImplementing Python’s `cmp_to_key` function
Sorting functions in programming languages often take a parameter which allows the user to control how comparison is done when sorting. The most direct way to do this is to have the parameter be a...
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The trajectory of a bouncing ball
You can also view this article at my Github Pages website (the formatting might be better there). Consider an idealized model of a ball bouncing on the ground, where the ball is falling along a line...
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A note on the correct definition of a proportion space
Recently I’ve been learning about torsors. A torsor is, in informal terms, an algebraic structure that’s like a group with the identity element “forgotten”. The precise definition (whose connection to...
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