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Most cited
Posts that commenters keep linking to, ranked by how many different people have cited them in Hacker News comments. 42 posts.
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The Categories Were Made for Man, Not Man for the Categories (2014)
I. “Silliest internet atheist argument” is a hotly contested title, but I have a special place in my heart for the people who occasionally try to prove Biblical fallibility by pointing …
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A collection of things that are Turing-complete by accident (2013)
A list of things that were not supposed to be Turing-complete, but are.
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Consistency Models
Jepsen analyzes the safety properties of distributed systems–most notably, identifying violations of consistency models. But what are consistency models? What phenomena do they allow? What kind of consistency does a given program really need?
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Probabilistic Programming and Bayesian Methods for Hackers (2013)
Bayesian Methods for Hackers : An intro to Bayesian methods + probabilistic programming with a computation/understanding-first, mathematics-second point of view.
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Let’s talk about PAKE
The first rule of PAKE is: nobody ever wants to talk about PAKE. The second rule of PAKE is that this is a shame, because PAKE — which stands for Password Authenticated Key Exchange — is actually o…
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1/0 = 0
Disclaimer 2024-12-02 Since this rant is going viral again (again (again)), some disclaimers: I wrote this six years ago to “”“win”“” an ugly Twitter fight. I’m glad I anonymized the inciting tweet and hope I’m the only person still getting grief over this. The core interesting bit— why it’s mathematically valid to…
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The Superinvestors of Graham-and-Doddsville (1984)
Columbia Business School. For over 100 years, we’ve helped develop leaders who create value for business and society at large.
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On Vigna's PCG Critique
On 14 May 2018, Sebastiano Vigna added a page to his website (archived here) entitled “The wrap-up on PCG generators” that attempts to persuade readers to avoid various PCG generators. That day, he al
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Visualizing quaternions: An explorable video series
Explaining how quaternions, a four-dimensional number system, describe 3d rotation.
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Intuitive Understanding of Euler’s Formula
Euler's identity seems baffling: It emerges from a more general formula: Yowza -- we're relating an imaginary exponent to sine and cosine! And somehow…
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Efficiently Generating a Number in a Range
The vast majority of my posts about random number generation have focused on looking at the properties of different generation schemes. But, perhaps surprisingly, the performance of your randomized a
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Why FTL implies time travel
In science fiction, it is pretty standard fare to introduce some form of faster-than-light communication or travel. After all, space is big, and you can't write your swashbuckling Hornblower-in-space novel if you have to wait for a generation ship to crawl painfully slowly between the nearest stars,
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The algebra and calculus of algebraic data types
Just as algebra is fundamental to the whole of mathematics, algebraic data types (ADTs) are fundamental to many common functional programming languages. They’re the primitives upon which all of our richer data structures are built, including everything from sets, maps, and queues, to bloom filters and neural networks .
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So You Think You Have a Power Law (2007)
Regular readers who care about such things — I think there are about three of you — will recall that I have long had a thing about just how unsound many of the claims for the presence of power law distributions in real data are, especially those made by theoretical physicists, who, with some honorable exceptions…
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Prospecting for Hash Functions
Update 2022 : TheIronBorn has found even better permutations using a smarter technique. That thread completely eclipses my efforts in this article.
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The 8000th Busy Beaver number eludes ZF set theory (2016)
I’ve supervised a lot of great student projects in my nine years at MIT, but my inner nerdy teenager has never been as personally delighted by a project as it is right now. To…
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Markov Chains Explained Visually (2014)
Markov chains, named after Andrey Markov , are mathematical systems that hop from one "state" (a situation or set of values) to another. For example, if you made a Markov chain model of a baby's behavior, you might include "playing," "eating", "sleeping," and "crying" as states, which together with other behaviors…
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Modern SAT solvers: fast, neat and underused
Before I started doing research, I saw SAT solvers as academically interesting but without practical uses ouside of other academic applications. I've since then changed my mind, and I want to change yours, because modern SAT solvers are neat, fast and almost criminally underused by the industry.
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Compressing and enhancing hand-written notes (2016)
I wrote a program to clean up scans of handwritten notes while simultaneously reducing file size.
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The Matrix Calculus You Need for Deep Learning
Most of us last saw calculus in school, but derivatives are a critical part of machine learning, particularly deep neural networks, which are trained by optimizing a loss function. This article is an attempt to explain all the matrix calculus you need in order to understand the training of deep neural networks. We…
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Introduction to A* (1997)
Movement for a single object seems easy. Pathfinding is complex. Why bother with pathfinding? Consider the following situation:
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The Wavefunction Collapse Algorithm Explained
The Wavefunction Collapse Algorithm teaches your computer how to riff. The algorithm takes in an archetypical input, and produces procedurally-generated outputs that look like it.
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A study of musical scales (2017)
PREFACE. This essay was written in 2009. Since then, a few errata have been corrected and I embedded the youtube videos, but there have been no substantial additions. I think it still holds up fairly well, but if I were to rewrite it today, I would change some things. My main regret is using the criteria of maximum…
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Understanding is a poor substitute for convexity (2012)
Something central, very central, is missing in historical accounts of scientific and technological discovery. The discourse and controversies focus on the role of luck as opposed to teleological programs (from telos, "aim"), that is, ones that rely on pre-set direction from formal science. This is a faux-debate: luck…
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The Waiting Time Paradox, Or, Why Is My Bus Always Late?
If you, like me, frequently commute via public transit, you may be familiar with the following situation:
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Jupyter, Mathematica, and the Future of the Research Paper
Economist. Policy Entrepreneur. Geek. Co-recipient of the 2018 Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel. University Professor at NYU. Focused on urbanization; cooperation at scale of millions and billions; science; technology; economic development; long-term growth; code as language.