Every blog post that took off on Hacker News, ranked by the votes of its best submission. 2,391 posts.
Roughly speaking, Gödel’s Incompleteness Theorem states that there are true mathematical statements that cannot be proven. When I was in 11-th grade, my geometry teacher Mr. Olsen, my friend …
A Japanese man came into the restaurant. I had seen him before, wandering around; he was trying to sell abacuses. He started to talk to the waiters, and challenged them: He said he could add numbers faster than any of them could do.
Most of us learn how to write technical articles by osmosis. After reading a hundred or so papers, we unconsciously pick up the common patterns of the field and begin to imitate them. This phenomenon is largely beneficial, but there are several truly odious habits that are almost universally and inadvertently adopted…
First of all, don’t panic. I’m going to try in this post to introduce you to quantum field theory, which is probably the deepest and most intimidating set of…
There are lots of interesting little subproblems in compilation like instruction selection, register allocation and instruction scheduling. These can be expressed in declarative interlinked way to constraint solvers.
Use these rules to avoid boring your audience
This is a Mega-Update ®™ that focuses on features that people have been requesting for a long time. I have a ton of stuff in Miegakure that could be shared with 4D Toys but what takes development time is not the cool 4D stuff but all the UI/UX around it. First of all because there is the VR version, the mouse/keyboard…
Gradient descent is a standard tool for optimizing complex functions iteratively within a computer program. Its goal is: given some arbitrary function, find a minumum. For some small subset of functions - those that are convex - there's just a single minumum which also happens to be global. For most realistic…
the forgotten data structure (not really)
The way that most physicists teach and talk about partial differential equations is horrible, and has surprisingly big costs for the typical understanding of the foundations of the field even among professionals. The chief victims are students of thermodynamics and analytical mechanics, and I've me
Authors: Yanbo Zhang, Benedikt Hartl, Hananel Hazan, Michael Levin
Proposing technology for pure mathematics: curating concepts, constructs, theorems, & literature; precise symbolic language; automated computations...
Since I’m a good mood today—at a beautiful science camp with my kids, high in the mountains near Big Bear Lake in California—I thought I’d blog about something positive. Las…
Edit 12/19/2017: added a new subsection on analyzing Chutes & Ladders as an Absorbing Markov Chain.
Many theorems in mathematics are important enough that they have been proved repeatedly in surprisingly many different ways. Examples of this include the existence of infinitely many prime numbers , the evaluation of \(\zeta(2)\), the fundamental theorem of algebra (polynomials have roots), quadratic reciprocity (a…
Just as I was complaining that we haven't seen many surprising breakthroughs in complexity recently, we get an earthquake of a result to st...
I hope you're not tired of the 8087, because I have another article about Intel's floating-point chip. 1 In 1980, Intel introduced the 8087...
The underpinning of information theory is Shannon’s formula for the entropy H of a discrete probability distribution P. Derived in 1948 in the paper “A mathematical theory of communication”, this is given by \[ H[P] = -\sum_{i=1}^N p_i \log p_i \] where \( p_i = p(x_i) \) is the probability of sampling \( x_i \) from…
<img src="http://zwibbler.com/shared/1024.png"/> jQuery creator John Resig needs a little help storing lists of words in his side project. Let's go overkill and explore a little known branch of computer science called Succinct Data Structures.
The Legendre transformation tells you how the inverse map acts on integrals
Greg Martin, a number theorist at UBC (the University of British Columbia in Vancouver) doesn’t think so, so he has decided to resign from the editorial board of Elsevier’s Journal of N…
For those who haven’t seen it: Aubrey de Grey, better known to the world as a radical life extension researcher, on Sunday posted a preprint on the arXiv claiming to prove that the chromatic …
Most schools teach number systems the same way. First they teach natural numbers (or “counting numbers”), then integers, rationals, reals, and complex numbers, in that order. Rationals are countable and reals are not, but that’s okay. Once you start taking roots, you need a number system more powerful than a countable…
The following tweet has been making the rounds on social media: Notice that since the Democrats won overall, it would be impossible for Republicans to win all seven districts, no matter how the map…
chordcat is a chord-naming application and a pet project of mine. It is written in C++ and SFML.
Out there in the wider world—OK, OK, among Luboš Motl, and a few others who comment on this blog—there appears to be a widespread opinion that P≠NP is just “a fashionable dogma of…
Ensemble averages and time averages
[This is a guest post by Jeremy Avigad. This blog post was initially written in a different file format and converted using AI. — T.] “Mathematics underwent, in the nineteenth century, …
How do we represent a rotation in 3 dimensions? This is a surprisingly deep question, and one we can learn a lot from. Our goal will be to “rediscover” quaternions from first principles, building them up by exploring different representations and fixing the failures along the way.
