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A while back, I posted the following on social media: If you’re unfamiliar, Conway’s Game of Life takes place on a two-dimensional grid of square cells, each cell either alive (1) or dead (0). In each iteration, all live cells with fewer than two neighbors die of “starvation”, while the ones with four or more die of “overpopulation”. Finally, any dead cell that has exactly three living neighbors comes alive — I guess that’s ménage à trois...

I have a complicated relationship with Hacker News . The site is the most important aggregator of geek news and a major source of traffic to this blog. At the same time, it has a fair number of toxic commenters, making it a dependable source of insults hurled in my general direction; if you want a taste, this article has been called “watered-down” and “slop”. The site is run by geeks and for geeks, so it’s not immune to tech trends; for example, around 2018, it had ...

In some of the earlier articles on this blog, we talked about the nature of real numbers and the meanings of infinity . The theory outlined in these posts is interesting but also hopelessly abstract. It’s as if we’re inventing make-believe worlds that have no discernible connection to reality. In today’s post, we’ll examine a cool counterexample: an outcome of an numerical experiment that can be backed up with fairly simple proofs, but that makes sense only if you take a ...

A while ago, I posted an article about the 19th and early 20th century quest to derive mathematics from the principles of formal logic . We kicked off with Peano arithmetic, which built natural numbers from two ad-hoc constructs: an element representing zero and an abstract “successor” function S(…) . Later, we leaned on set theory to encode the underlying structure of these symbols. This netted us a hierarchy of set-theoretic natural numbers known as ordinals . It also led to ...

If you follow information security discussions on the internet, you might have heard that blurring an image is not a good way of redacting its contents. This is supposedly because blurring algorithms are reversible. But then, it’s not wrong to scratch your head. Blurring amounts to averaging the underlying pixel values. If you average two numbers, there’s no way of knowing if you’ve started with 1 + 5 or 3 + 3. In both cases, the arithmetic mean is the same and the original inf...