I don't think raw machine code (not even assembly) is the best form to reason about program logic. It just takes too many steps to execute sophisticated processes to keep them all in your head at once - or keep them all in an LLM context at once.

Truth is some sort of value judgment that is outside the scope of formal systems. And looking at how bizarre Gödel statements are, it's unclear if there's any particular justification for declaring them to be true or false.

I've done both of those things. I know what you get taught. But I've kept my math education going for the 25 years since then. I've talked to practicing mathematicians about what they do. I've learned a lot about the scope of math.

As an aside: most people really dislike it when I say that they should be much more precise about different numerical systems. The integers are not a subset of the rationals. They are entirely different constructions, but there is an isomorphism between integers and a subset of the rationals that preserves the integers' ring structure within that subset of the rationals and a few other aesthetic concerns. You can see why no one wants to communicate like this, even if they acknowledge it's technically correct. So I know all about pushing symbols around.

But I also know that pushing symbols around isn't the whole story. Pushing symbols around is only useful as a final check. Do you want to validate that 1+2=3? Pushing symbols around can help. But how do you decide that the ideas behind 1, 2, 3, +, and = are worth having precise and compact representations?

Math doesn't just use formal systems to generate proofs. It's not enough for symbols to be arranged neatly according to some rules. Math is also the process of creating the sets of symbols and their rules and communicating to other people why this set of rules and symbols is interesting. What ideas get preserved when you are working with this system? What is it an abstraction over?

But mathematics is not a collection of facts. Mathematics is the study of abstraction. And what do you learn from a single data point? What can you abstract from that?

That's why just being a counterexample isn't really interesting. There has to be more than "counterexample" for there to be something to abstract. Was it generated from an analysis of the problem? Can the counterexample be generalized to explore the problem further? Is the counterexample a surprise in a way that suggests something is missing from current understanding?

Being a counterexample doesn't mean that something isn't interesting to a mathematician. But it's also not the interesting part.

The danger isn't everyone but you getting wealthy. The danger is that wealth tends towards concentration. And it tends to concentrate around people who are already wealthy. The danger is, bluntly, that things will get worse for all but a few and most people will be so caught up in a red queen's race that they can't see how to stop.

I don't believe it's possible to work in web dev and not run into this. It's a constant annoyance that semantic markup and declarative layout just have entire swaths of features unsupported on a garbage browser that you can't even say is unsupported because it's the only option on iphones. If it was up to me I'd just tell iPhone users that's what they chose. But businesses don't get to be as petty as I get to. And that's ultimately the point here. Apple is using their market dominance to force working around their poor support of standards on everyone else. They should be shamed for this shameful behavior.

What does a web dev have to do with Apple's choice? How do you improve Safari by not using features it doesn't support? What does your comment have to do with the argument the article is making?

I actually rely on this quite a lot, indirectly. Beat Saber is a fun game, but it doesn't support a lot of fun features, like walls with boundaries that aren't aligned to the coordinate system. There are mods to add those features, but they haven't all been ported over to the new line of the game since it upgraded the version of unity it's using. So I use one of the mod managers that supports, among other things, maintaining multiple versions of the game in parallel so you can choose what you want to do when you start a game. This feature very explicitly relies on downloading older versions from Steam. Because that's a thing you can do with Steam.

In the same week that Sony announced they'd be ending production of optical media, they also removed hundreds of movies from user libraries that they'd lost the right to sell. I don't really care if they have yet to do the same with games - they've demonstrated they're willing to remove access from paying customers for their own reasons. And there's nothing the owner of a locked-down console can do about it.

Steam exists in a different universe than digital-only games from PSN. Conflating the two because they use the same method of delivery is ridiculous.

If that's true, we're left with a question equivalent to "does everyone see the same red?". As far as I know, the pure version of that question cannot be answered because subjective experiences of sensation cannot be transferred. And at that point, I'd say the manner in which they're experienced differently is equivalent.

1. https://youtu.be/w3KswMaEBiI