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> Doing it properly requires a carefully designed interface to hide internal data structures, and splitting out the end user headers from the internal headers, but it works.

Out of curiosity, how? (If pointers or other mechanisms for memory indirection are allowed then it's pretty easy, so let's agree to ban those.)

Not a rhetorical question, despite the parenthetical.

Is this a regional thing? I've had interviewers inquire perplexedly from across the table as to why in the world someone with a mathematics degree was seeking a software development job.

Do employers in some parts really salivate over green math/phys grads with not so much as a github profile? Or do these anecdata leave off some key arc to the story where the grad does something substantial to bridge the gap between said major and software dev?

The burden is on the specifier of the compass and straight-edge algorithm to explain a reasonable mechanism for carrying out the steps exactly, to infinite precision. Without this, the procedure can only be expected to give a reliable approximation of sqrt(2) to some finite precision, which a Turing machine can match. A similar observation applies to chaotic systems.

In general, a chaotic system can only be said to "solve" a problem if it can be configured to reliably arrive at (an encoding of) the solution for a given instance, despite any chaotic behaviour exhibited by the system. It wouldn't be reasonable to, for example, claim that a Turing machine cannot solve the problem of predicting the behaviour of a chaotic system with some particular initial conditions, if you couldn't reasonably conifgure the chaotic system itself to do that either.

If key figures at the federal reserve didn't (don't?) believe in the self-regulating power of a free market, that's nice, but this article still seems to suggest that said belief is still alive and well among highly influential figures in the economy, i.e. those financial lobbyists or their interest groups must either

a) Believe that a free market can police itself; or

b) Be so myopically hell-bent on the bottom lines of their interest groups that they're willing to recklessly sabotage important regulatory impediments to meet their ends.

Either way, a lot of powerful people could still probably stand to be admonished on the importance of well-enforced regulations, even if those admonitions are embarrassingly basic. Am I missing something here?

The media's fondness of using the term "algorithms" for the products of rigor-allergic ML practitioners breaks my inner algorithm theorist's heart. :(

If your FAANG research team employs just as many PhDs as non-PhDs, and we accept that there are a couple of orders of magnitude more non-PhDs in software engineering than CS PhDs looking for work in industry, then I don't rate my chances of getting recognized within the company for my robust knowledge of theory of computation and abstract algebra later down the road if I take that grad job I was offered by a tech giant. Also, it's not all that likely that a job as an engineer is going to keep me my theory knowledge or research skills fresh.

The article makes a few resonant points, but overall I think the "you can get into research if you jump straight into industry" pitch it tries to make is very weak. As someone who very much wants to do research (mostly hard-to-monetize research about comparing exotic models of computation to one another, but I'll listen to a pitch for applied research too), I'd very much like to see someone lay out a highly plausible roadmap for getting into a research position without a PhD. I don't think this article is that.

It seems to lead to the sort of situation where the only kind of success a poor kid can hope for in life is the kind where they get into the kinds of degrees that offer pay, whereas privileged rich kids have the luxury of getting to go into "useless" humanities degrees and curate our culture and history and whatnot. Only enabling the leisurely well-to-do to enter the latter kind of career path is generally considered a bad thing.

As someone whose education straddled the four-pillars pure topics, mathematical foundations, and computer science, I share a bit of a wariness about AC with the constructivists, but I cannot let go of LEM. It's a big part of how I reason, and a number of results that I consider important and illuminating hinge upon it.

My (not entirely informed) sense is that constructive mathematics would have more success if more of an effort was made to pose it as a readily instantiable, tagged embedding within the larger classical mathematical corpus. Less Brouwer-style dogma about LEM being "wrong", more emphasis on streamlining a pathway for results to travel from the mathematical journal literature into software design specs.

Formal systems aren't just about mechanizing mathematical work to make it a subject of software engineering. (Though this is a useful and probably underdeveloped outcome.) They also turn mathematical work itself into mathematical objects which one can reason about, which opens the door to metamathematics, i.e. the ability to reason mathematically about aspects of mathematics itself.

|\N| = |\Z| = |\Q^2457000| != |\R| isn't wondrous to you? Nor are space-filling curves? Or unmeasurable sets and paradoxes like Banach-Tarski? You don't see beauty in the proofs of Cantor's theorem (|S| < |P(S)| for all S) or the Bernstein-Schroeder theorem (injections X -> Y and Y -> X imply |X| = |Y|)? What about the plethora of mathematical claims that are equivalent to the axiom of choice -- some of them very sought-after properties, some of them unnerving and seemingly broken? Does the status of the continuum hypothesis (and the mathematical legitimacy of, to say nothing of the resolution to, a bunch of oddball "infinitary combinatorics" problems) interest you?

What about where set theory interplays with model theory? Would it surprise you to hear that the dizzying towers of infinities that can be spoken of in ZF can be modeled by a merely countable universe? What about a computer program (albeit an impractical one) that decides whether sentences about the natural numbers with successor are true or false, whose existence is discovered by proving that large uncountable models are isomorphic?

To be clear, it's okay to be bored by this stuff. I tend to get bored pretty quickly when people start talking about systems of DEs or eigenvectors, or really much of anything that looks too much like calculus, and I'm not about to apologize for my apathy for those things. But most of the above stuff tends catch the attention of even the humanities types, who are easily deterred by most mathematical topics.

Out of curiosity, I might ask whether you're similarly bored by computability theory or complexity theory, which leverage many of the same tools as those used in set theory (mappings as preorders, diagonal arguments, etc), but add computable enumerability/uniformity/syntacticity requirements to the mix.

In a world where it's taken for granted that, say, countable additivity over measurable sets will work work out fine, and is felt to be a different beast than arbitrary use of a separation schema plus a set of all sets, sure. But it took early 20th-century mathematical logicians decades of exploration to get to a point where there was confidence that people doing the former thing could get by mostly "naively", without being in danger of doing the latter thing. (See e.g. Galileo's paradox.)

Modern pure mathematicians benefit from being able to conduct most of their reasoning on the lower footsteps of the von Neumann hierarchy (V_{omega + k} for small k), where they get access to higher-order tools like function spaces and topologies, countability as a fruitful property to impose and work with, and a playground where most of the disparate branches of mathematical research can be brought together without worrying about bindings between underlying frameworks. None of these things require a deep knowledge of the metamath of ZF to work with, but they'd all make many 19th-century mathematicians nervous.

Note also that the most frequently used tools of naive set theory today (e.g. unions/intersections, separations/replacements, powersets, cartesian products/relations/functions, a few notable infinite sets) roughy mirror ZF. That shouldn't seem like an accident. Having working mathematicians trained to think and organize their work in these constructs keeps them around those lower footsteps.

Also, Scott Aaronson gets pulled into speculation on these things a lot, e.g. https://www.scottaaronson.com/blog/?p=3208

There's a lot of subjectivity to the "we can simulate the universe" variety of claim stated by the article, but I'd tend to forgive considerations of the scale of the entire universe in that claim. It should be obvious that simulating the trajectory of every last particle is straight up impossible. (Even in principle: you could probably even show outright impossibility on logical grounds with a diagonal argument a la the time hierarchy theorem.)

Instead I'd take it as meaning something like, "any slice of physical phenomena there is to observe in the universe, we can simulate given reasonable resources to do so." So, put an imaginary box around some reasonably isolated plot of reality, pick your precision and your time scale, and you could replicate what happens in that space with a "reasonable" computational resource overhead. That's what elevates the quantum simulation overhead objection to number 1 in my mind.

Some combination of the holographic principle and the the information overhead of classically representing a quantum state? (I don't know, I'm not a physicist.)

Surprised that the currently exponential overhead for simulating a quantum system is a distant second in commenter objections here. "Simulating the universe" in the holistic manner that I feel such a phrase entails would require at least clearing that bar (say, with a quantum computer? Or at a longer shot, showing BQP = P?), and that's before thinking about the status of quantum gravity.