calf

I hate to use absolutely but it absolutely does matter from an education perspective. To put things in perspective to a layperson explain to them this particular galactic theorem requires full search below some titanic |x|. That should prompt some excellent lay questions that a serious computer scientist ought to reckon with.

It is not to say the approach has been wrong (as the author strongly believes in his reframing) but that this clarity does matter for purposes of, ahem, mathematical understanding.

I don't think the author's criticism of "Big-O Cabal" is very nuanced, but to the second half of the essay it is funny/concerning/sobering to find that the OpenAI proof is doing brute-force search for all instances below some super-astronomic threshold. It kind of buries the lede if the proof did not mention this clearly and the author has to comb through it to find out (I asked ChatGPT the same on the pdf file, but it came up with some other threshold number).

See, any serious Platonist would point out your line of reasoning is incoherent, because the conceit contained within such an accusation just covertly satisfies your own ego through a clever meta-discursive rhetorical maneuver. And they would be right

The truth of the matter is somewhere else but if people were actually a bit better rounded philosophically and politically.

And the material conditions for this "great man" narrative is again the particular form of intellectual clerisy.

In other words, the move to experience and cultural relativist statements about "math culture" is exactly the cynical ideological move that an academic cleric does. It deflects from examination of where these narratives come from and attribute it to e.g. "math history" or "math culture" or just "something specific to math itself" etc, etc.

Neoliberal intellectuals love to attribute toxicity to the Enlightenment. They can't help it.

It's inaccurate because nothing about math or CS is "existentially" toxic. Or "distinctly" as the other commenter.

If you accept that as a premise then obviously you agree with their truth.

If you don't accept toxicity as somehow intrinsic to the subject but rather the structural conditions, say, of post-neoliberalized academia at large, and certain faculties being only a bit more susceptible (if even, or just different, say compare humanities vs STEM reactions in the future), then the whole comment reads as a conflation.

Instead the simpler answer is that academia at large values freedom of inquiry and other Enlightenment notions which are privileged compared to outside of academia. But that is not toxicity, it's just intellectual integrity and the academic autonomy in pursuit of one's interests avocationally. The toxicity comes from the intellectual profession having to function as a modern clerisy for socioeconomic reasons under post-neoliberal capitalism.

People may conflate the two attributions and that is a deep mistake, in my opinion. In fact the conflation due to internalized oppression easily creates and explains this kind of resentful, dismissive tone response that is a distortion of the true situation.

Read JMS's eulogy for Koenig this week if you haven't already.

and Bester in Babylon 5, what a performance.

Mods obviously Star Wars people

Progressive neurological diseases suck. I would like to understand though since Stewart was filming up until this year? What was the time frame like, was it a more sudden onset or was he able to manage somewhat while continuing to film and work?

Macs are highly recommended by people in the know, like people in computer science departments over the last decade, by HN commenters and specialty magazines, etc. Those are better signals than the superficial market picture you paint.

They are great devices but they cost more. So you would not see them "recommend" in those industries you cite. Also it is reductive to accuse Apple of marketing.

What's much more plausible is that despite being a great product it also costs more. Not marketing nor the fact of being a minority.

It's also weird because your argument disingenuously ignores the wealth of HN commentary in the past about how great the M-chip MacBooks are. It's like ignoring actual reasons and doing this kind of armchair technical market analysis to prove one's point. It's a troll intellectual move to attribute success to marketing and failure to "because it is unpopular". Seeing your other replies in this thread suggests a common pattern of bias, not a one-off.

You must be trolling if half dozen replies in this thread you want me to think that Marvel films are better than art films because they have a bigger market share. This is not worthy of serious reply and it seems the problem is your worldview, not the merit of any argument about Apple.

So it cannot possibly be explained as Macs/macOS are still a better product despite some obvious flaws and drawbacks? It has to be cynical marketing and some kind of consumerist corporate conspiracy? I don't get your argument.

I don't agree with this narrative of the history.

It is true that CS had to make theoretical concessions. Just as Newton did. We don't criticize Newtonian mechanics (or, I don't know, Galileo?) for not being realistic about friction or idealized balls rolling down ramps. Their anti-empiricism was a starting point for physics.

To wit, the mathematical rationale for P-NP was precisely that that under polytime reduction this is still closed in P. That is a very reasonable starting point for a theory. This is also the reason why it is tripping up a large swath of the comment thread with people asking about subexponentials which do not preserve that closure

The opposite narrative is that in science, these idealizations yield profound intuitions and insights.

Furthermore, how far are you willing to go? Do you think Turing machines are invalid, that Turing-equivalent computation is pointless and unrealistic? So we are never allowed use that theory to convey the essential idea of in principle all computers being fundamentally functionally the same and that anything claimed to be more powerful than that had better have some incredible evidence, lest it be appeal to magic. Computability and complexity have a shared foundation so how much of that foundation are you going to reexamine?

Whereas if one wishes to be like a covert finitist/intuitionist, they ought to be open about that instead of accuse asymptotic complexity (or analgously, Turing computability as above) of trying to "cheese" complexity theory as a research programme. Meanwhile, it is true that P vs NP could have large exponents or constants in P, e.g. O(c*n^100...), literally this is the first thing taught in any complexity course or textbook. Students and practitioners remember this admonition well, maybe they just don't talk about it publicly as much.

One can go even further back. Cantor was marginalized for his ideas about infinities.

The difference with people like Lipton, and Knuth, is that the critique is coming from inside the house. In that they know the theory and the importance of it, but also hope/aspire for some new way to address the open question. They are keenly aware that P / NP is a problematic question, but not for lack of trying or stupidity on the part of other scientists. What is more likely is that the 21st/22nd century math to pose the better question does not exist yet (or may never be accessible to humans, another question well on the minds of many today).

Here's something to ponder. It is often observed that despite being NP-complete, many actual human instances are in fact tractable. E.g. routine airline scheduling depending on SAT solvers. But is this evidence that NP-completeness is a wrong theory or evidence that human-sized problems are a special case that becomes tractable (anthropocentrism), or some other more complicated reason?

But Apple laptops are still the best recommended laptops on the market, despite these extra costs and margins passed onto the consumer, why is that?

The technical result has nothing to do with "efficient" which according to textbook CS theory roughly defines it to be the set P, the strictly polynomial time algorithms. Has nothing to do with the ETH result which is still an exponential, 2^(n/ log n), just not a pure exponential.

I'm also confused what you are saying down here. Though I can see how "sub-exponential" is easily misleading terminology.

Yeah that would be interesting, like could there be different NP languages being sub-exponential but for entirely different reasons.

For example, a 2^√n subexponential algorithm then used on an input after a n^2 polynomial reduction from some other NP-complete language incurs a total running time of 2^√(n^2) = 2^n, which again is exponential.

In the Star Trek Voyager capstone episodes titled "Scorpion" parts 1 & 2, Chakotay explains to Janeway the behavior of the Borg through the parable of the fox and the scorpion making a transactional deal to cross the river together. After reaching the shore, the scorpion stings the fox, and the dying fox asks why. The scorpion replies, "Because it is in my nature."

(Upside of the story is Captain Janeway's trusted crew works together to formulate a trick up their sleeve precisely in case of predicted contingency...)

Of course, if anyone typed this take to any current LLM, it would immediately push back that you are making a sweeping generalization.

Yeah I took a look later and am still mystified as to why specifically he would find a site like that to be addictive writing and helpful to him.

Indeed, why in 2026 do we still see LLM Skinnerism as automatically a valid argument that has all four legs?

I mean it's just patently absurd. It is trendy to use this hubristic, and frankly bureaucratic, definition or requirement on math.

People can do math alone. They don't have to share it with others. Mochizuki was doing math (with errors). The aliens from Arrival could do math and we wouldn't necessarily be entitled to that. It doesn't make these things not valid math.

The only benefit of defining math through this vulgar social reductivism is to benefit the status quo. So you see some experts and expert-adjacent commentators using these math slogans but never critically justifying those words. Case in point, the above commenter asserts "...math is foremost about communication.", but the very next sentence does not explain why it is foremost, it just uses the sentence as a premise for the following sentence. Guess what, that is exactly how ideologues argue, by question begging fallacy. Ironic that STEM people who ought know better fail to catch this simple rational mistake.

Author says he reads Zvi Mowshowitz because he agrees with the writer's views on AI; anyone have a quick summary or good article to check out to get a sense?

The essay is conceptually muddled because it ignores all the basic computer science theory that is sorely needed to explain AI and LLMs, but underlying it is a valid issue of what it would take to "derandomize" an AI so it does what we would like it to do.

Unfortunately despite the author's indutrious optimism, this is equivalent to one or more open problems in theoretical CS. For example we don't know BPP = P or not. So are random algorithms tractable? We do not know. If you claim to know the answer, you could win a big math prize...

(It should be noted that despite the title, the author's main argument shows that output non-determinism does matter, since their entire effort is toward mitigating the stochastic behavior of LLMs.)

Rather the opposite, AIs are arguably deontically asymmetric because they are powerful and virtual digital entities, and thus should be very concerning to the human species. Not because they current have no internal abstract state for learning state like humans and animals.

I thought it was scaling

If not for Dijsktra you would not even exist to write the very sentence "Maybe there's a future for formal methods, but...".

What basic science does is open up new areas and new ideas. And you and I benefit from those foundations.

Dijkstra's proof methods derived from higher math sophistication than most programmers before, and today, have. Like, in one short essay he showed what invariants mean. He made formal theory accessible to mere non-pure-math mortals. You and I today know what invariants are. That's just one example.

How far are you willing to go? In your radical pragmatism, are you an intuitionist, or even a finitist? Oh wait, these are pure math topics that you do not care a about!

The irony being Dijkstra had lots of deep ideas about how programming ought to be done (from a basic science approach as informed by CS) and that the industry is still in the stone/bronze ages in terms of putting advanced ideas into practice.