The simplest explanation is that they don't give a flying flamingo about what you or I consider "improvements to products".

This report is an example.

There are several changes that modify CI behaviour, where the article gives no corresponding quality measurement.

They replaced type aware custom lint rules with AST-only static analysis. They don't say anything about what those new rules detect, didn't do old-vs-new rule comparison. They switched the TypeScript check from tsc to tsgo. Again, they are very proud of the performance improvement, but don't seem to care about diagnostic equivalence. The list goes on. They don't even report pass/fail agreement between the old and the new CI. They have 4x more tests, but no idea whether this big test suite works any better than the smaller old one, or even whether it works at all.

One post later, you answered that Nelson's IVT argument is in fact the "standard nested interval proof" with the limiting step rewritten in nonstandard language.

That claim is simply wrong. Why? Because Nelson's construction straightforwardly generalises to Brouwer, while the nested intervals proofs cannot. The discussion of reverse mathematics / computability is not a tangent, it explains precisely why Nelson's proof can generalise to give the BFPT in two dimensions, whereas the nested intervals proofs (which you claim is the same) cannot.

You then brought up that the BFPT generalisation of Nelson's argument needs the Sperner lemma as "crucial additional ingredient". Now you make the same point again:

> What I'm saying is that for the coloring proof of BFPT to work, whether clothed in standard or nonstandard language, you must perform a combinatorial argument that uses a topology of a triangle as a necessary ingredient, similar in shape to the proof of Sperner's lemma.

Presumably you keep pointing this out because you think it justifies some claim like '1D Nelson is actually nested intervals with the limiting step recast in nonstandard language, even if the 2D generalization of Nelson is not'.

But it does not. The combinatorial content is the same, the 1-dimensional interval case uses the topology of the domain just as much as the 2-dimensional triangle case does. The 2D argument wouldn't work on the annulus, and the 1D version would not work on the union of two disjoint intervals. The Sperner lemma is present in 1D, and present in 2D. If instead your point is only that proving the BFPT requires a harder case of the Sperner lemma than IVT, then of course it does. But what relevance does that have to the original claim that Nelson's IVT proof is the nested-interval proof? The proof of the Sperner lemma is pure combinatorics, it does not involve any (standard or nonstandard) analysis.

Or have you changed your mind on your earlier claim that Nelson's proof is "is the standard nested interval proof"?

If so, I think that's great, and closes the thread on whether NSA is largely the same arguments, since even the first proofs of the basic results are different.

If you still think that it's the nested interval proof, well, I am not sure what else to say, apart from linking the literature which studies this exact question, that I've already done, and that you dismissed as a tangent.

Either way, this discussion went on for too long at this point, so I won't monitor it further.

1. Two comments ago you confidently claimed that Nelson's IVT proof is "the standard nested interval proof" with the limiting step written in nonstandard language. That is a straightforward claim about the structure of the proof, one that you didn't bother to substantiate, and that is straightforwardly false.

2. After I explained why it's false (Nelson's construction proves BFPT, which no nested interval type proof can do), you changed your response: now the Sperner lemma was a "crucial additional ingredient". But Nelson's combinatorial step, that opposite endpoint colors force a blue-red interval, _is_ the one-dimensional instance of the Sperner lemma (and indeed the base case when you prove Sperner's lemma for arbitrary dimensional simplices by induction; the analytic part is independent of dimension, once you find an infinitesimal multicolored simplex, you take its common standard part and apply continuity exactly as before).

3. Then you wrote this:

> In the standard formulation, you apply the Sperner's lemma to find smaller and smaller triangles, and apply compactness, precisely as in the standard proof of intermediate value theorem.

There is a standard proof of Brouwer via the Sperner lemma, and it is _also_ not of the same form as the standard nested interval proof of the IVT. In the nested interval proof, you find a sign-change interval, then find a smaller sign-change interval inside it, and so on. The intersection of all of these contains a point, and that's your zero. Nelson's proof does not do this, and neither does the standard proof of Brouwer via the Sperner lemma: you do not, and cannot, take a 3-color interior triangle, then find a smaller 3-color interior triangle inside it and so on. Even the first step would not work, since the inherited labelling does not satisfy the right boundary condition relative to the small triangle!

And this is also why the computability paper I cited ("where I quote reverse mathematics stuff" ;) was very much relevant. There can be no effective "nested triangle" proofs of the Brouwer fixed point theorem at all, because such a proof would let you compute a Brouwer fixed point, and there are examples of computable maps on the triangle without computable fixed points. If Nelson's IVT proof was the nested interval proof, then swapping in the higher-dimensional Sperner step would give a nested-type proof of BFPT. No such proof can exist. Since Nelson's argument proves the BFPT without any change to the analytic part, it is not a nested interval type argument.

You first misidentified Nelson's proof as nested intervals, and then treated Sperner as an additional ingredient even though the coloring step in Nelson's proof is already the corresponding Sperner argument. Those are both fairly serious misunderstandings about these proof. Given this, I don't think our exchange leaves readers with much confidence in your assessment of NSA's drawbacks and benefits. That's a disappointing outcome, as far as I'm concerned. There are good arguments to make that NSA adds little value to undergraduate education, such as simplicity or the difficulty of the prerequisites, and good conversations to be had about them. But "NSA proofs are the same proofs wrapped in a different language" is not one, and I wish you had just narrowed it instead of doubling down.

They didn't have to, as there were no state grants or public research funds for mathematics during the 17th century.

Newton supported himself from his inheritance throughout the Great Plague, while he invented calculus, and then from flat salaries as teacher, then flat salaries from working at the London Mint. He later became fabulously wealthy after his appointment as Master of the Mint.

Leibniz was a diplomat, then a librarian.

It is not.

I'll be honest: your one sentence response tells me you did not read the proof above in any detail.

I chose Nelson's proof precisely because its construction is well-studied and well-understood. The same construction of a mesh containing all standard points, with the coloring forcing a tiny multicolored cell, extends from the interval to the triangle. In one dimension you get two adjacent differently colored points; in two dimensions you get an infinitesimal triangle whose three vertices have the three relevant colors. Taking their common standard part and applying continuity gives a short proof of Brouwer's fxied-point theorem on the triangle.

But it is well-understood (there's a whole field studying such questions [2]) that the nested interval proof of the Intermediate Value Theorem does not generalize to proving Brouwer's fixed point theorem on the triangle [1]. This fact can be derived from a computability argument as well [3].

Nelson's argument does generalize to prove Brouwer, so it's not the nested intervals argument. But really, nobody cares about these technical reasons. It's obvious to most math undergraduates that Nelson's proof is not the nested interval proof, the clear absence of any nested construction kinda gives it away. The only reason it was necessary to get technical is that you did not really inspect the proof before claiming it was nested intervals. The technical results cited above are just a formal way to show that any correspondence you might imagine between the two proofs is just not there.

[1] Shioji/Tanaka: "Fixed Point Theory in Weak Second-Order Arithmetic", Annals of Pure and Applied Logic v47, pp 167188 (1990).

[2] https://en.wikipedia.org/wiki/Reverse_mathematics

[3] Potgieter: "Computable counter-examples to the Brouwer fixed point theorem", https://arxiv.org/abs/0804.3199 (2008).

No. Let's take a nonstandard proof of the intermediate value theorem on [0,1] by Nelson.

By the transfer principle it is enough to prove this for a standard continuous function f on [0,1] with f(0)<0Take a finite subset of [0,1] containing every standard point. Colour its points blue, green, or red according to whether f is negative, zero, or positive at tha point.

The first point of the interval is blue and the last red. Hence either awe can find some green point, or we can find two neighbouring points that have different colours, the first blue and the second red.

In the first case there is a zero, so we are done. In the second, let the neighbouring points be p and q. By the completeness of the real numbers, every nonstandard real in [0,1] is infinitesimally close to exactly one standard real. So p and q are infinitesimally close to some standard real number, let's call it z.

Standard continuous functions send infinitesimally close points to infinitesimally close points. So f(p) and f(q) are both infinitesimally close to f(z). But f(p) is negative and f(q) is positive. The only standard number infinitesimally close to both positive and negative numbers is zero. Thus f(z) is zero. This proves the theorem.

You tell me, which standard proof is this? It's certainly not the nested interval proof. Not the supremum proof. Not the bisection proof in disguise. Which argument does it wrap in slightly different language? Can you point to a single textbook, course note or lecture that gives such an argument?

No. One could of course argue that this is not simpler/shorter than the usual arguments. But it is very different from them. Saying that it's the same arguments repackaged in a different language is just wrong, and detracts from an otherwise valid point.

Which numbers you believe is of course going to affect whether you think there's a moat or not. That's largely orthogonal to your TSMC/Samsung analogy I responded to. If you think the "moatists" are wrong because they believe the wrong numbers, that's fine, but then there's no need for the analogy.

[1] https://galileo.ai/blog/llm-model-training-cost

[2] https://medium.com/@theiand/how-can-deepseek-a-5-6-million-l...

Comparing the R&D costs of creating GPT-4o vs. DeepSeek V3 (the latest gen for which we already have good accurate numbers) it looks like the latter cost 1/20th as much to create.

If Samsung could catch up with TSMC for 1/20th of the cost, people definitely would say that TSMC has no moat.

I mean, that's exactly what a "slur" is. They are not coined with the recipient's comfort in mind, and saying them can sometimes provoke exactly the sort of fight you imagine.

But yes, violent consequences ought to be particularly unsurprising to an American, since the US is, by Western standards, an exceptionally violent country. [1]

[1] https://pmc.ncbi.nlm.nih.gov/articles/PMC9535176/ (table 2)

If you have time, can you elaborate or give some examples of mathematical nuances?

I am evaluating Sol and Fable on a fairly large dataset of subtly flawed informal mathematical arguments (task is to identify and name propositions with substantially incorrect proofs in a larger body of text), and Sol is saturating the benchmark, while Fable is below 50% even with the most generous grading.

I don't work in the natural sciences, so I suspect you mean something different by "mathematical nuance".

But one can already study the consequences of P=NP right now. You don't need to know that it's provably true in order to do that.

Knowing an actual proof would be useful, but an oracle revealing merely that it's true (or even provable) without telling you the proof does not let you do anything you couldn't do before.

As in, you were using state of the art generator X, and you couldn't see the performance bottleneck, but updating to a newer (faster, or same speed but higher quality) generator Y, and could subsequently identify the performance bottleneck?

If you're using PCG, not in the last 12 years.

(In a parallel comment I suggest trying AES-CTR for this use case)

Have you skimmed the linked thread?

> especially simulation/sampling approaches that are bound by the number and quality of uniform variates per second

Sorry, nobody does stochastic simulations where the number of uniform random numbers obtained per second is any sort of bottleneck. If you've spent considerable time on stochastic simulation, you already know this.

But even if you insist that you alone are doing some very weird stochastic simulation which is somehow bottlenecked on sourcing random numbers fast enough, the falling in planes phenomenon linked above would make xorshift-type generators a poor choice for most sorts of simulations. It introduces spatial correlations into any sort of lattice dynamics simulation (Ising model, percolation) and every high dimensional Monte Carlo integration. Beyond falling in the planes, since xorshift is linear over GF(2), it is also a particularly bad choice for nondeterministic cellular automata and Boolean dynamical systems which use parity, bit masks, or xors.

AES-CTR throughput on a modern CPU is higher than that of xoshiro256++, and much higher quality. No advances in non-CS PRNGs can beat that while maintaining the same quality. If your stochastic simulation is bottlenecked on random bits, CSPRNGs are still the way to go, and they don't interact in nasty ways with any dynamical system you can actually sinulate quickly.

There is no real use case for better non-CS generators, as explained by adrian_b back in 2021 [3].

[1] https://en.wikipedia.org/wiki/RANDU [2] https://arxiv.org/abs/1908.10020 [3] https://news.ycombinator.com/item?id=28886698