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As for police killings, you are comparing an event that’s extremely unlikely to happen with an event that’s multiple orders of magnitude more likely. Speech offense prosecutions in UK are more common today than political prosecutions were in post Kruschev Soviet Union. The only reason UK has fewer political prisoners than Soviet Union had is that the sentences are shorter and typically non-custodial.

If that makes it easier for you to understand it, in the standard proof, you also color every point of the rectangle, with the color of the edge it retracts to (picking the colors of the vertices of the big triangle arbitrarily, just making sure that the color of each vertex is a color of one of the edges it belongs to, not one of the opposite edges). Then, an easy argument from continuity shows that no interior point will have points of three different colors arbitrarily close to it. Finally, applying Sperner's lemma as above proves that such point must nevertheless exist, obtaining contradiction with the existence of the retraction.

and then treated Sperner as an additional ingredient even though the coloring step in Nelson's proof is already the corresponding Sperner argument.

I don't understand what are you saying here. 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.

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.

The rest of your post, where you quote reverse mathematics stuff, is completely irrelevant to the point I was making. Nothing I said is about what theorems follows from what axioms, but rather whether nonstandard analysis is meaningfully different, clearer, or more useful language than standard one. It is not.

The language used by mathematicians is subject to constant evolution. 18th and 19th century results in analysis are not expressed and taught in the same way their original authors did it. Newton, Leibniz, Euler, Lagrange, Fourier, Riemann -- none of them expressed their results in terms of epsilons and deltas. These only caught on in the second half of 19th century, and they did so, because they were a better tool to rigorously prove the ideas.

New terminology inventions that make the subjects easier to understand take the field by storm. Some of the relatively recent examples are category theory, homological algebra, or, for that matter, the notion of sheafs, popularized by J.P. Serre himself. Mathematicians are very open to innovation, and intransigence is not the reason why we're stuck with epsilon-delta.

The reason is that nobody has yet come up with a better way of talking about these concepts. I repeatedly observe many people who seem to believe that their difficulty in understanding math stems from mathematicians gatekeeping their results. I think that this belief is just a coping mechanism. Mathematics is genuinely hard, and when people have trouble understanding something, it's easier to think that it's someone else's fault, rather than accepting one's own deficiencies.

There is a certain view, popular among many people, that poor performing schools perform badly because they are underfunded. It has been vigorously spread by politicians, teachers, and activists. Well, we ran the experiment. With enormous federal and state spending, we managed to mostly equalize the spending, and, in many cases, fund the badly performing schools more than the good ones. The results of the experiment are very clear: additional funding, by and large, does not *cause* significantly improved educational outcomes. Any observed correlation between school funding and outcomes is a result of who lives in the area, and who attends the school. The casual arrow does not go from funding to outcomes, but from the school population to both funding, and, in parallel, to outcomes.

And somehow, none of the thousands of very smart mathematicians have done that, or if they had, it has not seen wide adoption. I recommend contemplating on this: if math could be made easier by changing notation, why hasn't this already happened?

The same is, of course, true about Google or Apple. Working at Apple will make it much easier for you to be productive than working at your own company. The nice thing about our industry is that the latter, while more difficult, is actually possible -- unlike steel mill workers, software engineers don't need as much capital investment, and can run highly successful companies that employ just one or a handful of people. It's just hard and risky to try that, hence people prefer to pretend that the productivity enabled by working as part of established, successful company is entirely due to their own merit.

The truth is that by working for a company, you get access to environment that makes you much more productive than you'd otherwise be on your own. You also are not on the hook for most of the risks. It is patently unfair, and extremely short-sighted, to claim that investors deserve no compensation for their investment.

Let’s actually talk more about this max - min example. Let’s say you calculated this delta. What is it useful for in your code? Typically, in real algorithms, what you want is to not only know the delta, but also which number is larger.