https://github.com/npm/cli/pull/8965

https://github.com/npm/cli/issues/8994

Its good that that they finally got there but....

I would be avoiding npm itself on principle in the JS ecosystem. Use a package manager that has a history of actually caring about these issues in a timely manner.

If you can find internal (rather than external) reasons to trust/believe in your own intelligence and capabilities, it makes it easier to be willing to look foolish. Also, a lack of knowledge/ability in a new area (or even a familiar area) is not a sign of a lack of capability. There's a difference between being a novice and being an idiot. So long as your source of intellectual self-confidence is strong enough (say, you have made great intellectual achievements in some other area of your life unrelated to the thing you're struggling with right now) its irrelevant if other people think you the fool: they're simply mistaken, and that's no skin off your back.

It is absolutely true that someone else would have come up with special relativity very soon after Einstein. All that would be necessary is someone else to have the wherewithal to say "perhaps the aether does not need to exist" for the equations already known at the time by others before Einstein to lead to the general theory.

General relativity is different. Witten contends that it is entirely possible that without Einstein, we may have had to wait for the early string theorists of the 1960s to discover GR as a classical limit of the first string theories in their quest to understand the strong nuclear force.

As opposed to SR, GR is one of the most singular innovative intellectual achievements in human history. It's definitely "out of distribution" in some sense.

- Albert Einstein

most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism https://en.wikipedia.org/wiki/Mathematical_Platonism

Mathematicians begrudgingly retreat to formalism and foundations when pressed because its easier to defend, but the day-to-day of contemporary mathematics is much more an explorative process of a "real" mathematical landscape. They aren't concerned with foundations because it "feels" self-evident that the mathematics they are discovering is true (because their means of discovery, rigour and proof, "guarantee" it to be so).

A lot of the comments here are making false assumptions like "but surely mathematicians all know that their field is ultimately justified as a symbol-pushing game from some axiomatic system right?" in the same way one might say "surely all computer scientists know that every language ultimately compiles down to 1s and 0s processed by a CPU" but that is not at all how most mathematicians think about doing mathematics.

Here's Gromov, one of the greatest geometers of the last 50 years, discussing his viewpoint on this.

The conclusion is that with a sufficiently large number of actors in the market all seeking profits by trying to find misevaluation of stock prices, the excess profits of any individual actor will (assuming they all have access to the same information) converge to zero.

Its less a paradox and more a matter of game theory. Every investment firm which gives up trying to look for alpha (believing it is fruitless) means the remaining firms have more opportunities to find stocks with available information not reflected in the price. There's no paradox here: each individual actor is incentivized to participate in order to not miss out on that potential for excess profits, and the net effect is the EMH.

Mathematicians actually do the same thing as scientists: hypothesis building by extensive investigation of examples. Looking for examples which catch the boundary of established knowledge and try to break existing assumptions, etc. The difference comes after that in the nature of the concluding argument. A scientist performs experiments to validate or refute the hypothesis, establishing scientific proof (a kind of conditional or statistical truth required only to hold up to certain conditions, those upon which the claim was tested). A mathematician finds and writes a proof or creates a counter example.

The failure of logical positivism and the rise of Popperian philosophy is obviously correct that we can't approach that end process in the natural sciences the way we do for maths, but the practical distinction between the subjects is not so clear.

This is all without mention the much tighter coupling between the two modes of investigation at the boundary between maths and science in subjects like theoretical physics. There the line blurs almost completely and a major tool used by genuine physicists is literally purusiing mathematical consistency in their theories. This has been used to tremendous success (GR, Yang-Mills, the weak force) and with some difficulties (string theory).

————

Einstein understood all this:

> If, then, it is true that the axiomatic basis of theoretical physics cannot be extracted from experience but must be freely invented, can we ever hope to find the right way? Nay, more, has this right way any existence outside our illusions? Can we hope to be guided safely by experience at all when there exist theories (such as classical mechanics) which to a large extent do justice to experience, without getting to the root of the matter? I answer without hesitation that there is, in my opinion, a right way, and that we are capable of finding it. Our experience hitherto justifies us in believing that nature is the realisation of the simplest conceivable mathematical ideas. I am convinced that we can discover by means of purely mathematical constructions the concepts and the laws connecting them with each other, which furnish the key to the understanding of natural phenomena. Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it. Experience remains, of course, the sole criterion of the physical utility of a mathematical construction. But the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed. - Albert Einstein

Pure mathematics is regarded as an abstract science, which it is by definition. Arnol'd argued vehemently and much more convincingly for the viewpoint that all mathematics is (and must be) linked to the natural sciences.

>On forums such as Stack Exchange, trained mathematicians may sneer at newcomers who ask for intuitive explanations of mathematical constructs.

Mathematicians use intuition routinely at all levels of investigation. This is captured for example by Tao's famous stages of rigour (https://terrytao.wordpress.com/career-advice/theres-more-to-...). Mathematicians require that their intuition is useful for mathematics: if intuition disagrees with rigour, the intuition must be discarded or modified so that it becomes a sharper, more useful razor. If intuition leads one to believe and pursue false mathematical statements, then it isn't (mathematical) intuition after all. Most beginners in mathematics do not have the knowledge to discern the difference (because mathematics is very subtle) and many experts lack the patience required to help navigate beginners through building (and appreciating the importance of) that intuition.

The next paragraph about how mathematics was closely coupled to reality for most of history and only recently with our understanding of infinite sets became too abstract is not really at all accurate of the history of mathematics. Euclid's Elements is 2300 years old and is presented in a completely abstract way.

The mainstream view in mathematics is that infinite sets, especially ones as pedestrian as the naturals or the reals, are not particularly weird after all. Once one develops the aforementioned mathematical intuition (that is, once one discards the naive, human-centric notion that our intuition about finite things should be the "correct" lens through which to understand infinite things, and instead allows our rigorous understanding of infinite sets to inform our intuition for what to expect) the confusion fades away like a mirage. That process occurs for all abstract parts of mathematics as one comes to appreciate them (expect, possibly, for things like spectral sequences).

As Tao puts it, the value of intuition becomes much higher in the post-rigorous stage once you have sufficiently developed your technical skills.

https://terrytao.wordpress.com/career-advice/theres-more-to-...

Unfortunately a chip (even a legitimately earned one) on ones shoulder about the bad parts of academia doesn't save you from being criticized for being crank-y.

Sabine, like many contrarians, takes advantage of the fact that there are smart and convincing criticisms of many mainstream ideas, and she does her best to rely on those criticisms. However like all contrarians she presents a biased and exaggerated view of things in order to stoke engagement, and unless you are an expert it can be difficult/impossible to determine whether the view she is giving is balanced.

This is a classic issue with string theory critics, because string theory has many legitimate problems with it, but many of the critics are intellectually dishonest and you probably shouldn't listen to their criticisms on principle (but even I must admit it's quite hard to find good quality intellectually honest criticism of string theory which is digestible, so these contrarians tend to be the only loud voice).

In Sabine's case it is not so bad, because it is clear from some of her other positions that she is basically a crank. MOND and superdeterminism are basically crank physics at this point but she supports them purely because she is a contrarian. On this evidence alone you should not trust anything she says on any other subject, otherwise you're falling for a kind of Gell-Mann amnesia.

That is, there are models of Peano arithmetic which contain all of the natural numbers we know and love, and some other ones on top of that and there are some Goodstein sequences using those extra "non-standard" natural numbers which do not terminate at zero.

https://en.wikipedia.org/wiki/Non-standard_model_of_arithmet...

Either what we know about black hole formation is basically complete (it goes gas -> star -> black hole -> accretion + collisions) but the environment in the early universe was sufficiently different/dense that parameters which rule out the formation of supermassive black holes now were different. Maybe there were many intermediate black holes just in the millions of years after the big bang and things were still close enough together that accretion could happen and collisions were "likely" at the rate needed to form SMBHs after just a billion years. If that is true we might expect to see many many active galactic nuclei as we get better telescopes and look further back, depending on how quickly such black holes formed.

The other option is there is a mechanism of black hole formation that bypasses the above chain which we understand. People talk about supermassive stars, gas clouds collapsing directly into black holes, or primordial black holes that existed due to essentially random distributions of density moments after the big bang causing some regions of space to collapse into massive black holes which then persisted. Such things are far more difficult to observe, but could be inferred if we don't see many many active nuclei as we get better telescopes but all other indications of the accuracy of the big bang + inflationary theory hold true.

Not to mention if supermassive black holes were being formed by accretion, you would expect to see many intermediate mass black holes (1000-1000000 solar masses) everywhere, but we see almost none.