spekcular

Thanks for the explanation! That sounds exactly what I'm looking for, and I will definitely read those Schoenberg books. Really appreciate the pointer.

Thanks, I think I basically agree. I guess my complaint is that these later composers generally involve more "bespoke" analysis because of the additional complexity, which is not treated adequately in the textbooks. (Whereas, for example, for a lot of Mozart piano sonatas you can just write down all the roman numerals and call it day -- I'm joking but only slightly.)

Also, I think you'd call your Mozart example a common tone diminished chord, notated CT7. I think this is pretty standard nomenclature now; see this book excerpt for example: https://viva.pressbooks.pub/openmusictheory/chapter/common-t.... Especially the example under the section "recognizing CT7 while analyzing."

Thanks for the book recommendations.

I used Aldwell-Schachter, Voice Leading and Harmony. Part VI is relevant to the composers I mentioned but didn't really explain many of the things I was wondering about. Sure, I can name the chords (most of the time), but on a compositional level there's much more going on.

Again, I am confused by your claims. The species system taught by Fux is, as far as I recall, the exact same way that counterpoint is introduced in, e.g., Gauldin's "A Practical Approach to 18th Century Counterpoint." (Gauldin goes into much more detail on other things, but the approaches are consistent.)

The fact that the upper voice goes 2-1 or 7-1 and the bass 5-1 at cadences is one of the first things taught in a counterpoint class. The rule of the octave is in most theory textbooks. The first example that comes to mind is Laitz, "The Complete Musician."

You're right that these textbooks don't teach how to improvise an accompaniment from a figured bass, but I'm not sure that's a good use of time for a theory class.

I'm thinking mainly harmony and the much more liberal use of chromaticism, unusual chord progressions, and so on.

I don't really understand your comment. Figured base and and common harmonic sequences are well covered in most standard theory texts (as is melodic/harmonic reduction).

I'm also amused by your remark that counterpoint is not taught in "the way it would've been understood by successful 18th-c. musicians." Most contemporary counterpoint instruction still follows the method of Fux's 1725 textbook (species counterpoint). Indeed, the fact that the pedagogy hasn't really changed for hundreds of years is often advanced as a criticism of current-day instructional practice!

I agree! But it seems that the books all stop at 1850. Or they toss in a few cursory chapters at the end. (If there's some text that doesn't do this, I'd love to read it.)

I have a more conservative critique of traditional music theory, along the same lines. Forget pop, jazz, and rock - standard music theory courses don't even equip students to properly analyze Western art from music the late romantic period, including things like impressionism. Most voice leading textbooks have very little material on post 1850-ish classical music, and some have essentially nothing. They seem to focus on Mozart, Beehoven, Haydn, etc., or aspects of later music amendable to the analytical tools they develop for those classical period composers.

Here's an exercise you can try. Choose a more harmonically adventurous French or Russian composer from around the turn of the 20th century (perhaps Debussy, Ravel, Scriabian, maybe even Shostakovitch), and get some sheet music for a solo piano piece off the internet. Grab any of the standard college harmony textbooks and attempt to write down a harmonic analysis of that piece. You're going to have trouble.

> If it wouldn't be the most used, would it be used as point of reference and called the tonic?

Yes. The tonic is determined by what the key is, not what chord is most used. (It's hard to establish the key without slipping the tonic in somewhere, but that doesn't mean it needs to appear frequently.)

> I-V and IV-V seems to be the same with the point of reference moved to IV. And IV-bVII-V is I-IV-V.

I don't think this is right. V-I has the root moving by a 5th, while IV-V moves by a second. And the second example can't be right because the relative distances of the roots of the first and last chords are different (again, a second versus a fifth).

> Is a fixed point of reference really needed? Doesn't it just add unnecessary complexity?

Yes, it's a useful analytical tool.

He was definitely aiming for a certain style, which involved following certain rules. There are surviving draft manuscripts where he makes edits to follow voice-writing rules like avoiding parallel fifths, etc. It seems pretty clear based on this (and his training, and even a cursory study of what he wrote) that his compositional practice was intensely theoretical. The dude was not out there freestyling.

In fact, the places where he broke certain fundamental rules are so rare that they form a rather short list, despite his voluminous output: https://www.bach-chorales.com/ConsecutivesInChorales.htm.

What forthcoming improvements to async are you referring to?

It was supposed to start at 5:30 ET.

This kind of decision is already routinely made by committees awarding NSF postdocs, Clay fellowships, etc. What's wrong with the process they use (and independent expert panel)? Do you think they select the wrong mathematicians?

> This might be true, but it's really really hard for someone who doesn't work with them day-to-day to distinguish.

It's really not (in mathematics, the field under discussion). It's not hard to look at the CV of a graduating PhD student in math and tell whether they're a semi-reasonable candidate for the Clay Fellowship, or the other fellowships listed above.

Among the small group who makes that cut, of course you need to rely on letters of recommendation, expert assessments, and so on. But the initial cut is fairly straightforward.

The author writes: "Is there a hypothetical world where I would be an academic working on the Birch and Swinnerton-Dyer conjecture right now? Sure. It's probably a world where high-flying students are given, upon graduation, some sort of "mini-Genius Grant". If I had been awarded a five-year $62,500/year grant with the sole condition of "do research", I would almost certainly have persevered in academia and — despite working on the more interesting but longer-term questions — have had enough publications after those five years to obtain a continuing academic position. But that's not how granting agencies work; they give out one or two year awards, with the understanding that those who are successful will apply for more funding later."

These awards certainly exist; the Miller institute at Berkeley, the Simons fellows in NYC, and the Clay Fellows are a few examples. (Some not quite for 5 years, but more than 2. Also, the pay is more like 90k.) The NSF math postdoc is also close to this (2 years full funding if you elect to not teach those years, plus a third year where you have to teach.) Regular NSF grants and NSF CAREER awards are also longer, IIRC, though perhaps less relevant to fresh PhDs.

So, in fact, these "mini-Genius Grants" do exist.

What empirical evidence exists for that claim? I agree with it, but I'm mostly generalizing from own experience.

No, that link will always point to the most recent version.

For example, https://arxiv.org/pdf/1909.03562.pdf gives version 5 of https://arxiv.org/abs/1909.03562, as noted in the left margin of the first page.

I also find this baffling.

To be fair, the link is about homotopy theory proper (very interesting!), not homotopy type theory (a somewhat different area of study, and less interesting, in my opinion). I actually don't remember any other links about homotopy theory that weren't related to type theory. So in my view, this is a welcome development.

> There are some toxic fields of research, but I don't think this attitude is particularly common in the academia.

The author was formerly an economics student, where this attitude is quite common. It might have colored his perspective.

Exactly. Proof writing is basically impossible to learn without this kind of social context and feedback.

You need a good tutor. I suggest finding a math major at a local university to teach you.

The problem is that, from what I can infer from your description, you don't have the fundamental skills necessary to self-tech effectively. You could try using some of the resources mentioned by other commenters, but chances are this process will much more tedious than if you had a mentor, and you'll probably come to believe various incorrect things that you'll have to unlearn later. Real-time feedback and correction would be more effective.

Also, I disagree with some of the advice given here. (Suggesting resources on Coq and ZFC to someone asking how exponents work? Really?) Tread carefully, and prefer the recommendations of people who have experience teaching high school students and undergraduates.

The original poster did not have the "for a novice" qualifier.

This is not great advice, according to what we know from both the research literature and people who are successful in (and successful coaches of) barbell sport. Namely, it's too much intensity, and not enough variety and volume.

One good reference to learn more is https://www.strongerbyscience.com/complete-strength-training..., and other articles on that site.

The top review on Amazon is devastating:

"The author purports to provide a blueprint to restoring a technological economy after a TEOTWAWKI event, but some his listed sources are from the realm of science fiction. Not an encouraging start.

He goes on to pretend that he knows more than he actually does. It's as if he skimmed a few sources but only superficially understood them. How else can he suggest that a collapsed society go direct to building blast furnaces, ignoring the bloomery method of reducing iron ore that provided mankind with workable metal for two millennia as a cottage industry? Then he goes on to suggest that we build Bessemer converters to decarbonize the pig iron. Does he not know that the Bessemer converter is all but obsolete? Did he miss the chapter about the (chemically) basic refining furnace, which is a lot easier to build?

He quotes a lot of interesting chemistry, then throws up a real laugher when he gets the simple and universally known formula for black powder exactly backwards!

While the book skims quite a potpourri of technologies we use today, he omits almost entirely the tools needed to implement them. Knowing how an electrical generator or motor is assembled is all well and good, but where will the impoverished builder get copper wire? Or the special steel sheet necessary for laminating magnet cores? Or the tooling for punching out the laminations?

He never even began to address the fundamentals of machine tools, on which about 99% of our modern technology rests, and without which you cannot build even an 18th century economy. .

As a high school science project, this would rate a solid C for effort, and something less for the end result."

This is a response to a recent front-page article on HN: https://news.ycombinator.com/item?id=31651086.

Indeed, a similar "secure" crush confession on an MIT facebook group was exploited in this way, by brute forcing every name in the student directory.

My QM knowledge is quite rusty, but given what I remember, I don't understand how certain claims in this article can be correct.

Hossenfelder seems to be referring to some well-known papers on Hyperion and decoherence. For a review, see: https://arxiv.org/abs/quant-ph/0605249

There are lots of references there that you can follow, including a debate on whether decoherence theory is really necessary to explain what's going on.

But as far as I can tell, no party in this debate ever claims the Hyperion example implies that "the chaotic motion of Hyperion tells us that we need the measurement collapse to actually be a physical process." The decoherence arguments about Hyperion do not assume this (as far as I know).

Can someone please explain? Am I misunderstanding?

Google the author and check their CV for the year they earned a PhD.

I think there are a lot of relevant dis-analogies between math and chess. A chess match at the professional level is a grueling, multi-hour contest, while math research is a lot more chill. Also, the time limit matters a lot in chess.

Anyway, you can look at the pages of Annals of Math and Inventiones and see that there is a good mix of ages.

Care to outline a precise argument for me to respond to? I'm honestly not sure one can even be extracted from pages 3 through 5, and I don't want to attack a strawman.

One basic problem is that using symmetry arguments to justify the assignment of subjective probabilities (cf. page 4, right column) is going to fail when there isn't a symmetry available, for example for irrational probabilities.

A deeper issue is the implicit assumption that one can coarse-grain into discrete "effectively distinguishable universes" for the purpose of doing these counting arguments. There are significant conceptual difficulties doing this, and this claim is much stronger than the bare MWI. Wallace discusses this at length; see for instance the citations in the blog post you responded to.