spekcular

I understand this desire but I think people who believe this should develop a new word or description for what you're attempting to do. See my comment here: https://news.ycombinator.com/item?id=30817964

I want to emphasize I'm not policing semantics just for the sake of being ornery. My reaction is approximately the same as it would be to someone who sends me a graph theory paper and calls it number theory. These distinctions are useful.

You write: "I don't think anybody is claiming that HoTT ought to replace ZFC for reverse mathematics in particular; it's "just" a capturing of some pervasive intuitions for concepts at the ground floor of the formalism itself. It takes a perspective that makes grappling with some concepts easier to approach. If you're already a deft hand with a different (very well-established) framework, maybe it's worth it to keep using the entirely viable tools you have!"

My claim is that "capturing [...] some pervasive intuitions" is not what ZFC is trying to do, or what people in foundations are interested in. Lurie puts the point well when he says a foundation "mean[s] a small number of concepts and assumptions from which the rest of mathematics can be derived." The purposes here are first, to figure out what exactly one must assume to get the rest of mainstream mathematics, and second, to provide a convenient base for doing metamathematics (since the fewer elementary operations/objects you have to reason with, the easier it is to write proofs). Capturing intuitions is completely orthogonal to this endeavor; no one is claiming that set-theoretic translations of everyday mathematical arguments capture the intuitive content of those arguments, because no one one intended ZFC to ever do that in the first place.

Similarly, no one is claiming that handing someone a Turing machine that performs quicksort is a great way to capture the intuition behind quicksort, or explain the algorithm in an understandable way. Turing machines exist as a special-purpose construct to probe the limits of computation, not to reproduce or communicate high-level algoritmhic reasoning. ZFC is like a Turing machine for (meta)math.

I'm sorry, but this is not gatekeeping; that term has an established meaning and mathematical history, and corresponds to a productive vein of research going back to Gödel (at least). If you'd like to argue that the study of foundations should be expanded to include the things that HoTT people are interested in, that's one thing. But can we at least agree it does not have a ton of relevance to the questions people have traditionally considered in foundational work? (This is almost by definition if we set it up so that it is mutually interpretable with ZFC.)

First, I'm not a set theorist. Second, one of the major motivations for introducing type theory in mathematics was to assist with computer-aided formalization efforts, which has had some success and has a promising future (e.g. the stuff the Lean community has been doing lately). This is great! I'm not complaining at all.

But I do think trumpeting HoTT as a new, improved foundation of mathematics evinces a misunderstanding of what people who work on foundations actually want out of a foundation of mathematics; or even more broadly, the mistaken belief that finding a new foundation of mathematics is an interesting project in the 21st century.

Lurie's comment here explains a bit more: https://mathematicswithoutapologies.wordpress.com/2015/05/13...

(And clearly, Lurie is not some ignorant traditionalist. There's a substantive disagreement that can't be reduced to people not wanting to learn "your stuff.")

In what sense is type theory "easier to work with"? For what purpose? ZFC exists largely because it's a useful framework for proving meta-mathematical theorems. In this sense, it is extremely convenient to work with.

Also, I'm not sure what "all of this stuff" refers to here re: your comment about inaccessible cardinals. You don't need large cardinals to do 99.99% of modern mathematics (essentially, everything but set/proof theory). ZFC is good enough (and you can usually get away with something weaker).

This is not "foundational mathematics." This is about the foundations of algebraic topology. The experts in the foundations of mathematics as a whole (set theorists, logicians, model theorists, etc.) don't care about homotopy type theory. It's sort of pointless for the study of the foundations of mathematics. That is, they really just care about ZF or ZFC, because those are already formulated so that they are maximally useful for foundational investigations. (And the e.g. reverse mathematics people who want to work outside of ZF/ZFC already have more direct ways of doing so.)

Some idea of what people in foundations think about day-to-day can be gleaned by searching the Foundations of Math mailing list archive: https://cs.nyu.edu/pipermail/fom/

In particular: don't confuse higher category theory (of interest to the algebraic topology community at large) with homotopy type theory (which is not). Jacob Lurie has a rather pointed conversation with Urs Schreiber about this, linked to here: https://mathematicswithoutapologies.wordpress.com/2015/05/18....

edit: I'd love to hear from the downvoters why they disagree with this take. It's pretty standard, afaik.

A multivariate Gaussian, for example.

Where does the the idea that PCA is optimal come from?

I understand the second example (girlfriend taking a phone screen): some people never took statistics past a linear models course, so they use what they know. Understandable, if inefficient. But they don't seem to be claiming linear models are optimal all the time, they're just stuck in a local maximum of trying to speed up the system they have.

But the PCA guy I really don't understand.

I'm sorry, but your description of how decoherence purportedly solves parts of the measurement problem is incorrect.

Even decoherence researchers agree that docoherence theory does not do this. You can find references and details in the Adler paper I linked, or in Schlosshauer's "Decoherence, the measurement problem, and interpretations of quantum mechanics." (Schlosshauer is the author of a main reference on docoherence: http://faculty.up.edu/schlosshauer/index.php?page=books.)

So, the reason that Griffiths avoids giving the explanation of measurement you prefer is that it is wrong. It's a virtue of the book, not a fault. He does discuss decoherence on page 462 of the third edition, though.

A) I don't consider the Feynman lectures a "standard textbook." I don't think there exists any university that uses them as the primary reference in their quantum course. They're fine, as far as they go, but I think modern pedagogy is better.

Concerning Griffiths, what do you feel it lacks? You've got the hydrogen atom, fermions, bosons, helium, and probably more stuff that I'm forgetting right now. What else would you stick in an intro course? Hartree-Fock?

B) Decoherence doesn't solve the measurement problem. Even the decoherence boosters admit this. See, for example, Adler's paper on this: https://arxiv.org/abs/quant-ph/0112095.

This isn't to say the decoherence program isn't important. I think it is. It just hasn't solved the measurement problem.

Yes. Whatever MIT uses for their OpenCourseWare is going to be fine, for example.

Can you point to a standard textbook that does this? The ones I'm familiar with definitely don't shortchange multi-particle problems.

And is the measurement problem not a mystery? If there's convincing explanation, that's news to me.

"Several"? Google Scholar shows two, one of which is an approximately 1-page comment on a rebuttal to the other one. Neither mentions "strand theory," the focus of the criticism in that link.

The author is a well-known crank. See post #10 here, for instance: https://www.physicsforums.com/threads/strand-model-published...

Proceed with caution.

Come on. We have plenty of good heuristics for why things are the way they are. Quantum theory is no more a bunch of "guessworks and heuristic" than Newtonian mechanics.

Also, physicists disagreeing on interpretational issues related to quantum foundations is not the same thing as disagreeing about the fundamentals of the theory.

Yes. It only tells you about variability within the sampled values. If you don't sample outliers, or get unlucky and same many non-representative vvalues, it can't tell you what you're missing out on (obviously).

Statistics is an applied science, and Solomonoff induction has had zero practical impact. So I feel it's not underrated at all, and perhaps overrated among a certain crowd.

Oh, if you check my comment history, you'll see me complaining a bit about Julia. I'm not saying it's perfect. But numba is so clunky and fickle that I think, just in terms of comparing the two, Julia is pretty clearly superior if you don't have a strong reason to use Python.

I've played with numba a bit, and my feeling is that if you're going to accept the disadvantages of just in time compilation, why not just write in Julia (where the experience is a lot more seamless)?

Julia was also faster in my experiments.

There are some old tutorials that show how to do concrete calculations with these objects with Macaulay. They were an immense help to me when learning algebraic geometry, because they showed me how all this sheaf blah blah blah connected to actual concrete polynomials.

Try these two: https://faculty.math.illinois.edu/Macaulay2/Book/Computation... https://faculty.math.illinois.edu/Macaulay2/Book/Computation...

There's more in the book "Computations in algebraic geometry with Macaulay 2," which is free online. See also Schenck's "Computational Algebraic Geometry."

edit: Also, topos theory is definitely not the way to go here. Getting your hands dirty with algebraic curves is. See e.g. Griffiths' Introduction to Algebraic Curves.

These are fairly disparate things that each require a separate degree to do well. I don't think any single degree would cover all of them, and any program is going to cover other things you're not interested in.

My recommendation would to pick the one you're most excited about and focus on it. You mention in another comment that you prefer to work with other people and not do books/MOOCs alone. In this case I suggest that you find a graduate student in the relevant area and ask him/her to tutor you (i.e. give you an independent course in the topic). This might cost $50 to $100 USD per hour if you shop around (perhaps more). The goal would be to learn enough of the basics that you can pick up the more advanced things yourself.

Good luck.

Can you give more details on why you want to learning about mathematical modeling? And what do you want to model? Given your educational background, there are probably more time- and cost-efficient ways to learn the same material (which is likely highly application-specific).

Also, applied mathematics programs teach proofs because they want to show that their applied methods are guaranteed to work, as opposed to being fragile, ad hoc guesses.

I was thinking of this paper in particular: https://www.jmlr.org/papers/volume15/miller14a/miller14a.pdf

Some other references:

https://www.jstor.org/stable/2241268

https://arxiv.org/abs/1308.6306

https://projecteuclid.org/journals/annals-of-mathematical-st...

Sure, I agree bad priors will give inaccurate inferences. My point is simply that to make a statement like, "an inaccurate prior generates many inaccurate inferences, and therefore it is garbage," one has to adopt a frequentist criterion for the quality of an estimator (like "gets good results most of the time").

Regarding "and third because there are good reasons why evaluations of outcomes and states of affairs ought to be based on lexicographic value comparisons, not just on a simple expected utility principle": do you have any suggested references that describe this in more detail?

Same question for "This is wrong on many levels, most notably by misunderstanding how theory discovery can and should work."

Also, do you have any suggestions for statistics books that you do like? Especially those with an applied bent (i.e. actually working with data, not philosophical discussions).

I mean "with simulations using a probability distribution [for the true parameter] different from the prior used in the Bayesian analysis." (The issue of model error is a separate question.)

Yes, in this case would should conclude there is something wrong with the Bayesian way. If you hand me a statistical method to e.g. estimate some parameter that frequently returns answers that are far from the truth, that is a problem. One cannot assume the prior exactly describes reality (or there would be no point in doing inference, because the prior already gives you the truth).

You asked, "Why don't you consider Cox's theorem - and related arguments - well-developed?" I consider Cox's argument not well-developed because D–S theory shows the postulates miss useful and important alternatives. So it fails as an argument for a particular interpretation of probability.

The issue isn't that Bayesian methods used incorrectly can have bad frequentist properties. It's that, according to many flavors of Bayesianism, having bad frequentist properties isn't a valid line of critique.

You may not believe in the particular stances I'm calling out, but if so, we don't disagree.

Dempster–Schafer theory is the obvious counterexample to "degrees of belief have to be represented by probability measures."

https://en.wikipedia.org/wiki/Dempster%E2%80%93Shafer_theory

Looks fun! Thanks for sharing. It seems like it covers complementary topics in a very concrete and clear way.