spekcular

Was ultraviolet catastrophe really an experimental inconsistency? It seems to best understood as a theoretical inconsistency: The theory predicts a diverging (infinite) value, which we know would be wrong regardless of what the experimental evidence is. It's a mathematical problem, not an empirical one. You also give the example of "the measured invariance of the speed of light in different rest frames," but Einstein claims SR was motivated by the invariance of Maxwell's equations (a theoretical consideration), not e.g. Michelson–Morley. So it sure seems that investigating theoretical inconsistencies has motivated a lot of good work.

I agree that, practically speaking, studying the black hole information paradox might not be so productive. Maybe there are special features of the problem that make it difficult to investigate productively through theoretical considerations alone. But this is not how I read Hossenfelder. Taking her blog post literally, she seems to be against theoretical investigations of any phenomenon (any inconsistency, etc.) where experimental tests aren't forthcoming. I think this is ridiculous.

Maybe she doesn't actually believe this, but then she needs to make an argument specifically about the black hole information paradox and why this particular problem is unproductive, not launch a broadside on non-empirical reasoning more generally.

The full paragraph is: "Why am I telling you this? I am not talking about this because I want to change the mind of my colleagues in physics. They have grown up thinking this is an important research question and I don’t think they’ll change their mind. But I want you to know that you can safely ignore headlines about black hole information loss. You’re not missing anything if you don’t those articles. Because no one can tell which solution is correct in the sense that it actually describes nature, and physicists will not agree on one anyway. Because if they did, they’d have to stop writing papers about it. "

It's hard for me to read this is as anything other than "I am not talking about this because I want to change the mind of my colleagues in physics [because they're too far gone]." That is, she believes the research is worthless and the problem shouldn't be investigated. She just doesn't think she can convince people of this.

Yes, but gravitational waves themselves were not testable. To get from the confirmation of the other predictions to belief in gravitational waves, for which you had no direct evidence (until recently), you need to apply exactly the non-empirical criteria (parsimony, consistency, etc.) that Hossenfelder (taken literally) seems to think is not sufficient for doing theoretical physics in the absence of direct evidence.

Sure. But this is a huge, incredibly novel prediction. It's not some trivial matter.

Some predictions of the model were tested soon after Einstein published the theory. Others, like gravitational waves, were not. Yet people still believed in gravitational waves long before they were empirically confirmed.

I don't see any possible basis for that belief (at the time) other than arguments based on non-empirical reasons like consistency and parsimony. (That is, it would be strange if other predictions of the model worked out but this one didn't.) Yet it is exactly this kind of theoretical reasoning that Hossenfelder, taken literally, seems to reject (in the absence of experimental data).

I'll also note that any other solution of the black hole information paradox has to be consistent with the the rest of what we know about physics, and any future empirical observations. So it can't be completely untethered from reality; it makes testable claims in this way. Further, it's not clear such solutions can never be directly tested, or will never be found to imply novel testable consequences.

General relativity predicts gravitational waves. For the theory to be correct, gravitational waves need to exist. You haven't fully confirmed the theory experimentally unless you confirm that consequence.

I disagree. The author is very pointedly saying that research on this problem is worthless and a waste of time. See the last paragraph, for example.

Sorry, I meant direct tests of gravitational waves. I think that claim is accurate.

Einstein had no positive evidence to suggest direct tests of gravitational waves would ever be possible. His arguments were entirely about consistency and theoretical parsimony, as are the arguments about the black hole information paradox. If we take the view that we shouldn't work on physics we can't test, or can't test for a very long time, then we are led to the conclusion that Einstein shouldn't have worked on gravitation waves. I find that conclusion untenable.

This seems like a fully general (and somewhat unconvincing) argument against doing theoretical physics. The key paragraphs are:

"What’s going to happen with this new solution? Most likely, someone’s going to find a problem with it, and everyone will continue working on their own solution. Indeed, there’s a good chance that by the time this video appears this has already happened. For me, the real paradox is why they keep doing it. I guess they do it because they have been told so often this is a big problem that they believe if they solve it they’ll be considered geniuses. But of course their colleagues will never agree that they solved the problem to begin with. So by all chances, half a year from now you’ll see another headline claiming that the problem has been solved.

And that’s why I stopped working on the black hole information loss paradox. Not because it’s unsolvable. But because you can’t solve this problem with mathematics alone, and experiments are not possible, not now and probably not in the next 10000 years."

First, let's grant that no experimental evidence will be forthcoming in thousands of years. (It's conceivable to me that some astronomers will get lucky and provide some indirect evidence of some sort, but ignore this for now.)

Why do we believe that this problem can't be solved, or at least profitably investigated, with mathematics (and physical intuition, and the rest of the experimental evidence we have about black holes – it's definitely not "mathematics alone")? At least in principle, one can imagine that there is a finite set of possible solutions (corresponding to dropping various assumptions, as she mentions earlier in the article), and all but one of those can be ruled out a priori via mathematical inconsistencies, a contradiction with physical evidence from non-black hole phenomena, or other undesirable properties.

Maybe there are special features of the black hole information problem that make this impossible. But this overall mode of mathematical investigation is how theoretical physics works and has always worked. Einstein discovered general relativity by tweaking assumptions and deducing the theory was likely to be true because it resolved various issues, but we had no direct test [edit: of gravitational waves] for about 100 years. It would have been unfortunate if he concluded the problem was pointless to work on because no experimental evidence would manifest within his lifetime.

(Example problem fixed by Einstein: https://aether.lbl.gov/www/classes/p10/gr/Precessionperiheli...)

What's wrong with a survey? Sometimes this is exactly what people want. Witness the popularity of Numerical Recipes, for example.

That blog post is cited in the paper the article discusses.

Sure, but I'm not talking about high energy physics in general, I'm talking about the estimates in this specific paper. Which the guy who is doing bleeding edge stuff seems a little suspicious of (see tweet).

See also: https://twitter.com/pietrovischia/status/1512174848558219270

This does not seems like bleeding edge, it seems like "Gaussian approximations for everything."

In fact, this is the reference for the technique they used: https://cds.cern.ch/record/183996/files/OUNP-88-05.pdf. (Although the criticism seems to be leveled at the way they estimate the correlations, not that linear estimator specifically?)

I admit I'm not familiar with the model used to aggregate the boson data. But there's an entire community of nonparametric/semiparametric statisticians that works on problems just like this. It seems crazy to me that that millions of dollars are spent to build the machines to collect this data, yet the papers are written using statistical models with distributional/independence assumptions known to be false. (The tweet linked above seems to be saying something similar.)

Is there a concrete reason we can't be naive and just bootstrap confidence intervals for example? Of course I defer to the physicists here – but I'm curious whether there's some simple high-level reason the usual tricks don't work.

Sort of. You don't need identical replications of the same experiment, just long run probabilities for any application of the method. See example two here: https://normaldeviate.wordpress.com/2012/11/17/what-is-bayes...

(The author is a stats professor at CMU.)

Quoting: "The plot shows the first 50 simulations. In the first simulation I picked some distribution {F_1}. Let {\theta_1} be the median of {F_1}. I generated {n=100} observations from {F_1} and then constructed the interval. The confidence interval is the first vertical line. The true value is the dot. For the second simulation, I chose a different distribution {F_2}. Then I generated the data and constructed the interval. I did this many times, each time using a different distribution with a different true median. The blue interval shows the one time that the confidence interval did not trap the median. I did this 10,000 times (only 50 are shown). The interval covered the true value 94.33 % of the time. I wanted to show this plot because, when some texts show confidence interval simulations like this they use the same distribution for each trial. This is unnecessary and it gives the false impression that you need to repeat the same experiment in order to discuss coverage."

It sounds like they know certain assumptions regarding parameters that are not of interest are wrong. So why explicitly model those, if we don't care about their distribution? We (apparently) only care about an accurate estimate of W boson mass.

I agree 102% is not a possible value for the efficiency of the detector. But if the confidence interval traps the true value of the efficiency 95% of the time upon repeated sampling, what's the problem? That's all that's required for a confidence interval to be valid. Some CI constructions do in general give intervals that include impossible parameter values, but if they contain the true value 95% of the time, there's no issue. The coverage guarantee is all that matters.

(One should not confuse a CI with a range of plausible values, in other words.)

What's wrong with saying "Our detector is 99%+-3% efficient," if they are giving the output of some procedure that constructs valid confidence intervals? The confidence intervals will trap the true value 95% of time (or whatever the confidence level is). If it does what it promises to do, I don't see the problem.

I agree with adding the "more extreme" part, but I'm not so sure about the infinite number of times part. Certainly, the p-value is (roughly speaking) the probability of seeing a result at least as extreme as the observed result, under the null hypothesis. But one doesn't really need to introduce hypothetical infinite sequences of replications to make sense of that definition.

Why don't people use nonparametric methods to get around the problem of assuming certain parameters are "nicely" distributed? (Not a physicist, but curious – this seems like the "obvious" solution.)

Issues with the license aside, there are substantial problems with the core algorithmic material of the book. The provided codes and approaches are often just plain incorrect.

Quoting from a review: "The authors of Numerical Recipes were not specialists in numerical analysis or mathematical software prior to publication of this book and its software, and this deficiency shows WHENEVER WE TAKE A CLOSE LOOK AT A TOPIC in the book [editor's emphasis]. The authors have attempted to cover a very extensive range of topics. They have basically found `some' way to approach each topic rather than finding one of the best contemporary ways. In some cases they were apparently not aware of standard theory and algorithms, and consequently devised approaches of their own. The MEDFIT code of section 14.6 is a particularly unfortunate example of this latter situation.

"One should independently check the validity of any information or codes obtained from `Numerical Recipes'...."

Source: https://www.uwyo.edu/buerkle/misc/wnotnr.html

1) Yeah, Lurie's books are advanced tracts aimed at specialists in algebraic topology. Of course there are going to be substantial prerequisites. This is sort of inevitable if you want to understand the motivating applications.

2) Why would we ever want to teach higher categorical notions of space to undergraduates? We can barely get them out in four years knowing real analysis and abstract algebra. The next stop would be the basic theory of manifolds, not higher topos theory.

3) I looked at Kock's book (the draft of the CUP book on his website). The definitions are more abstract than the ones I would give to undergrads, and consequently harder for them to understand. The appeal is not clear to me.

4) You don't have to change your foundations (meaning ZFC vs. type theory, etc.) to do higher category theory in a reasonable way. Again, go pick up Lurie's books. The exposition would not really be affected by whether you choose type theory or ZFC for a foundation. Let me know if you think there's a counterexample to this claim. (The exposition probably would be affected if you denied the law of excluded middle, for the worse; luckily Lurie doesn't do that.)

5) Foundations are sort of irrelevant here anyway. The point of ZFC is that you can forget about it when actually doing math. If you're thinking about foundations when reading Lurie's books, you're doing something wrong. (Consider: most mathematicians have zero opinion about foundational questions or concepts because they have no bearing on their day to day work.)

What's wrong with computer modern?

I'm not sure how that comment is responsive to my post. Sure, type theorists talk about universes, and sure, there are relative consistency results linking type and set theories. My claim is just that large cardinals (or some equivalent concept) are essentially necessary to order theories in consistency strength, so is hard to take a response of the form "it doesn't matter whether HoTT can talk about large cardinals" seriously.

I have no objections to claiming success in formalizing mathematics! As I've said, Lean is a big win for type theory.

But formalizing proofs with computers is essentially separate from the rigorous study of the foundations of math (consistency, relative strength of theories, etc.).

You've complained above about set theorists not learning "your stuff," and now it's my turn to complain. You write: 'Risk Assessment: depends how many "large cardinal" type things we want (unlike the 1970s category theorists I personally don't really care).' Large cardinal axioms are essential to the study of the relative consistency strength of mathematical theories. You personally might not care about them, but accommodating them is not a negotiable detail for anything that purports to be useful in the study of foundations.

And yeah, I'm aware the HoTT people have ways around this, fine. My comment is more about the perspective conveyed by your remarks: if you want to have a seat at the table in a discussion about foundational issues, you need to understand basic facts like this. My suspicion is that you might have had bad interactions with set theorists because you don't share certain knowledge they take for granted, and then they get frustrated and end the conversation early. I truly believe that if you went and read a book on set theory and logic, and learned, e.g., wny large cardinals are important, you might understand better where I and other people are coming from.

Here, from three years ago: http://www.socsci.uci.edu/~pjmaddy/bio/What%20do%20we%20want...

I don't have much to say here that I didn't already say above about the meta-mathematical issues. However, about object-level mathematical issues, you write: "It is a work in progress and might take some time before bearing substantive fruit, but note that the current alternative is that any student learning higher level mathematics at the level of Lurie's books has to go through a huge amount of foundational material while the ideas themselves may not be intrinsically complicated. Instead, this complexity is an artifact of embedding the theory in current foundations like the SDG example mentioned above."

Higher category theory is not the same thing as homotopy type theory. One does not need to learn anything about the foundations of mathematics to read and understand Lurie's work. As far as I know, issues of ZFC vs type theory simply do not arise. You can just pick up Higher Topos Theory and start reading, assuming you have a strong background in algebraic topology.

The linked article is only about higher category theory and making sure all of its claims are written down in a clear and rigorous fashion to be maximally useful to workers in the field. It has nothing to do with the foundations of mathematics as a whole.

That link talks almost entirely about implications for programming and not mathematics. I recommend the Penelope Maddy article linked in a comment above for a more in-depth discussion of the relevant history and aims of the field. She says things much more clearly than I can.

I absolutely agree that it's a wider, more sophisticated topic today. This doesn't mean that its main concerns have fundamentally changed. The Maddy article linked in a comment above is extremely clear explanation of this.

Thanks. I enjoyed the article you linked. I think the description on page 16 is a good summary of what most people working in foundations take the purpose of foundations to be.