Thought about math problem for 3-4 hours with no progress, ate food, immediately thought of trivial two-line solution to math problem
Hmmmmm maybe there is a lesson I can learn from this
I’m trying to prove that a series can’t have any of what my mind has started calling “bad coefficients” but I’m too tired to think very clearly about it so mostly I’m just repeating “bad coefficients … bad coefficients … bad coefficients … ” inside my head
I’ve recently had cause to revisit this paper and it’s still a gem
If a paper called “Why Philosophers Should Care About Computational Complexity” sounds like something you might be interested in reading, and you haven’t read it, I highly recommend it. (It’s by Scott Aaronson, which may be an enticement of its own)
[F]or many engineering and physics applications, a single term of an asymptotic series is sufficient. When more than one is needed, this usually means that the small parameter ε is not really small.
(John P. Boyd)
I’ve said stuff like this before but I think comments like Boyd’s are kind of profound, and a lot of how I think about things is influenced by this idea. Another way of putting it is: “if you’re making an approximation that isn’t a linearization, you should be worried.”
Beware of “approximating” anything you don’t know you’re already close to.
I would love to have a clear sense of exactly what it is E. T. Jaynes argues and where I disagree with it but that would take some real effort and I have other more things I’m more interested in doing with that effort, at least right now
From an “external view” it looks to me like Jaynes can’t possibly have resolved these issues in the clear-cut way people claim, because people who are clearly good at this kind of stuff don’t write as though he did. There would almost have to be this conspiracy of ignorance going on where a bunch of philosophy-of-math people just decided to ignore a completely perfect argument and spend all their time talking about Dutch Books and “Teller’s P” and all this other stuff. I don’t know where the flaw in Jaynes is, but if there is not one then something … unlikely is going on here
And that doesn’t make me want to put in the effort, but I’m pretty sure what I’d get out is “Jaynes doesn’t resolve the interpretation of probability once and for all” and that’s what all the experts on this stuff already seem to think, so for now I can just believe them because they’re experts
what is bayesianism? we (i) just don't know →
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I apologize for writing such a hasty response, but I think we’re largely talking past one another. I really don’t want to produce some sort of exhaustive and careful “point-by-point rebuttal” because that could get us into the whole thing people do on the internet where they go back and forth endlessly, providing some technical response to every single one of the other’s points while not convincing each other of anything, and that wouldn’t be appropriate here, where I think a lot of the distinction is just in terms of framing, language and perspective. So, here are just some quick sketches of points:
- You are reading my post from a perspective which I didn’t have in mind when writing it. The perspective I had in mind was “we are sure that there is such a thing as ‘probability’ in the sense that there is a well-defined mathematical object, and we can all agree about what this object is; the question we can’t agree on, and which we’re trying to decide, is how to interpret this object.” From what you’ve said it sounds like you think of probabilities (in the mathematical sense) as being purely justified by the subjectivist interpretation of probability – as if the subjectivist interpretation comes first, and “probability” is the math we get when we formalize our subjectivist intuitions, and without subjectivism we’d have no need for “probability.” This is not the framing you usually see in the academic literature on this topic; the academic literature treats the mathematical definition of object like P(B|A) as the most fundamental thing, because everyone can agree on it, and goes on to ask what a number like P(B|A) “means” as though this is an additional question to be decided after we’ve decided which mathematical rules are obeyed by P(B|A). When you object to me talking about P(B|A) without assuming a subjectivist interpretation, you’re making an objection that would also apply to almost any of the philosophical literature on the interpretation of probability. Go out and read academic writing on this subject if you don’t believe me: it is written from the perspective of my post, not from the perspective of yours. (Note that there are plenty of mathematicians and scientists who use probabilities in their daily lives without agreeing with subjectivist or Bayesian interpretations, so the two don’t always go together in practice, even if they should.)
- I didn’t make this clear in my post, but I’m not expecting Bayesianism to provide a description of reasoning rather than a norm; I’m just very, very skeptical of very alien norms that we can only approach by severe approximation, because I think in these cases it becomes too easy to focus on how great the norm is and ignore whichever errors are introduced by the approximation. I talk about this in this post, which is a follow-up to this one.
- TBH I haven’t gone through your description of Cox. I’ll do so tomorrow when I’m less tired. Maybe it will convince me. As I mention here I’m skeptical of claims from people who seem to have only read Jaynes that Jaynes’ presentation of what he calls “Cox’s theorem” resolves all of these issues. If it had, the rest of the literature on this issue would not look as it does; people who have clearly read Jaynes would not devote academic careers to wrangling with issues that Jaynes supposedly made crystal-clear to us all. (Look at the arguments discussed in this paper – these are supposed to be the standard arguments for diachronic Bayes. Where do your/Jaynes’ arguments fit in here?) This is an amateur interest for me, and I haven’t gone through Jaynes carefully and found exactly what I disagree with him about. But – provisionally, for now – there is something that makes me very wary about the fact that the only people pitching the “Jaynes resolved all of this stuff, just read his chapter on Cox” line are people who seem to have gotten their ideas about the philosophy of probability solely from Jaynes. I fear the man of one book.
- Note that (as I mention in the post linked in pt. 3) Jaynes himself acknowledges that many people (I am one of them) find diachronic Bayes intuitive in a sampling context but not in other contexts. Jaynes claims to demonstrate that it should be used in other contexts as well. But this distinction isn’t some bizarre pathology of mine in particular; it’s really a distinction so common that Jaynes assumes you will make it going in and sets out to disabuse you of it.
- The ideas here about how Bayes should work as a norm are interesting and I should think about them more when I’m less tired. I do want to note that while I have heard the ideas you’re talking about, I’ve also seen Bayesians quote numbers they appeared to really believe in. And also that, as I said before, when we’re approximating a norm we’ll never reach, we need to be careful about whether the direction we’re moving in is the right one, because reality is generally nonlinear unless you’re close to a thing. (The example I used in one of the posts linked above was “consuming elemental chlorine will not get you 'halfway’ to the effects of consuming nutritious NaCl.”)
I’m reading Edward Frenkel’s “Love And Math” right now and it’s a really wonderful attempt to really describe nontrivial math to non-mathematicians rather than just paint colorful metaphors about it
I’m just don’t think I can be anywhere near as good at it as Frenkel, and even then I don’t even know if Frenkel’s book is that easy to follow with no background
I hope I someday actually write more of that math methods book (I got about halfway through the first chapter and the awkwardness of the handwaving and need for nonstandard notation got too frustrating for the moment)
B/c I was really proud of the introduction I wrote for it and want there to be something that lives up to it, even if I’m not capable of writing it
[…] Here, let me get to the point. The crux of the matter is this: the kind of “math” that underlies basic theoretical physics, engineering, climate science, population biology, economics – and pretty much every other quantitative investigation the human race has taken into the world around us – is neither the flavorless gruel of high school math class nor the esoteric jewel of romantic quest-for-truth mathematics. It is a third thing. It is not actually very abstract or even, I think, very complicated. It lacks the vaulting, heavenly beauty of pure quest-for-truth mathematics, but it has its own brand of beauty: a grubby, fun, tangible sort of beauty. It is not aloof; it is a kind of earthy math you can almost feel in your hands, a fundamental vocabulary for human-scale concepts like motion and change, a logic that can dance.
This stuff is sometimes referred to as “applied mathematics” – as opposed to “pure,” romantic-quest-for-truth mathematics – although that term is really too broad for our purposes. More accurate, though less concise, is “the methods of mathematical physics” – as in the classic two-volume reference book Methods of Mathematical Physics, published in 1924 by mathematical luminaries Richard Courant and David Hilbert. For brevity, I’ll just shorten this unwieldy term to “The Methods” from here on. […]
Tropical geometry
There is an exciting relatively new area in mathematics called tropical geometry. If I understand it correctly, the idea here is to redefine the “sum” of two real numbers as their minimum, and the “product” as their usual sum (it is possible to use the maximum instead of the minimum as well). So:
- x ⊕ y = min(x, y)
- x ⊗ y = x + y
I agree this looks rather crazy, but that’s what makes the subject fascinating! For example, this beautiful drawings are graphs of polynomials of degree three, tropical cubic curves (in fact, they are elliptic curves).
Here is what a second-degree curve looks like:
Mathematicians have established tropical analogues of many classical theorems, such as Pappus’s theorem or Bézout’s theorem. I’d love to learn more about this topic! Oh, and it’s called “tropical” in honor of the Brazilian mathematician Imre Simon, who pioneered the field.
I met someone last year that was working on an undergraduate research project using tropical geometry and apparently it has connections to number theory and said person was actually getting a lot of attention for their project because it had the potential to give some insight on some of the big prime number conjectures that are out there right now
If the math people in the audience tonight have not read Everything And More and want to know just how bad it is, well, here’s an actual paragraph that appears in it:
It’s the Extreme Values Theorem [sic] (= if an f(x) is continuous in [a, b], then there must be at least one point in [a, b] where the f(x) has its absolute maximum value M and another point in [a, b] where f(x) has its absolute minimum value m), together with Weierstrass’s high-powered definition of continuity, that provides a mathematical way out of the no-next-instant crevasse. To wit: Since time is clearly a continuously flowing function, we can assume a finite interval [t_1, t_2] > 0 [sic] between any two instants t_1 and t_2, and now thanks to the E.V.T., prove that there is a point in [t_1, t_2] where the time-function has its absolute minimum value m, and therefore that this t_m will be, mathematically speaking, the very next instant after t_1.
(Since this is Wallace, this needs to be specified: the “sic"s are all mine.)
I’m not just surprised that this argument got published, I’m surprised that it even got thought up in the first place
just HOW high do you even have to BE just to DO something like that