ajkjk

It's funny how similar this is to a human reaction to the same thing. It's the right level of incredulity, to be sure, and the first thing you do is you start re-reading the conjecture to makes sure you didn't misunderstand the requirements.

well if he said it then it must be true

What do you expect from a person other than the generalizations they see from the experiences they have had...? What 'evidence' could possibly be given other than extensive anecdotes?

Happy for the individual, sad for society

Part of the problem is that the difficulty curve becomes, like, superexponential if you try to do the actual math. Fairly elementary atoms require the full theory of quantum mechanics to justify rigorously, and anything more complicated than that requires huge bodies of specialist knowledge on approximation schemes (I assume; I haven't studied them, but given that helium already requires approximations I'm assuming the trend continues..)

Of course, they could still do a much better job useful providing pointers into this knowledge, instead of just handwaving over it and insisting on rote memorization.

JS was popular despite its lack of types, not because of them. If a typed language had been bundled into browsers from the beginning then we would not be shimming types into it with TS today (although I'm sure there would be other difficulties).

When companies started making enterprise-scale JS applications they had to solve for the type story because it is unmaintainable otherwise. TS is the winner in a long line of iterating on that problem.

guess so. I still don't think their position is serious, but, that's just me.

was amusing to come back hours later and find out whether this comment ended up more upvoted or more downvoted, it was oscillating all day...

I think you have missed my point. Both are defined to integrate to one, over different measures.

It is the right question to ask. The idea that moral questions should have a market value is itself a moral failing, so assuming you want moral principles to rule over the design of your economy (which.. you'd better; otherwise slavery is permissible), you should not allow such things to be up for debate.

Although perhaps your disagreement is over whether this is a moral issue, in which case, fine, but let's be clear that that's what we're disagreeing over.

i could be wrong, but it was enforced as the default at several places I worked, and most editors now have the option of the tab key inserting spaces to bridge the gap. (I don't care about the actual debate; just, I thought I had noticed it had mostly gone in this direction)

completely agree. but I felt like even then it was clear that types were a good idea and the implementations were not. For instance I started programming on Java 4 or 5 and the types were pretty bad---but still it was obviously the right way to go compared to JS or, god forbid, shell.

I feel like I see all of these debates far less than I used to? Well I don't see anyone arguing about vim and emacs anymore at all, and spaces have mostly won over tabs, and static typing has mostly won over dynamic, with the holdouts being comparative novices and people who program in less modern environments, like in academia and at smaller companies.

well, as I said, I don't take them seriously :p

Some people would call it a Kronecker delta instead, but imo they are exactly the same concept. The Kronecker delta is the indicator for a single value, like 1_{x=0}, while the Dirac delta is the indicator for a single 'dx' partition, divided by the width of the partition: δ(x) = 1_{0 in (x, x+dx)}/|dx| which is why integrating it ∫ δ(x) dx = ∫ 1_{0 in (x, x+dx)}/|dx| = ±1 (depending on the orientation of the integral). The Kronecker can sorta be viewed as the same thing but with dx=1, although that is kinda silly because usually you would intentionally evaluate it on a discrete measure anyway.

I don't think ... serious people... argued that.

That's a bit hyperbolic so I'm sure I'm wrong, but I have an ace: if you point me at very smart people who argued against types I'm gonna say that they weren't serious. I think it's not possible, if you have the relevant experience of working on both typed and untyped codebases of at least moderate complexity with at least one collaborator, to come away seriously believing that the untyped way is superior (unless you were forced to use a really bad typed language, I guess). And arguing that untyped languages are better without that experience is also not serious, in the sense that anyone can unseriously say anything if they don't care about being well-informed enough to be right.

If you want to be detailed about it, https://www.reddit.com/r/chess/comments/1qjafcd/medical_exam... talks about the whole report; that sarcoidosis was the actual cause is speculative, and the "substance use as a contributing factor" part is widely believed to be in part due to the extreme stress he was under... but it is hard to say anything about causality definitively.

The implications are not significant..? the real world is messy enough that this will not ever apply.

that's fair, I was interpreting them differently.

This is also my stance. The fact that large numbers of people spend large amounts of publicly-funded time exploring what are essentially abstract puzzles is bizarre and not that different from, like, cloistered religious devotees who are supported in spending their time studying scripture and are considered to be the 'source' from which flows a certain kind of universal truth.

Not that it is wrong for them to be doing this---we do want a society where people get to devote their life to what interests them---but it is bizarre because of the framing. For some reason it is ambiently understood in our society that this work is of incontrovertible value, when in fact it is largely not. And the value-producing parts of the work, the parts that end up having applications to other fields, largely run contrary to the actual daily goals of the cloistered devotees: it is mostly the intuition and pedagogy and the compactification and refactoring of knowledge that have value at this point, not the production of esoteric theorems, yet that is expressly not rewarded in the incentive structures.

That latter point is more due to the sorry state of academic incentives in general than to a particular failing of mathematics, though. Were I somehow given the ability to restructure things by fiat I would immediately create journals which publish only useful articles that refactor knowledge, communicate intuition, better explain things, argue for structural improvements to notation and terminology, etc, and this would immediately create an incentive to do that kind of work for working researchers to do work which aligns with the actually-useful output of their fields. I suspect most fields could use something like this. New knowledge is just not that valuable if it is all dumped into a giant pile and unprocessed, and I have seen firsthand a bunch examples where entire subdisciplines are hamstrung in their actual application-heavy work because they don't have easy access to basic tools that are hidden behind hard-to-learn theory.

Of the three claims you just made, two are clearly false and the third is probably also...

You can prove something is created by AI by e.g. showing the transcripts, especially from the vendor side.

You cannot prove that something isn't created with AI, at least not if you require incontrovertible proof (outside of, like, working in some kind of verifiably AI-free clean room, or doing something that current models are provably unable to demonstrate). But you certainly might be able to prove it to the satisfaction of the legal system.

If AI generated content cannot be copyrighted, it does not follow at all that they can't infringe copyright; there is no deductive step there that I can think of.

people are very fond of it here -> there's no point arguing against it here?

Backwards logic. If they're fond of it then they're the people to be arguing against, no?

Meh. I'm fine with trying to rename things if it feels justified. And the more obscure the thing, the easier it is to rename in principle. Maybe the better name will help it be less obscure. But of course whether it works is proportional to how good of an argument there is for doing it, as it should be.

I don't think that argument is actually true? You can refactor for purely aesthetic reasons and it may well turn out to help. Maybe you don't even know, going in, what the goal is; just that the way it works now isn't good and maybe if you start looking you'll find a better approach. Happens all the time, I think. Of course you can also refactor for aesthetic reasons and not help, but that's a skill issue.

My clue-finding and pattern-matching and such is all based on philosophical aesthetics: something feels amiss when these patterns exist without being obvious; therefore they should be extracted and examined from various sides to see if a connection is found.

I happen to think that novel facts and theorems and proofs are way overrated. If you find a new fact it just goes into the giant pile of facts that are sitting around uselessly. The useful progress in math is comes from "refactoring" efforts to make things simpler and more intuitive.

I don't mean that this is necessarily the case, but that it is where we are now: we have found ourself in a situation where we have way too many facts and not enough simple perspectives that make them useful and accessible.

Just my opinion, though.

True, I guess you can just 'curry' exponentiation and say that's a baseless power. I couldn't find a clean notation for it so I gave up..

glad you liked it

I wonder if we should really just call them... vectors? Like the thing that torsors do, being defined only relative to a choice of origin in some space / group, is exactly what displacement vectors do. So really they are just generalizations of the concept of a vector. (In this scheme I would be careful to _not_ refer to points as vectors, so as to reserve the term for things that act like, well, torsors. I happen to think that much pedagogical harm has been done by not distinguishing the two concepts, points and displacements, early on.)

The two should be the same up to possibly some factors of k! depending on your definitions.

Just as an example, suppose you have two multivectors (abc) and (xyz) with all the vectors orthogonal (for simplicity). The geometric product

(abc)(xyz)

has its scalar part created by

(abc)(xyz) = (ab) (c.x) (yz) = (c.x) (ab) (yz) = (c.x) a (b.y) (z) = (c.x) (b.y) (a) (z) = (a.z) (b.y) (c.x)

You can see how the dot product (which uses the metric internally) is being applied "in-to-out" : the adjacent terms are dotted, at which point they become commuting scalars; then the next terms, etc. Which, frankly, is dumb. This is why the GA version of a scalar product has the "reverse" operation involved... because the GP is doing this in-to-out thing, the scalar product has to undo it by defining (abc) . (xyz) = (abc) (xyz)^~ = (abc) (zyx) = (a.x) (b.y) (c.z), with ^~ meaning reverse.

Whereas the standard exterior algebra inner product is always left-to-right, giving

(abc).(xyz) = (a.x) (b.y) (c.z)

IMO the GA version is a mess because it's conflating two concepts. When the GP works, it is composing operators, so AB = A ∘ B. But the inner product, at its core, is more like division---it wants to have (a).(a) = 1, since its job is to say say "how many copies of (a) are there in (a)?" To make this work for multivectors (ab).(ab), it needs to be left-to-right. GA does in-to-out to copy quaternions with their i^2 = -1, but that's not necessary -- i^2 = -1 follows from the fact that for a rotation, R ∘ R = -I, so it is composing two rotations, not measuring one in terms of the other. Really i^2 = -1 should not be interpreted as a dot product at all. This is very clear when a metric is involved: R_xy ∘ R_xy = -I is a degree-two tensor which transforms with two factors of the metric, whereas (xy).(xy) = 1 is a degree-zero tensor, a coordinate-invariant scalar. They are just different operations, which happen to overlap in simple cases.

Eh. If you can read Hestenes or the Cambridge papers without some red flags going off, then this is just a subjective thing you're not gonna understand and we'd best not try to litigate it :p

(more extreme examples can be found on the heaps of GA-based papers, youtube videos, and comment sections around the internet, but I think the original papers do contain some of the concerning stuff on their own)

I think of displacement vectors and position vectors as having different "types". Position vectors are geometric objects whose meaning comes from whatever physical system you're studying (this way of thinking is generally useful even if you're not doing physics; for instance if you're talking about a manifold you would think of that as the physical system independent of the coordinates you put on it).

In the same way I think of "vectors as operators" (rotations/scaling) as a displacement vector / torsor, but of a different type than their sense as translations. As far as I can tell, the geometric product between two displacement vectors is not so meaningful, whereas the geometric product between two "operator" vectors is (because it composes them as operators, in some sense). But in practice you're often rapidly switching between these representations so it's hard to tell which object you're actually talking about. For this reason I find it useful to distinguish their types explicitly.

Well, I think my points in the parent article here do stand (I'm aware that you do not). The fact that there is a good interpretation of the geometric product in some cases does not obviate the fact that everyone's writing crappy intuitive things about it. Anyway it has always been my stance that there is a _good_ version of GA, and people need to figure out what it is and write about that instead of bloviating about how good the current version is. Treating the GP as a composition of operators is a start, but it's not the whole picture --- why do operators compose in that particular way? why are these your operators in the first place? My hunch is that GA is really a subset of a larger and more intuitive algebra that has very obvious answers to these questions, probably from a representation-theory perspective.

(I meant to do this followup soon after the first article but I've been been having a lot of difficulty focusing / constant brain fog for the past few years so it's been on the to-do list instead; part of the problem is that to do it right I need to go read and digest everyone's different treatises / expositions on GA ... and that has felt taxing, to say the least.)