Who or what closes questions here on stackexchange? None of my questions are "homework" questions, I wish whoever closes them would actually "read them more closely". is it some dumb AI algorithm or what?
On the Home page of Physics Meta, the question “Why can I still no longer questions?” is attributed to user_250478, but on the Questions page it is attributed to a user who uses his name. Why do these pages differ like this when displaying the same question? Is it a bug or a feature?
@ACuriousMind I'm taking a second look at this. When you write $\int dx' \partial_x(\delta(x-x')f(x,x')$ is $f_1(x,x) + \partial_x f(x,x)$, did you mean $\int dx' \partial_x(\delta(x-x'))f(x,x')$ or $\int dx'\partial_x(\delta(x-x')f(x,x'))$? There is a missing parenthesis.
@ACuriousMind I have circumvented what I think is my suspicious step. In doing this, I am left with $\int dx' (\partial_x\delta(x-x'))f(x,x')$ as the only tricky term. You seem to suggest switching the integration and differentiation however, I don't understand that step because the differentiation doesn't act on the whole integrand.
@B.Brekke I think my core problem here is with the idea that you can differentiate a $\delta$ with respect to a different argument than you integrate it. Formally, $\delta$ is not a function, and the $x$ and $x'$ are actually not on the same footing, and the rigorous way to consider this object is either as the distribution that assigns the value $f(x')$ to functions of $x$ or the value $f(x)$ to functions of $x'$.
mathematicians usually write $\delta_{x_0}$ for the $\delta$ that assign the value $f(x_0)$ to functions
differentiation of the $\delta$ is defined via partial integration, i.e. we simply say that $\delta_{x_0}'[f] = -f'(x_0)$
it's not really clear to me what exactly applying the $\delta_{x'}$ to a function $f(x,x')$ means in mathematically rigorous terms
since you have an integral over $x'$, it seems we should interpret this as $(\partial_x \delta_x)[f_x]$, where $f_x(x' ) = f(x,x')$, so the question is, what is the derivative of the distribution-valued function $x\mapsto \delta_x$
the annoying thing here is that $f_x$ itself depends on $x$ - because normally, the derivative of such functions works in the weak sense, i.e. $(\partial_x\delta_x)[g] = \partial_x(\delta_x[g]) = \partial_x g(x)$
@RyderRude I usually keep a tab of this chat open with the intention of following along and keeping my physics knowledge at least somewhat active, but life often gets in the way!
"It is not true, however, as is so often stated, that this had to happen because mathematics uses the simplest possible concepts and these were bound to occur in any formalism. As we saw before, the concepts of mathematics are not chosen for their conceptual simplicity - even sequences of pairs of numbers are far from being the simplest concepts - but for their amenability to clever manipulations and to striking, brilliant arguments."
@Slereah this is pretty much how i think about the unreasonable effectiveness :P
@ACuriousMind i shall read this full.
I personally think that we hav extracted math from the universe. We came up with numbers becuz they wer there in our experience
@ACuriousMind i believe the axioms of finite set theory and first order logic were discovered from our experience with finite sets that were there in the universe
@Slereah I think that's not quite the point being made here: The miracle is not that we can count things. The miracle is that the things we count in physics follow such simple and universal mathematical laws like Newton's laws, or the Schrödinger equation. There is no reason at all to expect such regularity from the world, to believe that reductionism could work at all, and yet it is what we find.
The miracle is that induction is not only possible, but when done on the right level of mathematical formalism, exceedingly powerful
That's the unreasonable effectiveness: We can write down theorems about complex numbers (Sokhotski–Plemelj theorem), wholly disconnected from anything physics, and they turn out to describe dispersion relations everywhere from optics to quantum field theory (Kramers-Kronig relations)
But i still think that we were able to come up with mathematics only becuz the universe had mathematical regularity.
Like, counting cud b used to count sheep as well as cats. Logic cud b used on sets of sheep as well as cats
But i dont see any immediate connection of this with reductionism
still, we gotta admit that using the same logic to reason about different real world collections is a pre-historic form of physics
Let's imagine a chaotic world with no pattern in its time evolution, but this world still constitutes of collections at any moment of time. Wud we b able to discover basical arithmetic and logic in this universe?
Like, since the time evolution has no pattern, applying logic on these collections is useless. None of ur implications wud hold
Wigner is saying he believes biology is not reducible to physics
@ACuriousMind @Slereah Perhaps we could say that we saw regularity in the behavior of our everyday world, i.e. logic was applicable to everyday collections. This is how mathematics was born out of the regulariy in everyday world
But then we discovered that the regularity in our everyday world was a consequence of the regularity in the microscopic world. This is reductionism
And then we discovered that the regularity in the microscopic world was describable using the same language that we had created to explain the regularity in our everyday world, i.e. mathematics
Ultimately, all math does reduce to collections and logic. We have not used some alien math to formulate fundamental physics. It is the same type of math that we had created out of our everyday experience.
@ACuriousMind The link you offered about Radon-Nykodim to prove that all measures are expressed the same way on a manifold require some original measure I think
So I guess you need to prove first that such a measure exists in the first place
Trying to figure out what's the most obvious measure you could come up with without invoking the metric tensor
this is not a fully formed idea, but I feel the real "reason" n-forms define measures is just because they are "cohomological objects", i.e. duals to the cycles (~submanifolds) in homology
the "measure zero" sets are precisely the ones that aren't "real" n-cycles, but singular ones
@Slereah okay, maybe not homology, but something far more pedestrian: The measure zero sets need to be the ones that are Lebesgue measure zero in coordinate patches. Having Lebesgue measure zero is invariant under coordinate change and diffeomorphisms, so does that not give a definition of the "neglegible sets" on the manifold?
the MO answer I linked doesn't need a full measure, it just needs a definition of the sets of measure zero
note that that, however, essentially then makes the definition almost immediately equivalent to defining the density in terms of the local coordinate patches directly
Saying "A density is a measure on $\mathbb{R}^n$ that can be written as $f\mathrm{d}\lambda$ for $\mathrm{d}\lambda$ the Lebesgue measure" is equivalent by Radon-Nikodym to saying that a density is a measure that is zero on the sets of zero Lebesgue measure.
Transferring the definition of "set of zero Lebesgue measure" to the manifold via coordinate patches is the obvious generalization of that statement
not sure there is any advantage to generalizing the "zero Lebesgue measure" definition instead of the standard one by forming hte bundle
by applying Radon-Nikodym locally you can almost immediately reconstruct the density bundle from the measure definition anyway
in what you posted, it looks as if from the second to third line you wanted to use $$\frac{\mathrm{e}^t}{\mathrm{e}^{st}} = \frac{1}{\mathrm{e}^s}$$, which just isn't true
oh that's right, you don't apply the limit until after you take the antiderivative.
Okay I see
thank you
um wait so $$e^7 \int_0^{\infty} e^{t-st} dt = \frac{e^7}{1-s}\int_0^{\infty}e^u du \\ \frac{e^7}{1-s}[e^{t-st}|_0^{\infty}]$$ how do I evaluate $$e^{\infty - s\infty}$$?
I set $u = t(1-s)$ btw
I guess if you consider $s\infty = \infty$ it's just $1$
but that's not correct, then the whole thing becomes 0, in the answer key it's supposed to be the expression e^7/1-s so it somehow becomes 0..
I don't even know what you'd do for the inbetween integral of $$\int_0^{\infty}e^{-st}\sin(t)dt$$ like you can do $$\mathcal{L}\{\sin(t)\} = \int_0^{\infty}e^{-st}\sin(t)dt$$ and rearrange to get a form but if this whole thing is $\mathcal{L}\{t\cos(t)\} = $ then you'd have to use the definition of $\mathcal{L}\{\sin(t)\}$ which technically isn't really allowed in this section
Idk what style better suits humans, like a very intelligent self improving AI that's in a "black box" and we securely feed it questions so as not to let it free, or a dumber general intelligent AI that acts as a kind of assistant like cortana in halo or something
@SirCumference I still see this as a major positive, since we won't be dependent on this savage lifestyle on earth, it seems pretty brutal to be anything but the 1% of humans that aren't suffering every day
let alone the animals, not to say AI should cull every life form, but we could gradually lower the population & demands
or at least look to other forms of sustaining our population instead of war and mass farming and stuff
So we have the equation for local charge conservation $\nabla \cdot \vec{J} = -\frac{\partial \rho}{\partial t}$
Then Griffiths says the condition for magnetostatics is $\frac{\partial \vec{J}}{\partial t} = 0$
This then implies by the continuity equation that $\frac{\partial \rho}{\partial t} = 0$? right?
But I thought that the condition that the charge density does not change in time is the condition for electrostatics. but magnetostatics should not necessarily imply electrostatics?
Or I guess my interpretation is wrong. This is the rate of change of charge through a volume, so actually it allows for charge to move within a volume?
@SillyGoose you can have current without charge changing, for example a direct constant current in a wire: the total charge within every volume stays the same, but the charge is moving
@SirCumference this is only true if you haven't understood the purpose of writing essays, or if you think the main purpose of programmers is to produce boilerplate code
I deeply dislike the marketing trend of calling large language models (LLMs) "AI". LLMs are extremely good at producing output that is grammatically correct and sounds good, they are extremely bad at any sort of internal consistency: Just ask GPT for sources for its claims - it will invariably make up plausible-sounding literature references that...don't exist.
@Obliv This is ridiculous doomsaying, "AI" is not magic. Even if it's some sort of superintelligent overlord, you can just...pull the plug. Just because it's "smart" - whatever that means - doesn't mean it somehow transcends the boundaries of its substrate. Just like humans don't magically dominate other humans because they're smarter than other humans, AI will not magically escape the fact that it's running on servers that can be shut down, either
@Obliv The idea that suffering and war are somehow natural consequences of existence is classic Malthusianism - a deeply pessimistic view of human nature. I resent this view: Modern civilization is, for all its failures, essentially the history of humanity defeating Malthusian traps. The eradication of small pox, Borlaug's green revolution and a myriad of other feats show we are not slaves of competition for resources.
I don't think they're natural consequences of existence, but it may occupy a specific interval of time of existence, which we can get out of if we cooperate. I think AI and technological advancement is in the right direction, but if we fail to implement it or don't implement it at all, it might cause suffering & war to prolong.
The people who try to push the output of LLMs as "AI" are the same kind of empathy-deficient parasites who think the internet was invented to send spam emails.
@ACuriousMind Actually I wouldn't even relate my description of suffering to Malthusianism. I think it's multidimensional, and can't be ascribed to simply resources meeting demand. We could have an overabundance of resources and still suffer.
@Obliv Sure, but it is not some sort of intelligence deficit that results in that suffering. It's our choices. All our ideological conflicts and petty squabbles. AI - real or pretend - will do nothing to solve that.
The trapping and farming of animals, restricting their freewill is suffering, perhaps necessary right now. The hunting of "free" animals is probably better but in the end still invokes suffering. Veganism offers an arguable improvement, but I don't know where to base it on.
True!
I was thinking about this last week but if we never had access to fossil fuels and our technological progress was slowed, would we have more time to adapt to technology? Would this added time help us? (even though it could result in countless more wars since nuclear bombs would be delayed)
like a sid meier's civ situation with modified evolution time lol
@ACuriousMind petty squabbles spoken like a true AI lol. It's hard to judge the cycle of fighting, war, etc that's been around for tens of thousands of years and possibly encoded in us to some degree, when we're living a comfortable life in the first world, with no immediate dangers besides paying our bills.
@Obliv without fossil fuels we'd just have deforested the entire planet even faster to make charcoal
@Obliv the historical record isn't very comprehensive, but it is likely that we didn't know "war" before the advent of sedentary states after the invention of agriculture
Perhaps simply the existence of free will and laws of physics makes it nonsensical to imagine an existence where we live in peace, no matter how hard life may get. Not breaking down under stress but unifying
Maybe so, but territorial squabbles are basically of the same strain, and we've squabbled with each other, let alone other animals, I'd imagine, since we've existed.
Avoidance of conflict always trumps unnecessary violence for sure.
@Obliv you can argue one way or another: neither the historical record nor comparison to other great apes are conclusive. Some apes are very violent, others are pretty mellow and territorial conflicts are resolved by non-lethal competition
@ACuriousMind I should've been clearer, machines are not going to replace humans entirely in those areas — but IMO they will be able to take care of a lot of tasks that currently require humans, which can make it harder to find employment in certain fields
ChatGPT being able to produce essays isn't going to stop humans from writing them, but it may eventually lead to specific jobs being largely handled by bots
likewise, there isn't really a lot of evidence that homo sapiens resolved conflicts via lethal violence before the rise of sedentary states - like I said, the historical record is too spotty to tell, but it seems as if there's a rise in large-scale violent death by other humans only after sedentary civilizations meant that you could actually plunder significant amounts of resources by killing other humans
@SirCumference and how do you generate a newspaper article about a recent event from an LLM without a human having first written up a description of that event to feed it into the model?
@ACuriousMind Might've been wrong but I assumed there wasn't a precise definition of "artificial intelligence", insofar as there's not a precise definition of "intelligence"
@ACuriousMind Humans are certainly going to have some role in it, but it won't be as large or require all the same pay, experience and skills as reporters have now
@ACuriousMind In principle a bot could gather information from a greater variety of sources and use them to add details that a human would've missed or incorrectly represented
Humans make mistakes all the time. If you can make a bot that doesn't make mistakes as frequently as a human, then it could replace a lot of the human's tasks
Take research as another example. A bot could read up on a far greater number of papers than a human, write a good chunk of the code needed for the project, and write much of the paper to report the findings
@SirCumference A bot has no skin in the game - a crucial component of why we trust other humans is that their reputation is on the line if it turns out they were wrong
that this makes LLMs very good at producing syntatically correct language is - more or less- what we hoped, but no one has "designed" any kind of feature like "accurately summarize output that you're asked to summarize" into it
@SirCumference I am deeply skeptical of this claim
The fact that AI can perform art, play chess and do a lot of other skills that were assumed to require human creativity/brains, indicates to me they could replace other things in the future that we assume require humans
Also, crucially, we can mathematically prove that a chess engine like Stockfish plays chess "correctly"
we have no such proofs for the outputs of LLMs
you can explain what a chess engine does, it has an algorithm we can audit. LLMs are just giant black boxes you throw input into and get an output out of - this is a fundamentally different kind of technology with fundamentally different kinds of assurances
@ACuriousMind Would you agree that an AI can gather information on a far larger scale than a human could? E.g. find connections between different fields, which would otherwise require a human to specialize in multiple areas?
@SirCumference No, because you're already ascribing some sort of human-level understanding when you claim the AI "gathers information"
As an example, every time I've asked ChatGPT a question about string theory, the response as been garbage. Convincingly written, pseudo-supported by invented references, but ultimately completely non-sensical on a technical level
It's not "gathering information" if it can't even properly references its sources!
In principle an AI could always stay on top of the latest research, read up on far more than a human researcher could and find connections between different fields
@ACuriousMind What's impressive is the degree of improvement achieved within a few months :P
I'm saying they're not "AI", and the belief that human jobs could be largely replaced by this means that either a) the job was a bullshit job to begin with or b) you don't actually understand the human factor of the job.
I'm not claiming to be a prophet here, just saying that technological advances may reach that point
@ACuriousMind Well for instance, do you think there's a chance a lot of professional animation can largely be handled by AI in the future?
Given the time, cost and effort that humans have to put into it, it'd be much easier if a human can test out a bunch of outputs based on an input, and then fine tune it
@SirCumference To me, that's a pretty pointless statement: Unless you argue there's something special like a soul about humans, it's just obvious that technology can in principle do humans' jobs - at the very latest when technology is powerful enough to simulate the entirety of a human brain
(an idea to which I'll again link to Lena, one of my favourite short stories/fictional Wikipedia articles)
@SirCumference There is already a lot of automated animation, just look at these terrible autogenerated youtube vids for kids
@ACuriousMind That's a valid opinion, but if the average consumer just wants a story adapted into an animation with nice visuals, don't you think a company would heavily lean toward using bots for it?
@SirCumference so what are you arguing? That the soulless executives who have already bought up every franchise anyone ever liked and are running it into the ground are going to use "AI" to make even cheaper products that will have even less to say?
but see: In that regards, LLMs are just the evolution of Markov chains used to produce spam for decades - algorithms that produce meaningless output that'll fool enough people to be profitable but doesn't withstand any actual scrutiny
I was invited to watch an anime movie last night at a theater and I was thoroughly disappointed. They just don't makem like they used to.
I think the last movie I saw was a studio ghibli one or another classic
after the 2010s I'm pretty sure it got too mainstream and demanding and artists just had to put out crap
the use of CG was gross, like aesthetically
@ACuriousMind what was your method of reading/learning again? I think I remember you said you skim the pages and look at the theorems first, going back after ?
I'm trying to figure out how to finish 7 sections of D.E. before my test on wednesday
tl;dr: People mistaking GPT for "intelligent" are making the same category error as people who think that passing an exam demonstrates mastery of knowledge of a subject
@Obliv I don't really think I have a specific method so much as that I'm just fine with not understanding things on a first reading. Just...skim it, and take the stuff that doesn't make sense on faith during the first read, returning to it only later when you know where the text is going with the stuff you didn't understand
a lot of math and physics texts are written in this rather weird manner where you have to slog through heaps of technicalities before you get why we were doing all that stuff
but it's important to not just skip the details: often they are relevant, but the nature of the relevance is often only revealed somewhat retroactively
and the more you've read, the more you'll see stuff you already know
that's where skimming becomes really effective: if you recognize the general shape of an argument, you can appreciate that it's probably correct without getting into the details
On the Home page of Physics Meta, the question “Why can I still no longer questions?” is attributed to user_250478, but on the Questions page it is attributed to a user who uses his name. Why do these pages differ like this when displaying the same question? Is it a bug or a feature?
