And, of course, if they disagree with me about this and want to claim that they are, in fact, conscious, I'm not sure they can do that because... well the hard problem of consciousness.

Don't get me wrong, tech is great when it's a value-add, but TV tech has gotten out of control.

Any suggestions for a scanner meant for bulk scans of old family photos (think a few thousand images)? I bought, what I thought, was a reasonably solid scanner, the Pacific Image Powerfilm scanner but the software is so janky that it hangs every two strips and has to be restarted making the entire process super labor intensive. Also the entire "bulk feature" where it's meant to pull the strips one at a time iis not even close to working.

My interpretation of what you're saying is that since the next token is simply a function of the proceeding tokens, i.e. a Markov chain on steroids, then it can't come up with something novel. It's just regurgitating existing structures.

But let's take this to the extreme. Are you saying that systems that act in this kind of deterministic fashion can't be intelligent? Like if the next state of my system is simply some function of the current state, then there's no magic there, just unrolling into the future. That function may be complex but ultimately that's all it is, a "stochastic parrot"?

If so, I kind of feel like you're throwing the baby out with the bathwater. The laws of physics are deterministic (I don't want to get into a conversation about QM here, there are senses in which that's deterministic too and regardless I would hope that you wouldn't need to invoke QM to get to intelligence), but we know that there are physical systems that are intelligent.

If anything, I would say that the issue isn't that these are Markov chains on steroids, but rather that they might be Markov chains that haven't taken enough steroids. In other words, it comes down to how complex the next token generation function is. If it's too simple, then you don't have intelligence but if it's sufficiently complex then you basically get a human brain.

I go to a random house on a random street and knock on the door. A young girl opens the door. I ask how many siblings they have and they say one. What's the probability that they have a sister?

Now it's 50% even though cosmetically it seems like it'd be fair to say that the family has at least one daughter. The reason is that once I see a girl at the door, I'm slightly more confident in that it's a GG household since a GB or BG household would sometimes show a boy opening the door (assuming the two kids are equally likely to open the door).

P(GG | G at door) = P(G at door | GG) P(GG) / P(G at door)

P(G at door) = 1/2 (by symmetry)

So, P(GG | G at door) = 1 * 1/4 * 2 = 1/2

On the one hand you can imagine that work gets supercharged, allowing companies to produce 10x the number of widgets at 1/10th the cost. The economy would grow rapidly, wealth inequality would presumably be exacerbated, jobs would be automated, we might need some version of universal basic income, and so on. People debate whether or not this kind of transition is imminent or if it'd take decades.

On the other hand, it's conceivable that not much would happen in the "bulk" of the economy while at the same time the frontier of humanity might be pushed forward. We may see new treatments for diseases, new types of energy production, and so on. In this version of the world, jobs would mostly remain unchanged (at least in the short to intermediate term), perhaps with some small multiplicative efficiency factor, the economy wouldn't grow rapidly, there wouldn't be any mass unemployment, and so on.

In my mind, I'm much more excited about the second kind of impact that AI might have than the first. I guess I don't really feel like I want to have 10x the stuff that I already have while I'm really excited about someone curing cancer, Alzheimer's, Parkinson's, MS, and so on.

In other words, the power of Strasssens algorithm comes from a strategy that's similar to / reminiscent of dynamic programming.

If they're trying to exfiltrate data, they might want to rotate through IP addresses in order to obfuscate what's going on or otherwise circumvent restrictions. Using a simple ip rotator like the post talks about would maybe be an approach they'd use. If they're not careful with the IP addresses, once in a while one might get caught due to some restriction like being outside the US. It'd maybe appear as though you're getting these weird requests from Russia, but that's just because you're not logging the requests that are not being flagged from the US.

Maybe I'm reading the post incorrectly though (if so, please correct me!)

For any dome-like shape, you can start a marble at the bottom and roll it up with some initial speed. If you roll it with insufficient initial speed it'll turn around and come back down. If you roll it too hard, it'll overshoot the peak. By continuity, there must exist some initial condition where it stops at the top.

Now, here's the thing that makes Norton's dome special: For a typical dome shape it'll take an infinite amount of time before that marble stops at the top. If you plot the position as a function of time it'll have some type of sigmoid-like shape. However, for the special case of Norton's dome, you can make it settle at the top in a finite amount of time where it'll sit for the rest of eternity. In other words, if you plot the position as a function of time, there will be some critical time after which its position is constant.

Now, the clever thing to do now is to realize that Newton's laws are time reversal symmetric which means that any motion forward in time could equally well happen backwards in time.

So, you're allowed to take any position plot and flip it horizontally; this is also going to be a valid trajectory.

For any typical dome shape this is not a problem. For a typical dome shape you have a sigmoid-like solution which, when flipped, is still sigmoid shaped. In particular this means that there is no finite time at which you can place the marble at the top of the dome and have it roll off. At any finite time, the marble will be slightly off the top and have a small nonzero speed.

Norton's dome is different. If you flip its trajectory horizontally you'll see that there are many moments in time where you can start the marble at the top to have it abruptly start rolling off the top at some later time. This is the paradox. You can choose to have it sit at the top for one second and then start rolling or sit at the top for one minute and then start rolling.

Unlike other domes, Norton's dome seems to violate our intuition for how initial conditions work. In all cases the marble starts at the top with zero initial speed and yet falls off the top att different moments.

Consider the function f(x) = Sum_{n=1}^\infty c^(-xn)

Then differentiate this k times. Each time you pull down a factor of n (as well as a log(c), but that's just a constant). So, the sum you're looking for is related to the kth derivative of this function.

Now, fortunately this function can be evaluated explicitly since it's just a geometric series: it's 1 / (c^x - 1) -- note that the sum starts at 1 and not 0. Then it's just a matter of calculating a bunch of derivatives of this function, keeping track of factors of log(c) etc. and then evaluating it at x = 1 at the very end. Very labor intensive, but (in my opinion) less mysterious than the approach shown here (although, of course the polylogarithm function is precisely this tower of derivatives for negative integer values).

https://youtu.be/dRBmavn6Wk0

In high school physics we learn that a 1kg mass accelerates as quickly as a 2kg mass when only subjected to the force of gravity. When I used to teach physics, the intuitive explanation I gave for this hinged on a thought experiment. Suppose that you have three 1kg masses falling side by side after being dropped from the same height. Clearly they are all going to fall at the same rate since they're equivalent. Now imagine redoing the experiment but this time taking two of the masses and placing them closer together. Does anything change? Clearly not, they're still all equivalent and ought to fall at the same rate. Now imagine doing this until those two masses are right next to each other, touching. Does anything change? Well no, all three should still fall at the same rate. But now, why not glue those two masses together and call it a 2kg mass? Once you do that you've shown that a 1kg mass and a 2kg mass fall at the same rate.

This usually convinces people, but there's actually a flaw in the argument that gets to the heart of why gravity is so different from the other forces.

To see the flaw, replace the above masses by three electrons falling next to each other in an electric field. Everything goes through in exactly the same way. You end up gluing together two of the electrons and these two electrons will accelerate at the same rate as the single electron. But if you're not careful you'd conclude that all electric charges fall at the same rate in an electric field, something we know is false.

Where's the flaw? Well, all of matter is built from some particles, and as long as you restrict yourself to particles that have the same "charge/mass ratio", the argument above works. It is true that one electron accelerates the same as 100 electrons tied together but that's just because e/m is the same for all those constituents.

So, the thing that's glossed over in my high school explanation for why 1kg and 2kg accelerate at the same rate is that the constituent particles all have the same "gravitational charge / inertial mass" ratio. Because this ratio is the same for all particles, we may as well absorbed that ratio into the gravitational constant and just use "m" in place of both of them. It's this "universal coupling" that's really responsible for the equivalence principle and what sets gravity apart from the other forces.

There are then two natural questions: 1. if the perturbation series diverges, why doesn't the universe explode? and 2. If the series diverges, why can we use it at all?

Let's first talk about the first part: why doesn't the universe explode? Well, it's because the perturbation series is not actually what is going on, the real answer is the solution to the full set of equations. It's just that we're using a perturbation expansion as a crutch. It's sort of like if the universe's function is 1/(1-x) but we constantly insist on using 1+x+x^2+... Clearly the first function is completely well behaved at x=2 but the second one is not. If we notice that our series explodes for x=2 we should not immediately assume that the universe also must explode, it's just that our representation of the true physics is not faithful. This is perhaps a bad example because the series in question is convergent for some x, just not for x=2. The perturbation expansions in question are more subtle since the never converge.

This then leads into the second question: if the series diverges, how can we even use it? Well the idea here is that it's not just any divergent series (like my silly example with 1/(1-x) above) but rather an asymptomatic series. This means that as long as you truncate the series at some point it is in fact reasonably close to the target function for a sufficiently small value of the parameter. It's just that the more terms you want to include, the sooner the approximation breaks in terms of the parameter. So, if you want to include 10 terms it might be a decent approximation until x ~0.1 but if you include 100 terms it might only be a good approximation until x~0.01. Now, within the overlapping range (x<0.01) it's better to have 100 terms than 10 terms, so it's not like including more terms is bad in all ways. But you see the issue: if you include 1000 terms you get a better approximation for your function for values x<0.001 than you had with 100 terms but now your approximation breaks much sooner. If you want to include all the terms your approximation breaks the moment you leave the point x=0.

Why do we think that QFT perturbation theories generally have zero radius of convergence? Well, look at QED, the quantum theory of E&M. If the theory had any nonzero radius of convergence, that also means that the theory would need to make sense for negative coupling constants. However, what would E&M look like for negative coupling? Well, we'd still have electron/positron virtual pair creation from the vacuum since the interactions of the theory are still the same. However this time around they wouldn't attract each other anymore but instead repel each other causing an instability in the vacuum of the theory. We would just constantly be producing these particle/anti-particle pairs and they'd form two separate clusters where all the electrons attract each other and all the positions attract each other but they pairwise repel. In other words, the vacuum would break. This suggests that QED with a negative coupling constants doesn't make sense. But this contradicts the fact that the radius of convergence of the perturbative expansion is nonzero.

That's not to say that all QFTs must have zero radius of convergence, but similar arguments can (I think) be made for the type of QFTs that we actually see in nature.