There is a purely geometric reason for why elliptic curves have group structure. A geometric shape which is also a group, such that the group operations are smooth maps, has to be homogeneous - it has to look the same from every point[1]. Not just that, if you have a vector at some point, there is a natural way to transport it to every other point on the shape. The only surface (curves over complex numbers are really 2d surfaces) which obeys this property is the torus[2].
[1] Why should the homogeneous property be true? Because in a group, multiplication by g, pushes the identity e to g. M_g(e)=g where. This is a continuous isomorphism of the shape. So the shape looks the same at g as it looks at a (a neigbhourhood of g looks the same as neighbourhood of e). So an 'X' or 'Y' shapes cant be groups, as there are points which are locally unique, but 'O' shape can be a group. Moreover, M_g can also push a fixed non-zero vector v at e to a vector v_g at g.
Don't want to get into low quality generalizations in your post except to note tahta casual Google search will show you that Tata group is one of the most philantropically oriented groups. Which of course, doesn't excuse this issue.
This is a great way to present the concepts. Something like this would have been useful some years back when I was trying to use the Haskell library for lens.
For someone new, this was a really interesting read. Thanks for the effort. The picture of how the sky looks from underwater was so surprising to learn.
Regarding the 'first-principles' discussion, it is a relative term in usage despite the name. Explaining from a layer of abstraction below normal explanations.
Someone explaining how a computer works starting with machine instructions is doing first-priniciples, even if they wont explain details of atoms. The literal meaning would be almost impossible to implement, for instance none of Newton's work would be first principles given today's knowledge. Similarly, current physics can be subsumed in future principles.
There is an important difference between Atman and Soul. Thoughts, emotions, decisions are seen as part of nature/prakriti not Atman, whereas Soul is usually intended to include these things.
A better description would be that the atman is the consciousness in which physical things or mental constructs can appear and pass away. The nature of pure consciousness is also described as real(undisturbed by time) or ananda/contentment/bliss.
The disidentifcation from thoughts (for instance, seeing them pass by just like cars on road) is an important part of liberation.
Quoting examples without an effort to show that it is representative of Buddhist teachings is basically a smear. Like starting a discussion on liberalism, not with principles of individual freedom, but instead saying that the attempt to bring democracy to Iraq is the representative example of liberalism.
(Some on the left who oppose liberalism actually do some versions of this, quoting Mills on colonialism - but that is a genetic fallacy.)
It makes much more sense to say that anytime some teaching/philosophy becomes popular at a continental scale, the people who are involved in conflicts will try to appropriate it to justify their position.
If you want to evaluate the role of the teaching itself, one would have to compare it to alternatives and whether they would be more easily appropriated.
> Like starting a discussion on liberalism, not with principles of individual freedom, but instead saying that the attempt to bring democracy to Iraq is the representative example of liberalism.
Some prefer to discuss what a purported ideology or its adherents does out in the real world.
Sure, as long as your real world examination is careful about getting the causation right as practically any idea can be appropriated. For instance, someone makes false charge to lock up someone innocent in the name of 'reducing crime', is the issue the goal of justice and low crime or is it the problem with the standards of evidence used to lock up the criminal?
> For instance, someone makes false charge to lock up someone innocent in the name of 'reducing crime', is the issue the goal of justice and low crime or is it the problem with the standards of evidence used to lock up the criminal?
The immediate problem is the troll that is lying and hiding behind a purported agenda. Exposing their real agenda is the immediate fix.
You don’t rhetorically concede to the troll that “reducing crime” is good because they’re a troll. Conceding anything to them is a strategic blunder. They are trolling. It’s irrelevant to the case.
> we can keep on building theorems on top of theorems with increasing complexity
This is a somewhat bleak picture of math. We also have the other phenomena of increasing simplicity. Both statements and proofs becoming more straightforward and simple after one has access to deeper mathematical constructions.
For example : Bezout's theorem would like to state that two curves of degree m, degree n would intersect in mn points. Except that you have two parallel lines intersecting at 0 instead of 1.1 =1 point, two disjoint circles intersect at 0 instead of 2.2=4 points, a line tangent to a circle intersecting at 1 point instead of 1.2=2 points. These exceptions merge into a simple picture once one goes to projective space, complex numbers and schemes. Complex numbers lead to lots of other instances of simplicity.
Similarly, proofs can become simple where before one had complicated ad-hoc reasoning.
Feynman once made the same point of laws of physics where in contrast to someone figuring out rules of chess by looking at games where they first figure out basic rules(how pieces move) and then moves to complex exceptions(en passant, pawn promotion), what often happens in physics is that different sets of rules for apparently distinct phenomena become aspects of a unity (ex: heat, light, sound were seen as distinct things but now are all seen as movements of particles; unification of electricity and magnetism).
Of course, this unification pursuit is never complete. Mathematics books/papers constantly seem to pull a rabbit out of a hat. This leads to 'motivation' questions for why such a construction/expression/definition was made. For a few of those questions, the answer only becomes clear after more research.
> They were unable to locate the terrorists even after two or three weeks and needed a distraction.
This does not make sense. When France attacked Daesh in 2015 after the terrorist attacks in Paris or when the US attacked Afghanistan after 9/11, the objective wasn't to target the exact people who carried out the attacks, but the organization behind the attacks. People can always be found as long as the organization remains.
The goal of the attacks would be to make any future terrorist attack an expensive option for the Pakistani military as opposed to something which can be done routinely. There was a sharp drop in the terrorist attacks in Kashmir after the 2019 confrontation.
>when the US attacked Afghanistan after 9/11, the objective wasn't to target the exact people who carried out the attacks, but the organization behind the attacks
The mission in Afghanistan was very much to find Bin Laden. It was changed after he escaped.
The incentives of the Pakistani generals to permit organizations like LeT to commit further terrorist attacks is a different domain from whatever the local political situation is like in India. There has been a past regime where Pakistani generals were able to train and send militants regularly to conduct terror attacks in India. Without an effective response from India putting pressure on these generals, that can easily become the new normal again.
Do you think (plausibly) threatening to cut off water to large swathes of Pakistan, or blowing up some random terrorist camps, is the bigger actual threat?
Cutting off water supply is clearly the bigger threat. However, it involves a longer time frame - building infrastructure which one expects not to use in a normal situation.
Importantly, even once built, it selects the wrong targets, not terrorists or military bases - but regular people who will be faced with scarcity of water and food, as the crops use Indus water. This would be something highly unethical, and also not something sustainable - once visuals of hunger start reaching screens across the world, the force to restart the supply would be strong.
What force, though? Israel does something similar, and how much actual pushback (as opposed to sternly worded statements) did it actually get?
Even if it were to translate to sanctions, I'm not so sure BJP wouldn't welcome it. "The Fatherland is besieged, let's unite together around Dear Leader and fight back like one" tends to be a very popular take in authoritarian countries for a reason, and it's that much easier to pull off when you can actually point at some way in which your country is targeted.
Dont buy your description of India. Elections matter, BJP can and does lose many elections, India is dependent on oil from Gulf countries, it doesn't have US to shield it from actions which it shouldn't even be doing in the first place, there are much better options against Pakistan etc.
India is also buying oil from Russia - at a discount, or used to - because it is one of the few that could do so openly.
Attacking Pakistan over this also strengthens BJP’s hand and distracts everyone from the complaints that have been eroding their support, like ongoing corruption, high taxes with lower quality of life, etc. etc.
Don’t expect the gulf countries to come to Pakistan’s aid over this, especially if it comes to money. Muslim countries in general like to pretend to be friends, and they certainly talk a big game.
When it comes to actually doing anything though, they just use anything going on as a chance to stab each other in the back. Even when there is a chance for going after ‘the common enemy’ like Israel.
India hasn’t even been close to interesting to any of them since the Mughals. The minority Muslim population in India (about 20%) is also just a little less than Pakistans entire population, and almost half of the entire Middle East’s population, so it’s not like it would be a clean ‘attack the Hindu’s’ type situation anyway.
Also, Indians in general are not particularly warmongering, but this is about as righteous a cause as anyone has been able to come up with for awhile to ‘make someone pay’, is total rage bait for the hardliners/Hindutva contingent and is a good distraction for BJP.
As long as it doesn’t get too expensive, or look like it will escalate to Nuclear war, I’d expect it to go on awhile.
India in general loves to get all worked up about Pakistan (and to a lesser extent Bangladesh, though in that case it’s often about illegal immigrants).
It’s a trope like getting worked up about Cartels and/or ‘the illegals’ from down South in the US.
Source: westerner living in India for awhile now. मैं हिंदी नहीं बोल सकता, but I still get around eh?
> Attacking Pakistan over this also strengthens BJP’s hand and distracts everyone from the complaints that have been eroding their support, like ongoing corruption, high taxes with lower quality of life, etc. etc.
This won't last long, though. BJP lost the elections after Kargil War.
I'm not saying that India is full-on authoritarian today. But it's definitely edging in that direction under BJP, and I could see them embracing the war as their ultimate ticket to get there. I mean, if there is a shooting war with Pakistan, and they can credibly blame Pakistan for starting it, what will it do to BJP electoral standing?
As far as US, under Trump, I'd actually expect it to back Modi.
We could have gone after the people who actually did 9/11 but that was a bit of a non-starter. Also I think you're equivocating between multiple interpretations of "the terrorists" when most people absolutely wouldn't draw a distinguishing line between, using 9/11 as an example again, the actual hijackers and Osama bin Laden. There's absolutely no question that any time the phrase "the 9/11 terrorists" is used it means both the actual perpetrators and the people who planned and supported the attack.
The context was a reply to an assertion that the terrorists in Pahalgam were not found by Indian security. I interpreted this as people who physically did the attack.
If by terrorists, we mean the planners of the operation, that trail leads directly to Pakistan. Musharraf, the ex-army chief, is on record saying that the military has funded several militant organizations in Kashmir including LeT. (Osama's haveli in Abbotabad was incidentally also very close to the Pakistan Military Academy). The permission for the operations probably came all the way from the top as the attack came right after a strong statement on Kashmir by the army chief.
"funding militant organizations" isn't the same as committing acts of terrorism. Nobody would have said the US should have responded to the 9/11 attacks with airstrikes on the CIA headquarters.
We are not talking about re-targeting of training and weapons from Afghanistan to Manhattan, but direct planning of an attack with ability to restrain and release the groups on demand. Contra the truthers, even the CIA wouldn't go that far. Musharraf explicitly mentioned the groups operating in Kashmir. He wasn't talking about fighters in Afghanistan.
The CIA wouldn't go that far? Operation Northwoods [1] is obvious evidence to the contrary. The one and only reason that that operation wasn't carried out is because of a President who refused to sign on to it, who would shortly thereafter be assassinated by a 'deluded gunman.' [2]
> There was a sharp drop in the terrorist attacks in Kashmir after the 2019 confrontation.
There were fewer terrorist attacks, certainly. I'm sure the Indian government would like to believe that the 2019 strike had an effect, but far more likely causes are
- Money. Pakistan's economy has stagnated and the country has lurched from one IMF bailout to the next (2019, 2023, 2024). It got so bad at one point that politicians were asking people to drink less tea so they could conserve foreign currency.
- Covid. Affected everything, but certainly harder to think about waging conflict when such a massive problem is affecting the country.
- Internal political instability, especially when Imran Khan took on the military and lost. The military was actually in danger of losing their primacy for the first time in decades.
- Conflict with the Taliban and Pakistani Taliban. The ISI had nurtured the Taliban to be tame pets and it turned out not to be the case. Crushing these was the highest priority, not least because it made their policy of nurturing terrorists look idiotic.
All of these factors meant Pakistan wasn't and isn't in the best shape to wage war overtly or covertly with India. India's economy has continued to grow, in contrast to Pakistan. The official Indian policy of "benign neglect" towards Pakistan appeared to work well.
I'm sure these attacks will be spun as a success in the future. Safe to say a Bollywood movie dramatising the events is already in the works. But Pakistan's own economic and political problems are far more likely to influence its decisions to engage in this sort of behaviour.
If you are actually arguing that a country targeted by a terrorist attack does not gain deterrence with a counterstrike relative to letting things go on, then how uniform do you consider this prescription? Should the terror attacks in the US or France not have had a military response?
What happens to the incentives of terror groups in response to such a policy?
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The role of money only becomes an issue when conducting a terrorist attack becomes expensive. Missiles and jets consume much more money in comparison to training recruits via an intermediary organization like LeT and sending them across the border to carry out attacks.
A regime in which a terror attack leads to a high pressure, expensive situation for the Pakistani military is completely different from regularly scheduled, train and deploy terror attacks from militants which used to happen earlier.
In that situation, the military has to respond to economic pressure, pressure from allies and pressure from its own people.
The Pakistani military cares about itself, above all. It wants to maintain its role as the primary protector of the Pakistani people, answerable to no one but themselves. As long as the threat of India looms large, their primacy is guaranteed. As a reward generals are allowed to grow filthy rich.
Support for the Pakistani military was at its nadir during the era of benign neglect because there wasn't an Indian boogeyman to justify their interference in politics and economic exploitation. But now that India has attacked Pakistani targets this will quiet any internal criticism of the Pakistani Army.
In other words, the military absolutely loves it when India engages in so-called deterrence. No Pakistani army soldier died (according to both sides). Pakistani people support the Pakistani Army more strongly than ever. It's absolutely perfect for the Army. I fully expect that they'll fund more terrorists, leading to a constant cycle of violence.
> I fully expect that they'll fund more terrorists, leading to a constant cycle of violence.
Yes, that's the defining characteristic of all terrorist organizations. Get money, not through politics or production or economy, but by damaging others. Then get paid for not doing quite as much damage. This model has spread quite a bit in the past 5 years.
Yes, an outside target can be used to tackle internal strife. But, there is no sign that the Pakistani army is actually in any danger of being removed from power, barring a major military defeat, nor that it will lose its autonomy over military policy.
If say, India were to let this slide, the default outcome is another such attack. Given the above motivation of the military to create a conflict and the ideological bent seen in Gen.Munir's speech, the expected outcome would be to repeat till this they get a conflict.
Yes, the deterrence won't be perfect. The Pakistan Army might end up repeating an attack whenever there is a relief from economic constraints(it doesn't have money for frequent purchases of expensive weapons) or from pressure from its allies (who dont want their oil trade or pipelines to suffer). But this means that what India has to do to minimize the number of attacks is to not let an attack slide by with low cost for the army.
The best case scenario would be a peace deal, as was arrived in Vajpayee and Sharif's time, but it was sabotaged by the Kargil operation, for exactly the reason you mentioned - a peace deal marginalises the army.
I don’t think Pakistan orchestrated the last attack.
The structure was designed for being disavowed.
I expect it was more the army looked away, over condoned.
And yes, the expectations are to generate a response from the BJP.
By this rubric there are 4 actors on the stage.
- The people of Pakistan, the Pakistani army,
- The people of India, the BJP.
I’ve had this discussion with friends who are Pakistani and they concur that this makes the most sense.
The opinions of the Pakistani army have dramatically changed as per their interactions, having been at a nadir due to their domestic handling of events, and now these actions have reinvigorated public opinion.
The BJP has had its military credentials burnished.
I’d go a step deeper and suspect that there was a traditional response from India planned, and then at some point in the past 72 hours, a functionary on the BJP side raised the potential of a massive PR coup and the old guard got sidelined.
This has worked. This means this behavior will be repeated.
> I expect it was more the army looked away, over condoned.
Right, in the exact same way they "looked away" while OBL lived half a mile away from the most prestigious military academy in Pakistan and the same way they "over condoned" the Taj hotel shootings. You are not disagreeing with me here.
> The opinions of the Pakistani army have dramatically changed as per their interactions, having been at a nadir due to their domestic handling of events, and now these actions have reinvigorated public opinion.
This line describes Hamas just as well as the Pakistani state, unfortunately.
> The BJP has had its military credentials burnished.
Indian media always has been and always will be jingoistic and no matter how the government responded, it would gleefully report on the power of the Indian military.
> This has worked. This means this behavior will be repeated.
WHO has this worked for? The BJP isn't more popular because of this. Pakistan hasn't achieved any strategic goals beyond the continuing destabilization of J&K - which is already well on the way to integration.
> Should the terror attacks in the US or France not have had a military response?
Probably, yes. Military responses to terrorism are almost always counterproductive. I don't know which specific attacks you're talking about, but the ones I can think of the US did far more damage to itself with the blowback than the original attack ever achieved.
Note that I am not referring to the prolonged occupation of Afghanistan, much less of Iraq here. Rather, something like a strike which targets bin Laden and other organizers of the terrorist attack.
So it doesn't matter that military responses that have actually been tried in the real world have been counterproductive, because you can imagine other kinds of military responses and you imagine that those kinds of military responses would have gone better?
So maybe say which specific military responses you want to talk about in the first place rather than just saying "the terror attacks in the US or France" and expecting everyone to read your mind.
> What happens to the incentives of terror groups in response to such a policy?
You're imagining these people to be some sort of loyalists, rather than something closer to anarchists. Triggering military responses is going to be viewed as a bonus.
I'll note people rather frequently claimed Bin Laden wanted the US to be tied up in the military quagmire that their terror attacks produced. There's certainly some logic to that idea. His organization was too small to do much direct damage.
reagent (which you'd put re-frame on top of) is what handles rendering (via React) when we're using re-frame, and is the library you can say does "UI as a pure function of state". re-frame is basically like redux (over-simplification) but for the CLJS ecosystem, focuses solely on state and how to mutate it.
The Turing test is meant to see if computers can functionally replicate human thought (just focussing on textual conversations aspect for simplicity).
The implications of passing the Turing Test are profound. Tricking a user with a 5 minute conversation isn't the same. The examiner is allowed to be an expert questionner.
For one thing, software engineering would quickly be taken over as a client can just chat with an AI to get the code. That is far from happening, even with all the recent LLM advances.
Similarly for literature, science etc. Currently, you can detect a difference between a competent human & machine in all these fields just by text chat. Another way of saying this is that the Turing Test is AI-complete and is a test for AGI.
Whether machines can think, and whether we believe so, are distinct.
If we're just a kind of machine, we can just say that LLMs have not surpassed us.
While LLMs still give away obvious signatures (their flaws are LLM-y and not humanlike), they play the imitation game better than any other machines before them. I lived long enough and played enough with simpler chatbots to believe that the simple text-based imitation game that Turing suggested will be won by computers before machines achieve reasoning ability that surpass the smartest humans.
> [...] the Turing Test is AI-complete and is a test for AGI.
Since Turing never used the term AGI, this is an interpretation.
Since we cannot determine whether a machine can think by comparing their anatomy to ours and drawing a conclusion that whatever we do, they must do the same, we are bound to compare their ability to imitate us. Which means we can't ultimately know, we can just feel ourselves convinced.
So there's no objective point at which to place the goalpost.
So constantly moving it is an expression of acclimatisation.
I'm impressed by LLMs, in spite of the hype and the homework cheating.
If anything, the ability of LLMs to imitate human labor shows how far from thinking a lot of human behavior is.
That is not true. Even our cells, with features that look a lot like machinery such as a proton pump, is on the whole several orders of magnitude more complex than any machine. Even a single human cell is more like an ecosystem than a machine. Let alone entire humans or even just the human brain. Consider that both cells and humans are capable of reproduction.
> If anything, the ability of LLMs to imitate human labor
Within extremely narrow confines and quite often going over into just conjuring up nonsense.
Why? I'd say it's been self-evidently true for at least three decades now.
> Even our cells, with features that look a lot like machinery such as a proton pump, is on the whole several orders of magnitude more complex than any machine.
How many is several? I think that, at the level of a proton pump, individual components comes close to the order of complexity humanity deals with in man-made machines.
Also, at this level, things really look like machines, act like machines, quack like machines - there's no reason to not call them machines, given they obviously are. It's naturally originating molecular nanotech.
> Even a single human cell is more like an ecosystem than a machine.
Certainly. But then, an ecosystem is defined as a system made of bunch of varied stuff interacting with each other, finding balance through a set of feedback loops. An ecosystem of machines is still an ecosystem, and is arguably a machine in itself, too - after all, the term "machine" also applies to self-balancing / feedback driven systems, ever since we invented control theory and formalism to describe feedback loops.
> Let alone entire humans or even just the human brain.
Complex machines don't stop being machines when you keep adding moving parts to increase complexity. Or, at which point between a protein pump and a human being you believe the assemblage of molecular nanotech stops being a machine?
> Consider that both cells and humans are capable of reproduction.
What's that supposed to tell us? Human reproduction involves cell reproduction.
> a single human cell is more like an ecosystem than a machine
Good point. Not to mention the massive reliance on organisms like mitochondria and bacteria that don’t even share the host’s DNA.
> [imitate human labor] within extremely narrow confines and quite often going over into just conjuring up nonsense
Much like human labor.
So the intelligent, reflective and thoroughly iterated work is hardly replicated at all, and the poorly imitated, easily repeatable coursework and boring paper sludgework excellently so.
So we don’t just get to criticise LLMs for not actually being intelligent. We similarly get to criticise humans for not being so when we might think we are, either.
> Good point. Not to mention the massive reliance on organisms like mitochondria and bacteria that don’t even share the host’s DNA.
So? Does your car stop being a machine just because it's a complex systems of moving parts, many of which are dynamically balanced through feedback loops, which involve components sourced from different vendors, and substances that are not part of the original manufacturing data sheet?
Exactly what insight does this give us? I feel this is trying to contrast a single machine "unit" against a complex system, while also sneakily committing a naturalistic fallacy by using "machine" vs. "ecosystem" to imply "machine" vs. "life" in the magical sense (i.e. as if life was something beyond a physical process).
Yes, the Turing Test tries to give an operational definition of 'think'- can a computer give the same response as a thinking human.
Further, we can even disallow decisions based on the human/machine signature. The computers don't need to imitate human frailty and examiners decide based on the quality of the output, not whether it feels human or machine. I think that this is also why one of the stipulations is that the human/machine is not physically visible.
But even then, this goal is far away. We are not asking just about simple short chat imitation games but whether a computer can operationally do everything humans are capable of in the textual medium. This often requires experts.
The chat game version can be solved. But, the foundational goal is far away.
> Yes, the Turing Test tries to give an operational definition of 'think'- can a computer give the same response as a thinking human.
So it's quite neat that we've managed to beat the test by brute-forcing the right NN architecture on a goal function that literally is just "give the same response as a thinking human", in full generality of that phrase.
> But even then, this goal is far away. We are not asking just about simple short chat imitation games but whether a computer can operationally do everything humans are capable of in the textual medium. This often requires experts.
That is moving the goalposts. Or just another game entirely - which is fine on its own, as a further milestone.
Just like the original test, there are 3 participants in a test: an examiner, a human, and an AI.
The examiner may ask any question or series of questions to the human &/or AI, and receive responses. This is done via text or other means which prevent the examiner from directly seeing the body of responder.
Both AI and human are to attempt to convince the examiner that they're the human.
Unlike the original test, it then needs to repeat with different examiners.
If there exists an examiner which can distinguish an AI from a human (for all humans) in a Turing test with probability non-negligibly different from 50%, then the AI fails this meta-Turing Test.
Just because an AI can fool some people doesn't mean it's a human-equivalent intelligence.
Your "meta-Turing Test" is rigged. There are 8 billion humans on this planet, statistically someone is bound to be able to distinguish AI from a human with near-100% accuracy across a reasonable amount of rounds through sheer chance.
Existence is too strong a criterion. Even correcting for random chance, if there's 1, 10, 100, even 1 000 000 examiners who could distinguish an AI from a human with high probability, that steal means the AI passed the test in practice, because even a million examiners is still a rounding error compared to the human population.
[1] Why should the homogeneous property be true? Because in a group, multiplication by g, pushes the identity e to g. M_g(e)=g where. This is a continuous isomorphism of the shape. So the shape looks the same at g as it looks at a (a neigbhourhood of g looks the same as neighbourhood of e). So an 'X' or 'Y' shapes cant be groups, as there are points which are locally unique, but 'O' shape can be a group. Moreover, M_g can also push a fixed non-zero vector v at e to a vector v_g at g.
[2] The Euler characteristic of the torus is 0. A non-zero vector field has index 0. https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Hopf_the... See the special case of the sphere https://en.wikipedia.org/wiki/Hairy_ball_theorem
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