consilient

> Tensor product V⊗V is space of all bilinear forms on V*

This is only true for finite-dimensional V. In general bilinear forms on V* are linear maps `V*⊗V* -> F` (where F is the base field) which is isomorphic to `(V*⊗V*)*`. For infinite-dimensional V this is much larger than `V⊗V`.

The general characterization is that `V⊗W` is the vector space (and associated linear map `⊗: V x W -> V⊗W`) such that for every bilinear map `h: V x W -> T` there's a unique linear g such that `h(v,w) = g(v⊗w)`. Intuitively, it adds "just enough" elements to `V x W` to encode arbitrary bilinear maps as linear ones.

> Tensor product is basically the mathematical way to express if condition.

Not really, no. The way mathematicians actually express if conditions is with the word "if". The obvious pointlessly formal way to do it is with a pair of functions `ThingConditionedOn -> {0, 1}` and `{0, 1} -> Result`, but why would you?

> dealing with tensor matrix is simply a waste of memory since 90% of the time people are dealing with sparse system.

Tensors are not their components, any more than locations are their coordinates. Whether you choose a sparse or dense (or symbolic) encoding does not change the object being encoded.

> or seeking of risk

Sure. I very much doubt anyone seeks risk because it's risk. They do it for the thrill, or for social reasons, or because calculating risk is hard, etc.

> having any other computable utility function on any other measurable that we’d have to accept as equally valid if we want to convince any non-economist that we aren’t just using utility as a weasely word for money

Almost anything else will do just as well. If you can buy it (or the means to acquire it) on the market, then there's always some point at which twice as much money buys less than twice as much. Maybe 10 million is not enough to hit that point, or 5 million is too much, but the curve flattens out somewhere. And if you can't, then the money makes no direct difference, but can still buy you food and shelter and free time with which to pursue your other projects.

I suppose in principle you could actively want to not have money (and not want to give it away either), in which case getting rid of 10 million is probably not twice as hard as getting rid of 5, but that hardly seems realistic.

> You say “restricted” I say “better defined”.

Sure, whatever - the point is that they're different objects, and results about one are not results about the other.

> What do you mean by “restricted” when you are characterising a function?

I mean whatever the word "function" means in a given setting. A classical function X -> Y is a relation such that if `(x, y1)` and `(x, y2)` hold then `y1 = y2`. A function in intuitionistic type theory is a well-typed lambda term. A function in the categorical semantics of a type theory is an exponential object. And so on.

> Show me the decider… for “classical” and “non-classical” objects.

No such thing: it's a metatheoretical judgement, not a theorem. Same story as type errors: within the language, `stringToUpperCase (5 :: Int) :: String` isn't a false statement, it's just inexpressible nonsense. There is no such object as `stringToUpperCase 5` and so nothing can be said about it internally. When we talk about it, we're talking, from outside the language, about a syntactic construct.

> That is the definition of information; is it not? The answer to a yes/no questions

No. Self-information of a given outcome with respect to a random variable, which is probably the most common sense of the word, is the negative log of its probability. Shannon entropy, also often called information, is the expected self-information of a random variable. Mutual information is the KL divergence of a joint distribution and the product of the respective marginals. There are other notions.

Scandinavian countries did not establish successful profitable colonies (except perhaps Iceland, if you want to count that). It's ambiguous enough whether the average Briton really benefited from their colonial adventures: the Scandinavian peasantry clearly did not. Finland in particular was an impoverished backwater until the early 20th century.

PhDs are apprentices doing skilled labor, not students in the usual sense. An unfunded PhD offer, even in the US, is a polite rejection.

> You forgot the additional 13% in employers NI.

Only the part of that that would actually be passed through to you as salary in the absence of the tax could reasonably be considered part of your tax burden. Hard to say how much that is without any information about your field or employer, but I guarantee it's less than 100%.

Yes, but even homo Economicus isn't trying to maximize expected income, they're trying to maximize expected utility. Unless you're completely indifferent to risk, the sure thing is better.

> despite implementing various welfare programs similar to Scandinavian ones.

There are many countries with welfare regimes similar in size to those of Scandinavia. They are qualitatively quite different. A rough categorization:

- Liberal regimes (means tested but with relatively equal benefits between participants, market oriented) are typical of the Anglosphere. Food stamps in the United States are an archetypical liberal program.

- Corporatist (aka conservative, Christian-democratic) regimes (many recipients with highly unequal benefits, often tied to family status or employment) are typical of continental Europe, especially. Unemployment insurance is an archetypical corporatist program.

- Social-democratic regimes (extensive universal benefits, direct provision of public services, full employment as an explicit policy goal) are typical of Scandinavia. The Finnish national pension system is an archetypical social-democratic program.

Of course no currently existing states are purely one of these ideal types, but they do cluster. Social-democratic welfare states (Sweden, Norway, Finland, Denmark, and to some extent the Netherlands) significantly outperform conservative ones with comparable or higher levels of public spending (e.g. France, Austria, Germany).

> How about other GPs?

Depends on how good your insurance is. PPOs, sure - though most of the country does not have as many doctors per capita as SF. Typically an HMO will only let you have one primary care provider.

It has nothing to do with Marxism, it's just hyped-up mutual insurance.

> also I don't see it very similar to insurance, since insurance is withholding wealth in order to distribute it occasionally when needed because it's zero-sum. But if distributing it all the time everywhere is a net positive gain, that's entirely different than the insurance game.

Insurance is zero-sum (negative-sum, actually) when it comes to money. It is not zero-sum overall, because reducing risk is valuable. A 50/50 shot at either 10 million dollars or nothing is significantly less valuable (to anyone who isn't already very rich) than a guaranteed 5 million.

> Just your luck. Under Univalence identity is equivalent to equivalence and it's computationally decidable. So you should be able to produce a decider which determines whether any alternative encoding/representation is equivalent to the halting problem; or not.

Moving to a constructive setting lets you say things like "all functions are computable" because it has a restricted notion of function. It does not give you any new information about classical objects.

> I am bothered by the fact that most mathematicians aren’t bothered by that! Am I missing something?

Outside of logic and set theory, most mathematicians aren't really working in any particular foundation: they're working directly with the relevant structures. An ordered pair `(x,y)` can be modeled as the set `{{x},{x,y}}`, but that doesn't mean it is that set. What it is, to the extent the question is even coherent, is a thing that has the data ordered pairs are supposed to have: a first element, and a second element.

That's no longer the halting problem: formally, the input to a halting oracle is an index for some fixed choice of admissible numbering of the set of Turing machines, meaning one that can be computed from the standard numbering induced by some universal Turing machine. Your encoding is not admissible.

> then gatekeeping prevents the introduction of evolutionary bottlenecks, meaning the population stagnates in the absence of any external pressure to mutate.

That's the point! The pressures of mass culture point ever TikTok-towards. If you want something else, you need to keep them out.

> There are lots of more nuanced ways to technically tell the truth but still significantly warp the information to your benefit. I’m kind of surprised that China seems to be less savvy to this approach.

That is the national government's approach, for the most part: tweak the basket of goods you use for inflation here, pick the best looking among several measures there, and so on. The overt manipulation occurs at the subnational level: Chinese statistical work is heavily decentralized and officials are promoted (or not) partially on the basis of meeting growth targets. The national government corrects for this to some degree - but estimating just how manipulated the data is is hard, and provides another lever you can pull to push it one way or another.

Federal statistical agencies in the US conduct more and better direct research and report through fewer (though not zero) layers of politically motivated actors.

> Noah has become a political pundit who tends to straw man the opponents to his preferred political ideology of centrist us democrat ... But it’s important to keep in mind when reading articles like this where he’s hitting an extremely reputable magazine that isn’t politically aligned to him.

Huh? The Economist is as liberal-centrist as it gets. Whatever the rift forming here is (and I agree there is one), it's happening within the neoliberal bloc.

Why would they? All of the probability-theoretic structure comes from the fact that a measure is not just an arbitrary one-form, and has lots of extra properties. And then if you want to do probability over arbitrary manifolds, you need to talk about differential forms in general: there's no point at which it makes sense to generalize to one-forms but no further.

Yes, high energy physics probably has the highest author counts of any field. Astro papers average ~5-10, iirc.

> Avi Loeb, head of the astronomy department at Harvard.

He stepped down as chair a few years ago.

> What is Loeb's reputation among astrophysicists?

Not an astrophysicist but several friends are. The consensus seems to be that he was a good scientist at one point but has turned into a sensationalist attention monger.

> He's written "over 800 scientific papers". Is that... a good thing in his field?

He's been an author on over 800 scientific papers: that includes his work, work he's been a significant collaborator on, and student work he's supervised. 800 is a lot for someone his age but not unheard of. People care much more about paper quality (or less charitably, citation count) than quantity.

> I've never come across a name for this development style.

Libraries designed like this are sometimes called "combinator libraries".

You're looking at the chart for "ordinary" poverty, which they've pegged at $30 equivalent per day. Their threshold for extreme poverty is $2.15/day.

The point of abstraction is to reduce the number of concepts in play. If you're still tracking which old concept is "really" being used every time, you haven't actually abstracted over anything, you're just naming things badly.

No. Most were in the $1-10 million net worth range before taking office. Former presidents, of course, can easily make tens of millions on book sales and speaking fees. But that's still small time compared to the owners of major firms.

> In the Culture all real agency humanity has is lost to machines (Minds). Humans have about as much say in important matters in the Culture as dogs do in ours.

The Culture is explicitly a direct democracy, it's just that the books are mostly about what is effectively a somewhat-rogue intelligence agency's activities in other civilizations.

> Politics in the Culture consists of referenda on issues whenever they are raised; generally, anyone may propose a ballot on any issue at any time; all citizens have one vote. Where issues concern some sub-division or part of a total habitat, all those - human and machine - who may reasonably claim to be affected by the outcome of a poll may cast a vote. Opinions are expressed and positions on issues outlined mostly via the information network (freely available, naturally), and it is here that an individual may exercise the most personal influence, given that the decisions reached as a result of those votes are usually implemented and monitored through a Hub or other supervisory machine, with humans acting (usually on a rota basis) more as liaison officers than in any sort of decision-making executive capacity; one of the few rules the Culture adheres to with any exactitude at all is that a person's access to power should be in inverse proportion to their desire for it. The sad fact for the aspiring politico in the Culture is that the levers of power are extremely widely distributed, and very short (see entry on megalomaniacs, above). The intellectual-structural cohesion of a starship of course limits the sort of viable votes possible on such vessels, though as a rule even the most arrogant craft at least pretend to listen when their guests suggest - say - making a detour to watch a supernova, or increasing the area of parkland on-board.

> is everything just function application all the way down?

No non-toy programming language is actually implemented this way, but you can do it if you really want: SK combinator calculus is an example.

> HFT replaces massive floors of human traders doing basically the same job but at much lower cost.

HFT is just one form of algorithmic trading, they're not the same thing.

> Then we're doing the same thing, applying a minimum tick rate to trading, except to achieve this tick rate you have to spend X to get the technology.

Not quite, because the next step is being physically closer to the exchange, and there's only so much real estate to go around.