In my experience those debates are usually between experts who deeply understand the difference between ABC and XYZ widgets (the example I'm thinking of in my head is whether manifolds should be paracompact). The decision between the two is usually an aesthetic one. For example, certain theorems might be streamlined if you use the ABC definition instead of the XYZ one, at the cost of generality.
But the key is that proponents of both definitions can convert freely between the two in their understandings.
IMO it's not far off how most python or javascript devs don't care about registers or cache misses. Someone's thought deeply about those things so you don't have to.
Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study). For example, Wiles' initial proof of Fermat's last theorem used quite exotic machinery called "inaccessible cardinals", which lie outside of ZFC. Subsequent work showed they weren't needed.
Another good example of mathematicians caring which 'house of cards' their results are built on is the search for an "elementary" proof of the prime number theorem (i.e. showing it doesn't rely on complex analysis).
> Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study). For example, Wiles' initial proof of Fermat's last theorem used quite exotic machinery called "inaccessible cardinals", which lie outside of ZFC. Subsequent work showed they weren't needed.
In a way mathematicians can afford to do this more readily than people in software development, because if something is actually proven, then you can 100% rely on that. With software not so much. Or rather: Software usually is not proven to be correct, because that's usually expensive. In mathematics they don't have to consider the runtime of an algorithm, when they "merely" need to prove correctness. The time it needs to run is irrelevant for its correctness. And so they can stack and stack and stack, provided that each piece is proven correct, and it won't have negative consequences. Well, almost. There is some negative consequence in that another human being, wanting to understand a proof, needs to know perhaps many concepts and other proofs, in order to be able to do so. But that's probably the only reason to pursue simplicity in mathematics.
> The time it needs to run is irrelevant for its correctness. And so they can stack and stack and stack
This is a very naive take - the very direct translation of what you're saying doesn't happen does in happen in analysis all the time: there are many inequalities which can be "stacked" to prove a bound on something but their factors are too large so you cannot just stack them if you need a fixed bound for your proof to go through. Unsurprisingly this is exactly how actual runtime analysis also works (it's unsurprising because they're both literally math).
I think you are taking it a bit out of context here. Obviously, I assumed in that phrase, that mathematicians are stacking suitable methods and proofs. If some factors are too large to not fit in some bound, then obviously that's not something you would stack. But once you have suitable proofs and proven correct and suitable methods, you can stack, and correctness does not go out of the window. Correctness remaining, you will be able to get to a proven correct result.
Of course proving things in mathematics is also a lot harder, usually, than computer programming, and it is probably still easy to make mistakes.
> Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study).
I'd be careful about generalizing that to all or most 'mathematicians'. E.g., people working in a lot of fields won't bat an eye at invoking the real numbers when the rational or algebraic numbers would do.
I'm sure some python devs care about cache misses too. I guess my point was that the big results will be picked over again and again to understand _exactly_ which conditions are needed for them to hold.
Anyone have a good explanation for why elliptic curves have a 'natural' group law? I've seen the definition of the group law in R before, where you draw a line through two points, find the third point, and mirror-image. I feel like there's something deeper going on though.
As far as I've seen, the group law is what makes elliptic curves special. Are they the _only_ flavour of curve that has a nice geometric group law? (let's say aside from really simple cases like lines through the origin, where you can just port over the additive group from R)
I find a lot of motivation from topology. If you plot a smooth degree d curve over the complex numbers, it forms a surface of degree g=(d-1)(d-2)/2. In the case of a cubic, we get genus 1, i.e. a torus. Now tori admit a very natural group action, namely addition in (R/Z)^2. And sure enough, if you pick the right homeomorphism, this corresponds to the group action given by the elliptical curve.
Of course, the homeomorphism to (R/Z)^2 does not respect the geometry (it is not conformal). If we want the map to preserve angles, we need our fundamental domain to be a parallelogram instead of a rigid square. The shape of the parallelogram depends on the coefficients of the cubic, and the isomorphism is uniquely defined up to choice of a base point O (mapping to the identity element; for elliptic curves, this is normally taken to be the point at infinity). You still get a group law on the parallelogram from vector addition in the same way, and this pulls back to the precise group action on the elliptic curve.
The real magic is that the resulting group law is algebraic, meaning that a*b can be written as an algebraic function of a and b. This means you can do the same arithmetic over any field, not just the complex numbers, and still get a group action.
>
Anyone have a good explanation for why elliptic curves have a 'natural' group law? [...]
As far as I've seen, the group law is what makes elliptic curves special. Are they the _only_ flavour of curve that has a nice geometric group law?
I asked the same question to a professor who works in topics related to algebraic geometry. His answer was very simple: it's because elliptic curves form Abelian varieties
Being an algebraic group means that the group law on the variety can be defined by regular functions.
Basically, he told to read good textbooks about abelian varieties if one is interested in this topic.
> Are they the _only_ flavour of curve that has a nice geometric group law?
The Jacobian of a hyperelliptic curve (which generalize elliptic curves) also forms an abelian variety. Its use in cryptography is named "hyperelliptic curve cryptography":
If X is a smooth projective curve over an algebraically closed field k, then we can make a (huge, useless) abelian group Div(X) which is the set of formal sums of points on X. (The "free abelian group" on X).
It would be flippant to say Div(X) is an answer to your question, since it has nothing to do with geometry at all (we can form the free abelian group on any set). An element of Div(X) looks like \sum n_i P_i where n_i are integers and P_i are points on X, and they "add" in the obvious way. The sum doesn't "mean" anything. But we can get to geometry from it.
Inside Div(X) there is a subgroup, Div^0(X), of formal sums of points such that the set of coefficients is zero. Still nothing to do with geometry.
Inside Div^0(X), there is a very interesting subgroup, which is the set of "divisors of functions." Namely, if f is a rational function on X (meaning it's locally a quotient of polynomials), we get an element of Div^0(X) by taking \sum P_i - \sum Q_i where P_i are the zeroes of f and Q_i are the poles (caveat - you have to count them with multiplicity). This is an element of Div(X) but is not obviously an element of Div^0(X) -- this uses the fact that X is projective. Let's call the subgroup that comes this way Princ(X) (for "principal" divisors).
Now we get an interesting group that does have something to do with geometry, which is called Pic^0(X), by taking the quotient Div^0(X)/Princ(X).
Amazing theorem: there is a natural isomorphism from X to Pic^0(X) if and only if X is of genus one, i.e. an elliptic curve. (In general, Pic^0(X) is an abelian variety whose dimension is the genus of the corresponding curve.) This is why only elliptic curves (among the projective ones) are "naturally" groups. The relationship with the usual picture with the lines is that the intersection locus of the lines is the principal divisor associated with a functional that vanishes along the line.
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.
A second special case of this theorem is Pascal's theorem, which says (roughly) that a variant of the elliptic curve group law also works on the union of a conic C and a line L (this union, like an elliptic curve, is cubic), where the group elements are on the conic. One point O on the conic is marked as the identity. To add points A+B, you draw a line AB between them, intersect that with the fixed line L in a point C, draw a second line CO back through the marked identity point, and intersect again with the conic in D:=A+B. This procedure obviously commutes and satisfies the identity law, and according to Pascal's theorem it associates.
Under a projective transformation, if the conic and line don't intersect, you can send the line to infinity and the conic to the units in (IIRC) a quadratic extension of F (e.g. the complex unit circle, if -1 isn't square in F). Since the group structure is defined by intersections of lines and conics, projective transformations don't change it. So the group is isomorphic to the group of units in an extension of F. If they do intersect ... not sure, but I would guess it instead becomes the multiplicative group in F itself.
The multiplicative group of F can be used for cryptography (this is classic Diffie-Hellman), as can the group of units in an extension field (this is LUCDIF, or in the 6th-degree case it's called XTR). These methods are slightly simpler than elliptic curves, but there are subexponential "index calculus" attacks against them, just like the ones against the original Diffie-Hellman. The attack on extension fields got a lot stronger with Joux's 2013 improvements. Since no such attack is known against properly chosen elliptic curves, those are used instead.
> Are they the _only_ flavour of curve that has a nice geometric group law?
For affine conics over the real numbers, the non-degenerate ones are ellipses (affine transform to complex unit circle), hyperbolas (affine transform to y=1/x and use the group law (x,y)(x',y')=(xx',yy')) and parabolas (affine transform to y=x^2 and use (x,y)(x',y')=(xx',yy')).
I was thinking about projective conics, but it turns out there are no algebraic group laws on those, because they're always ill-defined over an algebraically-closed field. Moreover, over the reals and other non-algebraically closed fields k, the definition of a "regular map" needs to consider points with coordinates taking values in the algebraic closure of k.
That's more like the false positive rate and false negative rate.
If we're being literal, accuracy is (number correct guesses) / (total number of guesses). Maybe the folks at turnitin don't actually mean 'accuracy', but if they're selling an AI/ML product they should at least know their metrics.
The word “some” in the quote from Box is doing a lot of heavy lifting.
If a model is useful, I’d like to see it being used (outside academia, where there’s minimal penalty for complexity and a high emphasis on novelty).
If models like these are widely adopted at social media companies or news agencies, it’s fair to say OP’s take isn’t valid. Otherwise they may have a point.
Agreed. But these kinds of models almost always start in academia, that's one of the big reasons we have academia, to explore ideas that may (or may not) be useful. My point was that you can't prejudge the usefulness of a model simply because it doesn't fully replicate all the complexity of the phenomenon being modeled.
These ideas are used, and they influence what policy is crafted.
You can’t predict what an individual will do, but work like this kills many inaccurate ideological positions that we inherited.
There’s a paper from 2016 that shows how posts saturate/cascade through conspiracy communities and that it has distinct cascade dynamics. This wasn’t a model, it was a description of observed behavior.
Or take some relatively recent work from Harvard, which suggests that while our capacity to create misinformation has increased in both quantity and quality, its consumption rate seems to be stable.
> kills many inaccurate ideological positions that we inherited.
It doesn't, which is part of the point the OP is making. And now my point, it's ok that these pseudo-scientific "revelations" don't kill those "inaccurate ideological positions", because that's the whole point of human free will, there's no "accurate ideological position" when it comes to the day-to-day life, or to societal life in general.
> That said, I suspect you haven’t read the paper and are arguing from the headline.
You're correct on that, as I find that applying the word "science", or "research", or "paper", to the day-to-day life, like to news article in this case, is not science per se, and unfortunately I don't have time to lose on drivel that paints itself as science (but which it is not science, as I mentioned above).
A metre of sea-level rise is painful for a rural cottage by the sea. But if you're in a city - particularly a wealthy city - it's something that can be engineered around.
An expensive liability? Definitely. A civilization or nation ending event? Unlikely.
And how will these engineered workarounds be paid for ? It is known workarounds will cost trillions today, NYC alone could cost $1T+. And these workarounds should have been started 5 years ago when it became very clear we will never get off fossil fuels.
I fully expect no workarounds will be done just like Climate Change Mitigations. Getting off fossil fuels should have been seriously started 30 years ago, and maybe even 50 years ago. Instead the politicians have been adding hot air talking and fighting instead of doing real work.
We are now seeing this repeat with "engineered workarounds", no one wants to pay for it, so yes I call BS on the article.
All I can say is I feel real bad the past generations did nothing to really reverse CC, people being born now are looking at a very bleak future.
$1 trillion is one year of Manhattan's GDP. Painfully expensive? Absolutely, but it's absolutely affordable over the course of a few decades.
The sooner we start, the cheaper it will be, so we shouldn't put it off, but it's not going to kill everyone or even convince everyone to leave NYC in the foreseeable future.
besides the fact that 40% of the world's population lives near the coast - and that 2-3 feet of sea level rise is not a uniform "the tide used to be 8 feet, now it's 11 feet" - Entire islands in the pacific will disappear - How do you think global trade works? What do you think happens to ports? AMOC collapsing (a byproduct of sea level rise) will have profound effects on climate, despite this author claiming without any evidence whatsoever that "actually it isn't a big deal."
Ports get retrofitted, redesigned, and rebuilt. The AMOC collapsing is a serious thing, but I'm not saying climate change isn't real or isn't a threat. My original point is that three feet of sea level rise is manageable, if expensive. Simply that, nothing else.
If you draw the line at the year 2100, things are uncomfortable but maneagable. If your horizon is 2300 or 2500, you get a different story. But you would hope that in tha sort of time frame, we have time to adapt.
Anyone with a bit of common sense can understand that that "massively retrofitting cities in a way unseen in centuries due to climate risk" is just an off-the-charts level of reconditioning of society that I don't understand how that doesn't fall into "extremely alarming" levels of concern.
I didn't say it wasn't alarming, I said it wasn't civilization- or nation-ending. Unless you're a tiny island nation, in which case I'll happily retract what I'm saying.
There are degrees of awfulness between "the end of all mankind" and "nothing to see here", but it seems like there's a taboo on calling those shades of grey out when it comes to climate change.
So after many decades of wildly under estimating the rate of climate change, the same people in the same institutions, answering to the same money, have it sussed out?
This simple fact that ALL global shipping happens at sea level, and ALL shipping infrastructue is designed and built to operate in a rarrow range, and that this
whole edifice, minutly complicated, can be
adjusted continiously along with the million miles of coastal roads and bridges.
?Londan just walled off, all of NYC's wharfs
jacked up a bit, sure, sure, whats a few dozen cubic miles of equipment refit worth anyway, phffff
Canada, this year committed to spending $3.9 billion dollars to hopefully have just completed plans for a high-speed train line in six years [1]! The number of years and dollars to actually build the line are unknown at the moment. This is a project that has humongous potential economic upside.
Would Canada be able to build a seawall to protect Vancouver? I am not sure.
Fun fact - sea level rose 120m since 20,000 years ago but people seem to have largely not noticed. If you don't have large buildings and planning laws you could just move your shack a few yards inland.
He's comparing AGW, which drives a trend, with weather-based events, which are noise around the trend. He conveniently cuts his analysis off at the year 2100, by which we'll all probably be dead. But he's probably right that the trend itself doesn't cause insurmountable problems by that point.
But what about the year 2200, or 2300? At three degrees warming per century, the earth looks like a pretty hostile place to live in a few centuries.
"A society grows great when old men plant trees in whose shade they will never sit", and all that...
His idea of it being "not insurmountable" is essentially us not starving to death in a mass scale.
I care, in my comfortable life as an office worker, about the fact that chocolate, coffee, and wine will become luxuries as yields and quality drastically drop off.
I care about the fact that many places I visit frequently will need A/C to be _survivable_.
Those are not civilization-ending events but the hubris you need to have to just hand-wave this away are beyond my understanding.
But the key is that proponents of both definitions can convert freely between the two in their understandings.
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