Oct 30, 2019 06:59 PM
(This post was last modified: Oct 31, 2019 05:58 PM by C C.)
Do you believe in rectangles?
https://physicstoday.scitation.org/do/10...018a/full/
EXCERPT: In my previous installment of this column, I wrote about a fundamental difference between mathematics and physics: The former is a deductive endeavor in which the conclusions follow with certainty from the premises, and the latter is grounded in observations, measurements, and inductive reasoning.
[...] if you haven’t heard this story before, you might think it’s getting a bit ridiculous. Isn’t it obvious that rectangles exist? They’re everywhere, after all. ... Any material rectangle is at best an approximation of the Platonic ideal rectangle—a figure with four perfectly right angles—but surely that’s just because humans are imperfect at drawing rectangles, not because the Platonic form doesn’t exist, right? If you have heard this story before, then you know what happens next: Over the course of the 19th century, it came to light that the parallel postulate—and the existence of rectangles—could never be proved because there are alternative geometries, just as self-consistent as Euclidean geometry, in which it’s false. There’s hyperbolic geometry, in which there are infinitely many lines (or as mathematicians sometimes put it, “at least two”) through P that are parallel to ℓ. And there’s elliptic geometry, which contains no parallel lines at all.
Elliptic geometry is a bit of a headache to make rigorous, but it’s the easier of the two to visualize: It’s just geometry on the surface of a sphere. Hyperbolic geometry, although beautiful, can be more difficult to intuitively grasp, but you can think of it as the opposite of elliptic geometry. In the elliptic plane, lines get closer together as you extend them; in the hyperbolic plane, they get farther apart. In elliptic geometry, the sum of the angles of a triangle is more than 180°; in hyperbolic geometry, it’s less. In neither geometry do rectangles exist, although in elliptic geometry there are triangles with three right angles, and in hyperbolic geometry there are pentagons with five right angles (and hexagons with six, and so on).
It’s hard to overstate what a paradigm shift the new geometries represented. Euclid’s geometry may not have been motivated by its practical utility—the Elements doesn’t even discuss applications—but it was always meant to be about shapes that exist in the real world. But now there was not just one geometry but three, and they contradicted one another. At most one of them could be the true geometry of physical space. But the other two, whichever ones they were, were just as internally valid and just as “real” in the realm of ideas.
The structure of geometry, therefore, was neither logically inevitable nor constrained by anything in the physical world—it all depended on the assumptions you chose to start with. Mathematicians were free to dream up whatever abstractions they liked, limited only by their imaginations. But with that freedom came responsibility. Because they could no longer fall back on physical intuition, as Euclid had, to fill in the gaps in their lists of axioms, mathematicians needed to take much greater care in declaring their assumptions. As mathematics grew more inventive, it also became more rigorous.
At the same time, the new geometries raised the idea that physical space might not be Euclidean after all. For shapes on a human scale, Euclidean geometry is at least an excellent approximation, but on astrophysical or cosmic scales, space might be curved, and that curvature might mean something. The theory of general relativity—which Albert Einstein said he could never have developed if he hadn’t known about non-Euclidean geometry—holds that spacetime is locally curved in the vicinity of matter or energy. It also allows for the possibility that the universe as a whole might be flat (Euclidean), positively curved (elliptic), or negatively curved (hyperbolic). (MORE - details)
An Unpredictable Universe: A Deep Dive Into Chaos Theory
https://www.space.com/chaos-theory-expla...stems.html
EXCERPT: . . . This is the signature sign of a chaotic system, as first identified by Henri Poincaré. Normally, when you start a system off with very small changes in the initial conditions, you get only very small changes in the output. But this is not the case with the weather. One tiny change ... can lead to a giant difference in the weather... Chaotic systems are everywhere and, in fact, dominate the universe. Stick a pendulum on the end of another pendulum, and you have a very simple but very chaotic system. The three-body problem puzzled over by Poincaré is a chaotic system. The population of species over time is a chaotic system. Chaos is everywhere. This sensitivity to initial conditions means that with chaotic systems, it's impossible to make firm predictions, because you can never know exactly, precisely, to the infinite decimal point the state of the system. And if you're off by even the tiniest bit, after enough time, you'll have no idea what the system is doing. This is why it's impossible to perfectly predict the weather.
There are a number of surprising features buried in this unpredictability and chaos. They appear mostly in something called phase space, a map that describes the state of a system at various points in time. If you know the properties of a system at a specific "snapshot," you can describe a point in phase space. As a system evolves and changes its state and properties, you can take another snapshot and describe a new point in phase space, over time building up a collection of points. With enough such points, you can see how the system has behaved over time.
Some systems exhibit a pattern called attractors. This means that no matter where you start the system, it ends up evolving into a particular state that it is especially fond of. For example, no matter where you drop a ball in a valley it will end up at the bottom of the valley. That bottom is the attractor of this system. When Lorenz looked at the phase space of his simple weather model, he found an attractor. But that attractor didn't look like anything that had been seen before. His weather system had regular patterns, but the same state was never ever repeated twice. No two points in phase space ever overlapped. Ever. This seemed like an obvious contradiction. There was an attractor; i.e., the system had preferred set of states. But the same state was never repeated. The only way to describe this structure is as a fractal.
If you look at the phase space of Lorenz's simple weather system and zoom in on a small piece of it, you will see a tiny version of the exact same phase space. And if you take a smaller portion of that and zoom in again, you will see a tinier version of the exact same attractor. And so on and so on to infinity. Things that look the same the closer you look at them are fractals. So the weather system has an attractor, but it's strange. That's why they're literally called strange attractors. And they crop up not just in weather but in all sorts of chaotic systems.
We don't fully understand the nature of strange attractors, their significance, or how to use them to work with chaotic and unpredictable systems. This is a relatively new field of mathematics and science, and we're still trying to wrap our heads around it. It's possible that these chaotic systems are, in some sense, deterministic and predictable. But that's yet to be figured out... (MORE - details)
Nonsense lives on in the citations
https://blogs.sciencemag.org/pipeline/ar...-citations
INTRO: It’s apparent to anyone who’s familiar with the scientific literature that citations to other papers are not exactly an ideal system. It’s long been one of the currencies of publication, since highly-cited work clearly stands out as having been useful to others and more visible in the scientific community (the great majority of papers do actually get cited eventually by someone, by the way). But anything that be measured will be managed, and managed includes the darker meanings “gamed” and “manipulated”. The classic method is to cite your own work to hell and gone, but readers will have heard of reviewers who demand that their own work be cited, of citation rings where everyone gets together to boost each other’s numbers, of citations for sale, of publishers packing their own journals with internal references, and more schemes besides.
Now, outside of this sort of chicanery, you see many other problems: (1) people citing things because other people have cited them, not because they’ve actually looked over said reference themselves, (2) people just flat missing things, relevant papers (or patents!) that really would shore up their own arguments but don’t even get a look, and (3) people citing things that don’t necessarily do the job that they seem to think it does.
In that last category I put a special irritating feature of the synthetic organic chemistry literature, one that every bench chemist sees coming before I reach the end of this sentence. I refer to the nesting-doll method of referencing the preparation of some compound: instead of telling everyone how you made it, you just say that it was prepared by the method of Arglebargle, reference 15. So you go look up the Arglebargle paper and find that they don’t tell you how to make the damn thing, either, but refer you to Dingflinger et al. in the even earlier literature. I have had the Dingflinger-level papers themselves send me to yet a third reference, by now something written during the Weimar Republic and of course containing the finest spectral characterization data available in 1931, which ain’t much. Would-it-have-killed-you-to-put-in-the-procedure-and-the-NMR-data, etc.
So let’s make sure not to forget the major influences of laziness and stupidity on citation behavior.... (MORE)
https://physicstoday.scitation.org/do/10...018a/full/
EXCERPT: In my previous installment of this column, I wrote about a fundamental difference between mathematics and physics: The former is a deductive endeavor in which the conclusions follow with certainty from the premises, and the latter is grounded in observations, measurements, and inductive reasoning.
[...] if you haven’t heard this story before, you might think it’s getting a bit ridiculous. Isn’t it obvious that rectangles exist? They’re everywhere, after all. ... Any material rectangle is at best an approximation of the Platonic ideal rectangle—a figure with four perfectly right angles—but surely that’s just because humans are imperfect at drawing rectangles, not because the Platonic form doesn’t exist, right? If you have heard this story before, then you know what happens next: Over the course of the 19th century, it came to light that the parallel postulate—and the existence of rectangles—could never be proved because there are alternative geometries, just as self-consistent as Euclidean geometry, in which it’s false. There’s hyperbolic geometry, in which there are infinitely many lines (or as mathematicians sometimes put it, “at least two”) through P that are parallel to ℓ. And there’s elliptic geometry, which contains no parallel lines at all.
Elliptic geometry is a bit of a headache to make rigorous, but it’s the easier of the two to visualize: It’s just geometry on the surface of a sphere. Hyperbolic geometry, although beautiful, can be more difficult to intuitively grasp, but you can think of it as the opposite of elliptic geometry. In the elliptic plane, lines get closer together as you extend them; in the hyperbolic plane, they get farther apart. In elliptic geometry, the sum of the angles of a triangle is more than 180°; in hyperbolic geometry, it’s less. In neither geometry do rectangles exist, although in elliptic geometry there are triangles with three right angles, and in hyperbolic geometry there are pentagons with five right angles (and hexagons with six, and so on).
It’s hard to overstate what a paradigm shift the new geometries represented. Euclid’s geometry may not have been motivated by its practical utility—the Elements doesn’t even discuss applications—but it was always meant to be about shapes that exist in the real world. But now there was not just one geometry but three, and they contradicted one another. At most one of them could be the true geometry of physical space. But the other two, whichever ones they were, were just as internally valid and just as “real” in the realm of ideas.
The structure of geometry, therefore, was neither logically inevitable nor constrained by anything in the physical world—it all depended on the assumptions you chose to start with. Mathematicians were free to dream up whatever abstractions they liked, limited only by their imaginations. But with that freedom came responsibility. Because they could no longer fall back on physical intuition, as Euclid had, to fill in the gaps in their lists of axioms, mathematicians needed to take much greater care in declaring their assumptions. As mathematics grew more inventive, it also became more rigorous.
At the same time, the new geometries raised the idea that physical space might not be Euclidean after all. For shapes on a human scale, Euclidean geometry is at least an excellent approximation, but on astrophysical or cosmic scales, space might be curved, and that curvature might mean something. The theory of general relativity—which Albert Einstein said he could never have developed if he hadn’t known about non-Euclidean geometry—holds that spacetime is locally curved in the vicinity of matter or energy. It also allows for the possibility that the universe as a whole might be flat (Euclidean), positively curved (elliptic), or negatively curved (hyperbolic). (MORE - details)
An Unpredictable Universe: A Deep Dive Into Chaos Theory
https://www.space.com/chaos-theory-expla...stems.html
EXCERPT: . . . This is the signature sign of a chaotic system, as first identified by Henri Poincaré. Normally, when you start a system off with very small changes in the initial conditions, you get only very small changes in the output. But this is not the case with the weather. One tiny change ... can lead to a giant difference in the weather... Chaotic systems are everywhere and, in fact, dominate the universe. Stick a pendulum on the end of another pendulum, and you have a very simple but very chaotic system. The three-body problem puzzled over by Poincaré is a chaotic system. The population of species over time is a chaotic system. Chaos is everywhere. This sensitivity to initial conditions means that with chaotic systems, it's impossible to make firm predictions, because you can never know exactly, precisely, to the infinite decimal point the state of the system. And if you're off by even the tiniest bit, after enough time, you'll have no idea what the system is doing. This is why it's impossible to perfectly predict the weather.
There are a number of surprising features buried in this unpredictability and chaos. They appear mostly in something called phase space, a map that describes the state of a system at various points in time. If you know the properties of a system at a specific "snapshot," you can describe a point in phase space. As a system evolves and changes its state and properties, you can take another snapshot and describe a new point in phase space, over time building up a collection of points. With enough such points, you can see how the system has behaved over time.
Some systems exhibit a pattern called attractors. This means that no matter where you start the system, it ends up evolving into a particular state that it is especially fond of. For example, no matter where you drop a ball in a valley it will end up at the bottom of the valley. That bottom is the attractor of this system. When Lorenz looked at the phase space of his simple weather model, he found an attractor. But that attractor didn't look like anything that had been seen before. His weather system had regular patterns, but the same state was never ever repeated twice. No two points in phase space ever overlapped. Ever. This seemed like an obvious contradiction. There was an attractor; i.e., the system had preferred set of states. But the same state was never repeated. The only way to describe this structure is as a fractal.
If you look at the phase space of Lorenz's simple weather system and zoom in on a small piece of it, you will see a tiny version of the exact same phase space. And if you take a smaller portion of that and zoom in again, you will see a tinier version of the exact same attractor. And so on and so on to infinity. Things that look the same the closer you look at them are fractals. So the weather system has an attractor, but it's strange. That's why they're literally called strange attractors. And they crop up not just in weather but in all sorts of chaotic systems.
We don't fully understand the nature of strange attractors, their significance, or how to use them to work with chaotic and unpredictable systems. This is a relatively new field of mathematics and science, and we're still trying to wrap our heads around it. It's possible that these chaotic systems are, in some sense, deterministic and predictable. But that's yet to be figured out... (MORE - details)
Nonsense lives on in the citations
https://blogs.sciencemag.org/pipeline/ar...-citations
INTRO: It’s apparent to anyone who’s familiar with the scientific literature that citations to other papers are not exactly an ideal system. It’s long been one of the currencies of publication, since highly-cited work clearly stands out as having been useful to others and more visible in the scientific community (the great majority of papers do actually get cited eventually by someone, by the way). But anything that be measured will be managed, and managed includes the darker meanings “gamed” and “manipulated”. The classic method is to cite your own work to hell and gone, but readers will have heard of reviewers who demand that their own work be cited, of citation rings where everyone gets together to boost each other’s numbers, of citations for sale, of publishers packing their own journals with internal references, and more schemes besides.
Now, outside of this sort of chicanery, you see many other problems: (1) people citing things because other people have cited them, not because they’ve actually looked over said reference themselves, (2) people just flat missing things, relevant papers (or patents!) that really would shore up their own arguments but don’t even get a look, and (3) people citing things that don’t necessarily do the job that they seem to think it does.
In that last category I put a special irritating feature of the synthetic organic chemistry literature, one that every bench chemist sees coming before I reach the end of this sentence. I refer to the nesting-doll method of referencing the preparation of some compound: instead of telling everyone how you made it, you just say that it was prepared by the method of Arglebargle, reference 15. So you go look up the Arglebargle paper and find that they don’t tell you how to make the damn thing, either, but refer you to Dingflinger et al. in the even earlier literature. I have had the Dingflinger-level papers themselves send me to yet a third reference, by now something written during the Weimar Republic and of course containing the finest spectral characterization data available in 1931, which ain’t much. Would-it-have-killed-you-to-put-in-the-procedure-and-the-NMR-data, etc.
So let’s make sure not to forget the major influences of laziness and stupidity on citation behavior.... (MORE)