That's not nessecarily true. Bigger (cardinality?) doesn't mean more or less inner structures. If anything, it's the assumptions you choose to make that determine how much you can say about an object.
A reason why the complex numbers is tame in certain contexts is that it's algebraically closed. Discrete theory also tends to be more difficult imo since by taking its limit, you should still recover the continuous theory.
It doesn't have much to do with how "large" the set is. The complex numbers have the same cardinality as the reals; the rational numbers have the same cardinality as the integers and whole numbers. Even talking about finite sets, the group of integers modulo 13 is a lot less "interesting" than the group of integers modulo 12, because how such a group acts depends heavily on the division structure of the modulus: the more divisors there are in the modulus, the more interesting things get. So the sets of integers modulo any prime are all basically the same (and boring), while the integers modulo highly composite numbers like 60 are wild, complex structures.
A field is more delicate if it needs to be enlarged (into a larger field) for all polynomials with coefficients in the field to have roots. There is rich and intricate structure in the possible successive enlargements.
For the field C of complex numbers, all polynomials already have roots so there is no need to enlarge. In that sense the complex field has much less structure than the smaller ones (Z, Q, R).
I expect that it is because in general, more useful functions and operations are closed in the complex plane. Things that may superficially appear complicated to a human being learning math, like analytic extension [1], from a theoretical perspective means that when you need the tool, it is there, and continues to expand the sets of problems you can address.
As you add restrictions to the set of numbers you want to use, you are importing those restrictions into every proof you want to do on those numbers. For instance, consider the question, "Does this polynomial of some high degree have roots, and if so, how many?"
The Fundamental Theorem of Algebra proves that the answer is yes, and the number of roots is the same as the degree (although some may be roots multiple times). See [2].
Now, as you watch 2, consider what happens to the proof if you confine it even to the reals, let alone the integers. While the complex plane is "more complex" from a human perspective, the proof of the Fundamental Theorem of Algebra using the complex plane is not that complicated. Consider trying to make equivalent statements about polynomials with only real roots, and not using the complex plane in the proofs. In cases where it is feasible, your proofs will inevitably be carrying around a lot more caveats about what polynomials it applies to, and there may be things that the caveats simply render infeasible. As you step down the number hierarchy, the caveats get worse and worse. There's more and more "holes" that every proof about those simpler numbers has to step around.
The same simplicity that makes it easier to start your education with just integers becomes a crippling limitation when trying to work with them, which is also in some sense the exact same reason why we have to step you up the number hierarchy even in non-specialist education. Math limited to just integers is so confining and difficult to work with that it isn't even enough for day-to-day life. The simplicity is a double-edged sword.
It's quite a long one (40 tweets)! As a non-Twitter user, I will never understand why this is preferable to a blog post. However, some people seem to prefer it, and it's getting engagement from non-Mathematicians, which is all that matters really.
The complex numbers ℂ have an extra symmetry: complex conjugation. Objects which respect the full structure of ℂ have to respect this symmetry, which leads to overall nicer behavior.
For example, if a complex function 𝑓 is differentiable once then...
· It's differentiable an infinite number of times (smooth)
· It has a Taylor series approximation at every point (analytic)
x^4 + y^4 = z^4 is itself a great example of this.
In C, we can just specify any pair from x,y,z and the third is determined by rearranging.
In R, it's not true! (y=2 and z=1 runs into trouble.) However, there are still infinitely many solutions and they're still easy to find.
In Z, we have the solution 0,0,0.
In N, we meet disaster!
To squeeze some meaning out of this, C is in some ways nicer because it is designed for the very purpose of containing solutions to things. We only got Q because ratios aren't integers so we added them in; we got R because sqrt(2) is famously irrational so we take the completion of Q... As we go on we just sort of smooth things out and get a better behaved, richer space. The "gaps" in N can make proving things feel impossible (sometimes rightly so).
If you try to build Calculus on top of the complex numbers, you get complex analysis where every differentiable function winds up locally a power series, and things just work out over and over again unreasonably nicely.
Build Calculus on top of the real numbers, and as you dive in you wind up with menagerie of counterexamples. For example Weierstrass produced a function which is everywhere differentiable and nowhere twice differentiable. These are not mere curiosities, for example wavelets (heavily used in data analysis) always are only differentiable a finite number of times. We can ask whether infinitely differentiable functions can be represented by a power series. e^(-1/x^2) is the classic example of one which can't be at 0. https://en.wikipedia.org/wiki/Non-analytic_smooth_function#A... offers one which is nowhere. And so on.
But now let's go from the real numbers to the natural numbers. The real numbers have an axiom system which is known to be consistent and complete. See https://math.stackexchange.com/a/362840/6708 for some of the details. However the natural numbers famously do not. (That's Gödel's Incompleteness Theorem.) What goes wrong? Philosophically it turns out that the natural numbers can encode computation, computation can encode reasoning, and reasoning about reasoning allows us to encode the liar's paradox. Everything blows up, hopelessly, after that.
At a conceptual level what is happening is this. The complex numbers have a tremendous amount of structure. As you go towards having less and less structure, we can use that freedom to write down more logically complex things. Logically complex things give ways for stuff to go wrong. And the more depth you study these things in, the more that turns out to matter.
Think of the following: there are many many paths to get from one location to another on a plane. If you imagine a function on the plane, it’s behavior as you walk along each part would have a sanity constraint. That (along with the constraints of behavior under complex conjugation, should you choose to impose “analyticity”) adds up to a lot of constraints, which is the “structure” in complex numbers — which makes complex analysis simple.
For example if you draw a loop, specify a continuous function on the loop, and say it has to be differentiable inside of that loop, you can prove from the 2-d structure of differentiable functions that the value of the function at every point must be given by a specific integral around that loop. And from that you can start to recover a lot of the structure.
Here is a trivial example. The value of a differentiable function inside of the loop has reach its maximum absolute value at the boundary. Now consider a polynomial p(z) which doesn't have 0 as a root. 1/p(z) is differentiable everywhere that p(z) is not zero. If we draw a large enough circle around the origin, the value of 1/p(z) on that boundary is arbitrarily close to 0. But it isn't arbitrarily close to 0 at the origin, and therefore somewhere in that circle it must not be differentiable. And that point is a root of p(z).
This is the first proof that I saw that every polynomial over the complex numbers has a root. And it heavily uses the geometric structure of differentiable functions over the complex numbers. Note, that the reals lack this 2-D structure, and it is likewise easy to come up with polynomials over the reals that do not have real roots.
Is it related to (high school) polnomials always having complex roots, but not necessarily real roots? But that constraint on reals seems like "structure" to me, and complex numbers lack this structure...?
(It may well just require a few more years math on ny part.)
Yes. Polynomials over the complex numbers can always be factored completely based on the roots. Polynomials over the real numbers cannot. The result is easy to prove using complex analysis, which uses the 2-D structure of the complex numbers to prove it.
Interesting, so you're telling me there is a mathematical basis for why things always work out over and over again in the imaginary world but not in the real world?
hmm, I've deleted a long reply elaborating the distinction between syntactic and semantic completeness (the terminology is a mess). Just wanted to add that your interesting point shines here:
"but they are filtered by an extensive blocklist ... That account had 16000 and no such filters."
Does he mean his account has 1800 followers and 16000 users/comments on blocklist? I do not use Twitter but this absurd ratio makes me doubt if my understanding is correct
I understood it that he did this communication thing on some organization's twitter account which was wild and different from his usual personal experience. Eg raising the question of the difference in interactions that you get from a 1800 follower account and a 16k one.
I interpreted it as an artistic choice, since the outreach Twitter account is all German. It's different from the article's actual title though - which I thought was an anti-pattern for HN.
26 comments
[ 2.6 ms ] story [ 71.9 ms ] threadAkin to “you cannot say much about a genus but you can say a huge lot about a species”.
Of course this is a very rough explanation if at all.
Edit: there is much more to it, obviously, than my lame comparison.
A reason why the complex numbers is tame in certain contexts is that it's algebraically closed. Discrete theory also tends to be more difficult imo since by taking its limit, you should still recover the continuous theory.
For the field C of complex numbers, all polynomials already have roots so there is no need to enlarge. In that sense the complex field has much less structure than the smaller ones (Z, Q, R).
As you add restrictions to the set of numbers you want to use, you are importing those restrictions into every proof you want to do on those numbers. For instance, consider the question, "Does this polynomial of some high degree have roots, and if so, how many?"
The Fundamental Theorem of Algebra proves that the answer is yes, and the number of roots is the same as the degree (although some may be roots multiple times). See [2].
Now, as you watch 2, consider what happens to the proof if you confine it even to the reals, let alone the integers. While the complex plane is "more complex" from a human perspective, the proof of the Fundamental Theorem of Algebra using the complex plane is not that complicated. Consider trying to make equivalent statements about polynomials with only real roots, and not using the complex plane in the proofs. In cases where it is feasible, your proofs will inevitably be carrying around a lot more caveats about what polynomials it applies to, and there may be things that the caveats simply render infeasible. As you step down the number hierarchy, the caveats get worse and worse. There's more and more "holes" that every proof about those simpler numbers has to step around.
The same simplicity that makes it easier to start your education with just integers becomes a crippling limitation when trying to work with them, which is also in some sense the exact same reason why we have to step you up the number hierarchy even in non-specialist education. Math limited to just integers is so confining and difficult to work with that it isn't even enough for day-to-day life. The simplicity is a double-edged sword.
[1]: 3Blue1Brown on the topic: https://www.youtube.com/watch?v=sD0NjbwqlYw
[2]: https://www.youtube.com/watch?v=shEk8sz1oOw
It's quite a long one (40 tweets)! As a non-Twitter user, I will never understand why this is preferable to a blog post. However, some people seem to prefer it, and it's getting engagement from non-Mathematicians, which is all that matters really.
For example, if a complex function 𝑓 is differentiable once then...
· It's differentiable an infinite number of times (smooth)
· It has a Taylor series approximation at every point (analytic)
· If γ is a smooth closed curve in the complex plane then the path integral over of 𝑓 around γ is 0, i.e., ∳ᵧ𝑓 = 0 (see https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem)
· The real part of 𝑓 viewed as a function of ℝ² is harmonic (likewise for the imaginary part). This is a result of the Cauchy-Riemann equations: https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equatio...
· The residue theorem is very powerful and can be used to prove things about non-complex functions/integrals: https://en.wikipedia.org/wiki/Residue_theorem
The list goes on.
Historically, complex analysis developed about 30 years before vector calculus as we know it today.
In C, we can just specify any pair from x,y,z and the third is determined by rearranging.
In R, it's not true! (y=2 and z=1 runs into trouble.) However, there are still infinitely many solutions and they're still easy to find.
In Z, we have the solution 0,0,0.
In N, we meet disaster!
To squeeze some meaning out of this, C is in some ways nicer because it is designed for the very purpose of containing solutions to things. We only got Q because ratios aren't integers so we added them in; we got R because sqrt(2) is famously irrational so we take the completion of Q... As we go on we just sort of smooth things out and get a better behaved, richer space. The "gaps" in N can make proving things feel impossible (sometimes rightly so).
If you try to build Calculus on top of the complex numbers, you get complex analysis where every differentiable function winds up locally a power series, and things just work out over and over again unreasonably nicely.
Build Calculus on top of the real numbers, and as you dive in you wind up with menagerie of counterexamples. For example Weierstrass produced a function which is everywhere differentiable and nowhere twice differentiable. These are not mere curiosities, for example wavelets (heavily used in data analysis) always are only differentiable a finite number of times. We can ask whether infinitely differentiable functions can be represented by a power series. e^(-1/x^2) is the classic example of one which can't be at 0. https://en.wikipedia.org/wiki/Non-analytic_smooth_function#A... offers one which is nowhere. And so on.
But now let's go from the real numbers to the natural numbers. The real numbers have an axiom system which is known to be consistent and complete. See https://math.stackexchange.com/a/362840/6708 for some of the details. However the natural numbers famously do not. (That's Gödel's Incompleteness Theorem.) What goes wrong? Philosophically it turns out that the natural numbers can encode computation, computation can encode reasoning, and reasoning about reasoning allows us to encode the liar's paradox. Everything blows up, hopelessly, after that.
At a conceptual level what is happening is this. The complex numbers have a tremendous amount of structure. As you go towards having less and less structure, we can use that freedom to write down more logically complex things. Logically complex things give ways for stuff to go wrong. And the more depth you study these things in, the more that turns out to matter.
For example if you draw a loop, specify a continuous function on the loop, and say it has to be differentiable inside of that loop, you can prove from the 2-d structure of differentiable functions that the value of the function at every point must be given by a specific integral around that loop. And from that you can start to recover a lot of the structure.
Here is a trivial example. The value of a differentiable function inside of the loop has reach its maximum absolute value at the boundary. Now consider a polynomial p(z) which doesn't have 0 as a root. 1/p(z) is differentiable everywhere that p(z) is not zero. If we draw a large enough circle around the origin, the value of 1/p(z) on that boundary is arbitrarily close to 0. But it isn't arbitrarily close to 0 at the origin, and therefore somewhere in that circle it must not be differentiable. And that point is a root of p(z).
This is the first proof that I saw that every polynomial over the complex numbers has a root. And it heavily uses the geometric structure of differentiable functions over the complex numbers. Note, that the reals lack this 2-D structure, and it is likewise easy to come up with polynomials over the reals that do not have real roots.
Is it related to (high school) polnomials always having complex roots, but not necessarily real roots? But that constraint on reals seems like "structure" to me, and complex numbers lack this structure...?
(It may well just require a few more years math on ny part.)
https://math.stackexchange.com/a/117914/150078
Does he mean his account has 1800 followers and 16000 users/comments on blocklist? I do not use Twitter but this absurd ratio makes me doubt if my understanding is correct
Just so. I have no problem if an author so titles their blog post, but, when that's not the actual title, it seems unlike HN to change it.