[HN Gopher] Mathematical outreach: The good, bad und ugly
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Mathematical outreach: The good, bad und ugly
Author : qualudeheart
Score : 52 points
Date : 2022-02-17 22:57 UTC (1 days ago)
(HTM) web link (florianfelix.net)
(TXT) w3m dump (florianfelix.net)
| posterboy wrote:
| This sounds very much to my liking, but I'm not getting the
| context of the presentation which it elaborates about.
| Spivakov wrote:
| "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
| lapinot wrote:
| 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.
| jesuslop wrote:
| That story about the complex nums being tame and not the others,
| where it is told?
| pfortuny wrote:
| It is so "philosophically" because the larger your set, the
| less inner structures it has (the larger your set the less you
| can say).
|
| Akin 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.
| autopilot23 wrote:
| 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.
| feoren wrote:
| 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.
| nimonian wrote:
| 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).
| jrh206 wrote:
| https://threadreaderapp.com/thread/1488484906237841411.html
|
| 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.
| jesuslop wrote:
| Unfortunately it is in German, but I have learned this thing
| of ordering math twitter threads that can be very nice.
| Thanks!
| btilly wrote:
| I can give examples of each.
|
| 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
| Godel'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.
| hyperpallium2 wrote:
| Could you elaborate on the "tremendous amount of structure"
| of complex numbers please?
| ssivark wrote:
| 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.
| itronitron wrote:
| 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?
| jknz wrote:
| 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).
| jfarmer wrote:
| 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)
|
| * If g is a smooth closed curve in the complex plane then the
| path integral over of f around g is 0, i.e., [?]f = 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.
| jerf wrote:
| 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.
|
| [1]: 3Blue1Brown on the topic:
| https://www.youtube.com/watch?v=sD0NjbwqlYw
|
| [2]: https://www.youtube.com/watch?v=shEk8sz1oOw
| JadeNB wrote:
| There's a typo in the title ('and' ugly, not 'und' ugly).
| [deleted]
| jrh206 wrote:
| 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.
| dang wrote:
| It's true but it's charming.
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