When CAN'T Math Be Generalized? | The Limits of Analytic Continuation
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- Опубліковано 9 тра 2024
- There's often a lot of emphasis in math on generalizing concepts beyond the domains where they were originally defined, but what are the limits of this process? Let's take a look at a small example from complex analysis where we actually have the tools to predict when this is impossible.
This video is a participant in the third Summer of Math Exposition (#SoME3) hosted by 3Blue1Brown to encourage more math content online. To learn more, see this:
3blue1brown.substack.com/p/so...
=Chapters=
0:00 - Intro
1:15 - Extending a Geometric Series
3:39 - Complex Power Series
6:23 - Analytic Continuation
8:30 - Analyzing the Gap Series
11:51 - Visualizing the Gap Series
19:21 - Gap Theorems
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This video was generously supported in part by these patrons on Patreon:
Marshall Harrison, Michael OConnor, Mfriend.
To support future videos, become a patron at / morphocular
Thank you for your support!
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CREDITS
The music tracks used in this video are (in order of first appearance): Icelandic Arpeggios, Checkmate, Ascending, Orient, Faultlines
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The animations in this video were mostly made with a homemade Python library called "Morpho". It's mostly a personal project, but if you want to play with it, you can find it here:
github.com/morpho-matters/mor...
Hey, thanks for watching! This video was made as part of the 3Blue1Brown third Summer of Math Exposition (#SoME3), a so far annual contest to encourage more math content online. If you've ever considered making a math video or other piece of math content, there's hardly a better opportunity to do so than this event. More details here:
3blue1brown.substack.com/p/some3-begins
Good luck in SoME3! Hope this video gets at least mentioned in the end
Summer of math is such a good event. Hope it will be as good as last summer. This video was also quite good btw.
Looks like SoME has came again!
Convergence (syntropy) is dual to divergence (entropy) -- the 4th law of thermodynamics!
Finite (localized, particles) is dual to infinite (non localized, waves).
Waves are dual to particles -- quantum duality.
"Always two there are" -- Yoda.
2:04
You can't use that formula out of the interval between - 1and 1,
The original formula was set on a limit to infinity is lim (1-x^n)/(1-x) as n approaches infinity and x is in the open interval (-1,1)
Even if mathematicians couldn't generalise the series, they made sure to generalise the notion of ungeneralisable series 😂
_There must be something here we can generalize, I just know it!_
@@kindlinThis is the math anthem essentially
@@kindlin they must hate having to write exceptions
@@whannabigeneralize the exceptions
@@whannabi The exceptions just tell you that there is more likely an even more fundamental truth lurking somewhere in the math, waiting to be found that covers not only that exception, but others you didn't even think of yet.
When? When I need it to the most
lmao, literally what it feels like all the time even though it's just me being too lazy do to a bit too much algebra
looks like a bias
Word
@@Fire_Axus looks like a joke
I need a formula for summation for hp or i will always be an empty husk of a mathematician
I wish animations are more widely used in school. Visually seeing relationships and interactions really helps me understand and learn topics. This would've been helpful in my Complex Analysis class.
They're really hard to make...
@@charbeleid193 and take a lot of time, time that usually isn't paid to teachers lol
It woll be dope with anime style.
@@nifftbatuff676 how tf are you supposed to write eqns and draw graphs in anime style??
@@sawc.ma.bals. Manga Guide to Math, Manga Guide to Physics, etc.
I'd be really interested to see a proof / deeper intuition for Fabry's and Polya's theorems in your style. This was a very nice introduction
Fabry's theorem might be a bit too hard to do a video on, but Hadamar Gap's might be simple enough to make visualizations for.
@@falquicao8331Don't underestimate the power of vulgarization 🤓.
@@falquicao8331 I'd say the intuition behind Hadamar's theorem is there already. The Polya's _existence_ theorem, on the other hand is another matter. I'd be excited to see a visulalisation of it!
Strongly agree!
I'de love a hard episode with proof of these theorems ! Thank you for your videos, this one was great again !
The first series of powers of 2 "equaling" -1 reminds me of the usual representation of -1 in computers as many 1s, which is effectively saying the same thing!
Right!? In binary, the sum of the powers of 2 = ...1111111. If you add 1 to such a number you can see how it would "overflow" to 0. The p-adics seem incredibly natural, actually.
@@yamsoxYeah, turns out we all develop some intuition for p-adic numbers in school. It'd be great if they were actually taught or at least mentioned to give the most curious students an idea of where to start looking. Instead, we're left with a big question mark 😢.
this is modular arithmetic though, but i think p-adic numbers are just modular arithmetic under p^infinite :)
@@drdilyor yeah none of the previous comments really works lol
Hmmm. That's a finite number of terms, rather than an infinite number, and the representation of -1 comes from two's complement arithmetic. But what we're really saying here is that that two-complement representation converges in a nice, regular way to the infinite-bits case.
About the end: I encountered meta-mathematics and reverse mathematics quite early in my undergraduate studies and it kept me forever. Making mathematics itself an object of mathematical reasoning is - yes - complicated at times because you have to keep track of two “mathematicses” - the one you study and the meta one that you use to do the studying -, but it has so many interesting and thought-provoking results, even early on. If it sounded complicated, think of a toolmaker: A toolmaker uses (meta-)tools to make (object-)tools; a meta-mathematician uses one set of (meta-)axioms to reason about some (object-)axioms. Maybe confusingly, these axioms can be the same, as a toolmaker can use some tool to make another one of that tool.
What is reverse about reverse mathematics? The idea is to reverse the reasoning process: Normal mathematicians (blindly) accept some set of axioms, they find examples for things, observe patterns, craft definitions, formulate propositions they hope to be true and then - hopefully - prove those propositions true, that is, logically derive them from the axioms. Reverse mathematics takes propositions and attempts to answer the question: How fundamental are they? This is the same question as: What axioms are necessarily needed to logically derive the proposition. The fewer and the more fundamental (in the eyes of the human) of axioms suffice, the more fundamental the proposition is.
I have a rather simple example for you: In classical logic, for every proposition _A_ it is true that _A_ or not _A._ (In formula: A ∨ ¬A.) Also, in classical logic, not not _A_ implies _A_ (in formula: ¬¬A ⇒ A) and in fact, if you take away any single these axioms, the other follows from it and the rest of the logical axioms as a theorem. Both of these are somewhat controversial, and you’ll understand why by me giving you a mathematically infallible investment strategy: Pick a stock and a time frame; after the time frame, if the stock is above its current value, buy it; otherwise, short it. The problem is, the stock will indeed be above or below (that is: not above) its current value, but - you can’t really know until then. The strategy I outlined can only look mathematically sound because of _A_ ∨ _¬A._ The logic without both of them is called “intuitionistic logic” and it can only derive statements that are computationally valid, that is, if you find a fool-proof investment strategy using intuitionistic logic, it can actually be followed. Of course, in intuitionistic logic, _A_ ∨ _¬A_ cannot be derived for every _A,_ but for some; and it is equivalent to _¬¬A_ ⇒ _A,_ but that is due to the assumption that from a contradiction, anything follows: For every statement _B,_ if we can derive _A_ and _¬A_ (a contradiction), we may conclude _B_ (in formula: A ∧ ¬A ⇒ B). If we do away with this as well, we have an logic system called “minimal logic” and it makes not assumptions about falsehoods/contradictions/negation. In minimal logic, finally, _A_ ∨ _¬A_ and _¬¬A ⇒ A_ are not equivalent anymore. Here, we have that _A_ ∨ _¬A_ for all _A_ follows from the assumption _¬¬X_ ⇒ _X_ for all _X,_ but _¬¬A_ ⇒ _A_ for all _A_ cannot be derived from _X_ ∨ _¬X_ for all _X._ That is, one of them is more fundamental than the other, but you need to visit a weak logic to see it.
I can give you a little motivation for meta-mathematics: The “ordinary” axioms of mathematics are - generally - the axioms of Zermelo-Fraenkel set theory (ZF) plus the Axiom of Choice (AC), called ZFC.
In ZF, i.e. without AC, one can prove that it is equivalent that
(a) every family of non-empty sets has a non-empty Cartesian product (this is trivial for finite families, but provably not provable in ZF without AC)
(b) every set can be well-ordered (again, this is trivial for finite sets, but not infinite ones, but provably not provable in ZF without AC)
If you need a rephrasing for (a) it is this: Consider sets _A₀, A₁, A₂, …_ that are not empty, i.e. there is _a₀_ ∈ _A₀, a₁_ ∈ _A₁, a₂_ ∈ _A₂,_ etc. in the Cartesian product _A₀_ × _A₁_ × _A₂_ …, there “obviously” is the element (a₀, a₁, a₂, …), however, in ZF without the axiom of choice, we cannot actually prove that the tuple exists!
Both, (a) and (b) are actually equivalent to AC, they are as good as the axiom itself. However, (a) looks like something that’s “obviously true,” like, how could a family of non-empty sets have an empty product? On the other hand, (b) looks like something that shouldn’t be true since - of course _some_ sets can be well-ordered - why should every single set admit a well-ordering?
As humans, we have no choice in the logical consequences of axioms, but we do have choice in what axioms we base our reasoning on. (In a sense, given a fixed, agreed-upon axiom system, you cannot meaningfully ask “why” some theorem is true - the answer always is that it’s derivable from the axioms -, but you can meaningfully ask “why” this or that axiom is part of the axiom system because here, human intention and choice is involved.
Now my personal conclusion is, if an axiom system (ZF in this case) derives the equivalence of two statements and one “obviously true” and the other is “obviously false,” then the axiom system is not good. It’s not that it’s inconsistent, that’s a technical term and subject to proof. What I mean is, maybe there’s a better system of axioms out there. One in which the “obviously true” is true and the the “obviously false” is false.
Without being exposed to meta-mathematics even good mathematicians don’t have thoughts about it. In introductory courses such as analysis or linear algebra and most of what sits on top of it: complex analysis, functional analysis, probability theory, and even numerics, students are generally assumed to work in ZFC almost religiously. We laugh at Blaise Pascal for not considering other potential gods in his famous wager, but most of today’s mathematicians work in ZFC without ever considering working in a different axiomatic system. If you ask for one, there are other set theories, but there’s other foundations of mathematics that don’t take sets as their primitive notion; one of them is called Homotopy Type Theory (or HoTT for short); I found it really interesting to read the introductory HoTT book, homotopytypetheory.org/book/ (it’s 100% free) in fact, I read it multiple times, and there’s complicated stuff, but after a while, it finally snapped; I don’t think it’s too important to understand everything you read in there, except for the first chapter (which explains the basic concepts and notation) and maybe the second, but I got very far in my first read where I didn’t really understand most of the second chapter.
I read like half of that 😅
But re: reverse math, and the intuitionistic logic axioms you took as examples.
This reminds me of a similar example, it can be problem 1 of a coolguy-enough intro to group theory:
Agroup is a set with binary operation which satisfies:
Associative, left&right identity element, and left&right inverse functions exist.
If G is associative, then left identity element + left inverses, implies right identity+right inverses. BUT IF YOU ASSUME G is associative, has left identity element, and RIGHT inverses, it NEEDNT BE A GROUP.
[Theres a 2 page paper written in like 1943 by an Austrian recent immigrant to America, he called them (l,r)-systems and he gave a 100% complete, imho, description of what new, nongroup (l,r)-systems u can have
Yeah category theory, homotopy type theories, string diagrams and even diagrammatic languages seem much more promising than the old-fashioned Russel-Frege logical schools of thought. No mathematician seriously uses model theory for real-life applications lmao, though it remains interesting. It seems that some kind of mathematical constructivism seems essential for solving actual practical problems, instead of finding out "by contradiction", that some solution exists, but you can't make it explicit lol.
Which mathematical logic / alternative mathematical foundations books are you reading by the way?
as someone who just finished a complex analysis course, this video felt like a nice application of a lot of things I learned
Although I'm not super familiar with complex analysis (it's been quite a while since I studied it in uni), the explanations were very clear and intuitive, and they subtly gave away where they were going to go next, making it very pleasant to actually see you explain it after you were wondering about that. Great job!
2:26 "No, we're not going to talk about p-adics today."
Does that mean you plan on making a video on them later 👀? I'd love to see a 3rd science/math channel talk about them (1. Eric Rowland, 2. Veritasium).
As a programmer, I mean, _duh. Obviously_ the sum of powers of 2 equals -1! That's how it works for literally every integer type. And even when it doesn't, the sum of powers of 2 plus 1 is either an overflow error, or 0, and naturally the number that gives 0 when added to 1 must be -1.
@@angeldude101 2-adic integers are just infinite-bit integers
I mean there's got to be a dozen intro to p-adic videos on UA-cam. If you're really just looking for like, from axioms to arithmetic, one is plenty imo but really you could just dig into those guys with accents and power points. I would love a p-adic video that didn't cover the exact same material as all the others, but nobody ever goes further than like a basic number theory proof in the first video, and nobody ever makes a second video.
3blue1brown did one a long while ago, it was actually the first one I watched, it's a bit dated (compared with the quality of his more recent videos) but still really good.
Awesome!!! It took me until graduate complex analysis before it really dawned on me that you can have these dense boundaries of singularities that would prevent analytic continuation, so seeing this in such a clear format is a treat!
"Complex Variables" by John W. Dettman is a great read: the first part covers the geometry/topology of the complex space from a Mathematician's perspective, and the second part covers application of complex analysis to differential equations and integral transformations, etc. from a Physicist's perspective.
Hey, I’ve never seen someone explaining the murky concept of analytic continuation so neatly, well done! I’m subscribing 😅
Wow! The animations really helped visualize the equations so well. Studying math, I had to do this in my head and it wasn't always feasible. Thanks for reminding me how cool math can be. Keep shining!
I love the symmetry of Fabry's and Polya's theorems
I had this problem as a homework assignment in my complex analysis class, very cool to see it animated here and explained so simply!
thats so cool man
What a great video. Wonderful explanations and beautiful + helpful animations! Well done!
Thanks UA-cam for this recommendation! This one particular video us one of the best math videos I've seen, in terms of math-noob-friendliness. Love it!
I can't believe it was only about 8 months ago I watched last year's SOME2 round-up and had a look at these videos. Great channel!
Honestly, quite the amazing video! Will be rooting for you in SoME3!
I think this problem also leads well into a discussion about why analytic functions are conformal maps.
very good video. my understanding of "analytical continuation" had always been very handwavy, but you put it very simply, and were able to apply it in a way that made it make a lot of sense.
I paid so little attention in my Complex Analysis class, to the extent that I can't even assess whether my lack of motivation was my own fault (quite likely) or that of the lecture content. Since then I occasionally come across a video/article/discussion that highlights just how conceptually fascinating it is, and this is my favourite so far - thank you! I'm excited to begin relearning it properly some day. Alongside some of the other fields that feature heavily in modern approaches to number theory, I feel it's one of those subjects that can make an algebraist love analysis too 🤩
You asked whether there would be interest in proofs of the theorems you mention, and my answer is definitely yes for completeness' sake, but beyond anything else I would love you to keep focusing on aspects of the subject that lend themselves to great intuitive dives like this one, whatever those might be.
The animations are great!
When I first studied complex analysis, conformal mapping, etc., one kept one eye on the z-plane and the other on the w-plane. We all looked like Marty Feldman after the final exam.
I had done this back in uni 2 years ago, but I was able to somewhat get the gist after watching your video. Damn, makes me want to pick up math again, but this time with no exam pressure, just pure interest
Subscribed for the high quality explanations and visuals!
Holy smokes that was truly insane (especially 3 theorems at the end)! I gotta check out complex analysis sometime...
Thanks for this video, I had heard about this before but had never seen an actual concrete example of when it happens!
Oh my god, that transformation of g(z) into a chaotic tangle was one of the most beautiful things I've seen in math in a long while. It just grabbed some deep part of me.
7:09 an easy way to think about complex differentiability is that if you have f(x+iy)=u(x,y)+iv(x,y) being differentiable, then the jacobian matrix of f looks like multiplication by a complex number.
yes! equivalently that means that the differential of this f(x,y) function is complex linear, not just real linear.
What a fantastic video. Being unable to thread the needle past the boundary was awesome. Thanks for making it!
This was a great video! I will watch any complex analysis videos you decide to make.
I am glad you mentioned p-adics there.
Convergence (syntropy) is dual to divergence (entropy) -- the 4th law of thermodynamics!
Finite (localized, particles) is dual to infinite (non localized, waves).
Waves are dual to particles -- quantum duality.
"Always two there are" -- Yoda.
Very interesting stuff! Thanks for the great content!
This is fascinating and very well explained! Thank you for making this video!
i love this channel so much… keep it up!
Just went over analytic continuation for graduate econometrics! This is right on point, great video!
I have to say, I love the visualization of complex power series you use at about 12 minutes in. It makes it so clear how complex power series are related to Fourier series. Specifically, the Cauchy integral formula can be thought of as saying that if a function f is holomorphic on a disk D centered at a, then the Taylor coefficients centered at a are equal to the Fourier coefficents of f restricted to the boundary circle of D. That is, complex Taylor series are really the same thing as Fourier series. The visualization here makes this painfully obvious, especially when paired with some of 3Blue1Brown's videos on Fourier series.
This also makes it a bit clearer why the gap series would be badly behaved. We're taking a function with well behaved Fourier coefficients, but dropping the contributions from many frequencies. Having very few high frequency Fourier coefficients all with the same weighting is also likely to lead to pretty sporadic behavior, as you would have large amounts of very fine structure but not enough to expect useful interference.
This is such a good video on the intuition of analytic continuation!
Visualization is the best way to understand anything. Great work 👏
Wow! Fascinating and fully understandable. Great job!
Thanks so much for this video. I was trying to settle this in my mind for a long time and finally I can put the pieces together.
THIS IS CRAZY MAN never heard about the function wanting to be "the center of mass" of the values it oscillates between
I was so excited to watch this! I love it.
Thanks for the clear graphical explanation of this interesting topic!
Thanks for the primer on analytic continuation! I've wondered about this before, as it is often mentioned when the Riemann Hypothesis is discussed, but sadly my mathematics education hadn't quite reached this area.
Thank you! I have struggled with Titchmarsh's excellent Theory of Functions book for years now. They have exercises about these gap series that I never understood how to solve. You have helped me to solve them.
the arrow visualization is also a really nice way of seeing that even though the series diverges, it's imaginary part converges, which i think is neat!
i love your style, thanks for making videos like this! 😄
i was able to understand most of it and get some new insights, but also, watching graphs "misbehave" is just inherently funny!
Man these animations make my mouth water, thanks this was a great visual explanation
Thanks for the video, I dont usually catch this kind of glimpse in complex analysis, it was pretty to remember things about power series.
can we just appreciate how slick the transition at 5:08 was?
Perfect. The right mix of motivation and rigor. It’s hard to get it right, but you did.
This is probably the coolest video I have ever seen on UA-cam. And nobody at my physics faculty get‘s why this is so interesting to me
This filled in some gaps I had after taking complex analysis so thanks!
Beautiful video, looking forward to more from you !
Great video! Please make one about the gap theorems!
Truly excellent explanation, thank you for making this.
Another fantastic video! This was really interesting!
I appreciate your video！I never learnt about Analytic Continuation before but my number theory course asks me to read a paper. I picked one about prime number theorem and this term is in it. Now I get the idea! Thank you again!
One of the best math explainers on the internet.
PURE GOLD THANK YOU FOR THIS GREAT VIDEO!!
I studied a few of the concepts used in the analysis on my Control Engineering classes, always from the engineering pov. Looking these concepts from the mathematical pov make things sound much more reasonable.
I’d love to see a video on the other gap theorems!
This was really an interesting topic. Thanks.
this one gonna be ranked well in the SOME contest, I can feel it
Great job! Would love to see as many visual proofs as possible
this is great❤ i would 💯% love to see proofs of those "gap" theorems of analytic extensibility! keep up the great work
That was mindblowing. Amazing video.
Great video! Thank you!
I’m in love with this channel. Hi from Colombia!
Very nicely explained. Good job !
Really Elegant! Thank you so much , grateful
Fun fact the "complex number" he called 'w' is the square root of i
there are 8 roots to the equation, 1, 1, i, -i, sqrt(2)(1+i), sqrt(2)(-1-i),sqrt(2)(1-i),sqrt(2)(i-1), 4 of them are complex numbers(the ones with both 1 and i)
you’re correct, partially, which is the square root of i part, but it can be something else(just real or imaginary)(and not the sqrt of i)
We love your content. Can we ask what next? Deep dive into an ocean of function progressions and series with all criterions and proofs? Or mountain hike to BirchSD conjecture? Or someone may ask to camp near statistics forrest?
I love this channel so much oml
This is beautifully executed!
I would love to see the proofs of Fabry’s and Polya’s theorems! 😎 great vid!
Outstanding explanations -- instant sub
that was great, I understood the analytic continueation much better than with the 3b1b video
Loved the video thank you so much for your content.
Can we see a proof of the theorems next please :))))) i'd love that
Thks & Well-Explained
Thank you very much
Very amazing and philosophical topic 🎉🎉❤❤
Thank you for this beautiful video❤
this is really well-made!
Such an original piece of informative math content. Love it!
Keep doing what you are doing, man! This is great math insight nicely presented! I'm an old retired engineer and wish to learn more math so perhaps I will prevent dementia in the future. 😁😁😁😁
Amazing video. Thank you!
THE best submission to #SoME3 yet.
This pretty much summarizes many techniques used in physics such as the usage of power series analysis in QM or rotations in the imaginary plane etc etc
Excellent video. You have 3b1b's talent. Grant would be proud ...they grow up so fast
Really helpful, thank you
I would love to see a video on Fabry's and Polya's theorems!
Now I'm waiting for the video about extending fractional calculus to the quaternions.
thanks for the content, and yeah please follow up with the proofs and intuitions behind them!
Yes! Do a follow up, please
Thank you this video changed my life
Congrats for HM! Great video
This is well put. bravoooo