math duels!
November 11, 2021 11:11 PM   Subscribe

How Imaginary Numbers Were Invented - "A general solution to the cubic equation was long considered impossible, until we gave up the requirement that math reflect reality."[1]

also btw...
  • Galois Groups and the Symmetries of Polynomials - "No one knows why Galois found himself on a Paris dueling ground early in the morning of May 30, 1832, but the night before, legend has it that he stayed up late finishing his last manuscripts. There he wrote:"[2]
  • Go to the roots of these calculations! Group the operations. Classify them according to their complexities rather than their appearances! This, I believe, is the mission of future mathematicians. This is the road on which I am embarking in this work.
  • New Shape Opens 'Wormhole' Between Numbers and Geometry - "Coherent sheaves correspond to representations of p-adic groups, and étale sheaves to representations of Galois groups. In their new paper, Fargues and Scholze prove that there's always a way to match a coherent sheaf with an étale sheaf, and as a result there's always a way to match a representation of a p-adic group with a representation of a Galois group. In this way, they finally proved one direction of the local Langlands correspondence. But the other direction remains an open question."
-Galois Theory Explained Simply[3]
-Galois-Free Guarantee! | The Insolubility of the Quintic
-Group theory, abstraction, and the 196,883-dimensional monster[4,5]
posted by kliuless (27 comments total) 40 users marked this as a favorite
Whenever I see Galois' name, a little voice goes off in my head and says "God damnit, Galois!"
posted by Alex404 at 12:06 AM on November 12, 2021 [4 favorites]

Metafilter: Coherent sheaves correspond to representations of p-adic groups, and étale sheaves to representations of Galois groups.
posted by sammyo at 3:49 AM on November 12, 2021 [3 favorites]

Speaking as someone who passed basic algebra in high school only because the teachers were tired of trying to get me to understand it, when I see stuff like this, I can feel my little brain collapse into itself just a bit.
posted by Thorzdad at 4:43 AM on November 12, 2021 [6 favorites]

I really like the series "Imaginary Numbers are Real" on youtube and use it to introduce imaginary numbers to my Algebra 2 high school class. The beginning parts of the series are very friendly to the math-adverse and they talk about how people were also skeptical of fractions, zero, and negative numbers, but those numbers have uses which is why we invented them too. It helps make the math more interesting and it connects with the students bc they also hate fractions and negatives, lol. The video talks about math duels between mathematicians being the main reason the imaginary numbers were invented which the students also find interesting.

This series is great because it starts out pretty easy to understand but by part 13 it talks about multidimensional space. I majored in physics and still learned things from this.
posted by subdee at 4:54 AM on November 12, 2021 [14 favorites]

I once had to mark some students' working of the cubic formula on an example equation and came out understanding why the details of it weren't on the syllabus when I was an undergrad.
posted by polytope subirb enby-of-piano-dice at 5:07 AM on November 12, 2021 [1 favorite]

I stumbled upon the Veritasium video just a week ago and thought it was really interesting.
I don't think I ever saw the geometric method he used for the x^2 + 26x = 27 problem & I like that.

I was hoping this would help me solve an old problem that I saw in a MENSA magazine decades ago. I got to the point where I knew the solution was [one root of] X^3-14X=12
I could figure this out to many decimal places, but I was trying to come up with an exact solution like the cube root of (139/2) or something like that only more complicated.
I've been through the depressed cubic equations before and still can't figure it out.
(Emphasis on the 'depressed')
posted by MtDewd at 5:28 AM on November 12, 2021

One of the first things I did as a math undergrad when I got access to Mathematica was to have it crank out the solution to the general quartic equation ax^4 + bx^3 + cx^2 + dx + e = 0.
posted by indexy at 6:20 AM on November 12, 2021 [8 favorites]

I taught high school math for a couple of years. I was a math major in college. I never liked completing the square, always saw it only as a necessary step to deriving the quadratic equation. Now I get it. And I feel like I cheated my students just a little by not showing them the geometry behind this. To be honest/fair, I hate geometric proofs. My prefect math textbook is Walter Rudin's Principles of Mathematical Analysis. Less than 200 pages without a single image. Compact, complete, dense, and wonderful. I never considered the geometric origin of much of math before these last ten years. I think that there should be a component of every geometry class that shows how it connects.
posted by Hactar at 6:33 AM on November 12, 2021 [5 favorites]

Okay, now I just curled up with my 5 year old and watched the Veritasium video. Cool video, and it held my boy's attention. I think some of his brain started leaking out of his ear, but that was clearly the runny, gooey stuff being replaced by crystalline lattices.
posted by Alex404 at 6:34 AM on November 12, 2021 [4 favorites]

If you've done maths to (say) undergrad level but you haven't done Galois Theory, this series from Richard Borcherds is a good overview (his channel has a wide range of math lecture courses at undergraduate and graduate level).
posted by crocomancer at 7:05 AM on November 12, 2021 [4 favorites]

I never liked completing the square

Try casting out nines, it's the bomb
posted by thelonius at 7:30 AM on November 12, 2021 [1 favorite]

Try casting out nines

I'm pretty sure this is illegal, as the drugs required to invent/appreciate it are controlled substances.
posted by justsomebodythatyouusedtoknow at 7:42 AM on November 12, 2021

I'm the opposite; I'm a visual thinker who works best using images and geometry. When I first took real analysis the professor used Rudin's text, and I absolutely hated it. I dropped that class and took it the next year from a prof who taught from a book that wasn't afraid to use explanations, examples, and images. Rudin reminds me of math Wikipedia--concise and elegant if you already know the material, but difficult to impossible to learn from if you don't and need your hand held. I can appreciate Rudin now, but only after I learned the material elsewhere.
posted by indexy at 8:25 AM on November 12, 2021 [6 favorites]

I was always a middling (at best) math student in spite of eventually becoming an engineer and I feel a little cheated that content like this didn't exist when I was in school. Math was always a case of applying a formula or method without realizing what you're actually doing. I never realized what a sin or cosine was until after I got my degree.

Anyway, I wish stuff like this had existed when I was pursuing my degree. I'd probably have been a much better math student.
posted by mikesch at 8:27 AM on November 12, 2021

500 years of NOT teaching THE CUBIC FORMULA. What is it they think you cant handle? - YouTube - learn to derive it yourself (and quadrics).
posted by zengargoyle at 9:38 AM on November 12, 2021 [2 favorites]

As someone who breezed through geometry and hit algebra like a brick wall, I appreciate the geometric approach even if it's a dead end.
posted by ChurchHatesTucker at 9:42 AM on November 12, 2021 [2 favorites]

There were a lot of useful ways to think about math that did not get taught in school, yeah. With complex numbers, it would have interested me to be told that I had already accepted the idea that a number could be represented by a pair of integers, when I started believing in the rational numbers, and now we just have a new kind of number, that's represented by a pair of real numbers. If I were smarter I'd have seen that myself, I guess.

And either they did not explain well, in high school, or I wasn't listening, or I forgot, that multiplying by i is rotating a vector 90 degrees on the complex plane. Start with 1 + 0i, the unit vector, which we'll just call "1", multiply by i twice, rotating it 180 degrees, and what do you have? You have -1, that is what. So you can see that there is a perfectly simple geometric meaning to the notion that i^2 = -1. Perhaps this is in TFV, I have not watched it yet.

I got that from the introductory chapter to a book on signal processing, and the author mentioned that it is not unusual that undergrad students with otherwise strong math backgrounds aren't so clear on complex numbers, since it seems to be not taught so well in secondary school math.
posted by thelonius at 10:01 AM on November 12, 2021 [3 favorites]

Also: once you admit the real number line, where every point on the line corresponds to a real number, how about looking at it on the Cartesian plane? Why should only points on that one line represent numbers?

Of course you'd need to have already given up on the idea that a number has to represent a metaphysically "real" object, to think this way - if you were hardcore about that position, I suppose you might not even admit that the real numbers are legit, on perhaps finitist grounds, as was discussed in a recent thread about physics.
posted by thelonius at 10:12 AM on November 12, 2021 [2 favorites]

Imagining Numbers by Barry Mazur is one of my favorite math adjacent books. It's a poetical exploration on the invention of imaginary numbers, and it's a wonderful step-wise, accessible explanation as well.
I have great affection for the book itself, because I wasn't looking for it, I just happened upon it on an afternoon, killing time in City Lights while visiting friends in San Francisco. The same friends I was with in Amsterdam a few years before, when we were all turned on to Flatland in a coffee shop by a guy who claimed that the thing he enjoyed the most was bringing interesting books to the shop and striking up conversations around them. Kind of a MeFi thing to do, don't you think?
posted by OHenryPacey at 10:41 AM on November 12, 2021 [4 favorites]

Is this going to be on the test?
posted by chavenet at 11:00 AM on November 12, 2021 [3 favorites]

Alas, I can't find the reference to cite but I read in a book on the history of mathematics that it was work on the cubics that legitimized imaginary and negative numbers more or less simultaneously.

Negative numbers had been around for a long time, notably in non-western math, and were grudgingly accepted in cases such as finance where they could be understood as debt, but until remarkably recently they were viewed as suspicious. Since negatives can often be eliminated by moving quantities to the other side of the equation, mathematicians went to great lengths to do so. If your could arrange your proof to be free of negatives you would absolutely do it.

While the concept of negative numbers had been around for a while as a trick or computational device to use in computation and imaginary numbers appeared much later, cubics forced the issue for negative numbers. It was around then that people started working on foundations and questions like "what is a number anyway?" It wasn't until imaginary numbers were accepted as perfectly cromulent that they stopped giving negatives the skunk eye.
posted by sjswitzer at 11:23 AM on November 12, 2021 [3 favorites]

I share thelonius's experience of not fully appreciating the magic of complex numbers until studying signal processing. Once you fully grasp complex multiplication as rotating and scaling, things like Euler's identity are just obvious. Hey guys, did you know that ei is simply the point one radian along the unit circle (distance of one along the unit circle)? When that finally dawned on me, I felt I'd been cheated my entire math education.

The whole menagerie of trigonometric formulas is encompassed in complex multiplication and the Pythagorean theorem. All. Of. It. If I had my way, they'd teach trigonometry as a historical curiosity or a specialized tool for navigators and land surveyors then proceed directly to complex numbers and Euler's formula.
posted by sjswitzer at 11:36 AM on November 12, 2021 [5 favorites]

I really wish math was taught from historical perspective like physics is taught. It makes stuff make more sense. I did a B.S. in physics and realized i was poor candidate to go on to more advance degree because I basically need an equivalent understanding in mathematical theory to actually understand what was going on and I wasn't going to learn it in physics class. I am totally jealous of kids these and things they can access to study online. Every single math professor I basically if you don't understand what I am saying you are not going to get it. Then these short youtube videos do a way better explanation.
posted by roguewraith at 2:52 PM on November 12, 2021 [4 favorites]

Why are there so many
stubborn equations?
How can we be so blind?
Can we now solve them
with radical methods,
a clever and rational mind?
So we've been told and some choose to believe it,
I know they're wrong, wait and see...
Someday we'll find it,
the Galois Connection:
the Field Extensions and G

posted by ContinuousWave at 6:00 PM on November 12, 2021 [10 favorites]

As much as I love the history of math, I'm not sure it's a good pedagogical approach. For one thing, you'd have to to teach the histories of math, a fascinating topic and worth studying, but not necessarily helpful in instilling mathematical aptitude. But sticking to Western math (to the extent that it can be separated from Islamic math) which isn't much between the Classical era and the Enlightenment, the tradition of teaching geometry before algebra has scarcely been questioned and maybe it should be. Is it necessary to understand any math before Descartes created the bridge between geometry and algebra? Is geometry really the best way to teach axiomatic deduction? Is it necessary to teach Euclidean geometry at all before the graduate level? I ask this in all seriousness.

Newton's Philosophiæ Naturalis Principia Mathematica was encumbered significantly because he insisted on providing parallel geometric proofs to everything he'd already proved via algebra and his new calculus. Clasicism was foundational to the spirit of the Enlightenment, though they conveniently erased the Islamic scholarship that transmitted the classics to them. And in the time since then, progress has been in fits and starts and sometimes leaps, but not every step is really helpful in terms of developing an understanding of math today.

Mathematical notation evolved slowly over time and the modern form is ridiculously recent. Thank your lucky stars that you can study algebra using modern notations rather than the evolving notations that came before it; for all practical purposes they are best forgotten.

Modernizers of math have been focussed on foundational issues and soundness, and that's great. But someone should really address how to place these ideas into a practical pedagogical sequence. My sense is that "New Math" was an attempt to do just that and also that it was a bust. But that doesn't mean it's a hopeless cause.

Similarly, learning Latin is a good thing but it's also great that you don't need to learn Latin to do Physics anymore. It could equally be the case that learning Euclidean geometry (outstanding intellectual achievement that it is) is not essential to learning modern math and possibly even an unnecessary roadblock for some learners.

Wikipedia, which should be a useful resource for people wanting to learn math, is an absolute shambles. Nothing is pitched at people who want to learn things (which is what encyclopedias are for) but rather to describe what experts already know. We've already got Wolfram Mathworld for that!

We need a systematic progression of concepts and tools for learning math, but it's far from clear that it should be based its historical development. As fascinating and occasionally illuminating (also contentious and too often western-centered) as that history might be, I'd argue that the traditional, broadly historical, pedagogical sequence is less help than hindrance.
posted by sjswitzer at 6:10 PM on November 12, 2021 [7 favorites]

Won't it differ depending on who's doing the learning? Personally, I've often found these historical framings to be quite helpful, for at least two reasons:
  • They often (not always, nothing here is "always") illustrate the kinds of problems that the mathematics were invented to solve, or the characteristics of existing techniques that mathematicians were responding to in their work. For me, these illustrations have tended to highlight key features that deserve particular attention.
  • They remind me that mathematics is a human endeavour and that complicated ideas were built over many years by people using deductive powers, imagination, intuition, collaboration, and so on, which can be a useful thing to recall whenever you find yourself having to deal with concepts that can often be presented in a fairly abstract and austere way.
I say set up all the instructional methods and let me choose. Some people like examples, some people like history, some people like formal derivations, and so on. I don't believe in a universal pedagogy.
posted by tss at 8:22 AM on November 13, 2021 [1 favorite]

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