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In grad school around 1980 I took a cab home from a midnight showing of the reggae film "The Harder They Come". The cab driver asked me out of the blue, "Is it

by Syzygies 10mo ago

In grad school around 1980 I took a cab home from a midnight showing of the reggae film "The Harder They Come". The cab driver asked me out of the blue, "Is it true you can't tell the difference between +i and -i?" Cambridge, MA but still ... unexpected. If someone hands you a blank board representing the complex numbers, and offers to tell you either the sum or the product of any two places you put your fingers, you can work out most of the board rather quickly. There remains which way to flip the board, which way is up? +i and -i both square to -1. This symmetry is the camel's nose under the tent of Galois theory, described in 1831 by Évariste Galois before he died in a duel at age twenty. This is one of the most amazing confluences of ideas in mathematics. It for example explains why we have the quadratic formula, and formulas solving degree 3 and 4 polynomials, but no general formula for degree 5. The symmetry of the complex plane is a toggle switch which corresponds to a square root. The symmetries of degree 3 and 4 polynomials are more involved, but can all be again translated to various square roots, cube roots... Degree 5 can exhibit an alien group of symmetries that defies such a translation. The Greeks couldn't trisect an angle using a ruler and compass. Turns out the quantity they needed exists, but couldn't be described in their notation. Integrating a bell curve from statistics doesn't have a closed form in the notation we study in calculus, but the function exists. Statisticians just said "oh, that function" and gave it a new name. Roots of a degree 5 polynomial exist, but again can't be described in the primitive notation of square roots, cube roots... One needs to make peace with the new "simple group" that Galois found. This is arguably the most mind blowing thing one learns in an undergraduate math education.

owyn 10mo ago
The original post is written by AI so I will read it briefly, but your comment is fascinating. I got through undergrad math by brute force memorization and taking the C. Or sometimes the C-. The underlying concepts were never really clear to me. I did take a good online calculus class later that helped. However, I have questions: "Turns out the quantity they needed exists, but couldn't be described in their notation" What is this about? Sounds interesting. "Statisticians just said "oh, that function" and gave it a new name." What is this? I never understood there is a relationship between quadratic equations and some kind of underlying mathematic geometric symmetry. Is there a good intro to this? I only memorized how to solve them. And the existential question. Is there a good way to teach this stuff?
- bayesnet 10mo ago
  What is so unbelievably frustrating about math education is that these interesting questions are not even hinted at until far, far down the line (and before people make the assumption I was educated outside the US). I avoided math like the plague until my PhD program. Real analysis was a program requirement so I had to quickly teach myself calculus and get up to speed—and I found I really, really liked it. These high level questions are just so interesting and beyond the rote calculation I thought math was. I hope I can give my daughter a glimpse of the interesting parts before the school system manages to kill her interest altogether (and I would welcome tips to that end if anyone has them).
  - Terr_ 10mo ago
    Reminds me of "A Mathematician's Lament" [0]. I'd prefer to link a non-PDF copy, but nothing came up with a casual search. [0] https://en.wikipedia.org/wiki/A_Mathematician's_Lament https://en.wikipedia.org/wiki/A_Mathematician's_Lament
  - noelwelsh 10mo ago
    Geometric proofs are really accessible. You don't need any algebra to prove Pythagoras' theorem, or that the sum of the inner angles of a triangle is 180 degrees, for example. Compass and straight-edge construction of simple figures is also fun.
- defen 10mo ago
  > I never understood there is a relationship between quadratic equations and some kind of underlying mathematic geometric symmetry. In a polynomial equation, the coefficients can be written as symmetric functions of the roots: https://en.wikipedia.org/wiki/Vieta%27s_formulas https://en.wikipedia.org/wiki/Vieta%27s_formulas - symmetric means it doesn't matter how you label the roots, because it would not make sense if you could say "r1 is 3, r2 is 7" and get a different set of coefficients compared to "r1 is 7, r2 is 3". Since the coefficients are symmetric functions of the roots, that means that you can't write the roots as a function of the coefficients - there's no way to break that symmetry. This is where root extraction comes in - it's not a function. A function has to return 1 answer for a given input, but root extraction gives you N answers for the nth root of a given input. So that's how we're able to "choose" roots - consider the expression (r1 - r2) for a quadratic equation. That's not symmetric (the answer depends on which one we label as r1 and which we label as r2), so we can't write that expression as a function of the coefficients. But what about (r1 - r2)^2? That expression IS symmetric - you get the same answer regardless of how you label the roots. If we expand that out we get r1^2 - 2r1r2 + r2^2, which is symmetric, which means we can write it as a function of the coefficients. So we've come up with an expression whose square root depends on the way we've labeled the roots (using Vieta's formulas you can show it's b^2-4c, which you might recognize from the quadratic equation). Galois theory is used to show that root extraction can only break certain types of symmetries, and that fifth degree polynomials can exhibit root symmetries that are not breakable by radicals.
hurturue 10mo ago
can you tell if you flip -1 and 1? or just the i and -i axis is special
- purplesyringa 10mo ago
  You can: the equation x^2 = x holds for 1, but not for -1, so you can separate them. There is no way to write an equation without mentioning i (excluding cheating with Im, which again can't be defined without knowing i) that holds for i, but not -i.

owyn 10mo ago

The original post is written by AI so I will read it briefly, but your comment is fascinating. I got through undergrad math by brute force memorization and taking the C. Or sometimes the C-. The underlying concepts were never really clear to me. I did take a good online calculus class later that helped. However, I have questions: "Turns out the quantity they needed exists, but couldn't be described in their notation" What is this about? Sounds interesting. "Statisticians just said "oh, that function" and gave it a new name." What is this? I never understood there is a relationship between quadratic equations and some kind of underlying mathematic geometric symmetry. Is there a good intro to this? I only memorized how to solve them. And the existential question. Is there a good way to teach this stuff?

bayesnet 10mo ago
What is so unbelievably frustrating about math education is that these interesting questions are not even hinted at until far, far down the line (and before people make the assumption I was educated outside the US). I avoided math like the plague until my PhD program. Real analysis was a program requirement so I had to quickly teach myself calculus and get up to speed—and I found I really, really liked it. These high level questions are just so interesting and beyond the rote calculation I thought math was. I hope I can give my daughter a glimpse of the interesting parts before the school system manages to kill her interest altogether (and I would welcome tips to that end if anyone has them).
- Terr_ 10mo ago
  Reminds me of "A Mathematician's Lament" [0]. I'd prefer to link a non-PDF copy, but nothing came up with a casual search. [0] https://en.wikipedia.org/wiki/A_Mathematician's_Lament https://en.wikipedia.org/wiki/A_Mathematician's_Lament
- noelwelsh 10mo ago
  Geometric proofs are really accessible. You don't need any algebra to prove Pythagoras' theorem, or that the sum of the inner angles of a triangle is 180 degrees, for example. Compass and straight-edge construction of simple figures is also fun.
defen 10mo ago
> I never understood there is a relationship between quadratic equations and some kind of underlying mathematic geometric symmetry. In a polynomial equation, the coefficients can be written as symmetric functions of the roots: https://en.wikipedia.org/wiki/Vieta%27s_formulas https://en.wikipedia.org/wiki/Vieta%27s_formulas - symmetric means it doesn't matter how you label the roots, because it would not make sense if you could say "r1 is 3, r2 is 7" and get a different set of coefficients compared to "r1 is 7, r2 is 3". Since the coefficients are symmetric functions of the roots, that means that you can't write the roots as a function of the coefficients - there's no way to break that symmetry. This is where root extraction comes in - it's not a function. A function has to return 1 answer for a given input, but root extraction gives you N answers for the nth root of a given input. So that's how we're able to "choose" roots - consider the expression (r1 - r2) for a quadratic equation. That's not symmetric (the answer depends on which one we label as r1 and which we label as r2), so we can't write that expression as a function of the coefficients. But what about (r1 - r2)^2? That expression IS symmetric - you get the same answer regardless of how you label the roots. If we expand that out we get r1^2 - 2r1r2 + r2^2, which is symmetric, which means we can write it as a function of the coefficients. So we've come up with an expression whose square root depends on the way we've labeled the roots (using Vieta's formulas you can show it's b^2-4c, which you might recognize from the quadratic equation). Galois theory is used to show that root extraction can only break certain types of symmetries, and that fifth degree polynomials can exhibit root symmetries that are not breakable by radicals.

hurturue 10mo ago

can you tell if you flip -1 and 1? or just the i and -i axis is special

purplesyringa 10mo ago
You can: the equation x^2 = x holds for 1, but not for -1, so you can separate them. There is no way to write an equation without mentioning i (excluding cheating with Im, which again can't be defined without knowing i) that holds for i, but not -i.