5 ms·

> they haven’t learned the fundamental ideas of algebra I'd very much love to know exactly what Knuth considers the "fundamental ideas of algebra"!

Hasz 4y ago
Perhaps the eponymous fundamental theorem of algebra is a good place to start. https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra
- thomasahle 4y ago
  From the same page: "the Fundamental Theorem of Algebra is neither fundamental, nor a theorem of algebra"
  - jacobolus 4y ago
    For more on this theme, see: https://www.math.nyu.edu/faculty/edwardsd/carnegie.pdf https://www.math.nyu.edu/faculty/edwardsd/carnegie.pdf
bo1024 4y ago
It sounds like he's just talking about using variables to solve equations. I think the point he's making in that section is just about learning a good process so you can solve harder and harder problems, e.g. Problem 1: Two numbers add to 100, one is 20 larger. Smart student: oh, I see, 60 and 40. Dumb student Knuth: x + y = 100 and x = y + 20, solves to x=60, y=40. .. Problem 2: Four numbers sum to 1024, one is half the sum of the other three less 17, one of the others... Smart student: uh, I don't see the answer. Dumb student Knuth: w + x + y + z = 1024, w = (x + y + z)/2 - 17, ... solved it.
- anthk 4y ago
  x + y = 100 x - y = 20 2x + y - y = 120 2x = 120 x = 60 y = 100 - x y = 40.
- TimTheTinker 4y ago
  Algebra goes so much deeper than this. I had so much trouble with my compilers class because my professor expressed all the ideas and principles of compilation using algebra. So I second OP's comment - I wish there were a course one could take on algebra itself - not merely numeric expressions with variables.
  - dllthomas 4y ago
    > I wish there were a course one could take on algebra itself Depending a little on just what you mean, that's likely the upper-division undergraduate math department course called "Algebra" at most universities. Groups and rings and such.
    - commandlinefan 4y ago
      > the upper-division undergraduate math department course called "Algebra" And be careful what you wish for - the way eighth-grade algebra hurt your brain when you were 13 will be nothing compared to the way abstract algebra will hurt your brain when you're 20.
      - dllthomas 4y ago
        ... in a good way. I liked "high school" algebra okay, but group theory was fun, even if homework was sometimes spending hours and pages to wind up with a 10 line proof.
      - anthk 4y ago
        In my case abstract algebra was much easier to grasp than polynomials with no context.
    - deleted 4y ago
      [deleted]
    - TimTheTinker 4y ago
      I thought of that, but it sounded like it was just studying particular mathematical structures -- not learning "how to read arbitrary algebraic expressions". Or perhaps algebraic expressions in different mathematical specialties are differentiated enough to make knowledge about how to read a given expression mostly non-transferable?
dboreham 4y ago
After reading the relevant article section, I'd guess he's talking about the idea that problems can be mapped onto mathematical structures, allowing use of pre-known rules within said structure, for transformation and identity and so on, such that the problem can then be solved. He's saying that for simple problems if you're clever you can intuit the solution without that mapping/manipulation/solve process, but as a result you can never see how to solve more complex problems. Implication being that if you had been slightly less smart, you'd end up understanding mathematical structure earlier in life, with associated benefits in terms of success in certain fields. Like: https://en.wikipedia.org/wiki/Abstract_algebra https://en.wikipedia.org/wiki/Abstract_algebra

Hasz 4y ago

Perhaps the eponymous fundamental theorem of algebra is a good place to start. https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra

thomasahle 4y ago
From the same page: "the Fundamental Theorem of Algebra is neither fundamental, nor a theorem of algebra"
- jacobolus 4y ago
  For more on this theme, see: https://www.math.nyu.edu/faculty/edwardsd/carnegie.pdf https://www.math.nyu.edu/faculty/edwardsd/carnegie.pdf

bo1024 4y ago

It sounds like he's just talking about using variables to solve equations. I think the point he's making in that section is just about learning a good process so you can solve harder and harder problems, e.g. Problem 1: Two numbers add to 100, one is 20 larger. Smart student: oh, I see, 60 and 40. Dumb student Knuth: x + y = 100 and x = y + 20, solves to x=60, y=40. .. Problem 2: Four numbers sum to 1024, one is half the sum of the other three less 17, one of the others... Smart student: uh, I don't see the answer. Dumb student Knuth: w + x + y + z = 1024, w = (x + y + z)/2 - 17, ... solved it.

anthk 4y ago
x + y = 100 x - y = 20 2x + y - y = 120 2x = 120 x = 60 y = 100 - x y = 40.
TimTheTinker 4y ago
Algebra goes so much deeper than this. I had so much trouble with my compilers class because my professor expressed all the ideas and principles of compilation using algebra. So I second OP's comment - I wish there were a course one could take on algebra itself - not merely numeric expressions with variables.
- dllthomas 4y ago
  > I wish there were a course one could take on algebra itself Depending a little on just what you mean, that's likely the upper-division undergraduate math department course called "Algebra" at most universities. Groups and rings and such.
  - commandlinefan 4y ago
    > the upper-division undergraduate math department course called "Algebra" And be careful what you wish for - the way eighth-grade algebra hurt your brain when you were 13 will be nothing compared to the way abstract algebra will hurt your brain when you're 20.
    - dllthomas 4y ago
      ... in a good way. I liked "high school" algebra okay, but group theory was fun, even if homework was sometimes spending hours and pages to wind up with a 10 line proof.
    - anthk 4y ago
      In my case abstract algebra was much easier to grasp than polynomials with no context.
  - deleted 4y ago
    [deleted]
  - TimTheTinker 4y ago
    I thought of that, but it sounded like it was just studying particular mathematical structures -- not learning "how to read arbitrary algebraic expressions". Or perhaps algebraic expressions in different mathematical specialties are differentiated enough to make knowledge about how to read a given expression mostly non-transferable?

dboreham 4y ago

After reading the relevant article section, I'd guess he's talking about the idea that problems can be mapped onto mathematical structures, allowing use of pre-known rules within said structure, for transformation and identity and so on, such that the problem can then be solved. He's saying that for simple problems if you're clever you can intuit the solution without that mapping/manipulation/solve process, but as a result you can never see how to solve more complex problems. Implication being that if you had been slightly less smart, you'd end up understanding mathematical structure earlier in life, with associated benefits in terms of success in certain fields. Like: https://en.wikipedia.org/wiki/Abstract_algebra https://en.wikipedia.org/wiki/Abstract_algebra