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I'd rather hope that basic linear algebra and the notion of vectorized function application becomes universal math knowledge like addition and multiplication, a

by ansgri 4y ago

I'd rather hope that basic linear algebra and the notion of vectorized function application becomes universal math knowledge like addition and multiplication, as basic programming slowly becomes. We need languages and compilers with first-class optimized differential programming support, sure, but increasing baseline education would benefit wider population.

imiric 4y ago
I agree with your last point, but most people have no practical need to learn high-level math. We struggle globally with even basic education rates, and programming definitely isn't for everyone. What is needed are higher-level abstractions over ML principles that make ML more accessible for people who are interested in programming, but lack the math background to be productive. This doesn't necessarily mean that the lower-level details should be entirely hidden, but that the tools are friendly enough for beginners to easily pick up, and flexible enough for advanced users to tweak to their needs.
- beepbooptheory 4y ago
  Why though? Aren't everyday users more interested in finished models themselves? Why would they want to make one, especially if they, in this example, don't really know how they work anyway? What is the value here?
  - imiric 4y ago
    "Everyday users" aren't programmers, but people who would use the apps powered by ML. Think ChatGPT. There are programmers, though, who are not fluent in the math that makes ML possible, but would be empowered by friendlier tools.
- mananaysiempre 4y ago
  Note that “high-level math” by your definition covers things that were basically understood by 1900 (a good part even by 1850) and polished by 1940. It’s an incredible failure of the education system that any of this still counts as “advanced” in any way in 2023[1]. So abstract away, by all means, ensuring the user doesn’t need to know the Biot-Savart law to use a doorbell is the engineer’s job after all, and as semiconductor electronics show, users can include other engineers. (Hell, even the humble resistor is a fairly advanced piece of technology dependent a good chunk of 20th-century physics.) But sooner or later we’re going to have to pay back this “ed debt”. (“Soc debt”?) [1] See my recent comment at https://news.ycombinator.com/item?id=34389684 https://news.ycombinator.com/item?id=34389684 for a starting point for a literature dive.
  - kgwgk 4y ago
    It’s 2023! There is no reason why high-school students couldn’t learn everything that was known to humanity in 1940, right?
    - mananaysiempre 4y ago
      Hell no, that goes beyond sarcasm and well into egregious goalpost shifting. For reference, things known to humanity in 1940 (though not necessarily understood, that was an important difference in my comment) include the Schwarzschild (stationary black hole) and FLRW (homogeneous expanding or contracting universe) solutions to the equations of general relativity (1917 and 1924 respectively), the Darwin term for the effect of special relativity on the spectrum of hydrogen (1928), the severely non-classical exchange interaction responsible for the existence of ferromagnetism (same year), the London equations describing the (non-)penetration of the magnetic field into type-I superconductors (1935), and a way to produce superfluid helium (1937). And that only includes things I was supposed to ( :( ) know at some point. Also Gödel’s incompleteness theorems (1931) and Turing’s proof of undecidability of the halting problem (1936). I’m certainly not proposing to demand high schoolers know all of those. The only reason I had to go as far as 1940 or so is because linear algebra came relatively late in the history of mathematics, even though it should logically serve as the foundation of multivariable calculus—and was enthusiastically adopted as such once it finally entered the collective mathematical unconscious in the late 19th century. Working out the pedagogy took a couple more decades. If we were only talking about the “advanced” mathematics of calculus, I’d say Newton’s knowledge as of 1690 would be way overkill. (That would include the convergence speed of power series, a decent theory of ordinary differential equations, the beginnings of the calculus of variations, and even the “Newton polygons”, a theory of formal-series solutions to polynomial equations that properly belongs to algebraic geometry.) At that point J S Bach was five, the city of Philadelphia was a town founded eight years ago, and Peter Romanov (later called the Great) was still twelve years away from founding St Petersburg and thirty years away from proclaiming Russia an empire. To be honest, I think even if school taught no mathematics beyond basic numeracy, I wouldn’t complain as much. I would still complain, mind you, but only to the extent I do about being able to grow up without knowing what a fugue or the twelve-bar blues is or who Giotto or Niccolò de' Niccoli were. Instead we torture people with years of alleged mathematics bearing no resemblance to the real thing, and thus give them license to think they are not “maths people” (whatever that means) and that whatever maths they did not hear about in school is “advanced” and obscure. At least the average high-school graduate knows he doesn’t have a clue about music history or theory.

by ansgri 4y ago

imiric 4y ago

I agree with your last point, but most people have no practical need to learn high-level math. We struggle globally with even basic education rates, and programming definitely isn't for everyone. What is needed are higher-level abstractions over ML principles that make ML more accessible for people who are interested in programming, but lack the math background to be productive. This doesn't necessarily mean that the lower-level details should be entirely hidden, but that the tools are friendly enough for beginners to easily pick up, and flexible enough for advanced users to tweak to their needs.

beepbooptheory 4y ago
Why though? Aren't everyday users more interested in finished models themselves? Why would they want to make one, especially if they, in this example, don't really know how they work anyway? What is the value here?
- imiric 4y ago
  "Everyday users" aren't programmers, but people who would use the apps powered by ML. Think ChatGPT. There are programmers, though, who are not fluent in the math that makes ML possible, but would be empowered by friendlier tools.
mananaysiempre 4y ago
Note that “high-level math” by your definition covers things that were basically understood by 1900 (a good part even by 1850) and polished by 1940. It’s an incredible failure of the education system that any of this still counts as “advanced” in any way in 2023[1]. So abstract away, by all means, ensuring the user doesn’t need to know the Biot-Savart law to use a doorbell is the engineer’s job after all, and as semiconductor electronics show, users can include other engineers. (Hell, even the humble resistor is a fairly advanced piece of technology dependent a good chunk of 20th-century physics.) But sooner or later we’re going to have to pay back this “ed debt”. (“Soc debt”?) [1] See my recent comment at https://news.ycombinator.com/item?id=34389684 https://news.ycombinator.com/item?id=34389684 for a starting point for a literature dive.
- kgwgk 4y ago
  It’s 2023! There is no reason why high-school students couldn’t learn everything that was known to humanity in 1940, right?
  - mananaysiempre 4y ago
    Hell no, that goes beyond sarcasm and well into egregious goalpost shifting. For reference, things known to humanity in 1940 (though not necessarily understood, that was an important difference in my comment) include the Schwarzschild (stationary black hole) and FLRW (homogeneous expanding or contracting universe) solutions to the equations of general relativity (1917 and 1924 respectively), the Darwin term for the effect of special relativity on the spectrum of hydrogen (1928), the severely non-classical exchange interaction responsible for the existence of ferromagnetism (same year), the London equations describing the (non-)penetration of the magnetic field into type-I superconductors (1935), and a way to produce superfluid helium (1937). And that only includes things I was supposed to ( :( ) know at some point. Also Gödel’s incompleteness theorems (1931) and Turing’s proof of undecidability of the halting problem (1936). I’m certainly not proposing to demand high schoolers know all of those. The only reason I had to go as far as 1940 or so is because linear algebra came relatively late in the history of mathematics, even though it should logically serve as the foundation of multivariable calculus—and was enthusiastically adopted as such once it finally entered the collective mathematical unconscious in the late 19th century. Working out the pedagogy took a couple more decades. If we were only talking about the “advanced” mathematics of calculus, I’d say Newton’s knowledge as of 1690 would be way overkill. (That would include the convergence speed of power series, a decent theory of ordinary differential equations, the beginnings of the calculus of variations, and even the “Newton polygons”, a theory of formal-series solutions to polynomial equations that properly belongs to algebraic geometry.) At that point J S Bach was five, the city of Philadelphia was a town founded eight years ago, and Peter Romanov (later called the Great) was still twelve years away from founding St Petersburg and thirty years away from proclaiming Russia an empire. To be honest, I think even if school taught no mathematics beyond basic numeracy, I wouldn’t complain as much. I would still complain, mind you, but only to the extent I do about being able to grow up without knowing what a fugue or the twelve-bar blues is or who Giotto or Niccolò de' Niccoli were. Instead we torture people with years of alleged mathematics bearing no resemblance to the real thing, and thus give them license to think they are not “maths people” (whatever that means) and that whatever maths they did not hear about in school is “advanced” and obscure. At least the average high-school graduate knows he doesn’t have a clue about music history or theory.