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Yeah a lot of pure mathematics education very much emphasizes the learning the "spirit" or fundamental concepts of certain things rather than computation, somet

Yeah a lot of pure mathematics education very much emphasizes the learning the "spirit" or fundamental concepts of certain things rather than computation, sometimes to the extreme when compared even with applied mathematics. This is especially obvious in anything that even approached calculus. In the math courses I took in college, if we e.g. ever had a differential equation we needed to solve on a problem set "looked up the answer on Wolfram Alpha" was a perfectly reasonable response. Same for differentiation or integration. In some intro courses you'd be ask to prove that certain differentiation or integration techniques had mathematically rigorous backing, but never had to do the rote work of actually memorizing and using those techniques ever again. Again "looked up the answer on Wolfram Alpha" was a perfectly reasonable response. It was a far cry from the applied mathematics department. Another huge discrepancy was in linear algebra, where very little time was spent on matrix computation other than again proving that matrix operations and invariants preserved properties of linear maps and vector spaces and almost all the time of the course was spent on the linear maps and vector spaces themselves, whereas linear algebra in the applied math department was almost entirely about matrices and the intricacies of various matrix computations and decompositions and subjects like dual spaces or other things that couldn't be represented with normal matrix computation (e.g. infinite dimensional vector spaces) were omitted.

gjulianm 5y ago
To add to this comment, a lot of times we didn't even care what the solution was, only that it existed, was unique, or was bounded by some constant (and we didn't usually care about the value either). The more you advance into pure mathematics, the less you care about specific functions or values.
- raverbashing 5y ago
  And I suspect, this is what makes math grind its own gears. The way math is "built", with too generalizing theorems, non-constructive statements, etc makes it fall into traps like the Russel's Paradox (which literally cannot exist "in real life" - not even for the very wide definition of mathematical real life). Building math on saying "this exists" without actually building towards what it is, is a bit of a "sin".
  - gjulianm 5y ago
    > The way math is "built", with too generalizing theorems, non-constructive statements, etc makes it fall into traps like the Russel's Paradox I don't think Russel's paradox is a problem with ZFC set theory. Building sets out of "the set of everything" is not used, and even people are careful to specify when the axiom of choice is being used. > Building math on saying "this exists" without actually building towards what it is, is a bit of a "sin". It's actually not. The idea is that you prove only what you actually need for the next step. A nice example is again in the field of differential equations. Ultimately people are interested in a solution, but to get that solution you need an approximation method, and to get an approximation method you need to know when does it work and how fast does it work, and to know that you need to know when a solution can be found and what properties does it have. Constructivism is something that in theory might sound interesting, but forcing all mathematics to fit that model only servers to make certain proofs of true and useful statements far more complicated or even impossible.
    - lupire 5y ago
      It's rather unscientific to say that we believe things because we want them to be true and not because they are inherently true.
      - gjulianm 5y ago
        Nothing is inherently true, things are only true based on some axioms and hypothesis. What I meant with "results that are true" is things that are consistent with our physical world.
  - cryvate1284 5y ago
    You aren't very explicit what you mean, but I disagree I think. The main thing is willing things into existence (like ZFC, the Axiom of Choice) is equally consistent with not having it (ZF + not AC) or not deciding it (ZF) as shown by forcing. Russel's Paradox in particular was worked around by not allowing to "build sets" quantifying over all sets (see axiom schema of specification). If you are talking about avoiding "unprovable statements" (i.e. Godel's incompleteness theorem), you have to strip things way back further: it applies to systems that only have e.g. Peano arithmetic. An actual practical statement that is unprovable in Peano arithmetic is Goodstein's theorem: - write out a number in its base 2, including its exponents (i.e. recursive) - replace all the 2s with 3s - subtract 1 - write out a number in base 3, including its exponents - replace all the 3s with 4s - subtract 1 - ... The question is whether for all starting numbers this sequence eventually ends at 1. The answer is yes, trivially if you use ordinals (and replace all your base numbers with omega: the +1 will basically do nothing and -1 means it must be finite because ordinals well-ordered), but this cannot be used in Peano arithmetic. Fundamentally, like Paris-Harrington Theorem, I think the way to think about why this cannot be proved, is because these sequences get big before they end up at 1.
- sn41 5y ago
  From what I know, this happened in a large part due to the Three Body Problem. Once it was proved that there was no closed form solution in terms of elementary functions, Poincare suggested that we instead try to see if solutions existed at all, and whether they were unique. If so, we could at least hope for numerical solutions. I think this was partly responsible for the development of "analysis situs", i.e. topology. [1] This approach was fruitful in yielding concrete techniques - we now have efficient techniques like the fast multipole method by Rokhlin and Greengard, to numerically tackle the three body problem. Without existence and uniqueness results, there is no basis for the correctness of these algorithms. [1] https://en.wikipedia.org/wiki/Poincar%C3%A9_and_the_Three-Body_Problem https://en.wikipedia.org/wiki/Poincar%C3%A9_and_the_Three-Bo...
ls15 5y ago
My university had exams that involved arithmetic, matrix computation and differential equations, but no access to Wolfram Alpha though. It was impossible to pass the exams without prior training.
hn_go_brrrrr 5y ago
Once I got past calculus, it seemed like we stopped caring about numbers. Plenty of 0s and 1s, the occasion 2, but rarely a 3 or higher. I remember one theorem had a 24 in the derivation (Stone-Weierstrass, I want to say?) and we were all amazed. Given that, who needs arithmetic?

gjulianm 5y ago

To add to this comment, a lot of times we didn't even care what the solution was, only that it existed, was unique, or was bounded by some constant (and we didn't usually care about the value either). The more you advance into pure mathematics, the less you care about specific functions or values.

raverbashing 5y ago
And I suspect, this is what makes math grind its own gears. The way math is "built", with too generalizing theorems, non-constructive statements, etc makes it fall into traps like the Russel's Paradox (which literally cannot exist "in real life" - not even for the very wide definition of mathematical real life). Building math on saying "this exists" without actually building towards what it is, is a bit of a "sin".
- gjulianm 5y ago
  > The way math is "built", with too generalizing theorems, non-constructive statements, etc makes it fall into traps like the Russel's Paradox I don't think Russel's paradox is a problem with ZFC set theory. Building sets out of "the set of everything" is not used, and even people are careful to specify when the axiom of choice is being used. > Building math on saying "this exists" without actually building towards what it is, is a bit of a "sin". It's actually not. The idea is that you prove only what you actually need for the next step. A nice example is again in the field of differential equations. Ultimately people are interested in a solution, but to get that solution you need an approximation method, and to get an approximation method you need to know when does it work and how fast does it work, and to know that you need to know when a solution can be found and what properties does it have. Constructivism is something that in theory might sound interesting, but forcing all mathematics to fit that model only servers to make certain proofs of true and useful statements far more complicated or even impossible.
  - lupire 5y ago
    It's rather unscientific to say that we believe things because we want them to be true and not because they are inherently true.
    - gjulianm 5y ago
      Nothing is inherently true, things are only true based on some axioms and hypothesis. What I meant with "results that are true" is things that are consistent with our physical world.
- cryvate1284 5y ago
  You aren't very explicit what you mean, but I disagree I think. The main thing is willing things into existence (like ZFC, the Axiom of Choice) is equally consistent with not having it (ZF + not AC) or not deciding it (ZF) as shown by forcing. Russel's Paradox in particular was worked around by not allowing to "build sets" quantifying over all sets (see axiom schema of specification). If you are talking about avoiding "unprovable statements" (i.e. Godel's incompleteness theorem), you have to strip things way back further: it applies to systems that only have e.g. Peano arithmetic. An actual practical statement that is unprovable in Peano arithmetic is Goodstein's theorem: - write out a number in its base 2, including its exponents (i.e. recursive) - replace all the 2s with 3s - subtract 1 - write out a number in base 3, including its exponents - replace all the 3s with 4s - subtract 1 - ... The question is whether for all starting numbers this sequence eventually ends at 1. The answer is yes, trivially if you use ordinals (and replace all your base numbers with omega: the +1 will basically do nothing and -1 means it must be finite because ordinals well-ordered), but this cannot be used in Peano arithmetic. Fundamentally, like Paris-Harrington Theorem, I think the way to think about why this cannot be proved, is because these sequences get big before they end up at 1.
sn41 5y ago
From what I know, this happened in a large part due to the Three Body Problem. Once it was proved that there was no closed form solution in terms of elementary functions, Poincare suggested that we instead try to see if solutions existed at all, and whether they were unique. If so, we could at least hope for numerical solutions. I think this was partly responsible for the development of "analysis situs", i.e. topology. [1] This approach was fruitful in yielding concrete techniques - we now have efficient techniques like the fast multipole method by Rokhlin and Greengard, to numerically tackle the three body problem. Without existence and uniqueness results, there is no basis for the correctness of these algorithms. [1] https://en.wikipedia.org/wiki/Poincar%C3%A9_and_the_Three-Body_Problem https://en.wikipedia.org/wiki/Poincar%C3%A9_and_the_Three-Bo...

ls15 5y ago

My university had exams that involved arithmetic, matrix computation and differential equations, but no access to Wolfram Alpha though. It was impossible to pass the exams without prior training.

hn_go_brrrrr 5y ago

Once I got past calculus, it seemed like we stopped caring about numbers. Plenty of 0s and 1s, the occasion 2, but rarely a 3 or higher. I remember one theorem had a 24 in the derivation (Stone-Weierstrass, I want to say?) and we were all amazed. Given that, who needs arithmetic?