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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

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.
  - raverbashing 5y ago
    Thanks for your answer, I didn't know about Goodstein's theorem > Russel's Paradox in particular was worked around by not allowing to "build sets" quantifying over all sets (see axiom schema of specification). Ah I didn't know that by name but this is pretty much the gist of what I'm getting. ZFC and AC, while Axioms, seem like a better foundation than something that allows "set of all sets" which sounds like a vague specification even for a mathematician. And in the end, Russel's paradox raises the exact issue of (pardon my non-mathematical language) "Set of all sets is BS" Hence the axiom of specification which lets you build sets that exist. I'm not talking about Gödel or unprovable statements, because even if something is not provable it might be constructable.
    - yakubin 5y ago
      > I'm not talking about Gödel or unprovable statements, because even if something is not provable it might be constructable. What precisely do you mean by that statement? What is the thing that is not provable but constructible? A conjecture? If so, then what do you mean when you say that a conjecture is constructible? What does it mean to construct it? As far as I know, "construction" in maths usually means a finite process. And proofs are finite sequences of strings, satisfying certain rules specified by a proof framework (such as sequent calculus). So I'm curious in what way something could be constructible, but not provable. To be honest, I don't understand how those words could apply to the same things, because proving usually applies to conjectures, statements, formulas etc. But constructing usually applies to definitions and proofs, which cannot said to be true or false at all.
- auggierose 5y ago
  Time to read "Mathematics: Form and Function" by Saunders Mac Lane, my friend.
- wbsss4412 5y ago
  Mathematicians find what’s possible, then leave the algorithms to the computer scientists.
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...

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.
- raverbashing 5y ago
  Thanks for your answer, I didn't know about Goodstein's theorem > Russel's Paradox in particular was worked around by not allowing to "build sets" quantifying over all sets (see axiom schema of specification). Ah I didn't know that by name but this is pretty much the gist of what I'm getting. ZFC and AC, while Axioms, seem like a better foundation than something that allows "set of all sets" which sounds like a vague specification even for a mathematician. And in the end, Russel's paradox raises the exact issue of (pardon my non-mathematical language) "Set of all sets is BS" Hence the axiom of specification which lets you build sets that exist. I'm not talking about Gödel or unprovable statements, because even if something is not provable it might be constructable.
  - yakubin 5y ago
    > I'm not talking about Gödel or unprovable statements, because even if something is not provable it might be constructable. What precisely do you mean by that statement? What is the thing that is not provable but constructible? A conjecture? If so, then what do you mean when you say that a conjecture is constructible? What does it mean to construct it? As far as I know, "construction" in maths usually means a finite process. And proofs are finite sequences of strings, satisfying certain rules specified by a proof framework (such as sequent calculus). So I'm curious in what way something could be constructible, but not provable. To be honest, I don't understand how those words could apply to the same things, because proving usually applies to conjectures, statements, formulas etc. But constructing usually applies to definitions and proofs, which cannot said to be true or false at all.
auggierose 5y ago
Time to read "Mathematics: Form and Function" by Saunders Mac Lane, my friend.
wbsss4412 5y ago
Mathematicians find what’s possible, then leave the algorithms to the computer scientists.

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...