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They both share the view that explanations/hypotheses can either be retained or rejected (and better explanations can retained). Math is unique in that the sta

They both share the view that explanations/hypotheses can either be retained or rejected (and better explanations can retained). Math is unique in that the standard of proof is more comprehensive and the domain is bounded but this is a good example of someone employing the scientific method and exhibiting good scientific community values.

xyzzyz 9y ago
No, mathematics works completely different than science. In both math and science, you do some observations, and then formulate theory that explain some of these observations. However, in science, you then design experiments and predict the results based on your theory, taking care to ensure that the experiments distinguish between the reality in which your theory holds, and the one where it doesn't. In math you might also do that as a sanity check, but that's not the point. In math, the point is to actually _prove_ your theory to be correct, something which is absolutely impossible to do in science, where you can only prove theory to be _wrong_.
- deleted 9y ago
  [deleted]
- alexanderstears 9y ago
  >taking care to ensure that the experiments distinguish between the reality in which your theory holds, and the one where it doesn't. I get the impression that you're thinking of specific domains within science. Many theories are contextualized within something (e.g a theory of why there's lightning might be more domain specific than a theory of electromagnetism). I don't disagree that things are proved within mathematical systems and scientific claims / hypotheses are categorically different (and weaker, I suppose). But the author made a claim, discovered an error in their claim, and alerted the community to his mistake and signaled their intention to address it. That's not behavior that's exclusive to mathematics. Many scientists have made claims that they think the evidence supports, and then they realize that something wasn't correct, and they make the correction. Cartographers / mariners used to think that California was an island: https://en.wikipedia.org/wiki/Island_of_California https://en.wikipedia.org/wiki/Island_of_California They had solid evidence to think that's the case but then eventually they realized the mistake and they've corrected it.
  - thaumasiotes 9y ago
    > They had solid evidence to think that's the case Why do you say this? It is not supported by the link you provide.
- evincarofautumn 9y ago
  I like to think of it in terms of classical vs. constructive logic—the former is concerned with truth and falsehood, the latter with provability. In mathematics you can show that some hypothesis is true or false. In science you can only demonstrate that you have been unable to find an example of anything inconsistent with your hypothesis; that is, in the real world we don’t have the law of the excluded middle: ¬¬P is not the same as P. This also means that mathematics lets us talk about universal properties ∀x∈S.P(x) (“for all x in S, P is true of x”), but science really only gives you existential properties ∃x∈S.P(x) (“there is some x in S for which P is true of x”). The scientific method is based on the idea that enough statements of the form ¬K∃x∈S.¬P(x) (“it is not known that there is an x in S for which P is not true of x”) resulting from a large number of convincing experiments can serve as evidence of ¬∃x∈S.¬P(x) (“there is no x in S for which P is not true of x”) which gives us an approximation of ∀x∈S.P(x).
- mistercow 9y ago
  You can't prove a theory wrong in science either. All you can do is to collect evidence that pushes you in one direction or the other.
  - bovine3dom 9y ago
    No, you can prove things wrong. In fact, a lot of science is in making sure that your claims are falsifiable. E.g, "the moon is made of cheese". cf. Bertrand Russell's teapot.
    - notfed 9y ago
      If this is true then you can also prove things right: "the Moon is not made of cheese".
    - savanaly 9y ago
      We can't prove that wrong. We can just say it's almost certainly wrong given the body of evidence against it. It's distinctly possible that every single measurement and observation of the moon that observed it to not be made of cheese was mistaken for one reason or another, however.
      - bovine3dom 9y ago
        Ok, but then you're so far into pedantics that I'd suggest that you can't mathematically prove anything either, because you're using your brain to do so, which might be suffering from some flaw that stops you from realising that actually 1+1=3.
        mistercow 9y ago
        That is only as pedantic as it is pedantic to say that science can't prove things right. There is no fundamental quality that makes a statement "positive" or "negative", so "proving X wrong" is always "proving ⌐X right". It's possible to be very precise about what we mean when we talk about evidence and logic, so coming up with these wishy-washy principles like "science can only prove things wrong; oh, you know what I mean" just confuses things. As for applying that to math, yes that's true to some extent. And on rare occasion, mathematical proofs are found to have flaws in them years after they have become commonly accepted. I think there's an important distinction to be made, though, between the certainty of deductive reasoning, and the uncertainty of deductive reasoning as interpreted by humans.
codeulike 9y ago
Maths has proof. Science only has models that fit evidence collected so far
- gertef 9y ago
  How do you check a (human-created) math proof for correctness? By comparing it to evidence collected so far. If human-performed math were not a science, a retraction would never happen.
  - ColinWright 9y ago
    > How do you check a (human-created) math proof for correctness? By comparing it to evidence collected so far. I don't understand this. You don't check a math proof against evidence, you check it for the correctness of the logical deductions from the axioms to the conclusions. Here's a proof that there are infinitely many primes: Take any finite collection of primes. Multiply them all together, and add one, giving a number X. Let P be the smallest prime that divides X (note that P might equal X). P cannot be in the original collection, so we have created a new prime. Therefore any finite collection of primes is incomplete, and so the collection of all primes must be infinite. Suppose I want to check that for correctness. No amount of evidence will show that it's correct, so I really don't know what you are saying. Perhaps you could expand on your thoughts using this as a specific example.
dekhn 9y ago
I think everybody replying to my original comment ("math, not science") misunderstood what I meant. It's my fault, I was ambiguous. The person I replied to said "this is how science works". I replied "math, not science", meaning in this case, it was an example of how math works (self-correcting through review). But that's also how science works. I didn't mean to imply there were huge differences in the ontology of math and science, as practiced, although I found the subsequent discussion of the differences in their pursuits stimulating. I see less difference in math and science than most, probably due to the influence of "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" iin which Wigner, one of the biggest players in quantum mechanics, points out that mathematical models/theories of physical systems often have unexpected extrapolative predictive ability, suggesting some relationship between math and physics (I believe some people would say that the universe has an underlying mathematical structure). https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.htm...

xyzzyz 9y ago

No, mathematics works completely different than science. In both math and science, you do some observations, and then formulate theory that explain some of these observations. However, in science, you then design experiments and predict the results based on your theory, taking care to ensure that the experiments distinguish between the reality in which your theory holds, and the one where it doesn't. In math you might also do that as a sanity check, but that's not the point. In math, the point is to actually _prove_ your theory to be correct, something which is absolutely impossible to do in science, where you can only prove theory to be _wrong_.

deleted 9y ago
[deleted]
alexanderstears 9y ago
>taking care to ensure that the experiments distinguish between the reality in which your theory holds, and the one where it doesn't. I get the impression that you're thinking of specific domains within science. Many theories are contextualized within something (e.g a theory of why there's lightning might be more domain specific than a theory of electromagnetism). I don't disagree that things are proved within mathematical systems and scientific claims / hypotheses are categorically different (and weaker, I suppose). But the author made a claim, discovered an error in their claim, and alerted the community to his mistake and signaled their intention to address it. That's not behavior that's exclusive to mathematics. Many scientists have made claims that they think the evidence supports, and then they realize that something wasn't correct, and they make the correction. Cartographers / mariners used to think that California was an island: https://en.wikipedia.org/wiki/Island_of_California https://en.wikipedia.org/wiki/Island_of_California They had solid evidence to think that's the case but then eventually they realized the mistake and they've corrected it.
- thaumasiotes 9y ago
  > They had solid evidence to think that's the case Why do you say this? It is not supported by the link you provide.
evincarofautumn 9y ago
I like to think of it in terms of classical vs. constructive logic—the former is concerned with truth and falsehood, the latter with provability. In mathematics you can show that some hypothesis is true or false. In science you can only demonstrate that you have been unable to find an example of anything inconsistent with your hypothesis; that is, in the real world we don’t have the law of the excluded middle: ¬¬P is not the same as P. This also means that mathematics lets us talk about universal properties ∀x∈S.P(x) (“for all x in S, P is true of x”), but science really only gives you existential properties ∃x∈S.P(x) (“there is some x in S for which P is true of x”). The scientific method is based on the idea that enough statements of the form ¬K∃x∈S.¬P(x) (“it is not known that there is an x in S for which P is not true of x”) resulting from a large number of convincing experiments can serve as evidence of ¬∃x∈S.¬P(x) (“there is no x in S for which P is not true of x”) which gives us an approximation of ∀x∈S.P(x).
mistercow 9y ago
You can't prove a theory wrong in science either. All you can do is to collect evidence that pushes you in one direction or the other.
- bovine3dom 9y ago
  No, you can prove things wrong. In fact, a lot of science is in making sure that your claims are falsifiable. E.g, "the moon is made of cheese". cf. Bertrand Russell's teapot.
  - notfed 9y ago
    If this is true then you can also prove things right: "the Moon is not made of cheese".
  - savanaly 9y ago
    We can't prove that wrong. We can just say it's almost certainly wrong given the body of evidence against it. It's distinctly possible that every single measurement and observation of the moon that observed it to not be made of cheese was mistaken for one reason or another, however.
    - bovine3dom 9y ago
      Ok, but then you're so far into pedantics that I'd suggest that you can't mathematically prove anything either, because you're using your brain to do so, which might be suffering from some flaw that stops you from realising that actually 1+1=3.
      - mistercow 9y ago
        That is only as pedantic as it is pedantic to say that science can't prove things right. There is no fundamental quality that makes a statement "positive" or "negative", so "proving X wrong" is always "proving ⌐X right". It's possible to be very precise about what we mean when we talk about evidence and logic, so coming up with these wishy-washy principles like "science can only prove things wrong; oh, you know what I mean" just confuses things. As for applying that to math, yes that's true to some extent. And on rare occasion, mathematical proofs are found to have flaws in them years after they have become commonly accepted. I think there's an important distinction to be made, though, between the certainty of deductive reasoning, and the uncertainty of deductive reasoning as interpreted by humans.

codeulike 9y ago

Maths has proof. Science only has models that fit evidence collected so far

gertef 9y ago
How do you check a (human-created) math proof for correctness? By comparing it to evidence collected so far. If human-performed math were not a science, a retraction would never happen.
- ColinWright 9y ago
  > How do you check a (human-created) math proof for correctness? By comparing it to evidence collected so far. I don't understand this. You don't check a math proof against evidence, you check it for the correctness of the logical deductions from the axioms to the conclusions. Here's a proof that there are infinitely many primes: Take any finite collection of primes. Multiply them all together, and add one, giving a number X. Let P be the smallest prime that divides X (note that P might equal X). P cannot be in the original collection, so we have created a new prime. Therefore any finite collection of primes is incomplete, and so the collection of all primes must be infinite. Suppose I want to check that for correctness. No amount of evidence will show that it's correct, so I really don't know what you are saying. Perhaps you could expand on your thoughts using this as a specific example.

dekhn 9y ago

I think everybody replying to my original comment ("math, not science") misunderstood what I meant. It's my fault, I was ambiguous. The person I replied to said "this is how science works". I replied "math, not science", meaning in this case, it was an example of how math works (self-correcting through review). But that's also how science works. I didn't mean to imply there were huge differences in the ontology of math and science, as practiced, although I found the subsequent discussion of the differences in their pursuits stimulating. I see less difference in math and science than most, probably due to the influence of "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" iin which Wigner, one of the biggest players in quantum mechanics, points out that mathematical models/theories of physical systems often have unexpected extrapolative predictive ability, suggesting some relationship between math and physics (I believe some people would say that the universe has an underlying mathematical structure). https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.htm...