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He’s right - intuitive physical reasoning is useful (and often valid in the applied sciences) but it isn’t rigorous mathematical proof. Some areas of mathemati

He’s right - intuitive physical reasoning is useful (and often valid in the applied sciences) but it isn’t rigorous mathematical proof. Some areas of mathematics are applicable to physical realities, but they aren’t defined by those realities.

ajkjk 3y ago
Yeah it's not rigorous. The tragedy is that the organizations responsible for teaching math, like how to do it and how to understand things mathematically, for other purposes, would rather waste their time on rigor. No problem with there being a rigorous math department. For mathematicians. But, quite literally, no one else cares. Mathematicians and everyone else are at crossed purposes, and the mathematicians are wrong to choose their purposes over their students' needs. Uh... Not that I'm defending the previous poster. Just griping about rigor in general. Physical arguments for mathematical problems are rarely sound, and I'd assume that an undergraduate who thinks they have made a sound one is wrong until reading it to see.
- eigenket 3y ago
  I quite strongly disagree with this. Mathematicians aren't rigourous just for the hell of it. We have examples where a lack of rigour caused a bunch of people to waste a bunch of time working of stuff that ultimately didn't work at all. For example the Italian school of algebraic geometry caused the waste of legitimately decades of work by many people because the foundations weren't right: https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge.... Part of the purpose of a maths degree is to teach the students the rigour required to be a mathematician. If you don't want to learn that rigour then thats completely fine, you can use physical or intuitive arguments, but the place to go and do that is in the physics or engineering departments. I agree mathematicians would be wrong if they were forcing their rigour on physics students or whatever, but I think they're emphatically right to teach it to maths students.
  - zozbot234 3y ago
    On the other hand, if you want full and complete rigor that means formalizing things in a computer-based proof checker. Now, that can often reveal ways to refactor the argument into a cleaner, more elegant, more intuitive proof, which is why full formalizations are often regarded as publishable work - but the other side of it is that dotting all the i's and crossing all the t's does take a whole lot of effort since some published proofs are not nearly as clear as they're supposed to be.
    - eigenket 3y ago
      I agree that some published proofs are not as clear as they're supposed to be, but I disagree that formalizing things in computer-based proof checkers is the way forward there. Computer proof checking is cool, but its an augment to what we already do and not not a substitute for it. The purpose of a published paper is to explain things to other humans, while that of a computer formalization is to explain things to computers. Just as it is generally not easy for a computer to understand a published proof, it is usually not easy for other mathematicians to understand the code you feed to Lean or whatever. The stuff you need to focus on to get a computer to accept your proof is usually quite different to the stuff you want to focus on when explaining things to colleagues. Roughly speaking we care about why you're doing things, while the computer cares about the intricacies of what you're doing.
- ink_13 3y ago
  > No problem with there being a rigorous math department. For mathematicians. But, quite literally, no one else cares As someone who did an undergrad in mathematics, I have to disagree with you here. People think math is about numbers and computation, but that's like saying literature is about letters and composition. Fundamentally, mathematics is the study of things that are provably true. Without rigor it's not math.
  - ajkjk 3y ago
    That's just false. What physicists (and engineers, etc) are doing is also math and it's way more useful and insightful.. and (imo) the world would be a better place if that's what everyone else was also learning in college. The idea that math is only about rigor isn't intrinsic; it's a historical accident that people seem to not realize is optional. It is also about understanding things and being able to wield concepts to accomplish goals or convey understanding to others.
    - ink_13 3y ago
      Novelists and lawyers both do writing, but I wouldn't call their output the same thing. You might call legal writing more useful or novelists more insightful, but that's a matter of opinion. One is not intrinsically more valuable than the other. > The idea that math is only about rigor isn't intrinsic; it's a historical accident that people seem to not realize is optional I mean, it is by definition. To a mathematician, doing calculations is not mathematics, no more than spelling is doing poetry. Which is not to say that doing calculations is without value! I think what we have here is (ironically) an unrigorous definition.
      - Jensson 3y ago
        > doing calculations is not mathematics I hope you realize that physics isn't about calculations either? You can put numbers or types in the equations, just like you can put numbers or types in the expressions or results you get in math.
      - RoyalHenOil 3y ago
nuancebydefault 3y ago
If some mathematics are not applicable to physical realities, to what are they applicable? Is there any un-applicable math and if so, why does it even exist?
- jampekka 3y ago
  Maybe depends on what's meant by application, but I'd say a lot of, probably most, research-level math has no (clear) practical applications and most research is not done from the PoV of applications or relationship to physical reality. Math is study of formal systems, sort of "what can and can't follow from some set of formal rules". It's more a surprise that some rulesets map to physical world as well as they do [1]. For some there may be very important ones in the future like e.g. number theory got in cryptography. But for example just knowing whether P=?NP or if the Riemann hypothesis is true probably has no "physical reality" direct applications. [1] https://en.m.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences https://en.m.wikipedia.org/wiki/The_Unreasonable_Effectivene...
  - nuancebydefault 3y ago
    So if the application is not visible today, it might appear tomorrow. In fact that makes sense.
- sampo 3y ago
  Up until about World War II and use in cryptography, number theory was pretty much an un-applicable field of mathematics.
  - nuancebydefault 3y ago
    Pretty amazing that someone discovered and explored number theory without knowing if there would be an application to it. Hopefully they lived to see its usage.
    - jampekka 3y ago
rendaw 3y ago
Then shouldn't the professor have said that it isn't rigorous? Saying it's perfect implies that it is rigorous, just that they took issue with the association.

ajkjk 3y ago

Yeah it's not rigorous. The tragedy is that the organizations responsible for teaching math, like how to do it and how to understand things mathematically, for other purposes, would rather waste their time on rigor. No problem with there being a rigorous math department. For mathematicians. But, quite literally, no one else cares. Mathematicians and everyone else are at crossed purposes, and the mathematicians are wrong to choose their purposes over their students' needs. Uh... Not that I'm defending the previous poster. Just griping about rigor in general. Physical arguments for mathematical problems are rarely sound, and I'd assume that an undergraduate who thinks they have made a sound one is wrong until reading it to see.

eigenket 3y ago
I quite strongly disagree with this. Mathematicians aren't rigourous just for the hell of it. We have examples where a lack of rigour caused a bunch of people to waste a bunch of time working of stuff that ultimately didn't work at all. For example the Italian school of algebraic geometry caused the waste of legitimately decades of work by many people because the foundations weren't right: https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge.... Part of the purpose of a maths degree is to teach the students the rigour required to be a mathematician. If you don't want to learn that rigour then thats completely fine, you can use physical or intuitive arguments, but the place to go and do that is in the physics or engineering departments. I agree mathematicians would be wrong if they were forcing their rigour on physics students or whatever, but I think they're emphatically right to teach it to maths students.
- zozbot234 3y ago
  On the other hand, if you want full and complete rigor that means formalizing things in a computer-based proof checker. Now, that can often reveal ways to refactor the argument into a cleaner, more elegant, more intuitive proof, which is why full formalizations are often regarded as publishable work - but the other side of it is that dotting all the i's and crossing all the t's does take a whole lot of effort since some published proofs are not nearly as clear as they're supposed to be.
  - eigenket 3y ago
    I agree that some published proofs are not as clear as they're supposed to be, but I disagree that formalizing things in computer-based proof checkers is the way forward there. Computer proof checking is cool, but its an augment to what we already do and not not a substitute for it. The purpose of a published paper is to explain things to other humans, while that of a computer formalization is to explain things to computers. Just as it is generally not easy for a computer to understand a published proof, it is usually not easy for other mathematicians to understand the code you feed to Lean or whatever. The stuff you need to focus on to get a computer to accept your proof is usually quite different to the stuff you want to focus on when explaining things to colleagues. Roughly speaking we care about why you're doing things, while the computer cares about the intricacies of what you're doing.
ink_13 3y ago
> No problem with there being a rigorous math department. For mathematicians. But, quite literally, no one else cares As someone who did an undergrad in mathematics, I have to disagree with you here. People think math is about numbers and computation, but that's like saying literature is about letters and composition. Fundamentally, mathematics is the study of things that are provably true. Without rigor it's not math.
- ajkjk 3y ago
  That's just false. What physicists (and engineers, etc) are doing is also math and it's way more useful and insightful.. and (imo) the world would be a better place if that's what everyone else was also learning in college. The idea that math is only about rigor isn't intrinsic; it's a historical accident that people seem to not realize is optional. It is also about understanding things and being able to wield concepts to accomplish goals or convey understanding to others.
  - ink_13 3y ago
    Novelists and lawyers both do writing, but I wouldn't call their output the same thing. You might call legal writing more useful or novelists more insightful, but that's a matter of opinion. One is not intrinsically more valuable than the other. > The idea that math is only about rigor isn't intrinsic; it's a historical accident that people seem to not realize is optional I mean, it is by definition. To a mathematician, doing calculations is not mathematics, no more than spelling is doing poetry. Which is not to say that doing calculations is without value! I think what we have here is (ironically) an unrigorous definition.
    - Jensson 3y ago
      > doing calculations is not mathematics I hope you realize that physics isn't about calculations either? You can put numbers or types in the equations, just like you can put numbers or types in the expressions or results you get in math.
    - RoyalHenOil 3y ago

nuancebydefault 3y ago

If some mathematics are not applicable to physical realities, to what are they applicable? Is there any un-applicable math and if so, why does it even exist?

jampekka 3y ago
Maybe depends on what's meant by application, but I'd say a lot of, probably most, research-level math has no (clear) practical applications and most research is not done from the PoV of applications or relationship to physical reality. Math is study of formal systems, sort of "what can and can't follow from some set of formal rules". It's more a surprise that some rulesets map to physical world as well as they do [1]. For some there may be very important ones in the future like e.g. number theory got in cryptography. But for example just knowing whether P=?NP or if the Riemann hypothesis is true probably has no "physical reality" direct applications. [1] https://en.m.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences https://en.m.wikipedia.org/wiki/The_Unreasonable_Effectivene...
- nuancebydefault 3y ago
  So if the application is not visible today, it might appear tomorrow. In fact that makes sense.
sampo 3y ago
Up until about World War II and use in cryptography, number theory was pretty much an un-applicable field of mathematics.
- nuancebydefault 3y ago
  Pretty amazing that someone discovered and explored number theory without knowing if there would be an application to it. Hopefully they lived to see its usage.
  - jampekka 3y ago

rendaw 3y ago

Then shouldn't the professor have said that it isn't rigorous? Saying it's perfect implies that it is rigorous, just that they took issue with the association.