3 ms·

Most mathematics has no application to science whatsoever. It's a huge parts bin which scientists delve into when they build their models. And then much of the

Most mathematics has no application to science whatsoever. It's a huge parts bin which scientists delve into when they build their models. And then much of the work is in trying to shoehorn the mathematics into being tractable. Mathematics is also not provably internally consistent. This was famously shown by Gödel [1]. [1] https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theorems https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theor...

tsimionescu 3y ago
Most mathematics originates from trying to solve physical or engineering problems. Typically physicists have been on the forefront of mathematical research - this has only really changed significantly in the last few decades. Also, mathematics as practiced is internally consistent. It is incomplete, though. That is how it stays afloat of Godel's result. Basically Godel's results showed that no matter how much we strive, there will always be propositions which might be true, but which we will not be able to prove are true. Unless of course we start using methods that sometimes prove false propositions, which we have not done.
- gumby 3y ago
  > Most mathematics originates from trying to solve physical or engineering problems. Has this been true since the early 20th century? I have no feel for what constitutes "most" in the vast corpus of pure mathematics, so am not challenging your claim but rather am curious.
  - deleted 3y ago
    [deleted]
  - tsimionescu 3y ago
    You're right, that might actually be wrong. However, the claim I was actually thinking of, which is right I think, is that the maths used in the physical revolutions of the turn of the century (SR, QM, GR, and probably QFT, QED, and QCD as well) was invented by physicists or by mathematicians working with physicists for the express purpose of developing this theories, not the other way around. Also, the basis of mathematics and the first few thousand years were indeed motivated by these kinds of concerns.
    - defrost 3y ago
      I wouldn't agree - consider the hyperbolic transforms used to describe space time "bending" wrt relativity: https://en.wikipedia.org/wiki/History_of_Lorentz_transformations https://en.wikipedia.org/wiki/History_of_Lorentz_transformat... In mathematics, transformations equivalent to what was later known as Lorentz transformations in various dimensions were discussed in the 19th century in relation to the theory of quadratic forms, hyperbolic geometry, Möbius geometry, and sphere geometry, which is connected to the fact that the group of motions in hyperbolic space, the Möbius group or projective special linear group, and the Laguerre group are isomorphic to the Lorentz group. Mathematicians were following up on "what happens when you discard one of Eucilids Axioms" and discovering there was an entire world of consistent hyperbolic geometry and more. Some time later: In physics, Lorentz transformations became known at the beginning of the 20th century, when it was discovered that they exhibit the symmetry of Maxwell's equations. Subsequently, they became fundamental to all of physics, because they formed the basis of special relativity in which they exhibit the symmetry of Minkowski spacetime, making the speed of light invariant between different inertial frames. If you read mathematics histories it's a common complaint that it's nigh on impossible to discover something new and esoteric that doesn't soon end up with a military application; the ongoing search for interesting but useless mathematics is akin to the search for the fountain of youth. It is the case (IIRC) that quaterions arose directly from Hamilton's search for a better way to describe mechanical motions in three dimension spaces - ie created to be useful from the outset.
    - gumby 3y ago
      Yes, I agree to some extent with the restricted claim, which only (slowly) started to break down in the 17th century in the west. A lot of Indian mathematics was rather abstract going back to Vedic times, but since they didn't develop the concept of proof, it sadly had little impact on other mathematics practice (except as inspiration to Persian and Arab scholars) other than the the famous cases of zero and positional notation. The mathematical documents I've seen from that practice have been in the form of essays. I know little of Chinese or Mesoamerican mathematics and wonder where they were on this axis. It seems pretty likely that maths started in support of astronomy/planting predictions in the cultures I know of so likely also for East Asia and the Americas, but whither thence did it go?
- eternityforest 3y ago
  Is there no use for some kind of branch of mathematics that can prove false things? You would think that with how much math there is, there would be a whole field of working with uncertain proofs. I have no idea what for, but then again I'm not a math guy.

tsimionescu 3y ago

Most mathematics originates from trying to solve physical or engineering problems. Typically physicists have been on the forefront of mathematical research - this has only really changed significantly in the last few decades. Also, mathematics as practiced is internally consistent. It is incomplete, though. That is how it stays afloat of Godel's result. Basically Godel's results showed that no matter how much we strive, there will always be propositions which might be true, but which we will not be able to prove are true. Unless of course we start using methods that sometimes prove false propositions, which we have not done.

gumby 3y ago
> Most mathematics originates from trying to solve physical or engineering problems. Has this been true since the early 20th century? I have no feel for what constitutes "most" in the vast corpus of pure mathematics, so am not challenging your claim but rather am curious.
- deleted 3y ago
  [deleted]
- tsimionescu 3y ago
  You're right, that might actually be wrong. However, the claim I was actually thinking of, which is right I think, is that the maths used in the physical revolutions of the turn of the century (SR, QM, GR, and probably QFT, QED, and QCD as well) was invented by physicists or by mathematicians working with physicists for the express purpose of developing this theories, not the other way around. Also, the basis of mathematics and the first few thousand years were indeed motivated by these kinds of concerns.
  - defrost 3y ago
    I wouldn't agree - consider the hyperbolic transforms used to describe space time "bending" wrt relativity: https://en.wikipedia.org/wiki/History_of_Lorentz_transformations https://en.wikipedia.org/wiki/History_of_Lorentz_transformat... In mathematics, transformations equivalent to what was later known as Lorentz transformations in various dimensions were discussed in the 19th century in relation to the theory of quadratic forms, hyperbolic geometry, Möbius geometry, and sphere geometry, which is connected to the fact that the group of motions in hyperbolic space, the Möbius group or projective special linear group, and the Laguerre group are isomorphic to the Lorentz group. Mathematicians were following up on "what happens when you discard one of Eucilids Axioms" and discovering there was an entire world of consistent hyperbolic geometry and more. Some time later: In physics, Lorentz transformations became known at the beginning of the 20th century, when it was discovered that they exhibit the symmetry of Maxwell's equations. Subsequently, they became fundamental to all of physics, because they formed the basis of special relativity in which they exhibit the symmetry of Minkowski spacetime, making the speed of light invariant between different inertial frames. If you read mathematics histories it's a common complaint that it's nigh on impossible to discover something new and esoteric that doesn't soon end up with a military application; the ongoing search for interesting but useless mathematics is akin to the search for the fountain of youth. It is the case (IIRC) that quaterions arose directly from Hamilton's search for a better way to describe mechanical motions in three dimension spaces - ie created to be useful from the outset.
  - gumby 3y ago
    Yes, I agree to some extent with the restricted claim, which only (slowly) started to break down in the 17th century in the west. A lot of Indian mathematics was rather abstract going back to Vedic times, but since they didn't develop the concept of proof, it sadly had little impact on other mathematics practice (except as inspiration to Persian and Arab scholars) other than the the famous cases of zero and positional notation. The mathematical documents I've seen from that practice have been in the form of essays. I know little of Chinese or Mesoamerican mathematics and wonder where they were on this axis. It seems pretty likely that maths started in support of astronomy/planting predictions in the cultures I know of so likely also for East Asia and the Americas, but whither thence did it go?
eternityforest 3y ago
Is there no use for some kind of branch of mathematics that can prove false things? You would think that with how much math there is, there would be a whole field of working with uncertain proofs. I have no idea what for, but then again I'm not a math guy.