4 ms·

It blows my mind to think of mathematics/logic almost like a huge cellular automaton. “axioms” don’t necessarily correspond to “truth”, to me they’re arbitrary

It blows my mind to think of mathematics/logic almost like a huge cellular automaton. “axioms” don’t necessarily correspond to “truth”, to me they’re arbitrary constraints that can give rise to complexity. And sometimes the resulting systems can be useful

gumby 3y ago
The whole of the puzzles of cosmology actually all might be obvious if we had a few different fundamental theorems. But because we hit on some that almost work, and then build upon them a hole edifice of mathematics that is internally consistent and almost fits the universe we keep beating on it, not realizing that backing up a little and then driving forward again at a slightly different angle might yield a simpler, and even more consistent and explanatory system.
- moring 3y ago
  This has happened, and is happening all the time. Many groundbraking theories in physics can be framed this way. The problem is that "slightly different angle" is a huge space, so scientists throw a lot of theories at it and see what sticks.
- chongli 3y ago
  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.
- one-another-dev 3y ago
  This reminds me of "The Road Not Taken" by Harry Turtledove. Awesome story!
- ChainOfFools 3y ago
  In other words, you're claiming that we may have built-in biases, invisible to us, which cause us to mistake certain premises for conclusions, and now we've got a definition of something like what a "unit" is, or what identity means, that work well enough to solidly discourage further investigation. yet if we just tried, oh, making the unit circle a unit... ellipse... all of the epiphenomenal complexity that comes from remediating the pervasively accumulated 0.01% error in that fundamental assumption would instantly vanish.
nylonstrung 3y ago
I love this idea. You might enjoy Stephen Wolfram's writing- it's exactly what you're talking about
thriftwy 3y ago
Axioms are not wrong if you can derive some math from them. They may not correspond to anything in our world, and then we usually discover something that does.
- tsimionescu 3y ago
  The point wasn't that they're wrong, but instead that they are arbitrary. You could create a mathematical system with entirely different axioms than what we explore typically, and it would only be different in how usefully it maps onto real world concepts.

gumby 3y ago

The whole of the puzzles of cosmology actually all might be obvious if we had a few different fundamental theorems. But because we hit on some that almost work, and then build upon them a hole edifice of mathematics that is internally consistent and almost fits the universe we keep beating on it, not realizing that backing up a little and then driving forward again at a slightly different angle might yield a simpler, and even more consistent and explanatory system.

moring 3y ago
This has happened, and is happening all the time. Many groundbraking theories in physics can be framed this way. The problem is that "slightly different angle" is a huge space, so scientists throw a lot of theories at it and see what sticks.
chongli 3y ago
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.
one-another-dev 3y ago
This reminds me of "The Road Not Taken" by Harry Turtledove. Awesome story!
ChainOfFools 3y ago
In other words, you're claiming that we may have built-in biases, invisible to us, which cause us to mistake certain premises for conclusions, and now we've got a definition of something like what a "unit" is, or what identity means, that work well enough to solidly discourage further investigation. yet if we just tried, oh, making the unit circle a unit... ellipse... all of the epiphenomenal complexity that comes from remediating the pervasively accumulated 0.01% error in that fundamental assumption would instantly vanish.

nylonstrung 3y ago

I love this idea. You might enjoy Stephen Wolfram's writing- it's exactly what you're talking about

thriftwy 3y ago

Axioms are not wrong if you can derive some math from them. They may not correspond to anything in our world, and then we usually discover something that does.

tsimionescu 3y ago
The point wasn't that they're wrong, but instead that they are arbitrary. You could create a mathematical system with entirely different axioms than what we explore typically, and it would only be different in how usefully it maps onto real world concepts.