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> Mathematics can be seen as a logic game. You start with a set of assumptions and you come up with all the logical conclusions you can from that. Then, if some

by IanCal 3y ago

> Mathematics can be seen as a logic game. You start with a set of assumptions and you come up with all the logical conclusions you can from that. Then, if someone else finds a situation that fits those assumptions, they can benefit from the pre-discovered logical conclusions. This means that if some conclusions require fewer assumptions, then those conclusions are more generally applicable This is a really, really nice expression of something my mind's been hovering around for a while.

greydius 3y ago
The same principle applies to programming. Functions that know less about their arguments are more generally applicable.
lachlan_gray 3y ago
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.
  - 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.
tetha 3y ago
This is also a part of why I am somewhat fascinated by the idea and the state of Lean4 and mathlib in Lean4. People put more and more formally verified proofs into mathlib, which in turn makes formally proving further theorems in mathlib easier. If you start with nothing (like in the numbers game), simple proofs are a lot of ... just effort, because you have specify a lot of rewrites and overall work. In mathlib, however, systems like simp (the simplification system) or linarith ("There is a solution by linear arithmetic") seem to do a lot of heavy, repetitive lifting by now. It's a really interesting snowball effect. Sadly, everything I understand is most likely already in there, so I doubt I could contribute meaningfully, haha.
- IanCal 3y ago
  That's very interesting, I'm no mathematician but I should have a play around with it. > Sadly, everything I understand is most likely already in there, so I doubt I could contribute meaningfully, haha. I wouldn't be so sure - and even if so then remember there's enormous benefit to improving tooling around a system. If you want to be involved somehow, better devx, tutorials, output, packaging, error messages all make a big difference to end users. Edit - As another thought, is there benefit in going through papers and translating that work into lean4? I'm not really familiar enough with it but if so that may 1. Find issues in current work, like Tao did in his own work 2. Add to a reusable body of work
- lmpdev 3y ago
  Former computational mathematics major You absolutely can contribute meaningfully The maths world is incomprehensibly broad and deep, even if you just take the Erdos approach and go for interesting but shallow problems
_ahxg 3y ago
A good example of that is how the Axiom of choice impacts the measure/probability theory. It imply the existence of some sets that cannot be Lebesgue measured (which is an generalization of width, volume, etc for arbitrary sets, also generalization of probability for arbitrary sets)... but it's not possible to present a single example of those non measurable sets, only prove that they exist. And it's possible to construct an alternative theory with the axiom of determinacy, then any subset of R is measurable. * https://en.wikipedia.org/wiki/Axiom_of_choice https://en.wikipedia.org/wiki/Axiom_of_choice * https://en.wikipedia.org/wiki/Axiom_of_determinacy https://en.wikipedia.org/wiki/Axiom_of_determinacy * https://en.wikipedia.org/wiki/Lebesgue_measure https://en.wikipedia.org/wiki/Lebesgue_measure
jerf 3y ago
Mathematics is humanity's longest running, largest-scoped, most complicated game. It also happens to be useful, and you can dive into a lot of philosophy about that which is all very interesting. The utility itself is a large thing on its own. But I think of that utility as something separate from the game itself. The game is just a game. You can do whatever you want with it. If you want to convert your cookbook to hexadecimal just for fun, you can. The fact that it is (broadly speaking) useless, that it will produce no new knowledge, and if anything negative utility in general, doesn't mean you can't do it. That's the game. You can also try to play the game to prove the Twin Prime conjecture. That's a much harder level. This game is scalable to all ages and skill levels, has the best level variety, and can be done with anything from just your personal noggin, to a pencil & paper, to the largest computing cluster in the world. Technically all other games you play are a subset of this game; that may not always be a useful way to think of it, but it is technically true. And while there are a few rules, generally, nobody can tell you how to play it. You want to color pretty pictures? The game has lots of ways of doing that. You want to smash atoms together? The game can help with that. You want to simply count to the highest number you possibly can? Go for it. It's a very popular play with the younger players, but anyone can do it.

by IanCal 3y ago

greydius 3y ago

The same principle applies to programming. Functions that know less about their arguments are more generally applicable.

lachlan_gray 3y ago

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.
- 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.

tetha 3y ago

This is also a part of why I am somewhat fascinated by the idea and the state of Lean4 and mathlib in Lean4. People put more and more formally verified proofs into mathlib, which in turn makes formally proving further theorems in mathlib easier. If you start with nothing (like in the numbers game), simple proofs are a lot of ... just effort, because you have specify a lot of rewrites and overall work. In mathlib, however, systems like simp (the simplification system) or linarith ("There is a solution by linear arithmetic") seem to do a lot of heavy, repetitive lifting by now. It's a really interesting snowball effect. Sadly, everything I understand is most likely already in there, so I doubt I could contribute meaningfully, haha.

IanCal 3y ago
That's very interesting, I'm no mathematician but I should have a play around with it. > Sadly, everything I understand is most likely already in there, so I doubt I could contribute meaningfully, haha. I wouldn't be so sure - and even if so then remember there's enormous benefit to improving tooling around a system. If you want to be involved somehow, better devx, tutorials, output, packaging, error messages all make a big difference to end users. Edit - As another thought, is there benefit in going through papers and translating that work into lean4? I'm not really familiar enough with it but if so that may 1. Find issues in current work, like Tao did in his own work 2. Add to a reusable body of work
lmpdev 3y ago
Former computational mathematics major You absolutely can contribute meaningfully The maths world is incomprehensibly broad and deep, even if you just take the Erdos approach and go for interesting but shallow problems

_ahxg 3y ago

A good example of that is how the Axiom of choice impacts the measure/probability theory. It imply the existence of some sets that cannot be Lebesgue measured (which is an generalization of width, volume, etc for arbitrary sets, also generalization of probability for arbitrary sets)... but it's not possible to present a single example of those non measurable sets, only prove that they exist. And it's possible to construct an alternative theory with the axiom of determinacy, then any subset of R is measurable. * https://en.wikipedia.org/wiki/Axiom_of_choice https://en.wikipedia.org/wiki/Axiom_of_choice * https://en.wikipedia.org/wiki/Axiom_of_determinacy https://en.wikipedia.org/wiki/Axiom_of_determinacy * https://en.wikipedia.org/wiki/Lebesgue_measure https://en.wikipedia.org/wiki/Lebesgue_measure

jerf 3y ago

Mathematics is humanity's longest running, largest-scoped, most complicated game. It also happens to be useful, and you can dive into a lot of philosophy about that which is all very interesting. The utility itself is a large thing on its own. But I think of that utility as something separate from the game itself. The game is just a game. You can do whatever you want with it. If you want to convert your cookbook to hexadecimal just for fun, you can. The fact that it is (broadly speaking) useless, that it will produce no new knowledge, and if anything negative utility in general, doesn't mean you can't do it. That's the game. You can also try to play the game to prove the Twin Prime conjecture. That's a much harder level. This game is scalable to all ages and skill levels, has the best level variety, and can be done with anything from just your personal noggin, to a pencil & paper, to the largest computing cluster in the world. Technically all other games you play are a subset of this game; that may not always be a useful way to think of it, but it is technically true. And while there are a few rules, generally, nobody can tell you how to play it. You want to color pretty pictures? The game has lots of ways of doing that. You want to smash atoms together? The game can help with that. You want to simply count to the highest number you possibly can? Go for it. It's a very popular play with the younger players, but anyone can do it.