Thread

The Relation Between Mathematics and Physics by Paul Dirac (1939)

172 points · 84 comments · rramadass

GoblinSlayer · 1789637250 · focus · HN ↗
>but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen
This is quantifiable, right? Should be possible to measure how many mathematicians favor ideas like turbulence, macroscopic quantum physics, anthropic principle, analog geometry, finitism. I think, mathematics doesn't really try to match physics yet.
1. huurtehoog · 1789648350 · focus · HN ↗
  I think this passage is more about the fact that mathematics is a form of symbolic computation people create and it is weirdly congruent to the physical measurements and models of reality. Not that mathematicians see certain models and objects used in physical models in a favorable way. What favor even means here?
  It is quantifiable: several mathematical objects and categories people invented to do symbolic computation in letters and published papers are somehow very useful to model and understand data from physical measurements of reality.
  I highly recommend reading the "Unreasonable effectiveness of mathematics" essay by Wigner:
  <a href="https://www.hep.upenn.edu/~johnda/Papers/wignerUnreasonableEffectiveness.pdf" rel="nofollow">https://www.hep.upenn.edu/~johnda/Papers/wignerUnreasonableE...
  1. GoblinSlayer · 1789677405 · focus · HN ↗
    I don't see unreasonable effectiveness of mathematics. On one hand, it's a self fulfilling prophecy: if your math doesn't work, just make different math, and so on until it works (gravity was invented 6 times or so), so it's as effective as logic. On the other hand, it's fundamentally approximation limited by precision, so for any model we know when it breaks. And it's basically unworkable for chaotic or exponentially complex phenomena.
    podocarp · 1789690076 · focus · HN ↗
    You're just claiming it's effective. We know it's effective. But there's no reason it should be effective. It's primarily because philosophy has not caught up and we still can't explain the foundations of mathematics and causality. These things seem to be baked into our very nature. It is possible to image an alien race with different conceptions of logic and causality that are just unable however hard they try to formulate either proper mathematics or proper physical laws from mathematics. (Take for example the Pythagoreans who could not believe irrational numbers existed, so the famous cube root of 2 was fake news to them. They could never build their little cubic altar to whomever.)
    GoblinSlayer · 1789723253 · focus · HN ↗
    Mathematics can model simple parts of physics, because mathematics is simple and can do simple things, like a puddle fits in the hole. It doesn't model complicated parts of physics, because it can't to complicated things, because mathematics is invented by humies and thus limited by humie cognitive abilities.
    huurtehoog · 1789730107 · focus · HN ↗
    > Mathematics can model simple parts of physics, because mathematics is simple and can do simple things, like a puddle fits in the hole. It doesn't model complicated parts of physics, because it can't to complicated things,
    You ought to learn some advanced mathematics to avoid making such absurd claims as "mathematics is too simple to model complex physics"
    Pray tell what parts of physics are too complicated for mathematics
    [deleted] · 1789752259 · focus · HN ↗
    [deleted]
    GoblinSlayer · 1789803522 · focus · HN ↗
    Chaos, biology and exponential complexity.
2. DonaldFisk · 1789648777 · focus · HN ↗
  Dirac was talking about the fundamental laws. In favour of his point, and not known at the time he gave the talk, the standard model of particle physics is based on the symmetry groups U(1), SU(2), and SU(3), so Lie groups (which mathematicians find interesting) appear to have been chosen by nature.
3. empath75 · 1789651647 · focus · HN ↗
  I sort of have this suspicion that the kinds of phenomena that we can even apprehend are pretty closely related to what we can reason about mathematically, and the reason that math seems to work so well to model behavior that we understand says more about how our brain works than how the universe works.
  If a phenomenon isn’t model-able by relatively simple math it doesn’t even look like a phenomenon to us, it looks like a chaotic mess.
  Which is to say that unless something has mathematical coherence it is more or less invisible to our experience.
  1. sigbottle · 1789655130 · focus · HN ↗
    Well there are counting arguments to say that a higher level intelligence that somehow achieves say, 100000× brain efficiency of humans, still can't do that much more work than humans, if humans found the "best abstraction". Think about computability for example - that means you could feasibly solve problem instances of n+20 relative to what a human can solve.
    Of course, I think putting numbers to wishy washy meta-quantities like "how efficiently does a certain conceptual scheme help you" are super loaded and hard to properly talk about (incommensurability). I've been toying with trying to make a repository of all the possible "moves" one can make in this kind of abstract analysis - constrain the problem statement, argue something like "the system is what it does", dissolving, etc. but even that seems hard
  2. axionbraid · 1789662389 · focus · HN ↗
    [dead]
  3. rramadass · 1789663649 · focus · HN ↗
    In case you haven't read it; see "Science and Method by Henri Poincare" - <a href="https://archive.org/details/sciencemethod00poinuoft" rel="nofollow">https://archive.org/details/sciencemethod00poinuoft
    Here is a summary - <a href="https://thetelos.org/science-and-method-science-et-methode-henri-poincare/" rel="nofollow">https://thetelos.org/science-and-method-science-et-methode-h...
    Here is a video discussion - <a href="https://www.youtube.com/watch?v=sQ-t8-igDZo" rel="nofollow">https://www.youtube.com/watch?v=sQ-t8-igDZo
    The thesis is that Science is made for efficiency and hence does not include everything (which is infinite). It only looks for a hierarchy of interesting facts and focuses on simple foundational recurring phenomena from nature. It then uses the language of Mathematics to economize and impose order on the complexity to make it tractable.
    Finally, our subconscious prefers aesthetic attributes and hence we often find/create harmony/symmetry/beauty in our mathematical products.
    podocarp · 1789690677 · focus · HN ↗
    Perhaps symmetry is overrated and it is not the beauty that works, but it's economy. If something is symmetric depending on how symmetrical you can remove half or more of the required work during calculation. It is also useful to work in abstractions where it does not related to the system 1:1 but is some transformed representation (e.g. phase space). And it may be useful when you find a way to transform it to some easily solvable (symmetrical?) state, solve it there, and reverse the transform. Think of working in Cartesian and polar coordinates. Just a change in coordinate representations can improve your quality of life immediately on some problems with different symmetry. There's no intrinsic reason other than its much simpler on one than the other. Beauty doesn't have to come in. If it's round it's round. If it's square it's square.
    So I think symmetry might just be one of the tricks that keep working and keeps on giving back and so we love it and call it "beautiful".

Open on Hacker News to reply ↗

Unofficial Hacker News client; not affiliated with Y Combinator.