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. 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".

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