Not the OP, but...

I was just reading Landau & Lifshitz' "Statistical Physics", and can reflect on a series of thoughts that may elaborate on how intuition plays a role in the enjoyment and understanding of complex material. I've been meaning to write it down anyway...

On page 3, the book says "A fundamental feature of this [closed system/open subsystem] approach is the fact that, because of the extreme complexity of the external interactions with the other parts of the system, during a sufficiently long time the subsystem considered will be many times in every possible state." When I first read this, I thought "non sequitur, but whatever, I'll continue..." Now, the context of this quote is that the authors are trying to explain why statistical methods work at all. And they said prior that we start with laws that apply to 'microscopic' particles and use statistics to generalize to 'macroscopic' systems.

However, the second time I read this, I kept thinking it must be backwards. We didn't understand the motion of (classical) protons before understanding the motion of macroscopic balls. So we had to have been operating under the assumption that the macroscopic laws must apply to microscopic objects, and then require that the must also be reproducible macroscopically through statistical methods. That is, we require that these laws be invariant across the microscopic-to-macroscopic transition. But to do that, we have to use a framework which expresses such a transition. So, for instance, if we are reasoning about the motion of a ball, we have to translate our laws into laws over the motion of some statistical model of the ball. Say, it's center of mass. And with this concept in hand, we could write laws that apply to both the macro and micro worlds, since a 'center of mass' is a macro-micro-scale-invariant abstraction. So we partition the space of all possible laws of nature, and chose to work only in that partition which encodes things we can actually know about nature -- the partition identified by the macro-micro-scale-invariant.

So re-reading that passage, it is now not a non-sequitur for me. Now it says "because the interactions with the outside world are so complex, we could not hope to predict their influence. Thus we are justified in using random variables to model their influence, and concepts that derive from the use of random variables to ensure we have complexity-scale invariance when we formulate our laws of physics." And this is not a non-sequitur to me. It follows directly from the meaning of the word "random." Of course statistical methods work when the complexity of a system is indistinguishable from randomness.

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And this whole line of thought generalizes (albeit informally.) For instance, the meaning of words is an invariant across a long thread of contextual translations, and these invariants are used the same way: to partition the space of all possible meanings in such a way that one partition contains all of the 'knowledge' imbued by that word, and thus you can navigate a narrower space of meaning to find the intended and/or correct one. Gives me a certain brand of appreciation for good poetry.

Or -- my girlfriend -- who recently told me that she loved algebra but couldn't understand trigonometry. I tried telling her that the algebraic transformations were invariant-preserving operations selected because they conformed to known laws about 'functions' like addition and multiplication which have commutativity and identity laws, and that trigonometry was no different: different functions, but all of the algebraic transformations you needed were selected from the laws of trigonometry with the purpose of maintaining the exact same invariants. (Not that she cared much, to be honest...)

And on and on... I could probably talk for days about all the different ways every subject can be reduced to transformations and invariants and how they are used to solve problems.