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Comment by dboreham | Hacker News Reader

Well...pretty much everything important turns out to be an eigenvector.

That has less to do with some mysterious fact about eigenvectors, and more to do with the fact that linear algebra is incredibly incredibly well studied and understood; with octopus arms sticking into every special case of applied mathematics imaginable. Give it a few decades and "everything" will be some other (today obscure) mathematical idea.

At least one part of the universe really does seem to prefer linear algebra: quantum mechanics. Every quantum state is a linear combination of eigenvectors, and that's unlikely to change within a few decades (it would be a revolution analogous to the discovery of quantum mechanics itself).

I would argue that the reason QM is expressed in linear algebra is also not a mysterious property of LA, but because that is what was part of the physicists curriculum when the field was developed.

Physicists would have preferred to do it with calculus -indeed, intro QM is usually taught with calculus, not with linear algebra. But you can't go very far without talking about operators and eigenstates. Linear algebra fits the underlying physical phenomenon. People have tried to develop nonlinear versions of quantum mechanics, with no success so far. In particular, if quantum mechanics was not exactly linear, then a quantum computer would be able to solve NP-complete problems in polynomial time!

To get an intuition of why quantum mechanics looks the way it does, Scott Aaronson is extremely helpful: https://www.scottaaronson.com/democritus/lec9.html

I think the main reason for the initial focus on calculus in intro QM is more to do with making sure you have a solid grasp of the ugly, complicated, hard way of doing things before diving into the more elegant, easy way (once armed with an appreciation of all the whacky machinery under the simplified proverbial hood). I recall my intro classical mechanics had a similar approach on a few topics, and my professor explicitly said this was the reason for it. That and the elegant way is generally more abstract, and it's important to have the foundation the abstraction was built on first.

The link you posted looks interesting and fun to read, I'm looking forward to going through it after work. Thank you!

I'm not a fan of the teaching method where you do something the hard way first so you'll appreciate it, but I was a pretty lazy student so that's understandable. Maybe it works for most students.

You can also express QM/QFT in terms of matrix algebra (S-Matrix formalism).

That's a subset of the linear algebra used in quantum mechanics, no? Any transformation between QM states can be expressed as a unitary matrix - this is not a particular characteristic of scattering phenomena with S-matrices. The quantum states are still vectors, as in the rest of QM.

On second thoughts, I expressed myself very poorly... it goes without saying that matrices and linear algebra are identical; what I meant was that one can compute quantum interactions with the S-Matrix alone and not need to ‘explicitly’ use linear algebra concepts to deal with superposition of states & cetra. The focus on the s-matrix as a fundamental formalism was spearheaded by Wheeler, Heisenberg, and Landau (and amongst others), if my memory serves.

P.S. Even this statement is formally quite weak, but hopefully I have clarified myself enough to transfer my intended meaning. Apologies.

Your right it's not some mysterious property of LA, but you're wrong about it being the curriculum. Linear algebra just works and is the best we've found.

You’re right: it’s a fairly circular argument.

We actually know that QM (or rather QFT) cannot be the whole story because it cannot be renormalised and reconciled with general relativity, so quite possibly the definitive theory will not be expressed in terms of linear algebra and thus maybe the Universe is not based on linear algebra.

I agree that this is possible, but as I said, I don't expect such a revolution within the next few decades. I would be happy to be proved wrong!

A category maybe?

Wait a sec... everything already is a category.

And a set.

I thought it was known that not everything is a set?

There’s more than one way to define what a set is, and many of these definitions always allow you to postulate there exist further collections that cannot meet that definition of a set ad infinitum. Most importantly ZFC (which is at least in theory the formal basis for the vast majority of mathematics) has this property, and so do many extensions like NBG and TG.

of course - it's the empty set :)

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