"Classification of mathematical problems as linear and nonlinear is like classification of the Universe as bananas and non-bananas. "

and everything turns around the same principles. For example dynamical models and PID controls.

yet solving a banana, is the only thing we really know how to do. So we end up fitting everything in our banana models.

I disagree with the implication that linearity is an unnatural concept, it appears whenever the changes being studied are small relative to the key parameters that determine the system. Every system is linear for small perturbations. Even logic gates; in negative feedback they can form passable inverting amplifiers. In a place as big as the universe it is rather common for two things to be very different in scale and yet interacting.

> Every system is linear for small perturbations.

every smooth system, sure, but even continuity is no guarantee of locally linearity.

I've never actually seen a physical example of a system without a continuous first derivative. For example phase transitions, commonly touted as an example of discontinuity, don't actually occur until the matter has gone a bit over the point and a transition nucleates somewhere. The probability of a phase transition is a continuous function of temperature, with continuous derivatives.

I'm skeptical that discontinuities can exist because, if they did, they'd serve as infinitely powerful microscopes. If there's a discontinuity in nature, it must exist at absolute zero. I don't have a similarly good argument for continuous first derivatives but I do think it's interesting that there are no examples AFAIK.

Any physical system that makes/breaks contact, such as walking robots. Sure, the foot is not perfectly rigid and technically is a stiff spring. But from a computational perspective, problems still bear all the hallmarks of a discontinuous system such as requiring a very short integration step.

Yes, that's another example of the discontinuity existing in an idealized model, but not real life.

Quantized space is absolutely discontinuous, and tunneling is a discontinuous system. In fact assuming the universe is quantum it’s discontinuous in reality but the appearance is continuous. But these distinctions aren’t super useful unless you’re dealing with these sorts of effects. Continuity is the approximation, discontinuity is the reality. But depending on what’s useful we use the mathematics that help us.

Tunneling currents are continuous in every parameter, although I admit that when you're dealing with particles you have continuous probability distributions with continuously varying means, rather than continua of matter. (But that should count, because all macroscopic variables are expectation values.)

Specifically at quantized space and time levels everything is discrete even distribution functions. There’s no sense in having a continuous spacial distribution sub Planck lengths.

If I wanted to understand and obtain your intuition about linearity, what would you recommend?

I question how strongly I would recommend the years of working with it. :-)

Like pure math? Engineering?

Study it in school?

There are a lot of linear problems that I'm interested in.

Do you use a computer algebra system?

Do you have a few books?

Isn’t the whole point of chaos sensitivity to initial conditions? Where the output even when minuscule differences varies wildly with input?

In chaotic systems, time becomes one of the parameters that has to be small for a linear model to make predictions, but they are still linear for brief times.

yeah, we use a lot of small linear systems to model a non linear system.

We just break the problem in small bananas because that's what we can solve, and then solve for those lil'bananas and call it done.

So, Lady Finger, not Cavendish. Got it.

Interesting aspect to see the world from, thanks

My favorite moment in university was in the first class of semester 2, where a prof said "lets look at a really small part of our thing, and assume we apply some force to it. This will make it stretch, lets assume the stretching is linear relative to the applied force". I raised my hand and asked "is this assumption supported empirically?" and he said "no, we know it's not always true, but if we don't make it we can't calculate anything".

At the time I was mad at engineers for being non-scientific, but after a few years I understood the deep wisdom in that. Nonlinear materials exist, and materials we use have nonlinear ranges. We just don't build things from those, because the math is too unwieldy. (except in very very specific edge cases where we spend a lot of money building a very limited thing)

It's funny: I had the EXACT SAME PROBLEM !!!! Like "how is it science when a prof says: "this equation is too complicated too solve so let's erase these third and forth derivatives and pretend that it's the same - even if it's not - because that's all that we can do"

So I left the "physics" courses and went the "math" courses... only to learn 3 years later how to prove that this kind of approximation is mathematically sound indeed :-D

Should have taken one more course about chaos theory which is the special case when this approach fails.

Can you give some examples of linear and non-linear materials?

Practically all materials behave nonlinearly when stretched or compressed a visible amount. For certain structural applications, though, if that happens we've already failed. Linear models work really well for designing big concrete structures and certain metal structures. Sometimes we try to apply linear models to other things, but that's always kind of fishy.

Linear algebra was the first math class I took in undergrad. I thought the next one was going to be non-linear algebra! But it wasn't.

Which means it really all just finding hyperplanes that are near the data.