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The use of Calculus for the analysis of real-world phenomenon depends entirely on our universe not only being continuous, but being differentiable.

By "real-world phenomenon" I mean things like the motion of a planet under the influence of a gravitational force (or just the action of forces upon objects in general), or fluid flow in a pipe. Just things that we can represent mathematically.

For every function modeling some real-life phenomenon, it seems that at a single point that function has a definite derivative, and doesn't just sharply turn.

In other words, we can consider almost any function modeling some real-world phenomenon locally linear.

It’s useful that we can: if small segments of a function are locally linear, then infinitely small segments become easy to analyze. We can then apply whatever properties we found to be true for the small segments of the function, to the entire function. That's what calculus is for.

But why is it that real life works this way - that movement, forces, etc are all differentiable functions? Or, more importantly than WHY this is (since I feel that "why" something it isn't really a question that can be discussed), are all real-world phenomenon differentiable?

Would a universe where things like (for example) the motion of a flower pot thrown from a plane and falling towards a planet due to gravity, would instead be modeled by a function that isn't differentiable, even be possible?

Thanks!

  • Could you please explain your passage starting with "But why is this?" Is the sentence in brackets syntactically correct? – Jo Wehler Apr 12 at 15:40
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    If you are asking why all "observed functions" are smooth, have you considered that the space of smooth or even analytic functions is (usually) dense in the space of continuous functions? Since observations are approximations, it will (in the standard way of doing things) never be possible to tell them apart. So choosing the better behaved one seems reasonable. – Jishin Noben Apr 12 at 20:28
  • Are you asking if all corporal bodies are dividable? – Geremia Apr 12 at 22:44
  • A lot of times we add noise in our problems (such as the movement of dust particles for example), and the most popular noise is Brownian motion, which happens to be continuous everywhere but differentiable nowhere. – K9Lucario Apr 13 at 2:29
  • "Calculus for the analysis of real-world phenomenon depends entirely on our universe not only being continuous, but being differentiable". No, it does depend on that at all. It is well-known that continuous/smooth systems well approximate discrete ones with large number of elements, and they are more tractable analytically, the central limit theorem of statistics is a simple example. That's why. And it makes no difference if the universe "really" is continuous. If what quantum and string theories suggest is right, it probably is not. – Conifold Apr 14 at 9:20
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Classical physics as well as quantum mechanics (QM) and the theory of general relativity use as basic equation differential equations. A differential equation assumes that the law in question can be expressed by differentiable functions.

This assumption has proven fruitful since the times of Newton.

And QM shows: Even when the basic differential equations have differentiable solutions (self-adjoint operators), at a fixed point of space and time these solutions may have a discrete series of eigenvalues. And it is the set of eigenvalues that we observe (e.g., phenomena like the wave-length of spectral lines), not the generating solution.

Nevertheless, again and again alternative approaches are proposed to explain the phenomena by a strict discrete approach without using differential equations, e.g. by a model from cellular automata.

QM shows that the real-world phenomena are not necessarily differentiable. Therefore, I would like to change your question into the more fundamental question:

"Are all basic equations in physics necessarily differentiable equations?"

I do not see any a priori reason for the answer "yes".

  • You equivocated the physical phenomena of the universe, which was what the OP asked about; with the "basic equations of physics." In other words there's (a) How the universe actually works; and (b) Our latest historically contingent theories of physics. Two entirely different things. I'm sure you know that. And surely the latter question is LESS fundamental. What's fundamental is how the universe actually works, wouldn't you agree? – user4894 Apr 12 at 21:08
  • @user4894 ad a) I gave a counter example against differentiability, even continuity. - Then, moving a step further I said: I would like to change(!) the question: ad b) see my last line: I do not see any a priori argument which makes differentiability a must. – Jo Wehler Apr 12 at 21:31
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I wouldn't say that physical phenomena universe is differentiable, rather differentiability is a property of functions which are abstract objects (unlike rocks, trees, concrete stuff we do physics on). Note that momentum, forces and the like are abstract objects that we use to get information about concrete things.

When we do physics, we take a phenomenon, such as the trajectory (imaginary line) of a hammer, turn it into a mathematical object (my professor called this a homomorphism from the real world to some mathematical space), do some math, and then send the result back into the real world.

Differentiability in this case would not be a feature of the universe anymore than uniform continuity, integrability or invertability would be, all of which are also very common to functions that appear often in physics.

Now just because there are differential equations floating around everwhere in physics does not mean other nastier functions aren't used. When using probability, we resort to Lebesgue integrals to handle ugly functions, and don't forget about the dirac delta function, which is fundamental in all sorts of problems.

[Note: In partial differential equations, there are what are called weak solutions, solutions that are not differentiable but satisfy the differential equations in some other way https://en.wikipedia.org/wiki/Weak_solution. ]

P.S An interesting idea would be if you were to believe that abstract objects do exist ontologically in some way (such as Platonic forms). Then perhaps in some sense the universe would indeed be differentiable.

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