0

I am not crazy. Hear me out.

I am not from a physics background but from maths. I have a really weird question in physics that is making me lose sleep. How can calculus describe physics? How is it possible? Nature is not continuous (or is it?). Everything eventually breaks down into atoms. The way we deal with objects in physics is to consider constituent parts of $dx$ length. But the point is that $dx$ doesn't exist! It is like infinity! An abstract concept which doesn't have existence in reality. Calculus on the other hand deals with the continuum. Real numbers don't have gaps in between them but atoms do. I understand calculus is used as an approximation but it shouldn't even work like that. How does a body of knowledge created for an entirely different kind of object work for something else?

I am not crazy.

Qmechanic
  • 201,751
Suraj
  • 57
  • 1
    You might want to look up "nonstandard analysis" which deals with so-called hyperreal numbers, filling-in the required "gaps" in the real numbers. – André May 02 '20 at 08:33
  • 3
    What do you mean by "nature isn't continuous"? The basic entities on which calculus is applied are space and time intervals--and to the best of our knowledge, space and time are continuous. In fact, we would run into big troubles if they weren't continuous, e.g., different speeds of light for different wavelengths of light. –  May 02 '20 at 08:46
  • 2
    Possible duplicates: https://physics.stackexchange.com/q/92925/2451 , https://physics.stackexchange.com/q/70376/2451 , https://physics.stackexchange.com/q/33273/2451 , https://physics.stackexchange.com/q/9720/2451 and links therein. – Qmechanic May 02 '20 at 08:47
  • I meant objects like matter Dvij D. C. – Suraj May 02 '20 at 09:19
  • 1
    I think it's a perfectly reasonable question. He is asking how we can trust in the notion of infinitesimals and the mathematics to apply to the real world. – tryst with freedom May 02 '20 at 10:25
  • @suraj Sorry, I didn't see your comment yesterday. In any case, we don't assume matter to be continuous in the sense you seem to be proposing we do. The subject that deals with the structure of matter is that of quantum field theory and it is phenomenally adequate to describe structures such as atoms. It uses calculus on entities such as length and time or more abstract entities such as phases of wavefunctions. It really doesn't assume that the structure of matter is continuous in the way space is continuous. –  May 03 '20 at 17:28

3 Answers3

3

Mathematical equations only ever provide models of physical situations. There are often limit conditions where the model breaks down and physicists are stuck for an answer. Black holes, where spacetime collapses to a singularity, are one well-known example.

Most physics is done at a relatively larger scale than the granular nature of reality. For example classical mechanics works only as long as you treat your material as a continuum; once you get down to the atomic level you have to drop those equations and use quantum mechanics instead. Go down even smaller to the fabric of spacetime and you will have to drop that in its turn and resort to loop quantum gravity or whatever.

So when you consider say some velocity as dx/dt, you are just modelling a physical boundary condition of "much smaller" by using the mathematical "infinitesimally small" or "limit as dt approaches zero" as an acceptable approximation. Once your experiment gets so small that the "much smaller" boundary condition is reached, the maths no longer predicts the lab results and physicists are forced to abandon the model for a more useful one.

Guy Inchbald
  • 7,372
3

I think it's a perfectly reasonable question. He is asking how we can trust in the notion of infinitesimals and the mathematics to apply to the real world. The problem is due to it being taught using non-standard analysis. Think of mathematics as a tool to solve some problems easily. Take for example finding center of mass of a thin rod with a density $\rho$. This would be impossible unless we split the the rod into some mass chunks and do apply the results we have on point masses/ on those and add em back up. Now if we make the mass chunks very small then we can just write this as an integral.

Anyways the point is that splitting things up i.e :integrals help us to apply results which are meant for one singular thing. That's the big picture , for me at least.

If you learn real analysis you'd probably more clear with what's going on. I really don't get why forum people discourage these type of questions as they indicate that the person is thinking very deeply about the topics he has learnt.

tryst with freedom
  • 7,940
2

Calculus shouldn't work for describing physics

It is amusing, because Newton was the pioneer in using calculus for physics, who used it to calculate successfully quantities in mechanics.

Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, calculus has widespread uses in science, engineering, and economics

You say:

I am not crazy.

You should read the link which goes into the history of calculus. Ignorance is not craziness.

When Newton developed his gravitational and mechanics theories using mathematical models and imposing on the solutions physical laws to pick up the correct functions, atoms were not even a dream. Matter was considered continuous and modeled that way, using the continuous variables of space and time.

Electricity and magnetism theory also assumed continuous space and time variables and charge and electric fields too. Classical physics is still well described in the dimensions for which it was developed.

As research continued atoms and molecules were found, but space and time still are continuous variables, event down to elementary particles. The mathematical models are very successful in describing data and predicting future behavior using mathematics and extra axioms to pick up the relevant solutions.

anna v
  • 233,453