Differences are not approximations of derivatives, but derivatives are approximations of differences - The difference between mathematics and physics -

What is important in physics is that every physical theory has a range of applicability. For example, in mechanics, we introduce the concept of a point mass. However, there should be no physicist today who explains this under the assumption that objects with zero size actually exist. It means that in the regions of interest for experiments and observations, it is acceptable to approximate them as having zero size.

For example, celestial bodies in space are often treated as point masses to solve their equations of motion, and this explains observational data with good precision. But in reality, each celestial body has a finite size, structure, and material composition. Properties independent of these become visible in physics at large scales where the celestial body appears as a point.

In mechanics, objects are often treated as point masses, and their mass density distribution is frequently written using the Dirac delta function δ(x). However, that density distribution is not truly localized only at the origin with its value diverging to infinity. In reality, there is a finite size ε that depends on the detailed structure of the object.

For example, for a sufficiently small ε within the range of applicability of the theory, in the experimental region being considered, the integral of the product of an arbitrary smooth function f(x) from Figure 1 and the function representing the density distribution of an object that can be approximated as a point mass means that the density function can be approximated by a delta function. But in reality, it is not a point mass or a delta function, but has a finite size. The essence of physics is that if an object is sufficiently small, it can be approximated as a point mass, and from there, the properties of small objects with universality can be deduced. However, if one takes the limit of ε truly to zero as in mathematics, in many physical theories, one exceeds the valid range of applicability, and the results obtained become meaningless and completely unrelated to the real world.

Even when considering quantum mechanics, due to relativistic field effects, a single-particle state strictly localized at a single point in space does not exist. Many particles are required to localize a state. Please refer to the article below regarding this.

Also, in fluid dynamics, because of the existence of the atoms and molecules that make up the fluid, the strict ε→0 limit has no physical meaning. The picture of a continuous fluid is merely an approximate view of a collection of discrete atoms and molecules. In reality, the condition for justifying continuous fluid dynamics requires that ε be taken to be sufficiently larger than the size of the atoms and molecules, while simultaneously being overwhelmingly smaller than the typical wavelength of the fluid waves being considered. In such physical theories, it can be said that "differences are not approximations of derivatives, but derivatives are approximations of differences."

Ultimately, when limits of infinity or infinitesimals appear in theoretical physics, they are merely approximations of being finite but sufficiently large or small within the range of applicability of the theory. The effort to treat strict infinite limits as pure mathematics, while completely ignoring the facts of the actual natural world such as the discreteness of atoms and molecules, ends in mere futility.

In a story from the bestseller "Surely You're Joking, Mr. Feynman!", when a math student told the physicist Richard Feynman about the famous Banach-Tarski paradox, explaining that one orange could be made into two identical oranges, Feynman reportedly laughed and said that was impossible because oranges are made of discrete atoms and molecules and cannot be infinitely divided. This also clearly shows the difference in how the world is viewed in mathematics and physics.

The mathematical concept of topology has recently been playing a major role in theoretical physics as well. However, that physical theory also, of course, has a range of applicability. Mathematicians often express that "a donut has a topology with one hole," but that is merely an expression resulting from abstraction that ignores the concrete reality of physics.

A donut is a collection of atoms and molecules, and its nuclei and electrons are overwhelmingly smaller than the size of the atoms and molecules, making the inside of the donut a sparse space. Therefore, rather than saying a donut has one hole, it is more realistic in the natural world to say it is full of holes. Even so, topology has sufficient meaning as physics within a certain range of applicability where energy regions are limited. In experiments at a scale large enough to view the donut as a single mass, rather than experiments at a size where the sparse space of scattered electrons and nuclei is visible, the concept of "continuous deformation of shape" of the donut can be introduced with the theory's range of applicability. Only by acknowledging this does the mathematical premise that "a donut has one hole" become shared as physics.

Those who think that "a donut has one hole" is an obvious, natural premise that does not need to be questioned need to further refine their sense of physics. If you feel that the premises written in this short text are simple and easy to understand, you are too naive as a physicist. Behind this premise statement, there are actually many physical details (concretes) that should be considered. Those empirical scientific concretes are simply rounded up within this short text.

Professor Koshiba, who won the Nobel Prize for neutrino observation from a supernova explosion, also reportedly said that a good physicist is someone who knows well the range of applicability of physical theories. I believe that physics students, in particular, should take this to heart.

I hope to share this view of physics with many people in various fields.