Does physics have some division schema which divide physical amounts into these two classes? Does physics have some division schema which divide amounts into these two classes? :


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*A) amounts which can be counted by natural numbers (for example many units can be counted by number of electrons, photons per second etc.)

*B) amounts which cannot be counted by natural numbers (time, speed, length)


It looks like we understand much more things from class A and the history of physics can be described as an evolving possibility of counting more and more things with natural numbers. 
What does this mean? Why are natural numbers so special? And why can't some things be counted with natural numbers?
 A: We have observed that the underlying level of nature is quantum mechanical. Quantum means  "a definite quantity" of something so definite quantities can be counted and so integral numbers play a role : a) in the number of particles , in the number of energy levels characterized by quantum numbers ( i.e. integer numbers).b) There are the fields  which are described by real numbers and there exists another level, c) the operators, which act on those real numbers with rules constrained by the quantum numbers (a)   in the system.
Classical physics emerges from this underlying level and has its distinct equations and solutions to those equations  in real numbers . Integers will appear in special solutions, as in harmonic motion and wave equations ( for example: sound, tides, waves in general), but are useful for describing observations of specific problems, not a generalized class.
For classical physics what has been generalized in the constants describing bulk matter are what are called extensive and intensive variables, which do have a connection with the countability, but not with integer number per se. These are the intensive and extensive properties of some constants entering the solutions of the differential equations of classical physics>

In thermodynamics and materials science, the physical properties of substances are often described as intensive or extensive, a classification that relates to the dependency of the properties upon the size or extent of the system or object in question.
The distinction is based on the concept that smaller, non-interacting identical subdivisions of the system may be identified so that the property of interest does or does not change when the system is divided, or combined.
An intensive property is a bulk property, meaning that it is a physical property of a system that does not depend on the system size or the amount of material in the system. Examples of intensive properties are the temperature,refractive index,density and the hardness of an object. No matter how small a diamond is cut, it maintains its intrinsic hardness.
By contrast, an extensive property is one that is additive for independent, non-intracting subsystems. The property is proportional to the amount of material in the system. For example, both the mass and the volume of a diamond are directly proportional to the amount that is left after cutting it from the raw mineral. Mass and volume are extensive properties, but hardness is intensive.

Extensive properties are additive , and if one goes down to the quantum mechanical level one sees that they are connected with the existence of particles, which are countable with integers.
You state:

It looks like we understand much more things from class A and the history of physics can be described as an evolving possibility of counting more and more things with natural numbers.

I will disagree. Integers are no more important in physics than they are in our general mathematical knowledge. Physics started evolving when it could model with mathematics the observations in nature. Some observations, as the atomic/ particle observations were on countable items, many not.

What does this mean? Why are natural numbers so special? And why can't some things be counted with natural numbers?

The only specialness is our ten fingers, where humans learned to count. Mathematics has progressed a lot since then, so physics can use it to describe the exciting observations of nature, integers where necessary, but mostly real numbers for physical quantities.
A: Generally speaking, particles are quantized:  you have $0,1,2,3,4\dots $ of them.  Even in quantum mechanics, you may not know how many you have, but if you measure it you will get a natural number.  Most other quantities are real numbers and can take any real value.  It is very hard to say why-some would say that is the definition of a particle.
