Integers and the Physics

 

Someone very rightly said: “God made the integers, every other number is that the work of man. Now after more than a century, no mathematician would deny the importance and utility of the developments that Kronecker railed against. Yet I think that a lot of harbors some sympathy for his statement. The integers hold a special place within the heart of mathematicians. 

Many of the foremost famous unsolved conjectures relate to the properties of the primes. More importantly, the integers are where we start mathematics:
they are how we count.

In this article, I might wish to view Kronecker’s quote through the lens of theoretical physics. Tested against our greatest theories of Nature, I will be able to argue that the statement is wrong. 

Experimentally, falsifiably, wrong. It is not obvious that the integers have anywhere in physics. The counting that's evident in mathematics isn't very easy within the world. I was taught in college that
there are 9 planets within our solar system. Now there are 8. or even 13. As this instance shows, the matter of finding the integers in Nature lies not within the counting, but rather within the defining. The Kuiper belt contains objects ranging in size from a few thousand kilometers to a couple of microns. you'll only decide which objects are
planets and which are merely lumps of rock if you use a reasonably arbitrary definition of what it means to be a planet. 

To seek out the integers in physics, we'd like Nature to provide us with objects which are naturally discrete.

Fortunately, such objects exist. 

While the definition of a planet could also be arbitrary, the definition of an atom, or a fundamental particle, is not. Historically, the first place that the integers appeared was within the table of elements. The integers
labeling atoms which, we now know, count the number of protons − are honest.

Regardless of what developments occur in physics, I'm sure that we'll never observe a stable element with √500 protons that sits between titanium and vanadium.

The integers in physics are here to remain.