Why we can safely treat objects like mathematical entities?

In the study of number systems we learn the axioms of real numbers.

For example:

The commutative axiom $$X.Y = Y.X$$

The distributive axiom $$X.Z + Y.Z = (X+Y).Z$$

Well, the thing is that we are working with mathematical entities, we can add $$X$$ with $$Y$$ and form a number $$Z$$, we can add $$X$$ $$y$$ times and call it $$X.Y$$, which equals to $$Y.X$$.

Why do we assume these properties extend to objects that aren't mathematical entities?

For example: $$2.apples + 4.apples = (2+4).apples = 6.apples$$, we are using the distributive axiom here.

We can't add a number $$X$$ to $$apple$$ and call it a number $$Z$$, or add $$X$$ apple times and make a number $$Z$$, that proves they don't work like normal numbers, but we assume it behaves like mathematical entities to make equations and calculations. Why we can safely assume it works out all the time?

• – Lord Shark the Unknown Oct 21 '18 at 10:23
• Your fruit example is perhaps closer to a vector space - so the distributive axiom is restricted to scalar multiplication (if you want to avoid squared apples) – Henry Oct 21 '18 at 10:23

Note that there are all kinds of mathematical structures which do not model the world well - for example, the cardinal $$\aleph_{\omega_{\omega}}$$, which is too big to have much to do with the world at all. But the blind forces of evolution have discovered the abstraction of "number", which resides in the world of mathematics. That particular abstraction survives in us because of two apparent facts: the universe apparently supports scientific induction (so abstraction is helpful), and there is a particular abstraction which captures the notion of size-of-a-set.