In Axler's Linear Algebra Done Right Example 1.24, we are asked to prove that the set of all functions from some set S to the set of real (or complex) numbers is a vector space.
I proved this by using the definitions of the vector sum and scalar product associated with this set of functions and by using the fact that the output of these function are elements in the set of real or complex numbers, and thus as elements in fields, the properties of fields such as the distributivity of multiplication over addition and the existence of an additive inverse apply.
My problem is that to me, this seems like a lot of reasoning just to say that the real-valued ouput of a function behaves in the way that we all know real numbers do, just from our rather fundamental human experience of the world as $\mathbb R^3$. It even seems a little tautological ("opium causes sleep by virtue of its dormitive power") = ("a vector space onto a field is a vector space by virtue of a field being such a thing that makes a vector space when you add the right operations to it"). Why is the naive, "intuitive", or "geometrical" understanding we can from experience insufficient and this style of proof illuminating? What does humanity gain from being able to give a demonstration like this as opposed to just asserting when convenient that such a thing as a vector space exists when we need to use it to perform a certain type of calculation?
I fully admit that I am probably so ignorant of the matter that I can't even begin to see how, but how can I come to see that if that is so?
We do not experience the world around us as the set $\mathbb R^3$. There are no coordinates floating around us. Moreover there is no obvious way to agreeing upon where $(0,0,0)$ is. Our world is clearly coordinate free.
Not only do we not experience the world around us as the set $\mathbb R^3$ but even less so as a vector space $\mathbb R^3$. Is it clear when looking around us that we have a operations $+$ and a multiplication giving a vector space structure ? I don't think so.
Additionally it is not because something seems "intuitive" or "obvious" that it is actually mathematically easy to prove. In fact it happens quite often that something which seems intuitive is actually just false.
An example of this is when dealing with infinite sums
$$ \Sigma = a_1 + a_2 + a_3 + ...$$
You might believe that switching the order in which you sum the terms $a_i$ will not affect the value of $\Sigma$, but that is not the case.