From wikipedia, pure mathematics is mathematics that studies entirely abstract concepts.
Good. But what are abstract concepts?
From wikipedia, Abstraction is a conceptual process where general rules and concepts are derived from the usage and classification of specific examples. "An abstraction" is the outcome of this process.
So an abstraction is a result of a generalization process of specific -- non-abstract -- examples.
Abstraction vs. specific examples; metaphysics vs. physics; mathematical world vs.real world. Philosophors since Plato to Immanuel Kant to Roger Penrose all ponders and stresses this distinction. As they ponder, they more or less all favors the existence of pure abstract entities, implying the pre-existence of abstract concepts.
But wait, did we just looked at the definition of abstraction? If abstraction comes from a process of generalizing specific examples of reality, how can they pre-exist? In fact, I argue here that the abstract concepts are ambiguous concepts defined by reality. They are ambiguous because our knowledge of reality is ambiguous, and only way for us to disambiguate is to go back to reality and find additional specific examples.
Since I argue that we won't understand any abstract concepts unless we go back to reality, let's examin some examples. Let's look at Euclid's geometry. Euclidean geometry starts with 5 axioms: 1. two points determins a line; 2. a line extends infinitely at both ends; 3. there is a distance between two points that is conserved upon rotation and translation -- rotational and traslational symmetry of space; 4 two lines can be arranged to be perpendicular -- forming four identical angles; 5 two lines can be arranged to be parallel -- on a same plane but never meet.
Where do these axioms come from? They do not come from logic reasoning. Logic reasoning is a consistency check, and cannot produce new results. Axioms are coming from our observations of reality. But I made a skip here. The first question should be: do we understand what the axioms are saying? What are points, lines, and planes? No mathematics books I read defines them. If we do not define they, how could we know what we are talking about?
We shouldn't know what we are talking about. I think this is the essence of pure mathematics. All we need understand is points, lines, planes are different concepts and they follow the rules prescribed by a set of axioms. And we can draw conclusions based on these rules. We will not be able to understand the meaning of those results. Of course, we can relate the concept of points, lines, planes back to reality by consulting the original abstraction process. Then we attach meanings to points, lines and planes, and we can attach meanings to all the conclusions of geometry. But that meaning is separate from mathematics. In fact, in another world, another reality, we may attach a different set of meanings to geometry. However, if we are in a different reality, the chances of having abstraction process yielding the same set of axioms are very slim.
In the library of babel, every possible 410-page books are stored. Each book is consisted of arrangement of characters just as a subject of pure mathematics consists a set of axioms. In a pure sense, the arrangement of characters has no meaning, just as the set of axioms itself has no meaning. Their existence is its only meaning. Now it is argued that amongh the collection of library of babel, there are shakepeare's works, and there are great sentences of great meaning to certain real world that has not even introduced to that world. Similarly, within all subjects of pure mathematics, there are a set of axioms that describes how a real world works. But the truth is, withing the library of bable, all books are equal, and none of the books has meaning. Within pure mathematics, none of it has meaning unless we attach some reality meaning to it. The act of attaching reality meaning to its axioms and theorems make the that discipline no longer pure mathematics. Here, I draw the conclusion: pure mathematics is meaningless and useless.
Of course pure mahtematicians do not think their works are meaningless. In fact, I vouch they do think in meanings of their axioms and theorems. I believe whan a mathematicians do geometry, they regard points just as points in reality and lines just as lines in reality. Only by considering meaning in reality, they ponder on what axioms to include. For example, the parallel postulate is no-question true in Euclid's mind, he just not sure if it is redudant. Euclidean geometry is important and meaningful because it is an abstraction of reality.
The same cannot be said to Cantor's infinity math. The only meaning we have for infinity is the abstraction of unreachable extension. Since it is unreachable, any axioms treating infinity as a reacheable entity does not have reality correspondence, or meaningless. Of course, Cantor and his followers all think they have some meanings attached to these infinities, except that meaning is ambiguous and probably conflict in each's mind.