https://en.m.wikipedia.org/wiki/Abstraction_(mathematics)
Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected.Mathematics must have no connection whatsoever with the real world. Therefore, good mathematics is fundamentally meaningless.
Good mathematics is useless
Godfrey Harold Hardy, English mathematician wrote:
I have never done anything "useful". No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.Claims that are meaningless are in principle also useless. It is a rather unwanted accident if these claims somehow turn out to be useful anyway. By design, they should not.
We have concluded that the trivial mathematics is, on the whole, useful, and that the real mathematics, on the whole, is not.
Good mathematics is ridiculous
There must still be some non-trivial gap between premises and conclusion.
In Critique of Pure Reason, Immanuel Kant distinguishes:
- analytic proposition: a proposition whose predicate concept is contained in its subject concept.
- synthetic proposition: a proposition whose predicate concept is not contained in its subject concept but related.
A mathematical theorem must be synthetic. It must argue a proposition that requires non-trivial symbol manipulation in order to derive it from its premises.
Therefore, legitimate mathematics must produce a meaningless and useless statement that is still different from the meaningless and useless statements from which it has been derived.
Hence, a good mathematical statement is ridiculous.
This is probably the reason why category theory is widely considered to be the flagship of mathematics:
Wikipedia on "abstract nonsense" wrote: In mathematics, abstract nonsense, general abstract nonsense, generalized abstract nonsense, and general nonsense are terms used by mathematicians to describe abstract methods related to category theory and homological algebra.Conclusion
Labeling an argument "abstract nonsense" is usually not intended to be derogatory,[1][2] and is instead used jokingly,[3] in a self-deprecating way,[4] affectionately,[5] or even as a compliment to the generality of the argument.
When an audience can be assumed to be familiar with the general form of such arguments, mathematicians will use the expression "Such and such is true by abstract nonsense" rather than provide an elaborate explanation of particulars.[2]
The term predates the foundation of category theory as a subject itself. Referring to a joint paper with Samuel Eilenberg that introduced the notion of a "category" in 1942, Saunders Mac Lane wrote the subject was 'then called "general abstract nonsense"'.
The true nature of mathematics is that it is meaningless and useless. Its only redeeming quality is that it is also ridiculous.
That is in fact the very reason why I like mathematics. It satisfies my need to indulge in absurdity.