A mathematician's apology

As in Plato's Apology of Socrates or G.H. Hardy's essay, the following blog series shall also represent a formal justification, especially within the context of mathematics. However, in contrast to Hardy's arguments, it is rather impossible to come up with concepts that ultimately can be classified as useless, or more specifically, non-applicable to our reality. To use the words of Lobachevsky:

There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.

Going further, every abstract or even esoteric concept within mathematics, is eventually going to have an application in a potential implementation of possible realities.

Hence, we start our discussions with the fundamentals, work our way up to the latest concepts in mathematics including their constraints and also invest in the impact they have had on the corresponding fields.