On mathematical theories

Perspective $\blacktriangleright$ On mathematical theories $\blacktriangleright$ On reading


Entries such as Logic discuss the framework for writing down theories and their models, formalizing mathematical entities and making definitions. Much of the process isn't captured by formalities though, especially things that relate to motivations and conceptural understandings.

Say you have a theory and define a new concept within it. There will be many definitions that end up the same concept in this thoery, but generalize the different ways when considering generalizations of the theory.

Is the real $\log$ to be defined and inverse function of another function, as a group homomorphism, by a particular power series, as the soltuion to a differential equation / via an integral, …? For all of those ways to define it, there will be general frameworks in which they make sense but they might not be compatible - except when you go back to plain analysis.


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