I'll admit this is a bit of a vague question, but I'm having trouble actually formulating it.
I understand
- definitional systems can have recursive definitions
- however this can turn into nonsense
- If I keep on searching for justification for a claim in a proof, I should eventually hit axioms and definitions
I found myself pondering this question when reading Wikipedia's page on Mathematical Objects. Wikipedia states
A mathematical object is an abstract concept arising in mathematics. In usual language of mathematics, an object is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs.
What if there is an alien mathematician who knows nothing about our language and is obsessed with rigor. This alien might ask "What is a rigorous definition of formal definition?" Is that question even possible to answer rigorously without using the alien's language?