Typically, much of mathematics today is ultimately based on pretty abstract axiomatisations in first order logic (or, rarely, some other logic).
But all of those axiomatisations, whether for arithmetic (PA and all its extensions examined in reverse mathematics), set theory (ZFC, NBG and their variations), group theory, geometry (with good old Euklid and its variations), order theory etc, all result from having a preconceived idea of a mathematical structure (such as a group or sets) and then translating the ideas and concepts into formal language.
That leads to question 1: Are there any significant results in any area that started the other way? Has there ever been a case where someone sat down, came up with random axioms (perhaps with some bias, especially trying to avoid contradictions), thought on what models those axioms could have and what theorems would be valid?
A very mild example would be the development of non-Euklidian geometries, only that even then, most axioms were already "derived" from previously existing mathematics, or perhaps the various ZF variants being studied - but what I'm really looking for is more extreme cases with much less previously existing conceptions.
Question 2: Assuming some guidance by mathematical knowledge in conceiving the axioms, is there a reason to suspect that all such theories and their described structures are "useless" or "not interesting"?
Edit to clarify and rephrase the question, since apparently the above can be misunderstood: What I'm really asking: most math starts from having some conception of a structure and then making it concrete in the language of mathematics. The question I'm wanting to ask is if there are any significant results that were achieved the other way around: Inventing some rules first a priori, that is, without some previous idea of a structure, then examining the results.