(adsbygoogle = window.adsbygoogle || []).push({}); do we assume "logic"?

Mathematics has no foundations on reality, it stands on its own. But to construct it we have to assume a number of axioms, like if I wanted to create a "science" where the only rules are all circles are red and all squares are blue, then from that we can build theories and prove them from the assumed axioms, like a particular square has to be blue because all squares are blue, right?

However, if the defining rules of my "science" are contradictory, like all circles are red and all circles are blue, then my "science" is "ill defined".

But what indicates that? If the cornerstones of mathematics have to be "logical", not in the sense of mathematical logic or propositional logic per se but in the sense that they have to be "well defined", does that mean that we are assuming a set of implicit primordial rules that the rest of mathematics have to abide to?

If even the most basic laws of mathematics have to be "logical", does that mean that we are assuming a set of rules that dictate whether something is logical or not? And could these rules be "logical" if they define logic? In other words, can they abide by their own rules? Is that even possible?

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# Do we assume logic ?

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