Inconsistent theories, lacking application in empirically successful science, fail the indispensability criterion that historically justifies treating classical and even intuitionistic mathematics as legitimate objects of study.

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Classical mathematics(as used in philosophy of mathematics): The traditional system of mathematics most people learn in school, which allows certain logical rules like the law of non-contradiction (something can't be both true and false).
Empirically successful science(as used in philosophy of science): Scientific theories that actually work and make accurate predictions when tested in real experiments and observations.
Inconsistent theories(as used in logic and philosophy of science): Sets of ideas or rules that contradict each other—where one part says something is true and another part says it's false.
Indispensability criterion(as used in philosophy of mathematics): A standard for judging whether something is truly necessary and useful—in this case, whether a mathematical system must exist because science actually needs it to work.
Intuitionistic mathematics(as used in philosophy of mathematics): An alternative system of mathematics that only accepts things as true if they can be explicitly constructed or proven, rather than just assumed to exist.
Legitimate objects of study(as used in epistemology and philosophy of mathematics): Things that philosophers and scientists have good reasons to take seriously and investigate, rather than dismiss as meaningless or unworthy of attention.

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