How can mathematics be so useful if it is based on unproved axioms ? And what is the nature of axioms that can be used in mathematics ?... I mean obviously I can't invent a set of axioms arbitrarily, and start developping new maths based on them. So what enables mathematics to be that useful ?
2026-03-25 12:48:46.1774442926
The Foundation of Mathematics
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The original motive for axioms was that if we can agree on some truths, we can deduce others from them. Euclid popularized the use of axioms for geometry, but alternatives to his axioms were developed in the 1800s. Since then, mathematics has moved on from axioms being "true" to their being something whose consequences we choose to investigate. Are the axioms of group theory "true"? That's the wrong question: they're true by definition of groups - in fact, they're the definition of groups.
A deductive system is a set of propositions, including certain axioms and closed under certain rules of inference. Mathematics doesn't have to settle on one "canonical" list of axioms, because we're interested in many deductive systems. For example, group theory is a deductive system that tells you what is true of whatever satisfies the definition of a group.
Which mathematics will be useful? It's tempting to say the mathematics that happens to describe the real world is useful. But I think it would be better to say that the claims made about the real world with such mathematics are useful insofar as they are true (or at least probably approximately true). When physicists concluded the universe's geometry isn't Euclidean, that wasn't Euclidean geometry's "fault"; if anything, it's the universe's fault for not being Euclidean.
Mathematicians do occasionally need to reconsider which axioms to use. I'm not enough of a historian of mathematics to know whether Russell's paradox led to a shift from one list of set theory axioms to another, or whether axioms weren't in use in the first place, but either way it led to the development of modern axiomatic systems such as ZFC. But it takes something pretty rare for this to happen, such as an inconsistency in Russell's case, or a realisation we had assumed more things than we'd admitted, such as when it was realised "Euclidean geometry" assumes more than his five explicit axioms.