Are Axioms in Mathematics Truly Reliable?

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The discussion centers on the foundations of logical arguments and the nature of trust in statements. It emphasizes that confidence in arguments stems from axioms of logic, which are accepted as foundational truths within specific logical systems. The classical laws of identity, non-contradiction, and excluded middle are highlighted as having empirical roots in natural observation, yet their validity cannot be formally proven beyond their axiomatic status. In mathematics, axioms are not classified as true or false; rather, they exist as frameworks within which various models can be constructed. The example of the parallel postulate illustrates this point, demonstrating that while some models adhere to it and others do not, all can be practically useful. This underscores the complexity of validating arguments and the reliance on accepted axioms in logical reasoning.
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What makes us so confident that a line of argument works at all? Why do you trust statements so much?
 
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Give an example.
 
Do you mean mathematical proof? It is the axioms of logic that tell us when an argument is a correct proof, and they are just axioms.
 
It depends on the logical system in which you prove a statement. Certain systems seem to be based on axioms that are empirically verifiable at least to some extent. The classical laws of identity, non-contradiction, and excluded middle all have some basis in the way we observe nature to operate. Beyond this observation, though, there isn't any formal way I can think of to prove their validity.
 
Axioms in mathematics are not really true, or false, they just are. Eg the parallel postulate isn't true, or false. What we do is work in models which satisfy the axioms. In the case of the parallel postulate there a models which satisfy it, and those that do not, and they are all useful in many ways.
 

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