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No, axioms are hypotheses. In general, a mathematical proof has the form "If [conditions] are met, then [result] is guaranteed." This requires no faith whatso
by boothby 3y ago
No, axioms are hypotheses. In general, a mathematical proof has the form "If [conditions] are met, then [result] is guaranteed." This requires no faith whatsoever that [conditions] will ever be met.
But those numbers that require the totality of my computer's N bits of storage to represent? I can, without reservation, believe that every integer between zero and 2^N-1 exists. But incrementing that biggest number will be challenging. I hear you scoff; surely there are bigger numbers, you can just buy more hard drive. And I can! But there are only finitely many hard drives available for purchase, many more than I can afford, so I'd rather save my money and believe in less.
- epgui 3y ago> No, axioms are hypotheses IMO they're neither "faith" nor "hypotheses", although "whether a set of axioms is self-consistent" is probably a hypothesis. Axioms are just building blocks. They don't even need to be "true" in any real sense. They're just arbitrary determinations or rules that we make up, and then we figure out what the consequences of these choices are. In practice we often try to come up with axioms that have some practical value in the real world, but this is not necessarily or always how we go about doing maths.
- mensetmanusman 3y agoWhether you believe the axioms are true is an act of faith, because by definition they are the lowest bits.