Practice question · Multiple choice
Axioms are accepted without proof. Why is that not an admission that mathematics rests on unjustified assumptions?
Hints
- Try to justify every statement by proving it. Where does the chain end?
- Ask what a theorem actually asserts. Is it "this is true", or something more conditional?
Show the answer
A. Because a proof must start somewhere, and axioms are the chosen start.
Why
A proof derives a statement from earlier ones, so demanding a proof for everything gives an infinite regress or a circle, recognising that is structural rather than a concession. It also sharpens what a theorem claims: Pythagoras follows from the Euclidean axioms, and change the parallel postulate and you get the geometry of general relativity instead.
Practise The Structure of Mathematical Reasoning
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