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I've come here after reading about proving SAS congruence, where the response I read was about how it was an axiom.

I'm not sure, however, what exactly is an axiom. And can it be proven?

And if it can't be proven, how can one just assume it true?

Mauro ALLEGRANZA
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Ethan Chan
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    Can you prove anything without making some assumptions. – fleablood Jun 18 '18 at 16:02
  • I wasn't aware it was an axiom and I had been under the impression he had proven it. But none the less he had proven it with more primitive axiom. An axiom is a fundamental statement assumed to be true that can not be proven but is a building block to prove less basic statement. It can not be proven. One can't know it is true but you can demonstrate it leads to a consistent coherent system. Which is what truth means in mathematics as mathematical objects do not exist in the physical universe. – fleablood Jun 18 '18 at 16:15
  • See axiom : "An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments." – Mauro ALLEGRANZA Jun 18 '18 at 16:22
  • Thus, in the context of the theory where the axiom is assumed as such, we cannot derive it from other assumptions. But in a different context, call it $T_2$, we can prove the axioms of the theory $T_1$: of course, for doing so we need new axioms: those specific of the new theory. – Mauro ALLEGRANZA Jun 18 '18 at 16:24
  • The point being, you can't not prove anything without assuming basic axioms because we do not have any basis for determining whether anything is true or not. We don't like it but it's undeniable. SO we keep the axioms to be a basic and as obvious as possible. – fleablood Jun 18 '18 at 16:26
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    Since understanding the answer of another question is apparently the key thing here, I’d recommend putting a link in this question to the previous answer. – David K Jun 18 '18 at 16:45

2 Answers2

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What exactly is this question, and can this question be answered?

Tommaso Seneci
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We hold these truths to be self-evident...axioms are a set of basic assumptions from which the rest of the field follows. Ideally axioms are obvious and few in number.

An axiom cannot be proven. If it could then we would call it a theorem. However, there may be two concepts that are equivalent. And we might state one as an axiom and the other as a theorem. But, we would be able to change that priority if we thought it would me more concise.

As for SAS... Euclid proposed 5 "postulates" and 5 "common notions," and the "SAS axiom" is neither.

Early in "The Elements," Euclid runs into a problem proving congruence. His solution is to introduce the concept of "superposition." Euclid is not explicit in what he means, but the suggestion seems to be we could "pick up" one triangle and move it and place the two on top of another, if there is an exact correspondence between points and line segments, the triangles are congruent.

Today we might call this the group of rigid motions. But group theory wouldn't be discovered for another 1800 years.

Is this allowed by the axioms? Not really. The easy solution is to make an additional axiom.

Doug M
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    p $\vdash$ p is often a valid rule of inference. Consequently, axioms can often get proven. In formal logic, well-formed formulas which are axioms often can sometimes also get proved without the use of the rule p $\vdash$ p given that substitutions are allowable in theorems. – Doug Spoonwood Jun 18 '18 at 17:29