Nature of Mathematics: Axioms

According to Merriam-Webster an axiom is "a rule or principle that many people accept to be true." The origin of the word comes from the latin word axioma, which means worthy.  In mathematics, axioms are statements which are accepted to be true without proof, because they seem to be evident.  I guess you could say that axioms need no proof, because they are worthy. We do not question them, because we accept them and acknowledge them as true.  One clear example of an axiom is the reflexive axiom used in algebra, which states "a number is equal to itself". This is easily accepted without argument or proof. I could show you that a=a, but by the definition of equality we know that a=a. If a was not equal to itself, then it would not be itself, and we would not be able to say anything for certain about a.

You could argue that axioms while not needing any proof themselves are the basic building blocks for mathematical proofs. In order to prove something, you must use axioms to support your work. Try to prove anything without using something that is already accepted to be true.  You may think that a proof by example would be possible without using axioms. While you may be able to create some sort of proof without a formal axiom used, you will still rely on a universal truth to that is accepted to make your proof true. Let's say that I wanted to prove that 4-3=1.  Maybe I would set out 4 apples on a table.  Then I would take 3 of the apple away and eat them.  This would leave one apple left over. I could claim that I did a proof by example without using any axioms. But would that really be true? The apples could reappear. That would contradict the proof, but we know that 3 apples will not magically appear from nowhere and present themselves on the table. Therefore, one rule that is accepted to be true without any proof is that objects do not spontaneously generate from no where. That statement could be viewed as an axiom in the apple proof.
Axioms are needed for creating and giving validity to proofs.  They are also informally used in everyday life. People do not walk around astonished that aren't turning into dogs.  We accept that we are humans and are not going to change into another animal. No one needs to prove to us that we are humans and are going to stay humans, we just accept this to be true. We further build proofs off of our accepted knowledge.  This relates back to my previous blog about what is math.  Axioms are not just used on paper, and neither is math. Math permeates into everyday life. If nothing is accepted to be true, then there is nothing that can be used to prove anything.