Hey - I appreciate your effort man! To be honest, this might currently be a bit over my head. I think I get the gist of what you might be saying, but of course it's hard to tell. You're saying that there are truths of math that other truths can be based on?
What I didn't get was what you meant by saying you can prove the quadratic formula with axioms. We did learn the quadratic formula in my classes thus far, but nothing about axioms.
OK, this is actually very interesting to me! I'm a HUGE LOTR fan!!!
Having said that, I honestly don't think I understand what you mean. You were saying that there are axioms that are true and that we base other math on earlier, but then you say later here that these don't have to be true? ...and that we can invent our own truths like 1 + 1 = 0 Did I hear that correctly??? You need to come talk to my math teacher!! hahaha
Thanks for the memorization info. though. Very intriguing!
Yes, you heard that correctly. Axioms are true because we take them to be true. So in the rest of what we do, the axioms are true. However, they might not be true in real life!
Since we both like Lord of the Rings, let me give another example. For example, we might ask ourselves if Tom Bombadil could keep the ring safe and away from Sauron. An answer like "none of this is real" clearly is lame and insufficient here. Indeed, for the sake of argument, we have assumed that everything in lord of the rings is real.
In the same way, when we propose ##6+6=0##, we assume for the sake of argument that this is true, and then we see what else we can deduce from this.
Of course, if the axioms aren't applicable to real life, then none of the deductions will be either. However, if the axioms happen to be applicable to real life, then so will the consequences. Clearly, something like the Pythagorean theorem of the quadratic theorem is something applicable to real life. This is because we use it in physics and engineering all the time and it yields good results.
Something like ##6+6=0## is not applicable to real life because it's clearly not true. Or is it? It is definitely not the usual arithmetic we work with in our daily life. But this is an arithmetic that we use in reading the clock. Indeed, if it is now 6 hours, then 6 hours later it will be 0 hours. And if it is now 7 hours, then 11 hours later it will be 6 hours. So 7+11=6. So something like 6+6=0 isn't nonsense at all, it is actually useful. It's just that its uses are clearly different from what we typically use arithmetic for (like counting money).
Then again, there are some weird versions of arithmetic which aren't useful in real life at all. Still, they make up satisfactory theories which are mathematically acceptable.
So math really gives a many different theories (all based on different axioms). But only one (or a few) will be applicable to real life. In the same sense that if I give you three books, namely "The Lord of the rings", "The Furies of Calderon" and "World History of ancient times". All three are fascinating to read, but only one will truly be about our world. The rest are about worlds which are internally consistent, but are not real.
Which math theory is real (and which axioms are real) is something we must find out through physical experimentation and common sense (both of which can be deceiving).