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# Lack of faith is not a kind of faith

## Lack of faith is not a kind of faith

Posted Aug 20, 2012 6:42 UTC (Mon) by Cyberax (✭ supporter ✭, #52523)
In reply to: Lack of faith is not a kind of faith by apoelstra
Parent article: Garzik: An Andre To Remember

You don't 'need' to do anything with math. Theorems do not require your belief to be derivable using some sets of axioms and inference rules.

Axioms are not some 'self-evident truths', they are just starting points. And math is just a way to see what can be derived from them, nothing more and nothing less.

Want to see what happens if it's possible to draw exactly 3 lines parallel to a given line through an arbitrary point? Go on, that would be interesting. But that doesn't affect any geometry theorems operating in our familiar boring Euclidean space.

Lack of faith is not a kind of faith

Posted Aug 20, 2012 9:07 UTC (Mon) by hppnq (guest, #14462) [Link]

But that doesn't affect any geometry theorems operating in our familiar boring Euclidean space.

Unfortunately, of course, our familiar space is not Euclidian. Well, mine isn't.

But, given a nicely-behaved Euclidian space, the challenge would be to prove that there can be at most one line through a point that is not on a second, parallel line. Now, that's hard enough even for aspiring mathematicians, but if you could prove that it is possible to actually draw two or more such lines, it would most certainly mean the end of Euclid's famous fifth postulate, and with that all of Euclidian geometry, not to mention the start of a hectic tour of talkshows in which you would have to patiently explain that you cannot explain what it all means.

Lack of faith is not a kind of faith

Posted Aug 20, 2012 15:27 UTC (Mon) by Cyberax (✭ supporter ✭, #52523) [Link]

> Unfortunately, of course, our familiar space is not Euclidian. Well, mine isn't.
And? Mathematics is in no way related with the real world. The fact that some mathematical structures can be used to model objects and behaviors from the real world is just a happy coincidence.

> But, given a nicely-behaved Euclidian space, the challenge would be to prove that there can be at most one line through a point that is not on a second, parallel line.

That's impossible. The Euclid's fifth axiom is independent from others and that has actually been proven. It's not possible to prove that in general case (see: http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompletenes... ), but Euclidean geometry is a complete theory (in GĂ¶edel sense).

So you have the following choices:
1) Leave it out entirely. You'll have much poorer set of theorems.
2) Use it. You'll get boring old geometry.
3) Replace it with another axiom. You'll get non-Euclidean geometries as a result.