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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 0:47 UTC (Mon) by apoelstra (subscriber, #75205)
In reply to: Lack of faith is not a kind of faith by man_ls
Parent article: Garzik: An Andre To Remember

Note that I am not the GP.

> Mathematics is a religion, you say? How can the derivation from a set of
axioms and rules to a number of theorems be a religion?

Because to avoid infinite regress (i.e., to get a theorem B from axiom A, you would need the axioms "A implies B", "(A implies B and A) implies B", "((A implies B and A) implies B) implies B)", and so on), you need to take the definition of "implies" from somewhere in your psyche. Philosophers argue endlessly about where this definition comes from. Mathematicians generally take it on faith.

You can certainly do mathematics without ever thinking about the religion or philosophy of it. But all of mathematics is built from logic and set theory, which I would say are religion.

>Mathematics says nothing about the nature of the Universe.

No, but as soon as you accept that the two in "I have two hands" is the same as the two that mathematicians use, you have indeed made a statement about the universe (and one that also requires faith!).

Still no faith required

Posted Aug 20, 2012 1:07 UTC (Mon) by man_ls (guest, #15091) [Link]

Yes, you need axioms and rules. No, rules are not mysterious entities that require belief; they are a well understood part of the system. Logic and set theory are also sets of axioms and rules, and in that they are very similar to mathematics. Again, no belief or faith necessary: if you don't believe e.g. that "a and ¬a cannot be true at the same time" you can still be happy, just accepting that the conclusions of first-order logic cannot be carried out to the physical world.
as soon as you accept that the two in "I have two hands" is the same as the two that mathematicians use, you have indeed made a statement about the universe (and one that also requires faith!).
That is precisely why the "I have two hands" part is firmly planted in reality, and outside the realm of mathematics. I am forced to quote Einstein again:
as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
Not that I dislike quoting father Einstein, but citing the same quote twice in a week is excessive and has the danger of becoming old.

Lack of faith is not a kind of faith

Posted Aug 20, 2012 6:42 UTC (Mon) by Cyberax (✭ supporter ✭, #52523) [Link]

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 Göedel 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.