Lack of faith is not a kind of faith [LWN.net]

Lack of faith is not a kind of faith

Posted Aug 19, 2012 21:55 UTC (Sun) by man_ls (subscriber, #15091)
In reply to: There speaks the true believer by Wol
Parent article: Garzik: An Andre To Remember

Sorry, too much for me; I will bite. I do not recognize that mathematics is any kind of religion.

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? Mathematics says nothing about the nature of the Universe. To use a common example, our space may be Euclidean, hyperbolic or elliptic; if it is Euclidean then Euclid's postulates and conclusions will hold, and not otherwise, but maths do not (and cannot) say which.

The fact that mathematics can accurately represent some phenomena of our physical world, or rather the fact that our physical laws can be represented mathematically, is just an exponent of some kind of objectivity in the world: things sometimes work in a way that can be deduced from a few axioms. Is it a lucky coincidence or some kind of superior powers at work that makes integers behave according to Peano axioms? Maths do not go there.

There is certainly nothing to believe or to worship in mathematical laws, and few people since Pythagoras have done so. If there are still some people who believe in mathematics with some kind of enthusiasm bordering in religious fervor, then good for them; axioms and theorems do not need people to believe in them to work, contrary to most religions.

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Lack of faith is not a kind of faith

Posted Aug 20, 2012 0:47 UTC (Mon) by apoelstra (subscriber, #75205) [Link]

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 (subscriber, #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.

Lack of faith is not a kind of faith

Posted Aug 20, 2012 12:25 UTC (Mon) by man_ls (subscriber, #15091) [Link]

If you think of a "chicken" as a two-legged vertebrate then we are all chickens :)

What you describe is more "theology" than "religion": the deduction of conclusions based on a set of postulates. Not coincidentally theology is a branch of philosophy, heavily based on logic. In that respect, theology is to religion what mathematics is to science: science does indeed apply mathematical laws to our physical universe, thereby implying that some sets of axioms apply in our world. Answering apoelstra above, physics equates the "2" in "2+2=4" with the "two" in "I have two hands".

What Wol missed above is that science is indeed akin to religion, in that there are no proofs that it works at all but still people believe in them with something resembling faith. See e.g. Wigner:

arguments could be found that might [...] put a heavy strain on our faith in our theories and on our belief in the reality of the concepts which we form.

There is a big difference though: religions usually demand faith from the practitioner, while science works independently of beliefs. This conceptual leap (pioneered by Galileo himself) is worth centuries of progress.