is mathematics a religion?

For the discussion of math. Duh.

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Re: is mathematics a religion?

Postby PsiSquared » Sat Aug 09, 2014 10:32 pm UTC

schapel wrote:Maybe this discussion is about something that confused some of my classmates in computer science: there are an infinite number of natural numbers, but all natural numbers are finite.

If you always get a natural number as a result of adding two natural numbers, then there is an endless supply of natural numbers. If I produce a natural number and claim it is the largest, you can always add 1 and prove me wrong. Therefore, there are an infinite number of natural numbers. However, each natural number can be written down in decimal form. There is no natural number that represents infinity.

If you have a set of natural numbers, it's either the case that:
1) Adding two natural numbers results in an element which is not in the set, meaning that the set is not closed with respect to addition, or
2) If the set is closed with respect to addition, there are an infinite number of natural numbers.

However, there is no natural number that represents infinity, so there is no such idea as infinity + 1 in this set with the + operator on natural numbers, because infinity is not a natural number.

I suppose the confusing concept is that the number of elements in the set of natural numbers cannot be expressed in terms of a natural number. If it were possible to do so, that number would be the largest element of the set and addition would not be closed over the natural numbers.

And thus the whole notion of "ordinal arithmetic" was born.

In ordinal arithmetic, you start with the same good old natural numbers: 0,1,2,3... . But you also have a rule which states that "every series of consecutive 'ordinal numbers' starting with 0, is in itself a an ordinal number.

So in the ordinal world, the infinite series 0,1,2,3.... is a number. Not a natural number, to sure, but an ordinal number named ω.

Of-course, once you have ω, there's nothing to stop you from continuing: ω+1, ω+2, ω+3...

Now, there's still no "largest ordinal". But every sequence of ordinals, finite or infinite, can be represented by an ordinal (unlike the set of natural numbers). This does come at a price (for one, the class of all ordinals is too large to be considered a "set"), but ordinal arithmetic does have its appeal (and its uses). It also retains one important feature of the natural numbers: for any ordinal k, k+1>k.

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Re: is mathematics a religion?

Postby Forest Goose » Sun Aug 10, 2014 10:04 am UTC

PsiSquared wrote:And thus the whole notion of "ordinal arithmetic" was born.

...And, also, the most beautiful branch of all of mathematics - for being such a simple and extension of counting, they are deep in a way that seems to baffle any true human insight. --Sorry if that's a weird interjection, the way that transfinite numbers factor into logic and sets never ceases to make me feel a sense of awe.
Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.

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Re: is mathematics a religion?

Postby Nicias » Sun Aug 10, 2014 1:15 pm UTC

schapel wrote:
Moole wrote:I hope that student was making a math joke and not being severely disappointed after lots of work (I mean, one could never live that sort of mistake down).

The way I heard the story, it was something the student spent many months on and gave a presentation on. But I heard the story second hand, so it could be an urban legend.

Something like this happened to a postdoc I knew when I was in grad school at her PhD defense. She proved some statements about manifolds that had properties X, Y and Z, and halfway through the defense someone said "Can you show that any such manifold exists?" She said "No" and they let her finish her talk. They didn't let her graduate until she found an example. She did by the end of the semester.

Something like this also happened to a whole room full of people too, almost. I professor of mine in grad school saw a session at a big conference on something like "Such and such objects that also have properties X, Y and Z" and went to the session. He sat in the back for the first two talks, and then stopped the organizer before the third and said, "Can anyone produce an example of one of these that isn't <one specific example>?" think I proved that anything fitting these criteria has to be just this, but I never published it." Noone could, and when he got back to his office he found his notes and published it. It was like going to a whole series of talks about "even primes" or something.

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