Now this is interesting: one of the ancient arguments for the existence of indivisibles (e.g., "atoms") is that infinite divisibility implies an actual infinity. Suppose you can infinitely divide both a mustard seed and a mountain. So they are both composed of the same number (an infinity) of parts. Therefore, they are both the same size. Or, by the same reasoning, a part of a mountain is the same size as the whole mountain.
This isn't considered a problem in mathematics, where a set is infinite just in case one of its subsets (parts) is the same size as the parent. (For instance, the set of integers and the set of squares.) But it's an open question in physics whether there is an actual infinity.
Some evidence seems to imply it. Inverse square laws (like Newton's equation for gravity or Coulomb's law for electrostatics) can evaluate to infinity. Some solutions of Einstein's equations (e.g., some black holes) evaluate to infinity. If the topology of the universe is flat (analysis of the cosmic background radiation thus far suggests it is), then space is an actual, physical infinite.
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4 comments:
"So they are both composed of the same number (an infinity) of parts. Therefore, they are both the same size."
This is why people prefer a word like "cardinality" that doesn't carry the same connotations of measure. In other words, you can put two infinite sets in bijection, but this has nothing to do with their measure.
Here's some slightly more mathematical examples:
There are just as many even numbers as there are numbers. How to see it? Just line them up side by side
0 -- 0
1 -- 2
2 -- 4
3 -- 6
...
are even numbers and natural numbers the same "size"? As sets they are, but not for any reasonable notion of measurement.
One more.
Place a sphere on a plane and turn on a light at the north pole. Every point on the plane is struck by exactly one ray, and that ray passes through the sphere in one place. So we just found out that this infinite plane is "the same size" as our little sphere with the north pole removed. Here's a picture:
http://imgur.com/yhu3F
Those are called stereographic coordinates, for what it's worth.
"But it's an open question in physics whether there is an actual infinity."
I strongly doubt that there is any physicist who would regard the question as you've worded it as open or even well-formed.
Earlier I showed you that a ball is just as infinite as boundless space (it even has an extra point!), anybody who has tried for a moment to interpret Newton's or Einstein's equations can see that we are working with infinite sets. What isn't obvious is whether space is infinite in the sense of measure -- is there a limit on how far apart two points can be? What do the singularities and divergences in mathematical models mean? But you can see this is very far from any of the classical concerns with infinity that all seem to miss the distinction between counting and measuring, which are just totally different, almost unrelated operations with infinite sets.
For what it's worth, one incredibly annoying place where infinities come up in physics is in quantum electrodynamics. This kind of physics "works" extremely well and has some well-understood limits. But I think anybody in their right mind considers it still an open problem to figure out what the hell is going on.
http://en.wikipedia.org/wiki/Quantum_electrodynamics
This is all interesting, but how does it differ in substance from what I already said? Take a look at the original post again. Here's an abridged version of what I said:
(1) Ancient philosophers deduced the existence of atoms because of an apparent absurdity arising from the contrary assumption.
(2) The absurdity disappears if you adopt a modern definition of sets. (I had in mind the sort of Cantorian notion of sets described here: http://plato.stanford.edu/entries/cosmological-argument/#5.2 , though if you go back through my blog posts, I think I've done a few posts here and there on infinity.)
(3) But even if you get the definition right, it's not clear that something in reality corresponds to it.
(4) Though it really seems as though infinities are part of physics. I give a few examples of this. You give more examples of this. (Thank you.)
So ... what's the problem?
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