What does Betrand Russell mean when he says the mathematicians refuted the philosophers on the existence of infinity? In the second-to-last chapter of his book "The Problems of Philosophy", Bertrand Russell writes that philosophers used to think that there can't be such a thing as infinity, but they were disproved by Georg Cantor. Here is the exact quote:

Most of the great ambitious attempts of metaphysicians have proceeded
  by the attempt to prove that such and such apparent features of the
  actual world were self-contradictory, and therefore could not be real.
  The whole tendency of modern thought, however, is more and more in the
  direction of showing that the supposed contradictions were illusory,
  and that very little can be proved a priori from considerations of
  what must be. A good illustration of this is afforded by space and
  time. Space and time appear to be infinite in extent, and infinitely
  divisible. If we travel along a straight line in either direction, it
  is difficult to believe that we shall finally reach a last point,
  beyond which there is nothing, not even empty space. Similarly, if in
  imagination we travel backwards or forwards in time, it is difficult
  to believe that we shall reach a first or last time, with not even
  empty time beyond it. Thus space and time appear to be infinite in
  extent.
Again, if we take any two points on a line, it seems evident that
  there must be other points between them, however small the distance
  between them may be: every distance can be halved, and the halves can
  be halved again, and so on ad infinitum. In time, similarly, however
  little time may elapse between two moments, it seems evident that
  there will be other moments between them. Thus space and time appear
  to be infinitely divisible. But as against these apparent facts
  infinite extent and infinite divisibility philosophers have advanced
  arguments tending to show that there could be no infinite collections
  of things, and that therefore the number of points in space, or of
  instants in time, must be finite. Thus a contradiction emerged between
  the apparent nature of space and time and the supposed impossibility
  of infinite collections.
Kant, who first emphasized this contradiction, deduced the
  impossibility of space and time, which he declared to be merely
  subjective; and since his time very many philosophers have believed
  that space and time are mere appearance, not characteristic of the
  world as it really is. Now, however, owing to the labours of the
  mathematicians, notably Georg Cantor, it has appeared that the
  impossibility of infinite collections was a mistake. They are not in
  fact self-contradictory, but only contradictory of certain rather
  obstinate mental prejudices. Hence the reasons for regarding space and
  time as unreal have become inoperative, and one of the great sources
  of metaphysical constructions is dried up. [The Problems of Philosophy, Chapter 14]

What Russell seems to be saying is that Kant based his theory of transcendental idealism on the idea that in reality there can't be such a thing as infinity. But along came Georg Cantor and proved that the whole thing was a mistake and that there can indeed be such a thing as infinity. (This is perhaps how Russell justifies his attack on Kant in Chapter 8.)
But my question is as follows. It seems to me that when mathematicians speak of infinity and when philosophers speak of infinity they are talking about different things. When a philosopher says there is no such thing as infinity, she means to say that in the real world there cannot be an infinite supply of things, be it cars, atoms, space or time. As many as you have you will never have infinity.
On the other hand, when a mathematician talks about infinity existing, she means abstractly. There is an abstract mathematical concept of infinity that we can talk about in math and it can have implications in theory. But it doesn't (and can't) actually exist.
So what is Bertrand Russell referring to? Am I wrong about mathematicians agreeing that infinity doesn't actually exist? I'm not familiar with the works of Georg Cantor, but how can he [Cantor] possibly prove such a thing?
 A: You are right that to a mathematician, things are said to 'exist' as long as there is no logical contradiction, which is not the same as 'physical existence'. 
But while these two notions are different, the two are nevertheless related: if something is logically self-contradictory, then it can't physically exist either.  And that, I believe, was what Russell was getting at.
Indeed, note that Bertrand starts out his first sentence with exactly the notion of a self-contradiction: 

Most of the great ambitious attempts of metaphysicians have proceeded by the attempt to prove that such and such apparent features of the actual world were self-contradictory, and therefore could not be real.

That is, Russell characterizes some of the philosophers as arguing that something was self-contradictory, and therefore impossible to exist, whether logically, mathematically or physically.
Russell then discusses two purported contradictions:

Thus a contradiction emerged between the apparent nature of space and time and the supposed impossibility of infinite collections.

So there are two purported contradictions here:


*

*The impossibility of infinity itself. That is, that the very notion of infinity is self-contradictory

*The impossibility of space and time: space and time being infinitely divisible, thus infinite, and thus (in the light of 1) contradictory
That is, Russell seems to be claiming that both of these self-contradictions were argued for by philosophers, and again: if they are self-contradictory, i.e. logically contradictory, then that would imply that they would be physically impossible as well, i.e. there can be no infinities in the real world, and no space and time (at least as we conceive of it) either.
And then Russell of course brings up Cantor to defuse the first contradiction (i.e. that the philosophers were wrong in their arguments against the very notion of infinity) ... and thereby defuse the second contradiction as well.
In other words, all of the arguments really were on the purely logical/mathematical side of things; they just had implications for the nature of reality.
At least, that is my reading of this ...
A: Does the set of positive integers exist? In an abstract sense, yes, It's just a conceptual entity. Does it exist as a physical object in the real world? Apparently not, based on our current understanding of physics.

The set of positive integers could exist (i.e., be representable) in nature if we could demonstrate the existence of a countable infinity of objects. How do we know that no such set of objects exists? Quantum theory asserts a "smallest" unit, that's why. Could that understanding change? Perhaps, who knows?

As far as Russell's apparent acceptance of "infinity", he alludes to the apparent "fact" that on a line, there's always a point strictly between any two points. How does he know that? As I see it, he's blurring real-world existence with conceptual existence.

Russell also asserts that it's not reasonable to believe that the universe is bounded, but that was just a belief on his part. In fact, current theory asserts that the universe is, in fact, bounded.

Similarly, with regard to Russell's assertion that it's reasonable to believe that time is infinite in both directions, again that was just a belief on his part. By current theory, the Big Bang marks the beginning. As far as whether or not there could be an "end" of time, who knows, but I doubt whether any such claim could be proved or disproved.
A: Georg Cantor's realist stance laced with both metaphysics and theology concerning set theory is well-known. What is less well-known is that Bertrand Russell was influenced by Cantor's realist position to a degree somewhat embarrassing for a philosopher.  Thus in his The principles of mathematics (1903, not to be confused with the later Latin title lifted from Newton) he makes numerous statements to the effect that "infinite sets really exist" that seem quaint to a scholar today.
What he means by his curious claim is that he considers Cantor to have established an alleged "existence" of infinite sets though it is not clear by dint of which magic wand Cantor may have accomplished such a thing.  Additional details concerning the philosophical errors of both Cantor and Russell can be found in this publication in the philosophy journal Erkenntnis. 
"how can he [Cantor] possibly prove such a thing?", indeed.
