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?