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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?

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    $\begingroup$ I suggest that you ask this on the philosophy stackexchange, since mathematicians tend not to be very well-versed in philosophy, as evidenced by quasi's answer. $\endgroup$ – Eli Bashwinger Jun 13 '17 at 20:14
  • $\begingroup$ Perhaps Russell means that earlier philosophers had claimed that infinity was a logical impossibility. This was surely refuted by Cantor's set theory. $\endgroup$ – TonyK Jun 13 '17 at 20:30
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    $\begingroup$ How does "philosophy" relate to real-world existence? If by existence, Russell means physical existence in the real world, I think "physics" is the more the appropriate venue for such a question. $\endgroup$ – quasi Jun 13 '17 at 20:32
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    $\begingroup$ existence is overrated... Russell is also overrated. $\endgroup$ – Masacroso Jun 13 '17 at 20:33
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    $\begingroup$ @Masacroso: Overratedness is overrated, as are meta jokes. $\endgroup$ – Asaf Karagila Jun 14 '17 at 13:09
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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:

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

  2. 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 ...

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    $\begingroup$ +1, I think this is spot-on. I'd add, though, the additional quote from Russell "The impossibility of infinite collections was a mistake. They are not in fact self-contradictory, but only contradictory of certain rather obstinate mental prejudices.", which I think nicely drives the point home. Incidentally, I'd also point out that an important aspect which Russell doesn't touch on here is that not only is Cantor's theory apparently consistent, but also nontrivial - there are interesting questions and theorems! So it's not even "morally contradictory," that is, contradicting worthwhile thought. $\endgroup$ – Noah Schweber Jun 13 '17 at 20:49
  • $\begingroup$ Nope - what note was that? $\endgroup$ – Noah Schweber Jun 13 '17 at 20:57
  • $\begingroup$ Aha! I don't think pinging works for someone not the OP or already in the comment thread. Answering ... $\endgroup$ – Noah Schweber Jun 13 '17 at 21:01
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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.

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  • $\begingroup$ But then, what does Russell mean? Russell seems to think that the mathematicians (specifically Cantor) refuted the philosophers on infinity. $\endgroup$ – Isaac D. Cohen Jun 13 '17 at 20:02
  • $\begingroup$ Obviously the positive integers are not going to exist in nature, given that they are abstract objects. The question would then be, are these abstract objects fictional entities (i.e., are we espousing some form of antirealism), or do they exist in some platonic realm. $\endgroup$ – Eli Bashwinger Jun 13 '17 at 20:03
  • $\begingroup$ The positive integers could exist 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. $\endgroup$ – quasi Jun 13 '17 at 20:05
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    $\begingroup$ @quasi But the positive integers are not identical to any collection of objects, especially physical objects, just as the number $2$ is not identical to a scrawling on a piece of paper. $\endgroup$ – Eli Bashwinger Jun 13 '17 at 20:10
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    $\begingroup$ @quasi "Quantum theory asserts a "smallest" unit, that's why." That's missing an important adjective, namely "observable" (or "physically meaningful") To my understanding, it is entirely consistent with quantum mechanics that smaller-than-Planck-length entities exist, it's just that they're rendered uninteresting to our existence. $\endgroup$ – Noah Schweber Jun 13 '17 at 20:56
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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.

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  • $\begingroup$ "he considers Cantor to have established an alleged "existence" of infinite sets". See Pinciples (1903), Ch.43, §339, page 362.63. $\endgroup$ – Mauro ALLEGRANZA Jun 15 '17 at 6:37

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