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As far as I know, no actual "infinity" exists in the universe. For example, the age of the observable universe is thought to be 10 - 15 billion years, while its size seems to be about 93 billion light years.

So a theory that comes to mind is that the concept of infinity is simply a byproduct of our brains being able to build abstraction upon abstraction from simple rules. There are two reasons why I find this plausible: 1) our ability to iterate a rule in an inductive way allows us to construct infinite sets, 2) if nothing infinite exists in the world, where else could we have gotten this idea of infinity, rather than our own brains?

So my question is whether the possibility that "infinity" simply comes as a byproduct of our brains, and not as a feature of reality has been given any thought in philosophical literature.

PS. Schopenhauer writes in "On the vanity of existence": "This vanity finds expression in the whole way in which things exist; in the infinite nature of Time and Space, as opposed to the finite nature of the individual in both". One reason I'm asking this question is that if infinity was a feature of the universe, then "vanity" would be a problem of man and his place in the universe. If infinity was simply a feature of human brains, then "vanity" is a problem of man in relation to his own mind.

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    By the perennial question "if our brain is entirely a natural evolution of reality, then how the brain can generate some byproduct which is not a feature of reality?" From the famous causal closure principle this sounds impossible... As for Turing machines which is basically a mathematical ideal concept called set of 7 tuples, how can it have accidents? Jul 27, 2022 at 1:02
  • Infinity is an abstraction of the familiar process we learned as children where we can count 1, 2, 3, 4, ... and no matter what number anyone can name, we can always count that number plus 1. Humans have the power of abstraction and imagination, and abstracting the concept of infinity from the process of counting is an example of that.
    – user4894
    Jul 27, 2022 at 4:45
  • @foxmulder A cognitive system needs to be able to decide that it has all the premises it needs in order to infer a proper conclusion, so the notion of finiteness is a fundamental requirement of any natural cognitive system. The notion of infinity is simply the negation of that. It is infinite it is not finite, as evidence by the etymology of the word "infinite". The question of whether things exist as we think of them is something else entirely, so we have to accept that we are ignorant as to whether there is any actual infinity. Jul 27, 2022 at 10:05
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    Holes don't exist in the universe. Limits, numbers, infinity don't exist in the universe. Knowledge is a model of the universe, it is not the universe itself. Knowledge, in addition, is much more than a trivial copy of reality. Knowledge is a complex representation of the universe, using limits, numbers, holes, etc, infinity included. Call it an accident, call it a hullaballoo, thing-a-ma-gig, a gobbledygook,... infinity is just part of knowledge.
    – RodolfoAP
    Jul 27, 2022 at 14:11

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Rather than saying no actual infinity exists in our universe, it would be more precise to say that we can only see a finite portion of the universe, so we can't tell if the full universe beyond our sight contains infinities or not.

We see a piece of the universe, but we can't see the edges. It extends out of sight in all directions. So we can only speculate as to what might be out there, beyond the circle of firelight. We can count atoms going 1, 2, 3, 4, ... and so on. What happens at the edge? Is there some specific number of atoms to which it is impossible to add another? How/why? Is there some specific number such that when you add another atom they all disappear, and you start again at zero? Or something stranger? Or might it simply keep going without end? We have no data, how do we decide?

Infinity is often the simplest option. Having to specify a limit, and then specify the behaviour at the limit that stops you going any further, introduces arbitrary complexities into our model of the world. We have no evidence on which to base such speculations or to test their truth. So we pick the one that is simplest to work with, that is sufficiently accurate in its predictions for our purposes, but remain open to other possibilities.

On your question "whether the possibility that "infinity" simply comes as a byproduct of our brains, and not as a feature of reality has been given any thought in philosophical literature" - the literature on infinity has tended to operate the other way round. Since the very beginning it has repeatedly been denounced as illusory and inconsistent, a mental construct so rife with paradoxes and contradictions that it is barely credible even as an idealised mental model of the world, or Platonic ideal, let alone as reality. It has taken centuries of effort by mathematicians to rehabilitate it - to find ways to understand it as a respectable and internally consistent 'number' that we can do maths with same as any other number. Indeed, it has only relatively recently been found that the ordinary numbers appear to be 'broken' or incomplete without it. (For example, projective geometry adds 'points at infinity' to Euclidean geometry, resulting in a much more elegant and symmetric theory.) We have only just recently got to the point where we can take seriously the possibility that reality might actually be infinite!

And it's worth pointing out that the rules brains and Turing machines operate by are themselves the rules of the universe. The universe itself makes computation possible, and makes concepts like infinity conceivable. Why, then, should it not make infinity possible? We have found out so much about the universe by examining a tiny piece of it, figuring out the rules it follows, and then extending the logical consequences of those rules to the rest. (E.g. it's how we found out about the Big Bang, and that the universe is only 10-15 billion years old.) It's true that the only place we can observe infinities is in brains and Turing machines, but it is a bold claim to say this is merely an 'accident', that tells us nothing about the way the universe works. We can't see infinity, but we can see the rules in the part we can see, and we can maybe deduce what that tells us about the rest.

Regarding Schopenhauer, the issue there is that the individual is limited - we only see a piece of reality, not the whole thing. For Schopenhauer's purposes, it matters not whether the universe is actually infinite as such or merely a googolplex of light years across. The point is that we'll never get to see all of it. We can never complete it; we can never win the game, and sit back satisfied at our Ultimate High Score. We just go on and on, fighting the slings and arrows of outrageous fortune, until we eventually and inevitably lose the fight and die. Our 'vanity' is our urge to see it all, to know and experience it all, to solve all problems, to survive all dangers, to live happily ever after. To imagine that we can. That the universe is effectively infinite from our finite perspective is more than sufficient to prevent that. That we'll never find out if it's actually infinite is just the sort of frustrating obstacle Schopenhauer was complaining about.

It might be worth noting that Schopenhauer in "Immortality: A Dialogue" talks some more about the relationship between the individual and the universe. Here he takes a panpsychist-sounding view that individuals are finite immanent pieces of the transcendental whole. As individuals they are finite: they will fail and die. But that 'Will to Live' they experience is shared by everything that exists, individuals feel it because they are parts of the universe, and when seen in that way, the Will to Live can succeed and we get to explore everything and live happily ever after. He's talking about the relationship between the individual and the universe. Whether or not the universe is actually technically infinite doesn't really make a difference. What matters is that the individual is only a small part of the whole.

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No infinity does not seem a brain accident. In fact, infinity is a useful tool in logic and mathematics: there are general statements about finite numbers that cannot be proved using only finite numbers, on notable example is Goodstein's theorem. This theorem due to Reuben Goodstein in 1944, is an example of a true statement that is unprovable in Peano arithmetic, but if we use the theory of infinite ordinals (definable from ZF set theory), then there is a relatively short and simple proof of theorem. There is a video explaining all, this in accessible terms.

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