Science a GoGo's Home Page Forums General Discussion Physics Forum Alice and Bob in infinity

Those intrepid adventurers Alice and Bob, in their respective space craft, are the only occupants of an infinite void. Suppose that Alice and Bob are an infinite distance apart; if each moves, say, a billion light years towards the other, are they still an infinite distance apart? Intuitively, it might feel as though they should be two billion light years closer to each other, but that would mean that the distance between them was less than infinite. If the distance between them is less than infinite, it must be finite. This means that something that was infinite has become finite. Reverse their journey and something finite suddenly becomes infinite. This is impossible.

Any thoughts?

Yes, I've got some thoughts, Bill. This is how I see it, right or wrong...

Alice and Bob are an infinite distance apart. This means that they could never meet however much they moved toward each other. Which means that although the distance between them decreases over time, they remain separated by an infinite space. In other words, there are an infinite number of infinite distances.

As you can see, I disagree with your view that they cease to be an infinite distance apart when they move toward each other.

Thanks, Good that someone is not afraid of infinity.

Cantor would agree with you about the infinite number of infinities, but these must be mathematical infinities. Actually, I agree with you that they remain an infinite distance apart, what I cannot accept is that that infinite distance could be any less than the original infinite distance. In other words, I think they cannot have moved relative to the infinite "background", only relative to each other, and only in their Fs of R.
I am open to being convinced, though.

I don't know, Bill; by which I suppose I mean that I have no satisfactory concept of infinity, beyond a glib dictionary definition and a few pretty lines on a chart. But it would bother me to be dismissive of mathematics simply because I don't understand it. It is, after all, a rigorous system of reason and logic, and a proven reliable means by which to gain understanding of the physical universe. I would be loath to seek a contradictory understanding of reality based solely upon what I feel ought to be true.

I take your point about mathematics, its pedigree is long and good, but John Barrow, "The Infinite Book" makes an interesting point about mathematical "existence": “Cantor’s most dramatic discovery was that infinities are not only uncountable, they are insuperable. He discovered that a never-ending ascending hierarchy of infinities must exist." ..... "Before we see how he did that, it is important to say a little about the meaning of the word ‘exists’ in this context. We are used to using the word on an everyday basis without any ambiguity. ‘Cambridge exists’, ‘inflation exists’, seem to be assertions that are clear enough. They are about physical existence. Up until the early nineteenth century, mathematical existence was rather similar. Euclid’s geometry existed because it was manifested in the physical world. Indeed, it was believed for thousands of years that there could not be another logically consistent and complete geometrical system. The discovery of non-Euclidean geometries which described the topography of curved surface changed that view. Gradually mathematicians lighted upon a new concept of existence. Mathematical ‘existence’ meant only logical self-consistency and this neither required nor needed physical existence to complete it. If a mathematician could write down a set of non-contradictory axioms and rules for deducing true statements from them, then those statements would be said to ‘exist’.”

This seems to suggest that mathematical existence, or reality, might not necessarily be the same thing as physical existence or reality.

Yes, it appears that mathematics may be used to provide descriptions of all possible universes. That a particular description applies to ours may be verified only by experiment and observation. When it is verified, it becomes evident that our universe conforms to particular mathematical model. We can then see that mathematicians have discovered something of the underlying mathematical structure to our universe. One might, therefore, conclude that mathematics is embedded in reality, and is merely awaiting discovery by mathematicians.

Alternatively we could reason that mathematics is simply the language in which we can best express both our ideas and our discoveries about the cosmos. We do much the same with words. We express a theory in words, and if experiment and observation support the theory, it might be imaginative to speculate that the words were embedded in reality, but that is philosophy rather than science, in fact, the words merely provide the medium by which we examine and develop our ideas and share them with others.

A colleague of mine who was a maths teacher said: "If there is a God, I believe it is a mathematical God." I have thought quite a lot about that in the intervening 40 years or so, and while I understand what he meant, I am left with the feeling that this sentiment is no more than an expression of the way in which we are able to use maths to describe the world around us, and the extent to which mathematicians appreciate that. Had he been a writer, or an artist, I wonder what he would have said.

Certainly we're philosophizing rather than conducting scientific analysis. And of course it's true that while, as your colleague surmised, there may be a mathematical God, this is also the God of everything else - words included - a pretty big pantheon rolled into one. Philosophy? Yes. Science? No; as is the case for our whole discussion in the thread. I wouldn't let that deter you though. It does make for interesting exchange of ideas.

Returning to Alice & Bob; if they were a few metres apart, then set off in opposite directions in this infinite void, my feeling is that they could never reach a point where they were an infinite distance apart. Would you agree?

If that is the case, then my original question, which involved their being an infinite distance apart was meaningless! In order to be an infinite distance apart, they would have to be eternal beings who had always been infinitely separated, and always would be.