I have to admit ignorance here; my math education only took me through 4 years of high school math and a smattering of it in college. Interestingly enough, I never needed anything professionally that I didn't learn in grammar school. On the other hand, I did use intermediate algebra, logarithms and trigonometry in my hobby.
Dr Adaquate, We can not intellectually understand any possible physical reality outside of our universe due to the limitations of our intellects. Human intellect evolved so that we could navigate within the relative universe of space, time, matter and energy so anything else that might possibly exist is beyond our capability.
When we try to think 'outside the universe', we only project what is inside the universe to that 'realm' -- that's all that we can do. My argument is that whatever it is that is non-universe can't be what is within the universe otherwise there would be no distinction between the two.
When we talk about "other universes" in the context of the Inflation Hypothesis, we do mean just more spacetime and energy and matter, having a common origin with our own universe.
What makes them "other universes" is causality: the geometry of spacetime is such that what's in the other universes can't affect what's in our universe, and vice versa. It's not that there's (necessarily) an essential qualitative distinction between this universe and the next, just that they're causally isolated from one another.
Yes they are possible. The problem is there isnt a shred of evidence that another universe exists.
There are, in fact, shreds of evidence.
I've referred to the Inflationary Hypothesis. There is some evidence that this is correct. If it is correct, then other universes are a corollary of its correctness.
I suppose another piece of evidence, though I'm not sure how much weight we should attach to this, is that it's extremely rare for there to be only one of anything. Can you think of a particular thing which is also the only instance of its type?
For some reason I am more hesitant to reason inductively on the basis of this observation than I usually am, and I don't quite understand the reason for my hesitance --- I just have a vague feeling that it might not be satisfactory.
OK. hypothetical to the hilt, though as with no possible interaction, that leaves us out of the picture.
Well, they needn't remain hypothetical, as I pointed out to Portillo.
If we can establish strong evidence for the mechanism by which our universe began, and if the mechanism for which we have evidence implies the existence of other universes, then we would in fact have evidence for their existence.