Thinking about infinity

[Ignorant]

Some sets of infinities certainly seem logically possible. 

For example, the infinite set of natural real numbers seems to exist, which is to say that there is an infinity of natural real numbers existing even now, and there is none of them which do not yet exist.

There are an infinity of points on a line segment.

It is logically possible that reality has an infinite history of cause and effect (i.e. extending "backward" through the big-bang), and that it will have an infinite future.

In what sense could an infinity of simultaneously existing things exist together as a finite thing?

[Alex K]

What does it actually mean for the set of natural numbers to exist? It's not like all these numbers are somehow represented explicitly by anything in the universe. What we have is the notion of taking another step to the next one, and a notion of that being always possible. Does the existence of these ideas already warrant the statement that "the natural numbers exist in the universe"?

[Whateverist]

So better to say "the natural numbers exist in potential"?

[Alex K]

So better to say "the natural numbers exist in potential"?

Careful with that language, or you'll attract Wooters

I don't know what you mean by that precisely, though. In potential meaning they could exist... where?

[Whateverist]

I mean you could get to any of them with enough time, just take another step.

[robvalue]

Interesting topic! We have to be careful about we mean by "exist". It can be a slippery concept.

I would personally use a definition something like this:

Something exists if it can be uniquely identified as a subsection of our (assumed) objective reality.

Thoughts? For things like numbers, I would say they only exist abstractly; minds are able to have images that represent the concept. They are a step removed from reality. They can represent arbitrary groups of things in reality, but they do not themselves uniquely identify something in reality. You can of course have abstract concepts that do identify something existent. We could also have abstract concepts that represent qualities of existent things, such as length.

Things can theoretically be dissected into infinitely parts. We can model them mathematically, and slice them up as we wish. Can they in practice? That depends on the nature of reality. Maybe, maybe not. If there is some "smallest measurement" actually possible in reality, then such a thing wouldn't work.

I see no logical problem with an infinite past. The obvious objection is the intuitive one, "well how did we ever get here"?

I have two possible counters to this, off the top off my head:

1) This is taking the naive, linear and "mobile" view of time, and of cause and effect. Our way of viewing what we call time may be entirely the wrong way to look at it. Viewed from the correct perspective, maybe an infinite past makes perfect sense.

2) Our reality may be artificial. It may be generated entirely by some other process in another reality. In this case, an arbitrarily powerful simulator could return to us whatever information we managed to request from the past (using scientific methods of observation). It could generate as far back into the past as is required, just as it could allow the present to continue into the future in an unbounded sense. So this could allow for an artificial unbounded past at least, if not an actual infinite one. This gets around the "how did we get there" problem by virtue of the simulation being started at whatever point the process see fit. All "history" is calculated to be consistent with this.

[SteveII]

I see no logical problem with an infinite past. The obvious objection is the intuitive one, "well how did we ever get here"?

I have two possible counters to this, off the top off my head:

1) This is taking the naive, linear and "mobile" view of time, and of cause and effect. Our way of viewing what we call time may be entirely the wrong way to look at it. Viewed from the correct perspective, maybe an infinite past makes perfect sense.

2) Our reality may be artificial. It may be generated entirely by some other process in another reality. In this case, an arbitrarily powerful simulator could return to us whatever information we managed to request from the past (using scientific methods of observation). It could generate as far back into the past as is required, just as it could allow the present to continue into the future in an unbounded sense. So this could allow for an artificial unbounded past at least, if not an actual infinite one. This gets around the "how did we get there" problem by virtue of the simulation being started at whatever point the process see fit. All "history" is calculated to be consistent with this.

Your reasons for assuming an infinite past are "Our way of viewing what we call time may be entirely the wrong way to look at it." and "Our reality may be artificial."? is based on what? With this line of reasoning (I use that term loosely) can we know anything? It seems to me that you are merely trying to avoid an infinite regress no matter what the intellectual cost may be.

[vorlon13]

Line are defined as a string of points, and any 2 points are defined as being separable by another point between, so the infinities associated with lines are there because humans decreed it.

[SteveII]

My understanding is that infinities are useful fictions in math equations. When used in physics, they are always converted back to finite quantities (unless you are Hawkings and then you don't).

[Alex K]

(unless you are Hawkings and then you don't).

???