LKS_Blade

It is clear that in mathematics that infinity is conceptually possible, but the point of the thread is to discuss if actual infinities exist within physical reality.

Wiploc

Irrational numbers were regarded that way too. hence the name. And fractals, too, were thought to be an abomination at first. And I remember my own consternation when first confronted with imaginary numbers.

It doesn't take mathematics to wonder whether space is continuous or made of discrete chunks.
crc

Deke

Thomaq asks: can actual infinites exist?

In my opinion, the answer to your first question is an unabashed, no. There is a puzzle that the human mind has made for itself:

Counting things has become a favorite endeavor amongst human beings. Early humans must have found it useful to communicate quantity to one another; “Grak, there are ‘this many’ wooly mammoths over the hill (holds up two fingers)”, or “Chud, this many days will pass before the next full moon (claps hands 5 times).” In this manner, “numbers” were not an abstraction, they were representations of real things, things that could be seen, felt, or otherwise detected. The origin of numbers it could be said, derived from the survival benefit of counting physical things.

But something changed, patterns were recognized, and numbers became abstractions. It became meaningful to simply state, 2 of a thing, and 2 more of that thing, will always result in 4 of that thing. It didn’t matter what the thing was; 2 bundles of 2 things always meant a total of 4 things. In this manner, numbers separated themselves from mere devices for counting, and became esoteric entities in and of themselves. It musn’t have been long before some curious George pondered, “I wonder if there is a biggest number?” In must’t have been long after that, that Curious George realized that no matter how big a number was, that he could always add one more to it.

And here the early mind had a real paradox. If numbers are for counting things, then there must be a biggest number since I can only see a limited number of buffalo, trees, mountains, birds, etc. But I can also imagine that no matter what that number was, I could always add one more to it, in my mind anyway. So which is it? Are there endless things? Or is there a limit?

Quite a puzzle indeed.

How much is there to count in the known universe? There is 1 me, 6 billion everyone else’s, and a ultimately, a gazillion quarks. So, from a human perspective, there is a universal “biggest number”, a numeric speed limit if you will. Let’s call this number BigNum. BigNum then will be the total number of all of smallest known particles in the universe (granted, the smallest known particle may continue to get smaller, and so BigNum may continue to get bigger.)

In this fashion, the only way an “actual” infinity could exist, would be if the smallest known particle is infinitely small, and by definition, infinitely small means having zero dimension, and if a thing has zero size, then it is not a thing, it is no thing, nonexistent.

Thomaq: or is the concept of infinity by definition always potential?

I would say no, and that in fact, it is only in concept that an infinity could exist. Time is a great example, we can imagine that time stretches infinitely backward, and will continue infinitely forward. Space is another, half of one inch is a half an inch, half of that is a quarter inch, half more an eighth, ad infinitum (pardon the pun).

MHO,

Deke

eh

And what about an infinite volume of space?

Bill

Black holes have been observed. Thus, they do exist. The matter density is large, but not infinite.

Even the quantum nugget that existed at the beginning of the Big Bang had a large, but not infinite, matter density. The true singularity cannot exist within the physical universe because the infinite gravitational force would collapse the entire universe into it within no time at all. But yes, space would need to be continuous to contain an infinitely small singularity.

Is space continuous in that way? Or, does space exist in discrete segments just like other quantum phenomena? The best that I can recall, this is a "live argument" within cosmology right now, and there is no widely accepted answer. Many things within quantum mechanics seem somewhat self-contradictory, and the evidence points in both directions (for continuous and discrete space; not to mention, continuous and discrete time).

But the key point is that the Big Bang was not a true singularity (there were no actual infinities involved). It had a finite (but, of course, extremely large) mass and took up a finite (but, of course, extremely small) volume of space.

The key question is thus: just where did that little "first nugget" of space/time come from, anyway? (Down that path lies the road to infinite regression.....)

== Bill

Bill

As I just posted in another thread, we find ourselves existing in a "finite (but unbounded)" space/time continuum (as per Einstein). Because the space/time continuum within which we find ourselves existing is, in fact, finite, we cannot have direct contact with any actual infinities because we cannot have (based upon current knowledge) contact with things that might exist outside of the space/time continuum within which we (currently) find ourselves existing.

Does this mean that actual infinities do not exist? Well, no, so long as you agree that, if they do exist, they must necessarily exist outside of our "finite (but unbounded)" space/time continuum.

So, do such actual infinities exist? It is my assertion (in this other thread) that actual infinities must necessarily exist or else we could not exist because the only alternative to the existence of actual infinities is a "sprang out of nothingness" event, and that (with all due respect to the amatuer quantum mechanics among us) is a totally illogical and unsupportable alternative.

So, yes, actual infinities must exist, but they must exist outside of our "finite (but unbounded)" space/time continuum.

== Bill

eh

Who says space-time is finite but unbounded? Infinite is also an option consistent with the evidence.

Stiletto One

Real world: Stupidity?

On infinite density of a singularity, I'm not sure that it applies because, by nature, black holes completely FUBAR space-time, so while you can find a mass, you can't find a volume because the volume is undefined. (Although, for our purposes, we treat it as zero)

*back to lurking

ex-xian

Not if you believe in the big bang theory, which virtually all cosmologists believe. If the universe is expanding, then it cannot be infinite.
I think that Hawking's work disagrees with this. I thought that the currect theory holds that it was a zero-dimensional point, with infinite mass and zero volume. Do you hold to this because the graviational force would be infinte at a point of infinite mass? Since we don't have a viable GUT, there's no way to know how the four forces would behave under such circumstances.

I do agree that it's improper to use the quantum principles of particles appearing out of the vaccuum to try to expain the origin of the universe. The probability of a particle spontaneously appearing increasing as time goes on. At t=0, there is a zero probablity of any fluctuation of a quantum vacuum. Also, don't the particles have to disappear extremely quickly? If I'm not mistaken, them mediator of the weak nuclear force behaves this way.

Witt

thomaq:
can actual infinites exist? or is the concept of infinity by definition always potential?

No, actual infinities are absurd.

We cannot verify that a particular collection is uncountable.
Our finite existence denies the possibility.

Cantor's 'identity is 'one to one correspondence', might be challenged.

Even the unending loop of computers ends when there are no people to notice!

All mental things require the existence of minds.

When there are no minds there is no: truth, number, infinitude..and so on...endlessly.

Witt