Our theories allow it, 'tis true, but no one here has suggested that mathematics has no concept of infinity, it seems to me, so it is unsurprising that a mathematical theory of the universe might have the odd infinity or three. I’m not sure where this takes us in the overall argument. (I might have lost the thread of this bit of the discussion.)
right. i think the spirit of the thread was looking for examples of infinity in real life, which i’ve already contended that it would either be too large to exist within our universe since space itself is finite, or would be too small for us to measure given that we exist and quantify things in terms of dimensions whereas something infinitely small would have no dimensions. if you wanted to enumerate something towards infinity though like the number of fractions between 1 and 2, it would be no more than a mathematical representation that we all are already familiar with.
Right, erislover and pancakes are on to what I was getting at.
Mathematics certainly allows for infinities in our current models of the universe, not surprisingly. Yet, they may not necessarily represent reality. Have we always just assumed infinities must exist, simply because that’s what our equations have always pointed to. Also, should we, or have we, tried to take a look at the areas we think they reside, and maybe try to assume they can’t possibly exist to see where that brings us? What would that do to our current understandings?
Meh; it depends on what system of numbers you are considering. I hate the phrasing “X is not a number”; see my previous discussion of that phrase.
How do you represent the number -3 + 4i in the real world?
Facile answer: Can’t be done. You can’t have “-3 + 4i of” anything. Complex numbers, while interesting and perhaps even useful in some ways, lack correspondence to the real world.
Better answer: One representation of complex numbers is as combinations of rotation and scaling; we have -3 + 4i whenever we have an angle of just-over-143 degrees, one leg of which is five times as large as the other (we have it just as much as we have just-over-143 and five). This is just one way to fruitfully correspond these abstract concepts with particular physical quantities; there are myriad others.
Does the infinite exist in real life? Well, it’s like asking if -3 + 4i exists in real life, if Boolean algebras exist in real life, if the rules of chess exist in real life… Sure, they all do, under some interpretation or another (e.g., rotation + scaling, digital circuits, and, uh, the human activity known as “chess”, respectively).
If you mean purely things like “Are there two non-zero distances whose ratio is infinite?” and “Is there a bag somewhere with infinitely many kittens in it?”, well, maybe, maybe not. But there are undoubtedly other ways in which the concept of the infinite corresponds to real-world phenomena. [But of course it does! Lines have slopes, don’t they? And vertical lines exist, don’t they? See, it’s that simple…]
Bolding mine. It probably doesn’t change anything of what you are saying, but it may be worth observing that apples aren’t, per se, finite or infinite, nor even are apples’ sizes finite or infinite. Strictly (perhaps unnecessarily pedantically) speaking, it’s only the ratio between an apple’s size and the size of, say, some standard meter bar which is actually finite. Of a size in itself, one can’t even determine whether it’s under or over 1; after all, every object has size 1, in size-of-this-object units.
Seen this way, it becomes clear that the problem isn’t one of finding an object in the universe with some extraordinary property of being infinite; the problem is one of finding two objects in the universe whose ratio happens to be infinite, though this does not, in itself, impute special status to either object. Perhaps such a perspective makes the idea of infinite real-world quantities less “magical”-sounding.
Why is infinite always thought of in terms of vastness, as in, something can be infinitely large.
Surely, something can also be infinitely non existent using the same concept.
In terms of vastness we say that we can not measure it, but in terms of non existence, measuring is futile. Thus, measuring anything infinite is futile and perhaps even irrelevant to the concept.
It’s like trying to figure out how much RAM is required to play street hockey.
I don’t think the universe could be infinitely full of stars. If that was the case the night sky would be white, since everywhere we look we would see stars. Unless the infinite stars are rather new, but then how does infinite energy just get created like that?
But, going back to my original point. Is it not true that a thing can be infinitely non existent?
Um, maybe? What does it mean to be “infinitely non-existent”?
(Ignoring the business about “infinitely”, my default response to this line of speaking would be to say “Nothing exists which is non-existent”.)
(Not that it isn’t useful to say things like “Santa Claus is non-existent” and so forth, but these are far from the sort of concrete physical objects the OP is looking for. (Though what exactly the OP is looking for is unclear… (e.g., I’m guessing he wouldn’t care for “the slope of a vertical line gives a concrete example of the infinite in the real world” either, but I’m not sure why not)))
This idea, known as Olber’s paradox, actually puzzled astronomers for years. It wasn’t resolved until the discovery that the universe is expanding. Since stars farther away are redshifted more, stars very far away are redshifted out of visible range. And stars even farther away become infinitely redshifted, so they can never be detected in any way.
Because of this, the entire universe could still be infinite, but our observable universe is definitely finite. Since anything outside our observable universe is physically unable to be interacted with in any way, you might even just say it doesn’t exist. So there we go again with the universe taking any infinity we might discover and shoving it under the rug.
Incidentally, something can be infinitely big and still expand. What’s really happening is that the matter is being more spread out in time. If you go back in time, all matter gets closer together, until the Planck time when our equations can’t go any further. So what’s really happened is that our universe has evolved from a state of incredible density into one that is less dense. We like to call it the beginning of the universe because we don’t have the tools to say what happened before then.
No, such things are imaginary. A concept in the mind. If something never existed, and never will, then it can’t have any physical properties. The question I’m posing is, is infinity a physical property, an abstract conceit, both, or something in between?
We’re spoiled, because mathematical and abstract concepts have been so good to us, and have brought us much in the way of knowledge and technology. I wonder if we forget some of the things math shows us, and even helps us with, has no analog in reality. Infinity isn’t the only of these, just the most curious to me (for now).
There is a difference between an abstract construct and the physical world. One can be a model for another, but what they represent isn’t the actual thing. I’m just wondering where infinity belongs, or if I’m just missing the obvious.
So, I ask, Does Infinity Really (i.e. physically) Exist? And, ironically, this brings us full circle. 
Infinity is an abstract concept (or, rather, a family of abstract concepts); but then, so are natural numbers, integers, real numbers, complex numbers, quaternions, vectors, tensors, angles, volume, slope, polynomials, trigonometric functions, differential equations, symmetry groups, topological spaces, and so on. To the same extent as any of the the latter have analogs in reality, so do all the others and so does infinity. There may be certain aspects of the universe which do not admit of the infinite; that does not mean it does not correspond to anything else.
Like I said before, one very simple example is just slopes of lines, which take values in the projectively extended real numbers, including an unsigned infinity for vertical lines; is this any less acceptable a way for a mathematical concept to correspond to a physical reality than other ratios of lengths?
Okay… I’m almost with ya.
Agreed. But, there’s a problem we run into with infinity. With the other things you mention, you can have a pure mathematical or geometrical counterpart, to the approximate physical counterpart.
You can’t approximate infinity. It simply is infinite, or it is not. That’s the kicker.
Let’s try an example for the hell of it…
Let’s take pi. The ratio of the circumference of a circle to its diameter. The decimal goes on forever – it’s infinite. The more decimals you tag on the end, the more perfect the circle is mathematically. In the real world, we can try to craft the most perfect circle possible, or even find one in nature, but that decimal is going to stop somewhere. At some point, a physical circle will be an approximation of the mathematical circle.
How can we conceive to make a physical circle perfectly and purely pi? Or if we made a circle and wondered if it was perfect, could we measure it to that degree?
I’m just a shlub liberal arts major moving into middle age, but thinking about the OP has been one of my stuck in traffic passtimes for several years. I haven’t got very far, but here goes:
Infinity is only a concept. Nothing is infinite except in concept. Sub-atomic particles, even photons are finite in number. You can always find more of them and add them in, but they exist in large and finite quantities. Unlike mathematical constructs, which by their nature can be defined as going on forever. For the real world, you can talk about there being more particles, but unless they actually exist, and can be refered to, they must be finite. (Ignoring the blather about infinite possible universes occupying the same space in parallel dimensions, which is nonsense, because the mass of even knowing about such universes in other than concept, being infinite, would make this universe too massive.)
As for infinity being a useful concept, calculus is based on an understanding of infinity (or so I am told). With it you can build everyday objects, do engineering, and calculate orbits, etc.
But you will never run into an infinity of anything in the real world. You’ve heard the saw that if you had an infinite number of monkeys typing on an infinite number of typewriters they would eventually come up with all the written works of Shakespeare? Yes, they would, but it would be almost immediately that they churned out these works because some would immediately spit out the works, while an infinite number of others did worthless stuff. So much worthless stuff that you would never find a single coherent piece of typing despite the fact that the monkeys produced an infinite number of copies of Shakespeare (and all other works). Without an infinite number of inspectors, all you would have is an infinity of garbage with an infinitely small fraction of useful stuff: so small that a single inspector would never find anything.
Interesting that infinity exists only in the human mind.
I missed this as I was typing up the above post. Does it address this in any way?
Same same for nouns, verbs, adjectives, adverbs, prepositions, and so on. These are abstract concepts, and yet, as you so eloquently said of numbers, we use them in ways we find convenient to facilitate our understanding. We even use them to discuss infinity.
Well, things simply are π or not also. But that doesn’t make you say you can’t approximate π.
Nitpick: Surely, rather than “that decimal is going to stop somewhere”, you mean only “that decimal is going to diverge from π somewhere”? Every number has infinitely many decimal places; some just have lots of 0s.
One easy way to approximate infinity: Get yourself some copper, form it into a wire, and try and make the cross-sectional area of that wire as small as possible while keeping the length constant. That is, make its cross-sectional area approximate 0. Well, resistance is inversely proportional to cross-sectional area, so it will approach infinity at the same time.
Here’s another: like I said, slopes of lines. If you measure angles by their tangent and approximate a right angle, then, bam, you’ve approximated an unsigned infinity.
Here’s another: Construct a quiet room. Now make it quieter. Keep making it quieter till you approximate absolute silence. How many decibels will that be? Negative infinity.
No, we probably could not measure it well enough to conclude that its circumference-to-diameter ratio was exactly π. But for that matter, we can’t measure any ratios of different lengths well enough to conclude that they are exactly 1 (or 7, or 2.5, or…), either [mathematically, we might say that length equality is co-semidecidable; we can affirmatively tell when two lengths differ, but not when they are equal. Another way of saying this is that lengths have a Hausdorff, non-discrete topology (properties modelled by the reals)]. But what of it? (Was this supposed to demonstrate something about π or something about infinity?)
Sorry, that second sentence ended up in the wrong place during editing. Pretend I didn’t write it. Instead, pretend I made the point I wanted to make, which was, if you’re sure that every measurable quantity eventually differs from π, then you probably, for similar reasons, should be sure that no measurable quantity has a decimal representation ending in an unending stream of 0s; i.e., that “no measurable quantity’s decimal ever stops”.
While I’m at it, another easy approximation of infinity: take some positive mass, compress it into as small a volume as possible. As the volume approaches 0, the density approaches, you can guess. To the same degree that you can approximate any other mathematical concept under some correspondence with physical quantities, so you can with the infinite.
Thanks, Ind.
Let’s put it this way then. You’ve kind of pried apart my thinking.
All the examples you give are different forms of the infinite. Even my example: pi. At some point, as said, they diverge from their inherent infinities, and become approximates.
Unfortunately, a pure definition of infinity can’t be approximated. And I suppose that’s the heart of the matter.
My pi example was supposed to demonstrate the intrinsic infinities within the ratio and how they disappear in the real world. This was just an example, not that I’m focusing on pi more than any other example you presented. I’m sure we could demonstrate the same for everything you’ve mentioned and more.
I’m not sure we’re really in disagreement here, just coming at it from opposite angles.
As for your physical arguments, I’m not sure how you can get a copper wire down to only 2 dimensions? And a quiet room is merely a space with either no vibrating air or it’s a perfect vacuum. Noise, like temperature is something that has a starting point. You can’t go below it, so there’s a limit there. Infinite describes no limit. So your “quiet room” analogy, would have to be swapped with a “loud room” analogy, and you’d have to keep making the room louder until you reached infinite decibels. Not possible.
Can you help me with the sloping lines though? I’m not sure I grasp that one.
Yes, but in the real world, unlike on paper, you can approach infinity all you want, but you’ll never get there. Due to myriad circumstances.
I’m saying the infinities we dream up on paper, collapse in the real world. This is why we can’t make a spaceship travel at C, or make a perfect circle.
We may never know the true nature of a singularity, or the universe itself, but my guess is it’s not infinite. Not where we reside.
Perhaps maybe way beyond the Planck Length and from whence the universe was borne, if these “non-spaces” and “non-places” somehow exist, than I suppose there you’ll find infinity. Or maybe we just need a new word?
To switch it up some, here’s a thought. You have two infinitely long intersecting lines: X and Y. They are both perpendicular to each other. Now, keep one line still (X), as you turn the other line (Y) at a 90º angle. Your goal is to make them both parallel to each other. At what point does Y not touch X?
I’m not sure where I’m going with that, but it seems relevant to the conversation somehow. It’s late…