But this is infinity in the abstract. Infinity, a symbol in math, exists in the abstract as all mathematical symbols exist.
I will never forget the book we used for this class. It was the smallest college book I'd ever seen. I think it was about 60 pages. Those were 60 pages of the most grueling math any of us had ever seen. I think it took two classes to get through the first page.
Sure. All math is abstract. However it often has real counterparts. And we derive predictions about the real world that are correct, using math. And there isn't one infinity. There are an infinite number of different infinite sets [different cardinality].
There was a famous [science and math famous] prediction made by math about subatomic particles. There was a graph showing the known particles. I don't remember exactly what the graph showed but the particles all fell on the right half a hyperbola. The physicists said hey look, this forms a hyperbola!!! But the mathematicians objected. No it doesn't! they declared. That is only half a hyperbola. Where is the other half? Some number of years later we discovered anti matter. And guess where the graph of those particles landed. Yes, it was the left half of the hyperbola! So in some fundamental sense, the math told us there was more to the story.It was hinting at the existence of anti matter.
So it might be argued that since math says there are a infinite number of different infinities, we might expect this in the real world. What comes to mind immediately is the multiverse. It may be an infinity of infinities. Like the hyperbolas, it is tempting to think that math predicts or hints at the existence of the multiverse.
Yes, Math exists in the abstract, but it has real world applications. I get that. But, does this suggest infinity, as an abstract math concept, suggest infinity must exist, therefore, in reality?
We don't know. As I said, based on the history and the intimate relationship between math and physics, it is temping to think it might be so. We know other abstract concepts have real world meaning. Here is another philosophical question: Why do we need imaginary numbers [and I mean complex numbers that have both real and imaginary parts] to solve real world problems? Essentially multiplication by i [the square root of negative one, an imaginary number] is represented by a phase rotation. We use it in virtually all electrical engineering. So here you have not only an abstraction, but an "imaginary" abstraction, that we need to solve real problems.
The sets don't map 1 to 1. This gets really difficult and I need to be careful. But for example, consider the set of all real numbers as compared to the set of all integers. I think this example works... For any integer there are an infinite number of real numbers. Between 1 and 2 we have 1.1, 1.01, 1.001, 1.000000000001... ad infinitum. So there are an infinite number of numbers in the real set, for any number in the set of integers. So we can see that while the set of all integers is infinite, the set of reals is infinite for each integer. Using very heavy math that is now beyond me, you can show that these two infinities are not the same.
So just to clarify what I said above. The set of all integers, ie 1, 2, 3, 4, 5...1000...a billion, a quintillion, and on, and on and on, is one infinity. These are the countable numbers Then we have the set of real numbers, an infinite set which includes the set of all integers in addition to other sets. So we take all of the countable numbers, 1, 2, 3, 4, etc, and for each one of those are an infinite number of fractions as well as decimal values that can't be represented by a fraction, in between. It can be proven that the infinite set of all real numbers is not the same as the infinite set of all integers. The set of all real numbers contains the infinite set of all integers as well as an infinite number of other infinite sets. This is a shotgun explanation of infinite sets having different cardinalities.
Physicists use math as a descriptive language. Like with natural languages, the existence of a noun does not mean that there exists an actual physical object of that name. I can say unicorn without there being a unicorn anywhere in the universe. There is plenty of math that is not used by physicists, as it doesn't particularly describe some part of nature they study. If there is something about our universe that is infinite, then that would surely be accompanied by an explanation concerning how it applies. But, physicists I have read have claimed that they have seen no use for the various infinite sets that math describes - that is, simply being infinite is enough for discussions of our physical universe.
Yet. Binary had no use for over 300 years. Nor did 10 dimensional mathematics. Name some examples. A trivial response. Who specifically said this?
I don't get your point. Maybe more of math will be used to describe this universe at some time in the future. Who knows? But, the existence of some math construct does not mean there is an analog in reality. I believe Sean Carroll has pointed out that not all math has a use in physics. And, why should it?
Just to point out what should have been obvious, you seem to have lost track of the context in which I gave you that reply. You obviously don't seem to understand what I meant, and so I will not be wasting any more of my time in this discussion with you. There can be a big difference between two relative infinities (especially when they are not actually literally infinite) and true infinity. I how don't see how you could be incapable of seeing that. I am going to refuse to have a conversation with you, since I know you will not be keeping track of the context of our discussion. It's not worth my effort to have to continually remind you what we are talking about.
In some effort to answer the OP I would say that the universe exists because physics happens. The "why" is asking for a moral answer such as purpose or end game. I don't think there is either. It is because physical properties of what it is made up of, determines its existence. IMO there doesn't have to be a "why". The question "how" is interesting and keeps a few people employed but the "why" has always been the reason religion happened. It attempts to answer that question by inserting moral instruction and combative power games into attempting to answer an unanswerable question. Since the dawn of thinking man, some have pretended they speak for God. They still do. Imagine the power that gives them.
The abstract, therefore, is a very necessary aspect of life in order for humans to function in the physical world. Without it, we would still be hunters and gatherers, and even then, we were probably using basic arithmetic, once upon a time.
My, what a lofty perch you occupy! Alright hot shot, I'm just a layman, struggling with these concepts, and I respond to a lot of posts, and no, I can't keep track of them all. sorry. ANd your last sentence I don't buy since you are all over this forum, a lot, for quite a spell, so, apparently, arrogance becomes you. FYI, I was just reiterating what I thought you said.
I think you have to be a little careful with that. Complex numbers don't have anything to do with the advent of human agriculture. That concept comes from the 1500's. Do you see trigonometry as part of "basic arithmetic"? Trig was developed in the ME several thousand years BC. I don't see this as indicating anything about "abstraction".
The universe exists to allow you to wonder why it exists. Nothing more simple than this. Jokes aside, the universe doesn't wonder why it exists. We wonder ...
I meant elementary arithmetic at the hunter gatherer level. You know, I'lld trade you four buffalo skins for a vase of your grain. That kind of thing.
OK, now you are going WAY back, it would seem. Money was created by 5,000 BC. And, the trade you speak of was surely in place for a long time before that. Chimpanzees don't barter, but that may have something to do with their view that they own only what they have with them, which is mostly the food they want/need. Maybe bartering came after concepts of ownership included storage.
Indeed, science is more interested in how than why.. Why is Plancks Constant what it is? Nobody knows, we just know what it is, and that's enough. We have a lot to learn...
Wooops, it's Debt the First 5,000 Years.. https://www.amazon.com/Debt-Updated...ber&qid=1649940157&sprefix=debt,aps,96&sr=8-1