Is the concept of infinity practical or just theoretical?
In: Mathematics
I tend to think of infinity and negative infinity like directions, like East and West. They tell you which way you are going, but they aren’t a place you’ll ever actually reach.
Practical in mathematics as things will go to infinity, common in things like calculus (but math is theoretical in a way).
In my opinion it’s theoretical everywhere else. Everything has a time or an end.
Something that you’re not really taught in school is that modern mathematics is predicated on there being some margin of error for your calculations. For what you’re going to be doing in most normal people applications, that margin of error is small enough to be irrelevant to you.
So, for example, the formula 1+1 has a margin of error that is so close to 0 that you can say, for all practical purposes, that the answer is 2 (even though there is a non-zero chance that it *might* not be). The same is true of 1/3. Its easy for you to take a pizza and divide it up into 3 “equal” parts, despite the fact that its isn’t actually possible to divide something into 3 *perfectly* equal parts.
The reason for that is that there is a margin of error when you divide something into 3 parts. That margin of error is small enough that for all practical purposes, the pizza has been divided into 3 equal parts – despite the fact that we know that one of the parts is microscopically larger than the others.
The same is true of calculations that incorporate the concept of infinity. Your calculation has a margin of error. If that margin of error is large enough that using any arbitrarily large number will result in the same outcome, then infinity can function as a stand in for any such number.
In other words, infinity is basically a concept that allows calculations to be made when you don’t actually care what the exact number that you’re working with is.
This is also why infinity is smaller than infinity+1. That’s just basically a fancy way of saying: “I don’t care what the first or second numbers are, so long as the second number is 1 higher than first.”
Quantum mechanics necessarily has infinities that have to be reconciled with. And some of the predictions there are some of the best tested and most accurate in physics. So to my understanding that seems like they are actual things that really exist.
Feynman diagrams are involved in a way of calculating the infinite possible paths that a particle can take to get from A to B. They are actually workable by humans because the more steps the less likely they are to happen, so if you calculate enough steps you get a very accurate answer. That seems to me like the infinity is involved in the actual underlying mechanics of how it works. And we just simplify it out when it becomes so small to not matter.
Errr, it’s everywhere? Think about anything that has growth. Eg when YouTube was young, you could say “it would take x hrs to watch all the videos on YouTube” at some early point, the answer became infinite because there is more content uploaded in a day than could be watched in a day. Same with a lot of practical things.another example is how long could I sail a boat in a particular direction before hitting land…. Well, from certain locations for certain directions the answer is a very real infinite.
We do not know. There are different schools of mathematical philosophy concerning this topic with some of them believing that infinity does not exist.
That’s difference between ‘actual infinity’ and ‘potential infinity’.
While it might seem weird that some mathematicians working with infinite stuff does not believe in existence of infinity, the point is that it does not matter for any application whether infinity exists or whether it is just theoretical.
Infinity lets us have an easier time with certain things. Convergence is one example. When for example, let say we have an equation that is hard to solve. With infinity, we can have an infinite amount of terms that at the limit approximate that hard-to-solve equation. Even if it is not exactly the same thing, it is in some way good enough. If you took math in college, you might remember things like Taylor series or holomorphic / analytic functions. They all use the idea of infinity to achieve some sense of convergence.
For a number to be infinite means that you can never count to it. The word “endless” also applies because there will literally never be an end to your counting, you’ll just go on and finishing is impossible. In fact, “infinite” means the exact same thing as “endless” from a etymological perspective. There are some actual things that are infinite in the real world, like how many years there will be or how far you can travel
A lot of times in the mathematics or physics it’s just easier to work with infity than actually big numbers, because infinities have some properties that make math simpler.
For example, imagine that you push a boat on a lake, and I want to calculate the precise point where it stops.
I write the equation that shows me the friction of the water against the speed of the boat, and how the initial impulse is going down with time. And I just need to iterate this over time. But what time? I care about point, not time. So I can just integrate the equation until “infitite time”, because the speed of the boat will go down and down to zero and my math is simpler with worrying about specifics.
Another example, you might want to calculate something like how light rays will curve in a glass. Sun generates rays in all random directions, but because for your practical purposes you might consider the distance to the Sun “infinite” and consider your rays to be parallel, and not slightly angled (like it would be with a close light source), making your calculations less precise, but still practical enough for your case.
The vacuum of space is the only infinity that exists in every direction. Otherwise, everything else is finite: the number of atoms in a galaxy, the rate of conversion from mass to energy, the number of photons emitted by phosphorescent sources, the paths of those photons and other particles, the timing of their release from an electron or neutron, the extent of the observable universe, the mass of black holes, etc.
Infinity is a mathematical abstraction that helps our equations in determining probabilities, but doesn’t adequately describe the universe as a real object.
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