OK, sorry for the woo-sounding thread title but it is relevant, as I’ll explain. By definition “infinite” means unending, meaning one cannot “arrive at” infinity. Or can you? It’s simple enough to graph a curve such that one can ask, “at this point X on the graph, what is the value of Y?” and the answer would not have a finite bound. So for that value X, is the value of Y “at” infinity? Or is there a more logically rigorous way of expressing this situation so that these conundrums are avoided?
I’m not positive I understand the question, I’ll answer that it is a journey. It is never really possible to evaluate as a number.
In high school when presented with the .99 repeating = 1 scenario which everyone was fighting as I was. I posited "There is no point of observation where these two values are equal. .9 != 1 nor is .99 .999 etc. They’re never the same and the difference is we need a .01 with the ZERO repeating and this is the value of the difference and it works at any point that you choose to evaluate the statement.
No idea on how the thinking on this is now but I’m the only arguer who didn’t get his point soundly rebutted so there’s that.
Since I can’t really argue the proof that he presented to me either it makes me realize that infinity can’t be evaluated as a destination since it never exists as a destination that can be reached. It has to be a journey.
“Follow this path forever…and then turn left.”
No, you can never “accomplish” any infinite task. So: journey, not destination.
Still, you’ll find a lot of interesting things, and make a few friends along the way. We’re all in the gutter, but some of us are looking at the stars.
It’s neither. People have the habit of treating “infinity” as just a really big number. it isn’t a number at all. Nor a range of numbers. It’s not a destination or a process.
Martin Gardner, in one of his Mathematical Games columns, described infinity-invoking “supertasks.” One was: at time t=0, turn the light switch on. After 1/2 second, turn if off. After 1/4 second, turn it on. After 1/8 second, turn it off. Continue, in a Zeno-esque fashion, until a full second has passed.
Is the light switch on…or off?
(Or just molten to a puddle…)
There are ways to create one-to-one correspondences like this between finite and infinite – um – things. (Don’t really want to call them numbers. Sets, perhaps.) But they always fail in practice. Even Superman or The Flash can’t flip that light switch more than maybe twenty times.
I would call it a “horizon” - one that is always receding as you move toward it, except it’s not really an “it”.
From a mathematical point of view it depends on context and definitions. You can have a function such as 1/x with an asymptote at zero, such that as x approaches zero from the right the values increase to infinity, but the value actually at x=0 is undefined. This follows your journey definition. On the other hand, there are places where a singular “number” called infinity is useful.
For example the a set is compact if for every sequence of numbers, there is a sub sequence that converges to a particular value. The set of Real numbers is not compact, since for example the sequence {0,1,2,3,…} doesn’t converge. But if we add a value at infinity then we can say that the sequence {0,1,2,3,…} converges to infinity and so the Reals plus a point at infinity is a compact set.
Also in set theory a set the number of elements in a set is an important concept, and sets can have infinite numbers of elements, and so there are real concrete definitions of an infinite value. In fact, certain infinite sets are larger than other infinite sets, and there is a sequence of larger and larger infinite sets.This observation would be impossible if infinity was treated as an unapproachable limit rather than a fixed concept.
Are you thinking of a type of graph like that of y = 1/(x^2)
If so:
Q: What is the value of y when x = 0?
A: y is undefined when x = 0
One can say, however, that y approached infinity as x approaches 0
Y is defined when x=0 on the affinely extended real number line, in which the values +∞ and -∞ have a well-defined meaning.
Neither, I think of infinity more as a direction.
“At any point that you choose to evaluate the statement.” That’s the flaw in the argument. If you stop at some point and put down a 1, you’re no longer talking about infinity. The line of zeroes must extend with no end. But that’s still equal to zero, and 1 - 0 = 1.
But you have trouble grasping that, so let’s look at it another way. Go back to the repeating line of nines. Let’s say you decide to stop after five places: 0.99999. But the next digit would be another nine, which means that you need to round up. So 0.99999 becomes 1. Whatever point you stop at, the next digit will be 9, so you need to round up to 1. You don’t just truncate a decimal if the next digit is 5 or more.
Okay. I suppose you’d say that for f(x) = 1/(x^2) that f(x) = +∞ ?
And for g(x) = 1/x what is g(0) ?
What happens at infinity is often an important question in general relativity and using some jiggery-pokery called a “conformal compactification”, which is somewhat akin to allowing objects to travel infinite distances and clocks to tick for an infinite amount of time, a convenient and rigorous definition of infinity can be arrived at for some situations.
That said the infinity itself can often be divided into different regions: for example in the standard conformal compactification of the flat spacetime of special relativity there are 5 regions: timelike past infinity, timelike future infinity, spacelike infinity, null past infinity and null future infinity. These regions can defined by what kind of particle (i.e. tardyon, tachyon or luxon) can reach that region and whether they originate or terminate in that region.
However though this should not be taken as signalling that things really do reach (or originate from) infinity as it is merely a convenient way of adding rigor to unphysical, but useful statements such “particle A escapes to infinity”.
I think the best answer is really both, and that it entirely depends upon the context. From a mathematical perspective this makes sense too. There are plenty of infinite series that have both convergent or divergent sums and they make sense in context; and then there are some that have sums that don’t make any intuitive sense.
There’s the obvious famous example of Hercules and the Hare, Zeno’s paradox, etc. where our natural sense is that, obviously, the faster moving object takes over the slower moving one, but when we break it into an infinite series it seems impossible. But that only seems impossible from a naive perspective of what an infinite series is. When we actually sum it all up, the mathematical result of the infinite series matches our intuitive understanding and it’s only unnecessarily complicating it that made it seem like a paradox to begin with.
And also inside of math, there’s plenty of other interesting things with infinities that defy our intuition. For instance, a famous one is that the sum of all positive integers isn’t divergent as one might intuitively assume it would be, but is -1/12 (at least for most common sets of axioms). Even more interestingly, as I understand, this is important for doing certain calculations related to quantum mechanics, but that’s definitely outside my realm of expertise, so I won’t even try to explain that as there’s tons of website and youtube videos that do a better job than I ever could.
Ultimately, though, for a given mathematical situation involving infinities, there’s many different ways to interpret and it really depends upon what axioms one accepts. In some cases, for the same calculation, it might make more sense to say undefined and in another to use versions of infinities or infinitesimals.
And since you mention woo, I’ll touch on that too, since I think it’s actually pretty relevant. One thing that’s interesting about infinities is precisely that there’s a distinct difference between our intuition and the underlying mathematics, and this is true of the physical universe too. From our perspective time, space, and matter appear to be continuous and infinitely divisible. I think this is why we have these paradoxes showing up. Ultimately though, the universe is quantized; eventually we get to a point where we can’t have a less of a thing or a shorter distance or less time. In this sense, something like the idea of an infinite sum of smaller and smaller tasks breaks down because it eventually gets small enough that it stops having any sort of meaning.
I’m in the “neither” camp. I fear the OP’s question is only marginally more useful than “Is infinity red or blue?”
It’s a mathematical concept, or several mathematical concepts actually.
It’s also a scientific concept and a philosophical concept.
Some of these may overlap, or not. It depends.
But none of them are journeys, and pretty much by definition they are not destinations since “you can always go further” is what infinite means.
Hm? The sum of a whole bunch of positive numbers somehow turns…negative? The sum of a whole bunch of integers somehow turns…fractional?
Pardon my skepticism, but I’d really like to see this drawn out in standard notation, 'cause my instincts are shrieking “Bogus!”
The -1/12 one is a bit tricky and requires several stages, but there are some simpler ones.
Start with this sequence:
f(x) = 1 + x + x^2 + x^3 + …
Some mild refactoring:
f(x) = 1 + x*(1 + x + x^2 + x^3 + …)
And we see that the bit on the right is our original sequence, so:
f(x) = 1 + x*f(x)
Rearrange and solve to get:
f(x) = 1 / (1 - x)
Ok, so what about this sequence:
1 - 1 + 1 - 1 + 1 - …
Well, that’s just the sequence above with x = -1. Plug into the formula and you get that the sequence equals 1/2. Weird, eh? A fraction where there was no fraction before.
Slightly more complicated: take the derivative of both sides. We then have:
1 + 2x + 3x^2 + 4x^3 + … = 1/(1-x)^2
Plug in x=-1 again, and we get:
1 - 2 + 3 - 4 + … = 1/4
Also pretty weird. It looks like we always subtract more than we add, and still we get a positive fraction.
One can go farther but I’ll let a resident mathie do that.
You’re welcome to read through this thread I started last year as well as the one before. Doesn’t make sense to me either but then again, I hit my mathematical wall somewhere not long after differential equations.
I won’t highjack this any further, then… But… Color me unconvinced.
ETA: I deleted the part where I lied and tried to highjack it further.