I think this is more suitable to Great Debates than GQ.
Colibri
General Questions Moderator
Infinity is a word which we use in certain ways, and to understand the meaning of infinity is just to use the word in the same way. There’s nothing more that need be done for the word to acquire a sense.
I don’t know what this business about “grasping infinity” is supposed to be, or why it would be necessary to counter the titular supposition of infinity as nonsense. I know what the term “unicorn” refers to, even though not only have I never physically grasped a unicorn, but there aren’t even unicorns in the real world for me to attempt grasping. I know what the term “rook” refers to, even though the only meaning of this term is to indicate a particular role in a particular game. And I know what the term “millenium” refers to even though I’ve never even lived through a full century. Why all this hand-wringing about the term “infinity”?
Yes, reasoning that for any prime number we can always find another, larger prime and therefore saying “There are infinitely many primes” is an example of correctly understanding infinity. Reasoning that one boy and no girls is an infinite boy:girl ratio is an example of correctly understanding infinity (and an infinity visible in the physical world regardless of the size of the universe). Infinity is not hard to grasp, in this sense of “grasp”. Any other sense of grasping it is unnecessary.
“Quote:
Originally Posted by MrQwertyasd View Post
“Why do you need to visualize an infinite line at all?”
Because that’s what we are asked to do, without - in my experience - it ever being explained exactly what it means.
Who is asking you to do this?”
Can’t pin down a specific instance, but I studied enough maths to know that we were being asked to do this at least sometime in my schooling - plus the way it’s used in the general culture. Sorry to sound like a fudger but it’s the best I can offer.
“Yes. All math is fiction. That’s easy to forget”
It usually seems that way to me, but I haven’t explored this fully at all. I think there is a tendency to regard maths in almost a theological way - it all hangs together, on it’s own terms, but what is it really based on ?
Questions like the origins of these mathematical objects - back in the history of maths, I was hoping some erudite person would clue me in and give me The Straight Dope, but I guess it’s back to the library. But did the first person to craft the word for infinity, and apply it to something he couldn’t even imagine, write down his whys and wherefores?
I think the guy who mentioned the higher dimensions and such has it right, that it’s none-sensical quite literally.
Probably right to be in great debates.
Maybe look at time as one possible 4th dimension. You can’t see it, but your 3D senses register changes due to its effect in the dimensions you do interact with.
No, I’m saying that “precisely defined” and “simply declared” are opposites and that we can see the differences easily.
If you want a history of the mathematical concept of infinity, go to the library or to Amazon. There are many. They take up entire books, though, and they can’t be easily summarized in a paragraph. (Well, not by me at least. Maybe one of our erudite mathematicians can.) Nor did you ask for a history of infinity at any time. We can’t guess what your question really means deep down; we can only go by the words you actually write.
If what you want is a history of or debate on the philosophy of mathematics, there are many books on that, and about a zillion threads on the subject. The search engine here is wonky, but you can also use Google to search.
And use the Quote button to quote, and the Multi-quote button next to it to quote multiple posts. You can then trim them to size if needed.
We know that the English word “infinite” comes via Old French from the Latin “infinitas”, which very directly means “no-end-ness” (or, in more idiomatic English, “endlessness” or “the state of being unlimited”).
Surely you can imagine how the idea of “endless” or “unlimited” might have come up; just imagine something with no end (it’s easy; make up a game with no rule for how to end it. That game is now endless! One such game is the counting game; another such game is “This is the song that never ends…”. Or consider a loop; it has no endpoints [indeed, this line of thought is where the lemniscate symbol for infinity comes from])
Sorry; I thought this was established fact, but it seems it’s actually unclear why John Wallis chose the symbol ∞.
Regardless, a loop is, in a natural sense, an endless (aka, infinite) object. There are plenty of situations in nature where one is compelled to observe endlessness.
We don’t need to be able to fully imagine something to have an idea of it.
Yes, and that’s the beauty part!
People want to treat infinity as a number or a quantity. It is neither. In mathematics, infinity is just a way of saying something has no limit.
It’s all three. We started with the idea of limitlessness, and then abstracted and reified it in various ways. Depending on what you’re doing, you may wish to use the term “infinity” in various different ways. One way would be to just say something has no limit, in whatever relevant sense of “limit” (and one might indicate this limitlessness via making up a number which means just that); other ways would be to say other more specific or more technical things or just plain different things.
(The only reason I retort, despite agreeing that the notion of infinity originates in the notion of limitlessness, is because I get annoyed by saying things aren’t numbers.)
The problem with calling infinity a number is that it won’t act like the other numbers if you try to do anything with it. Add anything to infinity, or subtract, or multiply, or divide and you still have infinity. So call it a number if you like, but don’t try to treat it like any other number.
Sure. And 0 doesn’t act like other numbers in that you can’t cancel out multiplication by 0, and 1/2 doesn’t act like other numbers in that you can multiply 1/2 by an even number and end up with an odd number, and -1 doesn’t act like other numbers in that multiplication by -1 reverses the direction of inequalities, and so on.
But 0 does act like other numbers in a lot of important ways, and so does 1/2, and so does -1, and so does infinity, even if the important ways aren’t exactly the same in each case.
How many numbers are there?
Well, let’s see, what can you do with it. Any arithmetic operation you do with any other number gives infinity. So what do you get if you add infinity to infinity? Same thing. Okay I guess you can subtract infinity from infinity and get zero. Divide infinity by infinity and get 1.
But wait a minute. That would mean:
2infinity/infinity would be:
(2infinity)/infinity = infinity/infinity = 1
or conversely:
2*(infinity/infinity) = 2 * 1 = 2
You get different answers depending on the order of operation and that is definitely something you don’t want numbers to do. So yeah, when you start trying to treat it like a number you run into all kinds of problems.
And then what would infinity * infinity be?
I don’t agree with the proposition that infinity is nonsense; I merely say you can’t grasp it the way you can grasp “three”. I think of it this way: We can easily grasp “one”–for example a single light bulb in the center of a room; similarly we can handle 100, in the form of a 10x10 matrix of bulbs.* Even 1000 isn’t too difficult because that only requires that we stack nine more 10x10 matrices on top of the first. Now I do understand that the number 1000 itself, like any number, is only an abstraction, and even our ability to visualize the 10[sup]3[/sup] matrix is arguably one or two levels of abstraction removed from a single sequence of 10 objects, because it rests on the notion of three of those 10[sup]1[/sup] rows of objects. If we up the ante to, say, a matrix of 25[sup]3[/sup] or one of 17x39x8, visualization becomes more difficult if not impossible for everyone. Just so with an infinitude of like objects, or an object of infinite extent, like a plane in geometry.
*Yes, like in the Mathematica exhibit that stood in the California Museum of Science and Industry for the last 40 years or so of the last century.
But wait a minute!
2 * 0/0 would be:
(2 * 0)/0 = 0/0 = 1
or conversely
2*(0/0) = 2 * 1 = 2
You get different answers depending on the order of operation and that is definitely something you don’t want numbers to do. So yeah, when you start trying to treat zero like a number you run into all kinds of problems…
Except they’re not dealbreaker problems at all. What we actually do is restrict our previously universal rule that x/x = 1, instead say 0/0 is undefined for natural reasons, and move on.
And similarly we can reconcile ourselves to the arithmetic behavior of ∞.
There are various different arithmetic systems which incorporate ∞, but perhaps the simplest in this vein would be the projectively extended numbers, which have a single unsigned ∞ such that x + ∞ = ∞ for finite x, x - ∞ = x + ∞, x * ∞ = ∞ for nonzero x, x/∞ = x * 0, x/0 = x * ∞, and ∞ + ∞ and ∞ * 0 are undefined (and thus ∞ - ∞ and ∞/∞ are also undefined).
Yes, undefined, in exactly the same way that we ordinarily take division by zero to be undefined. [Actually, it would be better to say indeterminate, or multivalued to the point of encompassing all possibilities]. It’s no more of a problem than that.
This system comes up all the time in mathematics, whether one acknowledges it or not; it describes the behavior of “limits” in the calculus sense (in particular, the undefined operations correspond to indeterminate forms for limits).
In this system, 2 * ∞/∞ would be indeterminate, because ∞/∞ = ∞ * 0 is indeterminate. Yes, this means we don’t have x/x = 1 unambiguously for all x [but then, we already didn’t have that; x = 0 served just as much as a counterexample as x = ∞ does]
∞, in this system.
I’m not thrilled with the particular post I wrote which you replied to, actually, but I will say this: I don’t think “grasp X” should mean “have X many distinct points of light in an image in one’s mind”, or some such thing. For one thing, this leaves us wondering what it would be to grasp something other than a counting number, like, say, a p-adic number.
But for another thing, I’ve rarely grasped any counting number above maybe twenty-ish in this particular way. It may be my own weakness, but I actually can’t picture a 10 x 10 matrix of bulbs all at once, as such (I can grasp the idea of ten times ten, of course. But I can only grasp it as “ten times ten”, not as
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That image, I have no direct access to. I have no mental experience tantamount to summoning that array of dots to my mind’s eye, distinct from just any random array of dots. Fundamentally, I understand 100 by saying to myself “ten times ten”, and not through any visualization [though of course others will be different])
Now, in the same sense as I can grasp “ten times ten”, I can grasp everything else you said; I can imagine them as built up from some basic values by operations such as addition and multiplication, in just the way you wrote them out. If that means only grasping the name and not the actual quantity, well, then, I’ve never grasped almost any sizable quantity. But it’s not any different for me, then, with infinity than with, as I said, any other number old enough to drink.
I guess my concern is that it is arbitrary to privilege a particular kind of visualization over other sorts of understanding, and bless it alone with the status of “grasping”. But whatever; it’s a minor thing to get hung up on. I’ll let it go.