During this very interesting exchange in the comments section of this post about infinity (The question remains: how could the universe be infinite), for a second, my world entirely disintegrated.
I wrote, as a comment:
I just thought of something – like in a “Zeno paradox” kind of way. Shouldn’t the definition of a point be impossible? Because the minute you say a point, a dot, you can think, well, it could be smaller, and if it could be smaller, then it could be even smaller, and so it goes, infinitely so, to the smallest of the smallest right before it’s nothing. But that is so small that it’s impossible for us to ever “reach” because we, humans, are limited. So this means a point could never be a definite or stable thing. It just keeps going smaller and smaller and smaller – see what I mean?
I’ve just realized I’ve been lied to – ever since primary school when they told me a point is a dot. LOL
Maffblogger (intheshadowofsacredtopology.wordpress.com) replied:
…It’s useful to imagine a point as a dot, but it certainly isn’t one, as you’ve pointed out (pun intended) it possesses no properties such as length, area, etc., but its does have one thing: position! And that’s really all it is, a position. But you’ve made a good point, since we’ve come this far, what do you think about lines? They seem to have length, but no width. You could always imagine a line being narrower and narrower, until perhaps it disappears altogether. …
And I went through years of school thinking it was a dot!!!! They lied to me! LOL My world is disintegrating 😉
“But you’ve made a good point, since we’ve come this far, what do you think about lines? They seem to have length, but no width. You could always imagine a line being narrower and narrower, until perhaps it disappears altogether.”
Yes, it’s the same problem as above. I had never thought of any of this through all my years in school. A new stage begins. You must know of the Zeno paradoxes, right? Actually, when I thought of the “dot problem” above, I also immediately associated it with Zeno’s racetrack paradox. http://www.iep.utm.edu/zeno-par/#H3
And you know what I just thought? There are infinite ways of tracing a path around a sphere (I mean, on the surface of the sphere).
I was picturing your sphere and a path going around the sphere. Actually I pictured a red party balloon (it’s just prettier). So then I went around the balloon, like in a perfect circle, looping back to the beginning. And then I thought, well, we could wiggle around the perfect circular path, like zig-zag and stuff, so you are covering a much longer path to loop back to the point where you started. Now you could do this, i.e., trace paths, in infinite ways. And then I thought: maybe you can’t. But you should be able to.
I can’t decide. Because we’re back to the dot/position problem. If you think that a point is just a position, you can’t trace it, because it has no width or length. But we can trace a path. So I guess you could trace infinite paths. But then you think, but what is a path? It’s a line. And what is a line, it’s a bunch of points. And what is a point? It’s nothing – or is it one of those weird things in math that “tends to nothing”? And what is a position? It’s math, it’s abstract, it’s not real. Everything just disintegrates.
Anything that has to do with infinity is mind-boggling to me. I’ll have to think about this path/line problem a bit more.
So in re-reading the above, I don’t know if I’m on firmer ground, but the answer must involve the “tends to nothing yet is not nothing.”
How did I get there? By thinking about this question:
Does 0.999… = 1 ?
No, it does not. It can never equal 1 because that contradicts the “…” at the end there – that is, the “…” involves infinity, and infinity as we have seen, is something that is often beyond the scope of math. The infinity here means there will always be a 9 at the end, and therefore, that is not sufficient to equal 1. That is the point!
So… this provides us with the beginning of an answer to our dot problem (and how many lines could you trace on a sphere).
A dot exists, but we can never “catch hold of it” because it’s too small, it is truly infinitely small. However it will never be nothing, that is the beauty of it. It is obviously the inverse of the above 0.999… issue. A dot is what exists just before we get to nothing – and we, as humans, can never “catch” that. It is the magic moment when you go from nothing to something, the very beginning, like the conception of light and life, if you will, in mathematics.
Very well – so now onto “how many lines could you trace on a sphere” question. This just might be unsolvable like the question “what is infinity + 1” or “what is infinity + 1 minus infinity + 1”. You can see an answer to the last question that makes sense to me here:
There’s generally no answer. In the typical number systems, there is no such thing as infinity.
In the extended real or complex numbers, infinity minus infinity is undefined (or indeterminate, if you prefer that way of thinking about it).
For cardinal numbers, there’s no well-defined subtraction between infinite cardinals either.
For ordinal numbers, let me substitute ω for infinity (the smallest infinite number), and assuming left-associativity of your operations, we get:
ω + 1 = ω + 1
(ω + 1) – ω = 1
(Only left subtraction works for ordinals, that is to say given x ≤ y there is a unique z such that x + z = y, but in general you can’t find a unique z such that z + x = y.)
((ω + 1) – ω) + 1 = 1 + 1 = 2
If you paranthesize your operations differently you obtain zero in the same manner, although that has nothing to do with infinity but more to do with how the problem is ambiguously stated. You will have the same ambiguity if you substitute 4 for infinity.
I can’t opine on all the comments about the different types of numbers, but what makes sense to me is that our regular math system often breaks down when you meander into infinity, that is, the answer is that there is no answer. “Break down” may not be the best term – math is simply ill-equipped to deal with infinity, it’s limited in that respect.
So if lines are a bunch of dots, and dots are infinitely small, how many lines can you trace on a sphere? You need to define the problem further to play with the concepts. If the line in question can be of infinite length, then I would guess the answer is an infinite number of lines. But what if you say that the lines can only go up to the circumference of the sphere X 2? What if the length of each line must be equal to the circumference? Maybe better start with that. And a sphere is already too complicated an object.
Take a square. How many lines can you trace parallel to one of the sides inside the square, without redrawing one line on top of the other? Is it an infinite number or is it less than infinite?
And my brain feels like a pretzel!