Infinite Misconception
The concept of infinity is often misinterpreted. Because we humans live in a finite world, anything that is conceptually too large to think about gets labelled as infinity.
For example let’s say I had a regular 8.5X11 piece of paper in a 10 trillion meter squared room. How many different positions could I place that paper in. The answer is not infinity because, while the number would be indescribably large, the fact that I am in a finite room, means that one day I will havev exhausted all options.
One would almost expect ‘laypeople’ to make this mistake, but when physics professors do it, you know there’s a problem. Today, my physics professor was talking to a student before class and I was able to overhear. I’m not sure what exactly he was talking about, but he mentioned something about a “nearly infinitely large sheet of paper.”
Something that is ‘near’ or ‘almost’ infinity means that it is still not infinity and that it is finite. It may be really big (or really small) but it is not infinity. I hope my professor was just using this term as a matter of speech, and didn’t imply the physically impossible by him language.
Uh oh.
Of course, this is physically impossible. Something cannot be of ‘near’ infinite size, because that implies that infinity is a definable size. If it has a definable size than it is finite. Something that is ‘approaching infinity’ is ok, because it implies that there is no upper bound limit. But something that is ‘near’ infinity implies that there is a definible end to its size, which precludes ability to be classified as ‘infinity.’
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February 20th, 2008 at 4:47 pm
I disagree. Although the room has a finite size, I can place the paper in an infinite number of different positions.
Consider the space R2 in the region -1<x<1 -1<y<1
This is clearly finite, but there are an infinite number of points in the region.
Physicists always use loose language like this. When he says the size of a piece of paper is ‘nearly infinite’, it only means that he will use the approximation that it has infinite size.
February 20th, 2008 at 6:08 pm
There are an infinite number of points, but the paper is clearly not a point.
How can you place something of finite matter in an infinte number of ways?
I understand that its an approximation, but one that fails, in my mind. Because when something is really large, it is finite and cannot be considered infinite, right?
February 21st, 2008 at 12:59 am
Let the center of the paper be a point (since it is). You can place it anywhere you’d like.
Notice the different between this case and “In how many positions can I place my mouse pointer?”
There are a hell of a lot (the number of pixels my monitor has). But it is a finite number. This is because there is a finite step size (one pixel).
In the world, or in your room, objects can move an arbitrarily small distance.
So even in a 1-meter by 1-meter room, I would say that the paper can have an infinite number of positions. The finite size of the paper is no issue. Consider this 1-dimensional analogy:
How many intervals of length 5 are there in (0,10) ?
Well, you can have (1,6), (3.2365,8.2365) etc… An infinite number.
As to whether or not an approximation “fails,” that depends. Say you want to calculate the magnetic field around a power line that’s 100 km long? If you want to know the field 30 meters away from the line, it’s probably safe to say the length is infinite, unless you care about the 15th decimal place. If you’re 30km away, probably not.
February 21st, 2008 at 2:20 am
not arbitrarily small though… space is quantized at subatomic levels, is it not?
February 21st, 2008 at 7:39 am
That’s yet to be confirmed, discrete space would be a consequence of a quantum gravity theory.
February 21st, 2008 at 3:19 pm
ahh… I stand correcting pending further investigation then.