Something That Keeps Me Up At Night

Since a commenter recently noted that Steve's been writing almost all of the Triablogue posts of late, I figure I can post this one on the T-Blog even though I'm not quite sure there's any practical apologetic use for it. On the other hand, it's stuff that I find "wicked kewl" and therefore is interesting to me. But it'll have a bit of math in it, so if you don't like that, well I'm sure Steve will write something new shortly :-)

One of the questions that cosmologists have pondered is whether the universe is open or closed. An open universe would extend infinitely in all directions, whereas a closed universe would have a "boundary." However, even a closed universe could still be infinite. If space was curved in such a manner that, just like you could always travel East on Earth and return to the point you started from, in the universe you could always pick a direction and travel long enough and you'd return to your starting point. In other words, you could travel infinitely in one direction yet always return to your starting point (this would assume space was curved in the fourth, or higher, dimension that we cannot physically see).

I have to admit that I have a strange attraction to these kinds of loops. I don't know why, but they appear "pleasing" to me. And therefore I find it no surprise that I've discovered one such loop within numbers themselves. In other words, just as we could say that the universe is infinite yet closed because it loops back (assuming that theory is correct, I must add—by far this is not proven!), I say that numbers themselves are infinite and yet closed because they loop back on themselves too.

For a simple proof (simple in that it requires nothing more than algebra), consider the following.

1. The number 1 (one) is that number which has no factors other than 1.

This can be restated as:

1'. If a number n has only 1 as a factor, then n = 1.

This seems fairly straightforward to me, yet by the end of this you'll see why it might be tempting to deny the above.

Now we need to give one other tidbit of information. I'll explain it below (and note that because we are dealing with factors, by definition we're only considering positive values and whole numbers, so all the numbers below are positive integers):

2. Let c be a factor of w.

3. Since c is a factor of w, the next integer greater than w that c can likewise be a factor of is w + c.

Since many people don't like thinking with letters instead of numbers, let me give a concrete example. Let's say that c = 7 and w = 21. 7 is a factor of 21, so (2) above is satisfied. (3) states that if 7 is a factor of 21, the next number greater than 21 that 7 could be a factor of would be 21 + 7, or 28. And this is obvious because 22, 23, 24, 25, 26, and 27 cannot have 7 as a factor. Indeed, (3) is really nothing more than restating the definition of a factor.

Now let my proof begin in earnest:

4. Let x be the product of all positive integers. That is x = 1 x 2 x 3 x 4 x … x infinity.

5. Since x is the product of all positive integers, then x has all positive integers as factors.

6. Let a be a factor of x.

7. The next number greater than a than will be a factor of x is x + a.

8. Consider x + 1.

9. Let a = 1.

10. a is a factor of x + 1 (per (7)).

11. Therefore, 1 is a factor of x + 1.

12. Let a be greater than 1.

13. a cannot be a factor of x + 1 because the next greatest number than x that a could be a factor of is x + a (per (7)), and a > 1 (per (12)).

14. Therefore, 1 is the only factor of x + 1 (per (11)).

15. Therefore, x + 1 = 1 (per (1')).

16. If x + 1 = 1, then x = 0 (algebra).

17. But(!) x = 1 x 2 x 3 x 4 x … x infinity (per (4)).

18. Therefore, 1 x 2 x 3 x 4 x … x infinity = 0.

Now the way that I see it, there are one of two options that mathematicians can take here. Either we can simply rule that when x is 1 x 2 x 3 x 4 x … x infinity, then x + 1 is undefined (similar to the way that division by zero is undefined), or we can say that numbers themselves contain some sort of looping mechanism, wherein by the time you reach infinity (the infinity defined as the product of all positive integers), you "loop back" to zero.

You already know which way I'll go because I like loops. :-) But there is more evidence. I think we can see the "loop back" when looking at a tangent graph. Since I don't want to throw in Greek symbols here, assume that a is an angle: tan(a) = sin(a)/cos(a). So, whenever cos(a) = 0, tan(a) is undefined because of division by zero.

The tangent graph looks like this:



That's with the classic orientation, where the origin (where the arms of the graph cross) is located at (0,0). You can tell that since the right-hand portion of the graph is running up toward infinity and the left-hand portion is running down toward negative infinity why there would be a sudden "jump" in the graph at pi/2 (since cos(pi/2) = 0). If the x value is just slightly less than pi/2, you have positive infinity, but if it's just slightly more than pi/2 you have negative infinity.

Instead of assuming these things just go off to infinity, what happens if we assume that they "connect" at infinity and redraw the graph from that perspective? If I did it correctly (and since it's late at night right now, I am subject to correction), you'd get something that looks like this:



For this graph, we're looking at how it relates to infinity. Basically, what I did was assume that the graph "rolls over" at infinity, and made the horizontal axis the point where positive infinity and negative infinity intersect. In essence, you move the lower left to the upper right on the tangent graph and vice-versa. Naturally, the graph is horrifically distorted since it's representing two infinities on the vertical axis—the lines would actually appear to be virtually synonymous with the vertical axis for most of the trip, with the hook out at the very end; but I think this is sufficient to at least give a faint picture. (Note: technically, the origin on this graph would still be undefined, since the origin in this view is the point where the division by zero takes place.)

In any case, note that this graph would continue in sequence, just like the tangent graph does. That means that you could print out a row of these figures. The interesting thing about them is that you can then take the top of the graph and "fold" it down so that the 0s appear on the same line (the graph would now be on a donut-shaped paper rather than a 2D screen). At this point, the line graphed would look continuous (bearing in mind that at the origin of each cross point (multiples of pi/2) the graph would still be undefined).

This would imply that the graph, represented flat on a 2D surface, takes on the characteristics of a bent 3D object. Though only two dimensions are present in the tangent graph, there is an assumed third dimension where the graph "rolls over" from positive to negative infinity. In this curved 3D representation, the graph no longer has an infinite jump from positive to negative infinity, but rather that jump is a mere point, more akin to switching from positive to negative numbers at 0.

In short, it would be a curved space of infinite length, curved in a higher dimension.

This might actually affect physics. If it is true that math on the number line itself assumes a higher dimension of curved "space" then one could question whether that means reality really is curved, or whether it means that our math will always make it appear to be curved regardless of what it really is. In other words, is the fact that the math involved in physics seems to indicate a curved universe the result of the way that the universe actually is, or is it because the only method by which we have of probing the universe on such levels is mathematically, and math itself is curved? To use an analogy, suppose you use a level and see that a board appears warped; is the board warped or is the level warped? If we define the level as being level, then the board is warped; but what if we begin to see evidence that the level itself shows a curve?