A Critique of “Imagining the Tenth Dimension”
Recently I saw a video called Imagining the Tenth Dimension (there is also a part 2). I suggest that you watch the video before reading this post. If you can get past the annoying sound effects, it’s actually pretty entertaining.
The beginning of the video is perfectly acceptable; it discusses some widely-known ideas popularized by Edwin Abbott’s Flatland.
The first problem is that this video assumes that time is the fourth dimension. Well… not really. Space and time are not actually dimensions, but rather are constructs that can be represented as dimensions. A dimension is simply a range through which something can change. In a square, you can either move up and down or you can move left and right; hence, two dimensions. Time is one-dimensional, as you can only move forward or backward (in practice you cannot move backward, but that’s irrelevant in this case). Dimensions can also be represented in other ways. Images of the Mandelbrot Set, for instance, often represent a third dimension using color rather than depth, because color is more convenient in that case. But it could just as easily be represented using depth, or time, or even something else.
A dimension is a mathematical construct. Space is not three dimensions. Space has three dimensions; that is, it’s three-dimensional. To say that time is the fourth dimension can be useful sometimes, but it is not inherently true. The only reason time is the fourth dimension is because we say it is. The fourth dimension could be saltiness, for instance. That may seem counter-intuitive, but think of it this way. A dimension is something that you can change though. Your geographical location can change, and we call this a dimension (actually, three dimensions). But your saltiness can also change. You can become more salty or less salty. If you place saltiness values on a line, you can even represent saltiness using a spatial dimension. In fact, we often represent other dimensions in two spatial dimensions by using graphs.
There is nothing wrong with representing the fourth dimension with time; but there is something wrong with assuming that time is the only way to represent the fourth dimension, which is exactly what this video does.
The next problem is a good deal more serious. After five minutes, the narrator states,
The long undulating snake that is us will feel like it is moving in a straight line in the fourth dimension, but there will actually be in the fifth dimension a multitude of paths that we could branch to at any given moment.
Wait, hold on. How did the fifth dimension get in there?
To be able to have a multitude of paths through time, we do not need five dimensions; we still only need four. Think about Flatland. A Flatlander can get a sense of us when we pass through the second dimension. But it can get a different sense of us if we pass through while walking sideways rather than forward. It can get a completely different sense if our friend passes through the Flatlander’s space. The Flatlander can see a multitude of paths branching out, but only needs three dimensions to do so. Similarly, we would not need five dimensions to see all the branching paths of time — only four.
At five minutes and thirty seconds, there is a reference to quantum physics which is used as some sort of completely unnecessary metaphor which only seems to serve to make the video seem more strange and interesting.
At 6:40, the narrator proposes yet another unnecessary dimension. For this one, remember the analogy of the Möbius strip. Your own timeline is like a two-dimensional strip of paper, except that it is in four dimensions. To visit your own past, you can wrap the paper around itself through the third dimension, forming a Möbius strip. To visit an alternate timeline, you do not need another dimension; you only need to attach your current strip of paper to a different strip, one that contains the timeline of preference. This means that we do not need six dimensions, nor do we even need five; we still are good enough at four.
If you are following along on YouTube right now, please go to the next video.
As soon as the second video starts, it attempts to compress three dimensions into a single point. You’re not allowed to do that. The fourth dimension does not join the Big Bang to one of the possible endings of our universe; rather, that is what our particular cross-section of the fourth dimension does. The fourth dimension is capable of containing all possible timelines; but we are not able to simultaneously perceive them all.
Now, as we enter the seventh dimension, we are about to imagine a line which treats the entire sixth dimension as if it were a single point. To do that, we have to imagine all the possible timelines which could have started from our Big Bang joined to all the possible endings for our universe (a concept which we often refer to as infinity) and treat them all as a single point. So for us, a point in the seventh dimension would be infinity.
All right, now the author of this video clearly has no idea what infinity is. Infinity “is an unbounded quantity that is greater than every real number.” The number of possible points along one dimension is not even infinity, but is aleph one. The number of points in two dimensions is also aleph one — they have the same cardinality. In fact, the number of points in any number of dimensions (greater than zero) will always be aleph one.
What the author is calling “infinity” really refers to everything that can possibly be perceived by humans. And, if I may point out, we’re still not in seven dimensions. We’re only in four.
But if you think about it, the first sentence of my last paragraph is actually not true. We are capable (in theory) of perceiving infinitely many degrees of saltiness; these cannot all be represented in four dimensions. It would require adding a fifth. The same logic applies to the other four tastes, as well as to the three colors of light, every volume and pitch of sound, and every other thing that we are capable of perceiving. By the time you’ve added these all up, there are more than a dozen major dimensions, as well as at least a hundred less noticeable ones.
The video goes on to propose that the set of possibilities stemming from our Big Bang is only one infinity. There are other infinities resulting from other initial conditions. The problem with this, though is that having these two infinities does not necessitate another dimension because adding two infinities results in the same infinity.
At this point, you should watch up to 2:00.
If we’re really talking about adding other initial conditions, though, then we’d end up with a lot more than eight dimensions. We would need one axis to cover all the universes with different speeds of light; an axis for all the universes with different gravitational constants; all the universes with different fundamental forces, different types of particles, the list goes on and on. So either we are in four dimensions, or way too many to count, depending on how you look at it. But we are certainly not in eight.
The tenth dimension, which is introduced at around 3:30, is just absolutely ridiculous. The author keeps extending dimensions further and further when all we really need is four. He claims that there is “no place left to go” after 10 dimensions; hmm, according to this author’s perverse logic, wasn’t that true after seven?
The reference to string theory near the end is essentially the same as the reference to quantum mechanics: only in there to sound cool, and without any relation to the actual content of the video.
Before I wrap up, allow me to get in one final point. Mathematics has no problem with defining however many dimensions it wants. These dimensions are not spatial; space is merely a way to represent these dimensions, albeit only three of them. There is absolutely no reason to be limited to ten dimensions (or rather, four, as I have demonstrated). But remember, no matter how many dimensions you are using, you can’t cover any more ground. You’re still looking at aleph one points.
Maybe you watched that video and thought it was amazingly cool, and I have just crushed your dreams. But there are plenty of mind-blowingly cool things out there that don’t rely on lies and pseudoscience. If your mind is blown by a lie, is it really worth it?