Home > Math, Science > Four Dimensions: Objection, Refutation, Objection, Refutation, Confusion

Four Dimensions: Objection, Refutation, Objection, Refutation, Confusion

This is a complication of a refutation of an objection to a refutation of an objection to my proof that all combinations of space and time can be represented in four dimensions. I apologize for not posting much recently, but I think that this behemoth of a post more than makes up for it.

A note on etymology: When I refer to “the universe”, I do not mean the actual physical universe that we live in. Rather, I refer to a more theoretical universe that has infinitely many points in space (it is argued by some that our universe is discrete, rather than continuous) and expands infinitely in all directions. Our physical universe is actually a subset of this much broader infinite universe. When we are talking about dimensions in a theoretical way, it makes more sense to refer to “the universe” as being infinitely large, because dimensions extend infinitely in either direction.

The objector concedes that any combination of space-time points is possible in four dimensional space. However, not every combination of space-times points is possible because the points are not all touching each other. In order to be able to move through every possible space-time, there must be a line connecting every combination of points. In mere four-dimensional space, there is no such line.

If we take space-time and drag it through a fourth spacial dimension, then each point of space will be capable of occupying infinitely many things. But this still limits each point of space to being able to move to the previous value or to the next value; it cannot move to just any value. So we need to drag space-time through a fifth spacial dimension to be able to represent every possible timeline in the universe — that is, a total of six dimensions.

If you are satisfied with this objection, please continue reading as I carry it to its absurd conclusion.

Unfortunately, six dimensions are not sufficient. If we drag 4d space-time through two more dimensions, it is still dragged as a whole unit. The different zero-dimensional points cannot move independently of each other. This may not be obvious, so allow me to use an analogy. Imagine that you have two red points. You can drag them through the first spatial dimension, and hit every possible shade of red. But what if you want the first point to darken twice as fast as the second point? Now you need a new direction. What if you want the first to darken three times faster? Now you need still another direction. It’s not hard to see that you need infinitely many directions. All these directions cannot be contained in one spatial dimension, but they can be contained in two.

But what if you want to add a third point? That one has to be able to change, too. So now you need three spatial dimensions.

What about a fourth point? You guessed it: four spatial dimensions.

Now let’s return to our six-dimensional space. For two points to be able to change brightness independently of each other, we need seven dimensions (one point can change along the sixth axis, while the other point changes along the seventh axis). For three points to be able to change independently of each other, we need eight dimensions. For four points, nine dimensions. So since there are more than infinite points, we need more than infinite dimensions. (If you’re confused, see this site for a fun explanation of how something can be bigger than infinity.)

But this itself is incorrect: it assumes that, to move through our dimensions, we must move in a straight line. Why can we not jump from point to point? We don’t jump from point to point in physical space, but there is no reason to assume that we cannot do it in other dimensions. Physically, that sort of jumping is impossible; but if we think of space not as something physical but rather as a model, then we can most certainly jump from point to point. In that case, to represent all possible brightnesses of the color red, we only need one additional dimension. Since we were just using color as an analogy, we might as well remove that dimension altogether — for now. (I’ll talk more about it later.) Additionally, the fifth and six dimensions were only added because we were assuming that we had to more in a line. Sure, a line is a convenient construct for some sorts of practical applications, but why limit ourselves to it? If we can jump around, then let’s remember that every possible point in space and time is represented within four dimensions. In that case, we can pick and choose which points we want, creating any possible timeline for any possible universe.

So now we’re back down to four dimensions. But I’ll bet I can add in some more.

You may remember that I said that color has a dimension, that saltiness has a dimension. Well, the universe contains color and saltiness. It also contains many other things. Any possible thing that can be measured independently could be considered another dimension, which means that then universe hardly requires four dimensions — it requires infinitely many. I addressed this point in my original critique.

But this objection can also be refuted, and considerably more easily. Remember that the four dimensions contain every possible point in space and time, but only space and time. Color or saltiness might be contained within space, but it’s not necessarily space. So if we are talking strictly about space and time, then there are only four dimensions.

Why, then, do we live in a universe with saltiness and color? Doesn’t that mean that we have to have more dimensions than just four? What happened here is that our model deviated from reality.

In the actual universe, color, saltiness, sound — every perceptible dimension — is really just a product of physical space. Saltiness is really just chemical reactions on your taste buds, which boils down to reactions between atoms, and even further, to reactions between quarks. Color is just the movement of light through space. Sound is just the vibration of particles in certain ways. According to the Standard Model, a point in space can only be one of so many things: one of the fundamental particles, or just empty.

But remember my previous argument about dragging color through space? Don’t we need infinitely many dimensions now?

If the universe is continuous, yes. But what if it is discrete — that is, the set of points in space is countable? In that case, it is easy to prove that we only need one more dimension.

I’m not sure how many fundamental particles (plus empty) the Standard Model predicts or how many particles there actually are, but if there are finitely many, then we can use logic similar to Cantor’s proof that rational numbers are countable to prove that we only need one additional dimension.

To make thing simpler, let’s match every particle (and the empty space) to a number. And, for the sake of convenience, let’s say that there are ten of them; the number doesn’t really matter. (If the number is one — that is, space is just space — then this proof doesn’t apply; we will still only be working with four dimensions.)

We will be moving through the fifth dimension by counting up through the numbers. If every possible combination of numbers is represented, then every possible universe can exist in five dimensions.

To start off with, let’s say that every point is set to 0. Now look at some point. Change it from a 0 to a 1. (We are now moving through an additional dimension, because this change cannot be done in space or in time.) Now change it from a 1 to a 2. Now 2 to 3. Keep counting up until you get to 9. Now you’ve covered every single combination of universes where every particle but that one is a zero.

Reset that particle to 0 and add 1 to the particle next to it. Now start counting up again with the original particle. The next time when you get to 9, add another 1 to the second particle, which now becomes 2. Do it again, and it becomes 3. Keep going until this second particle also gets to 9.

Now what do we do? Well, this time, reset the first two particles to zero and add 1 to a third particle. Keep adding numbers to the first particle again.

If we place these three particles in a row, we get a pattern that looks like this:



Gee, that sure looks familiar. We can continue to extrapolate this to every single point moving along a line, and we can represent every combination of particles using a number.

But wait: if we are moving along a line, we are missing all of the other particles that don’t fall on that line. How do we fix this problem? Well, it’s pretty simple. Instead of moving in a line, we move in a spiral. This way we hit every single particle in all of space and time. It will take a lot longer to reach the outer points, but it is impossible to name a point that won’t eventually be reached. By this process, every possible combination of particles — that is, every possible timeline of the universe — is assigned to a number.

Wait, did I say that we need five dimensions? Hardly. In this case, we can represent every possible timeline of every possible universe using a simple number line. And a number line isn’t called a line for nothing — it’s one-dimensional. That means that if our universe is discrete, then it can be represented using a one-dimensional line.

Notice that this isn’t always true. If the universe is continuous, then there is no way to enumerate all the possible points, which means that we can’t give a number to every single point in space and time. In this case, we need infinitely many dimensions to represent every possible combination of particles.

So let’s recap. If the universe is discrete, it can be represented with one dimension. If the universe is continuous but only represents space-time, then it can be represented with four dimensions. If the universe is continuous and also contains different particles, then it cannot be represented in any finite number of dimensions. If the universe is continuous and saltiness, color, etc are dimensions rather than physical constructs, then we need however many dimensions as there are measurable things — presumably infinite.

That was fun.

Categories: Math, Science
  1. Matt
    April 25, 2010 at 1:46 am

    No it wasn’t 😡

  2. Shea Kauffman
    October 25, 2012 at 9:30 pm

    I don’t particularly like the 10 dimensions video either, but I think your criticism is off base; I don’t think using the diagonal lemma to point out that it’s possible to encode all the dimensions in a single dimension is very meaningful. I /do/ agree with your conclusion but based on a different argument.

    What is a dimension:
    You define a dimension as “a range through which something can change”. That’s mostly correct, but more accurately a dimension is a a slot in a vector. For example, taste is five dimensional, specifically the five dimensions of taste are {sweet, bitter, sour, salty, and umami} (the neuroscience behind this actually is more complex as each slot int he vector has some bleed over into the other, but that’s neither here nor there). Now of course you can take the five dimensions of taste, and traverse them in a “spiral” (i.e. apply the Diagonal Lemma), but by doing so you are just equivocating.

    So when we talk about the dimensions of space we talk about there being three of them: {x,y,z} or {length, height, width}, or whatever. We all have a fine understanding of them. Filling in the three values represents a point in space, and we can make a line by creating a line vector, which is a vector of space vectors with a cardinality of 2 (i.e. line = {{a,b,c},{d,e,f}}).

    The author of 10 dimensions chose to increase the length of the vector to 10 points, and I agree that that confuses things. In this case he is equivocating, because he’s taking 3 separate vectors and merging them into one long vector and asserting pointlessly (no pun intended) that they’re all the same. Really, we tend to assert:
    time = {time}
    space = {x,y,z}
    or {time:{time}, space:{x,y,z}}

    Now lets look at the actual cardinality of the time vector:
    If time has no branches it is simply a single variable vector, where that single variable has a range from 0 to theta (theta being the point at which no change can be measured). This combined with space allows you to define a point along spacetime.
    However if there are branches the situation changes, suppose there are three points in time, and each point in time can have two branches, you end up with the possible permutations:

    Which as you can clearly see is most easily represented in 2 dimensions. Specifically time becomes a vector of {number of steps, path chosen}. This for the record is exactly how saves used to work in old Nintendo games, except the vector would tend to represent something like {level, path chosen}. Of course the more intuitive way to represent those choices is to create a number of dimensions equal to the number of steps, so if there is 3 time steps you have {t1,t2,t3}.

    One difference that’s immediately striking though, and exposes a problem with the 10 dimensions, it that the space vector is not orthogonal. Specifically we can’t draw a meaningful line between two points on the time plane because branching creates a Poset, instead of a totally ordered set. If in the example I created a line between {0,0} and {2,8}, I would not make a meaningful traversal, I would for no obvious reason have traveled through unrelated decision paths.

    So thus we /can/ have a vector with a cardinality of 2 for time. And with our space vector we can project every possible universe in our simple space-time. Thus we /can/ have 5-dimensions (though as you rightly point out we have a lot more since there’s the taste, and spin, etc…)

    Now for this tiny example we can assert a 5-dimensional vector, 4 orthogonal directions and an extra dimension that’s wonky. But lets assume that the number of choices is arbitrarily large and so is time. Now suddenly our schema is really problematic, because the permutations for the vector for time is steps^choices with no easy relationship between them. Our choice of coding it as a two dimensional vector has created a worse problem than simple having a vector with a cardinality equal to the number of steps. This is not a problem for space because the line relationship between x,y,z is useful; you can traverse any point to any other point seamlessly. In the case of time={steps, choices} each movement from one point to the next requires a complete search of the vector space. If instead we have a slot in the vector for each step you can simply alter the choice in the appropriate slot.

    This brings up the problem of arbitrarily long timelines, how many slots in the vector (which remember are dimensions) do we need. Well we need one for every point in time, and since that is potentially infinity, or at least perpetually growing, we no longer have a dimension, we have an array. And if we suppose as does the author of 10 dimensions that there are infinite choices in infinite time we have a transfinite (infinity^infinity) number of points in our “4th and 5th dimension”. This means that the author is by definition wrong, as there is no longer a discrete encoding of any point in branching space-time; none, can’t happen. The author has proposed a model that is by definition compact and arguing that it can be represented with a rational number; that he can fit pi into a pair of integers.

    So while I agree that there aren’t more than a single time dimension, it isn’t because we can encode the choices easily using diagnolization, it is because it would require an infinite number of dimensions to come close to properly encoding something he brushed aside as obvious.

  3. Shea Kauffman
    October 25, 2012 at 10:04 pm

    I wrote that off the cuff, there are a few mistakes. The Diagonal Lemma is not what is used to uniquely encode a countable pair, and it never provides savings to encode a permutation in two dimensions.

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