Preparatory to analyzing J. W. Dunne’s theory of multiple temporal dimensions, as laid out in his book *An Experiment with Time*, I am trying to clarify my understanding of the four “regular” dimensions of spacetime. This post represents my first stab.

I am very definitely in over my head on this particular topic and would greatly appreciate any helpful comments my readers can offer.

**What is a dimension?**

Dunne defines a dimension as “any *way* in which a thing can be measured that is *entirely different* from all other ways.” This definition requires some modification, since there is no way of measuring which is entirely different from *all* other ways. Consider, for example, the “ways” indicated by the different colored lines on the below diagram.

Each “way” consists of a pair of complementary directions. For example, the way indicated by the red line is that corresponding to the directions “up” and “down” on your monitor, while the blue line indicates the “left-and-right” way. We do not have convenient words for the ways indicated by the other three lines. In what follows, I will refer to these ways by the colors I have used to represent them above. Thus, *Red* and *Blue *refer to up-and-down and right-and-left, respectively; and *Green, Yellow,* and *Purple* refer to the other three ways.

If we imagine these lines extending arbitrarily far in each direction, then any motion which would bring something closer to one of the line’s endpoints and farther from the other, constitutes motion *in* that way.

For any two ways, A and B, A is considered to be “entirely different” from B if it is possible to move in way A without moving in way B at all. Therefore, Red is entirely different from Blue because it is possible to move up or down without moving right or left at all. However, none of the other ways is entirely different from Red, because any movement in the Green, Yellow, or Purple way entails moving “up” or “down.” Likewise, Green and Yellow are entirely different from each other, but neither is entirely different from any of the other ways.

In short, “entirely different” turns out to mean *perpendicular*. Two ways are entirely different if and only if they are at right angles to each other.

So, which of the lines in the diagram should be considered “dimensions”? According to the literal meaning of Dunne’s definition, none of them — because none of the lines is perpendicular to *all* the other lines in the diagram, let alone to all the other lines we might conceivably draw.

In fact, dimensionality is not a property of a way, but of a *set* of ways. Modifying Dunne’s definition, we might say that a set of ways constitutes a set of dimensions if and only if all of the ways in the set are mutually perpendicular. Thus, the set {Red, Blue} is a set of dimensions, and so is the set {Green, Yellow}. No other sets of dimensions are possible using only the lines in the diagram, but of course there is nothing to stop us from drawing other lines. If we drew a pink line perpendicular to the purple one, then {Pink, Purple} would also be a set of dimensions. Infinitely many such sets are possible. However, if we restrict ourselves to ways which correspond to lines on the surface of your monitor, no set of dimensions can ever contain more than two members. That is what is meant by saying that the monitor is a two-dimensional surface. Any motion on the monitor is motion in two dimensions — but if you ask *what* two dimensions those are, there is no one correct answer (though there are of course many *incorrect* answers, since no two non-perpendicular ways can constitute a set of dimensions). We tend to think of up-and-down and left-and-right (i.e., Red and Blue) as being the dimensions of the monitor — but this is simply a bias of human psychology; in geometric terms, there is nothing special about that particular set of perpendiculars.

**Our three-dimensional world**

Once we move off the surface of your monitor and into the world we inhabit, we are in a three-dimensional space, meaning that it is now possible to create a set of *three* mutually perpendicular ways.

As with the two-dimensional world of the monitor, there are infinitely many possible sets of dimensions, no one of which can be privileged over the others in strictly geometric terms. (Take the three mutually perpendicular ways defined by the edges of a cube. You can rotate the cube every which way, but the edges remain mutually perpendicular and therefore correspond to a set of dimensions.)

In practice, though, one of these infinitely many ways is singled out by us as “special” and is virtually always considered a dimension. This is the vertical dimension of up-and-down, the one defined (for each person) by a line which passes through that person’s body and the center of the earth. The other two dimensions are lumped together under the heading “horizontal,” and it is understood to be fairly arbitrary precisely *which* two horizontal ways are to be considered dimensions. We might think of the other two dimensions as north-and-south and east-and-west, or as forward-and-backward and left-and-right, or any number of other possibilities — but these ways lack the distinctive character of “up” and “down.”

The specialness of verticality has nothing to do with geometry, but is a consequence of the fact that we live in a gravity well. “Down” is the direction in which things fall, and “up” is its opposite. All of the things we interact with in daily life behave as if “up” and “down” were quite different from all other directions, and so we naturally think of them as different.

**Time as a fourth dimension**

Whatever set of three spatial dimensions we choose, *time* is perpendicular to them all — because something can extend from past-to-future without affecting its extension in any of the spatial dimensions. Time would thus appear to be a fourth dimension of our world, no *geometrically* different from the other three — though, like verticality, it is considered “special” or “different” for non-geometric reasons.

As H. G. Wells’s Time Traveller explains it,

any real body must have extension in

fourdirections: it must have Length, Breadth, Thickness, and–Duration. But through a natural infirmity of the flesh, which I will explain to you in a moment, we incline to overlook this fact. There are really four dimensions, three which we call the three planes of Space, and a fourth, Time. There is, however, a tendency to draw an unreal distinction between the former three dimensions and the latter, because it happens that our consciousness moves intermittently in one direction along the latter from the beginning to the end of our lives….There is no difference between Time and any of the three dimensions of Space except that our consciousness moves along it.

Is that really true, though? Time seems to be *really* different in a way that the vertical dimension is not.

Consider our cube. We can rotate the cube so that the edge which once extended north-to-south now extends east-to-west — or up-to-down, or northeast-to-southwest, or (to coin a couple of words) northup-to-southdown. But can we rotate it so that it extends *past-to-future*? We can turn a cube such that what was once called its “height” is now its “length” — but can we turn it such that its height becomes its *duration*? Of course not.

Another problem is that, while all other dimensions can be measured using the same units — inches or centimeters or whatever — time cannot. Can we conceive of a tesseract (four-dimensional analogue of a cube) for which time is the fourth dimension — that is, a shape whose length, width, height, *and duration* are all equal? What could that even mean? How many seconds or minutes or days should be considered equal to an inch? Duration appears to be incommensurable with all other dimensions, which casts serious doubt on the proposition that it is just another dimension.

Sir Arthur Eddington, in *Space, Time, and Gravitation* (quoted by Dunne), makes the implicit claim that time and space *are* commensurable:

An individual is a four-dimensional object of greatly elongated form; in ordinary language we say that he has considerable extension in time and insignificant extension in space. Practically he is represented by a line . . .

Eddington’s statement makes sense only if we take it for granted that threescore years and ten is *much longer* than six feet — implying that there is some common standard of measurement which is applicable to both space and time. But there is no such standard. We consider 70 years a fairly “long” duration as durations go, and six feet is a “short” distance, and so in a metaphorical way we can speak of them as if they were commensurable — just as we might say someone is “as honest as the day is long” or “even more intelligent than she is beautiful.” Eddington’s characterization of the four-dimensional individual as “greatly elongated,” while it presents itself as an objective scientific description, is in fact just an imprecise metaphor, a figure of speech. However, I will concede that it is a particularly *natural* metaphor, one that *seems* like it could be literally true in a way that “Milton was as puritanical as a pig is fat” does not.

But the theory of time put forward by Dunne seems to *require* that time and space be commensurable — that it be possible to compare a temporal duration and a spatial distance and say which is longer — for, according to Dunne, this difference in length is precisely what makes time time, what distinguishes it from space. In making “an artificial distinction between time and space,” any given observer

would regard Time as

stretching in the direction in which his body line extended. It would follow that his body line would seem to him to be runningstraightup this Time dimension of his, and not bending this way and that in Space —i.e., sitting in a railway train, he would seem to himself (until he began to speculate about it) to be at rest.

Dunne’s language is again somewhat imprecise here. We cannot speak of “the” direction in which a person’s “body line” extends, because it is not a true geometric line but rather a solid extended in four dimensions. However, looking at this four-dimensional object, we see that its extension in one particular direction is *very much greater* than its extension in any other direction, and we therefore identify that direction as being the time dimension as perceived by that person. This is meaningless unless all four dimensions can be measured in the same units.

Einstein’s idea of the relativity of simultaneity — which states that precisely which direction is “time” may be slightly different for different observers — also implies that time and space must be commensurable. Time and space are fundamentally the same thing and must therefore be measurable in like terms. I’m sure there’s a Nobel Prize waiting for whomever can discover precisely how many days there are to a cubit, because as far as I know no one has heretofore done so. (If I’m simply displaying my ignorance here, more physics-literate readers are encouraged to set me straight in the comments. I freely admit that my understanding of relativity is roughly three times as shallow as my eyes are blue.)

Clearly, then, time is very different from the other dimensions, and not just by virtue of the fact that “our consciousness moves [sic] along it.” The other three dimensions are arbitrary; provided only that they are mutually perpendicular (and perpendicular to the non-arbitrary dimension of time), any three ways we care to choose will serve equally well. But the fourth dimension *must* be time. North-and-south is an arbitrary dimension; northeast-and-southwest would serve just as well. Up-and-down is somewhat less arbitrary, but still we could in theory choose to use upeast-and-downwest or some similar “diagonal” way instead. But “spatiotemporally diagonal” dimensions are inconceivable. There can be no such dimension as northpast-and-southfuture, because things can’t be rotated that way and we have no units for measuring extension in those directions.

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If we want to preserve the idea of time as just another dimension, not fundamentally different from the other three, it seems we must commit to the assumption that time and space are commensurable *in theory* — but that for some extrageometric reason or reasons (e.g., some inconvenient facts of physics or psychology) we are unable *in practice* to measure them in the same units. Can we make sense of such a supposition?

Imagine a world in which, due to some obscure law of physics, the verticality of everything is fixed. In this world, things can be rotated only on a horizontal plane. You can turn a cube such that its north face becomes its west face, for example, but not such that its top and bottom cease to be its top and bottom. In such a world, it would be impossible to measure a cube’s height with the same ruler used to measure its horizontal dimensions. There would be vertical rulers and horizontal rulers, and neither could be so rotated as to become the other. It would follow that vertical and horizontal distances would have to be measured in different units. We could never discover how to convert horizontal units into vertical ones, because we could never lay the two rulers side by side and see how they lined up.

Still, it seems intuitively that even in such a world we would be able to “eyeball” things and get a rough sense of the ratio of height to width, without any need to use actual rulers. But maybe it would be harder than I imagine; maybe that “eyeballing” ability depends on the ability to mentally rotate things, which in turn depends on past experience *actually* rotating things. Maybe the inhabitants of our hypothetical world would have very little ability to compare vertical and horizontal lengths even in approximate terms.

So perhaps some analogous state of affairs holds in the real world. Perhaps some contingent fact of physics makes it impossible to rotate things spatiotemporally, and perhaps that in turn makes it impossible for us to compare temporal and spatial distances with any precision. And perhaps Eddington’s intuitive sense that a year is much longer than an inch represents a rudimentary “eyeball” measurement which is valid as far as it goes despite its lack of precision.

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Well, this post has gotten quite long enough, so I think I’ll take a breather. Next I’ll be going through Dunne’s argument that there is *more than one* dimension of time.

The speed of light is our conversion factor from time to distance. One second is equivalent to three hundred thousand kilometers, a distance that can be called a light-second. The Sun is some eight light-minutes away, and the next nearest star is more than four light-years away.

Certainly the speed of light (or any other speed you care to choose) is a “conversion factor from time to distance” in the sense that you can use it to derive a distance unit from a time unit — but is it a conversion factor in the stronger sense? Would you say that a light-year

just isa year, and that a foot is approximately equal to a nanosecond in precisely the same way that it is equal to 12 inches?(If that’s true, then we can finally understand what Han Solo was talking about in the first

Star Warsmovie. Apparently theMillennium Falconmade the Kessel Run in less than 39 years!)Dimension is well-defined only within some particular space; but most often it means the minimal number of scalars required to represent any point in the space such that as the distance between a pair of points approaches zero, so does the difference between each matched pair of scalars. (x,y,z,t) is a basis of spacetime that exhibits this property under most reasonable distance metrics, no matter what conversion is selected between meters and seconds, and is the smallest number of scalars that does, so spacetime has four dimensions.

Dimensionality does not suggest linearity, the existence of affine transformations, the existence of change-of-basis formulas, etc. It does suggest that any lower-dimensional representation is discontinuous (see, e.g., space-filling curves).

The speed of light is significant only in the Minkowski Space, the geometry suggested by the metric used in special relativity, where the distance between two points in space-time is ±((ct)^2 – (x^2 + y^2 + z^2)) (the sign depending on if you ask mathematicians or physicists). It does not allow rotations in the Euclidean sense, so saying “39 years = 1 parsec”, which suggests a 90° rotation, is mostly nonsensical.

In less strict conversation, “dimension” is used to mean all kinds of things. Dunne’s (from your summary) doesn’t seem to encode any information at all: a single time dimension contains all of the information contained in his infinite set. He is right that it “does not contradict […] modern physics” since it doesn’t say anything with any kind of mathematical meaning. My best guess at his meaning is “we describe time in terms of motion, which requires both space and time; so if time is space then there must be another time too.” Two simple solutions to that are either (a) don’t describe time in terms of motion or (b) acknowledge that the description need not *be* anything, it’s just a description.

Thanks, Luther. That’s very clear.

Dunne’s additional temporal dimensions do not and are not supposed to encode any information about the physical world. They relate to consciousness. More on this in future posts.