Here’s a few references regarding waves having trouble propagating in even dimensions. It’s a big enough concept that it’s been given a name: “Huygen’s principle”. I don’t claim to understand the WHY’s of this, but there sure is a lot of literature out there on the subject:
www.math.niu.edu/~rusin/known-math/99/huygens
Also I’m a bit puzzled about Dexter’s belief that there are any numbers of dimensions. If so, he should be able to draw a diagram showing lines going off in any number of directions, all at 90 degrees to each other. I can only do three.
Maybe he means in theoretical space, but I thought we were talking about REAL space.
[Physicist Nitpick]
The second of these references has nothing to do with Huygen’s Principle or solutions to the wave equation. Nor has the abstract “flow” it’s talking about anything to do with a fluid.
[/Physicist Nitpick]
Not exactly. It’s a fine point on what’s meant by “propagate”. First of all, both of your examples are really one-dimensional in a technical sense.
A “wave” is a solution of a certain class of differential equations on manifolds. If you consider an n-manifold on which the functions live, the equation is (essentially) the Laplacian of the manifold evaluated on the function set equal to the second time derivative of the function. “Propagation” comes down to a question related to the Cauchy problem of the manifold.
Almost no mathematicians today are “Platonists”. Platonism as a philosophy of mathematics has been outdated for decades, as has formalism. The terms are part of a debate that has been dead since shortly into the 20th century. The most prevalent view these days is closer to structuralism, which CDH doesn’t seem to be contradicting at all.
Now, to weigh in on the debate at hand: physicists have for a long time (since before GR) thought in terms of a “space-time”, which the GR model gave the structure of a manifold.
A manifold is a topological space that locally “looks like” n-dimensional coordinate space. Locally, the surface of Earth “looks like” a plane, so the surface is a 2-manifold. The question “how many dimensions are there?” becomes “for what n is space-time an n-manifold?”
The problem these days is that the answers have an even weirder meaning than this. When a string theorist says that the equations have ten dimensions, these are not all the same kind of dimension. Formally, they’re related to the familiar ones of space and time, but they behave very differently. Which “dimensions” should be counted? Oddly, by the time people understand the models well enough to work with them, they don’t seem to care about that question very much.
No, what I meant was, I was under the impression that to make the Feynman calculations come out to the right answer, you had to sum up over paths that included backwards time.
Sigh, I posted a reply and the moat monster ate it.
You sir, are a mathematician. And I mean that in the nicest possible way.
Your reply demonstrates the philosophical difference between physicists and mathematicians. To a physicist, a sound wave is not an approximate physical realization of an equation of a particular class. The equation is an approximation to the physical wave, which belongs to a particular class of “objects”. In my experience, the difference between mathematicians and physicists is less the mathematics that they work with, and more the goal. Physicists tend to view themselves as using mathematics to model material reality, and hence define a wave as a physical thing. The material seems to be less of a concern to the mathematician, hence your definition of a wave as a mathematical thing.
Waves do nicely point out the limits of labels such as “materialist”. After all, waves are a collective motion that do not require any net motion of a particle in the direction of propagation, so in a sense, waves are not a “material” thing. That didn’t stop a 19th century physicist, who thought of himself as a materialist, from viewing the world as made of particles and waves.
Obviously, one can make too much of any particular terms. I was trying to stick with terms that most people understand. I could be wrong, but I’m not sure how many people understand “structuralism”. I’m pretty sure I don’t.
You miss my point. I’m using the mathematical mode in order to explain in what sense it is true that “waves only propagate in an odd number of dimensions”.
Many people understood “ether” at the beginning of the 20th century. Your statement directly implied that most mathematicians are Platonists, which is blatantly false, and your tone of praise implied a dismissiveness toward mathematicians for their “Platonism”. In short: you impeach mathematicians for misdemeanors they haven’t committed in decades.
Working mathematicians do not believe that there exists “the number 2” floating out there in some real sense anymore. For most of them, what they believe “exists” (and in a different sense than Plato) is the structure of N, and “the number 2” can only be defined in relation to this structure. That’s the best quick example of structuralism. There are plenty of good books out there to read up on before slinging accusations.
I’m not exactly sure which Feynman calculations you are referring to, but since what you say is certainly true of some of them, perhaps we need not belabor the point.
I didn’t mean anything derogatory about the term, and so did not think I was “slinging accusations”. Honest. I was just trying to make the point that most physicist I know/knew are way closer to materialists than most mathematicians I know/knew.
No you can’t, you can draw three lines at what looks like 90 degree angles to your eyes, because they’re used to identifying three dimensional shapes from a two-dimensional picture of the world. (Well, at least it’s two-dimensional if you close one eye. With two there’s probably some sort of triangulation going on, but I don’t know much about how depth perception works.)
It’s just as easy (almost) to draw a two dimensional representation of a four dimensional object, by connecting the corresponding vertices of two “three dimensional” objects, just as you go from two 2-d objects to a 3-d object or from two 1-d objects to a 2-d object. It’s just that your brain won’t interpret it as a 4-d object, because you’ve never seen a four dimensional object and your mind isn’t trained (or hard-wired or whatever) to recognize them from 2-d input.
Also, there seems to be some discrepancy over whether we’re arguing about whether there are three dimensions or arguing about whether there are three spatial dimensions. Depending on your definition of a spatial dimensions, the fact that there are three of them can be built right in to the definition. Whereas it’s absurd to say that any system can only possibly be described in three (not-necessarily-spatial) dimensions, or even to say that that’s the most convenient way to describe any system. Clearly, in relativity it’s more convenient to work in a four-dimensional space, with time as the fourth dimension. Of course it’s not a spatial dimension, because we’ve defined spatial dimensions to be the other three.
