It seems that there are three concepts of continuity that we should keep separate: the continuity of the real numbers; the continuity of a geometrical line; the continuity of a physical line and of physical space generally. Some commenters seem to be conflating these.
The set of reals forms a mathematical continuum. Since there is a one-to-one order-preserving mapping of the reals onto the points of the geometrical line, the latter is a continuum too. But that doesn't tell us what physical space is like. Since motion occurs in the physical world, one could take Zeno's paradoxes of motion as reductiones ad absurdum of the assumption that physical space is continuous.
But there is an argument from Hermann Weyl that seems to show that physical space cannot be discrete. Take a gander at the chess board below. It is an 8 X 8 array of squares or tiles. (Hence "Weyl's tiles' with 'Weyl' pronounced like 'vile.')
Consider the right triangle whose legs are the a1-a8 file and the a1-h1 rank, and whose hypotenuse is the a8-h1 diagonal. Now we know from the theorem of Pythagoras that if the legs are each 8 units in length, then the length of the hypotenuse = the square root of 8
2 + 8
2 = 11.313708 . . . .
But if we count the tiles (squares) along the a8-h1 diagonal, we see that there are exactly eight of them. Notice that, no matter how small we make the squares, the hypotenuse will be equal to the length of either leg.
Therefore, if a two-dimensional space were composed of discrete space atoms, indivisible bits of space, then the theorem of Pythagoras would not be true of it. Yet we know that said theorem holds true of the physical space of our empirical acquaintance. The inference to be drawn is that physical space, or at least planes in physical space, are not composed of space atoms.
One will be tempted to object that the diagonal distance across an atomic space-tile must satisfy the Pythagorean theorem. But if the tiles are truly atomic, then they have no parts! And if they have no parts, then there is no way the diagonal distance across one of these tiles can be different from the distance in any other direction. To think of space-tiles as satisfying the theorem of Pythagoras one would to imagine them against the backdrop of an encompassing continuous space. But this would be wrongheaded since the whole point of introducing space atoms is to replace continuous space with discrete space.
But doesn't quantum mechanics imply that space is quantized? And surely the little exercise I've just run through doesn't refute QM!? We'll have to think about this some more.
REFERENCES
W. C. Salmon, Space, Time and Motion, pp. 64-66.
Hermann Weyl, Philosophy of Mathematics and Natural Science, p. 43.
(Weyl's argument reminds of Zeno's stadium argument, is that right?) You say that we know that Pythagoras's theorem holds true of the physical space of our empirical acquaintance, but I wonder if we do... We know empirically that physical space is approximately Euclidean, but surely not that Pythagoras's theorem holds exactly (?)
Or have I misread?
Also, does the discreteness of a space atom imply that its dimensions in every direction are the same? Suppose that I take two discrete space atoms and place one upon another, such that they conform precisely, then rotate either one, but not both, of the two by 45 degrees. Would the result show that discreteness is different from measurement since if the diagonal distance were no different from any other distance, then the superimposed tiles would create the same pattern no matter how much one of them in rotated?
I probably haven't expressed this well, but perhaps you can see my question.
Jeffery Hodges
* * *
Hope all is well on your end.
My notation is correct, and is familiar to chess players, but it doesn't matter. Imagine a triangle whose legs are the left side of the board and the bottom side. Now connect the uppermost point of the left side to the right most point of the bottom side. That diagonal makes the hypotenuse of the triangle.
What is fascinating is that all three sides are 8 squares in length, in violation of the Pythagorean theorem. No matter how small you make the board, and thus the squares, the result is the same. Even if you shrink the squares down to atoms of space.
Atoms are partless. (I am of course using 'atom' in the etymologically correct way, not the way 'atom' is used in physics.) So space atoms have no spatial parts. That implies that they are dimensionless and shapeless. They are not to be thought of as tiny spheres or tiny cubes, or, in 2-space, as tiny squares. If a space atom in 2-space were a tiny square, then it would be composed of two triangles which would imply that it is not an atom.
One cannot imagine a space atom, but one can conceive it. Think of a volume of space, whether empty of matter or filled, suppose it to be composed of parts, and suppose there are ultimate parts (parts that do not themselves have parts). These ultimate parts are atoms of space.
I don't think your question makes sense. You are thinking of space atoms as little things in space, when space atoms are not things in space but the ultimate indivisible constituents of space.
Can the enigmatic Enigman help me explain this better?
If time is discrete, then it is composed of time atoms; but these are distinct from events that occur at those times. Similarly, space atoms are not occupants of spatial positions.
Perhaps I should have said that the theorem of Pythagoras approximates very closely to the macrospace of our empirical acquaintance.
There are a lot of very difficult issues here that I am perhaps not competent to discuss properly, or even formulate properly.
The space of GTR is not Euclidean, but it is continuous. Am I right? So we don't need the 'vile tile' argument.
Could space be quantized at the microlevel but continuous ar the macrolevel?
Is there room for a priori argumentation when it comes to establishing the properties of physical space (as opposed to geometrical space as opposed to purely mathematical 'space')?
Is Zenonian argumentation rejectible on the ground that certain actual phyical constraints are not met? For example, a Thomson lamp is physically impossible. Is that relevant?
Yes, GTR is continuous, but I don't know that that is more than a mathematical convenience... about the rest, again I don't know (I find physics very difficult:)
Could space be quantized at the microlevel but continuous at the macrolevel? I think if space is quantized, it just is quantized, but if you mean can macrolevel space be approximated as continuous, yes. GTR and QM are extremely accurate theories, so the assumption of continuity works for all practical purposes.
Is there room for a priori argument? In general there is room for a priori argument in physics. I read a book that discussed about a dozen a priori derivations of the special theory of relativity. Regarding properties of space I don't know. In general it seems that physics tends to try to eliminate infinities and singularities, so Zeno was probably on to something.
I would not reject Zenonian argumentation just because we know that the conditions described in an argument can't be physically realized. The Thomson lamp shows that a switch with zero switching time leads to indeterminate results. We already know that a switch with zero switching time is impossible, but if this argument had been presented a couple of hundred years ago, it would not have been a trivial conclusion. And there appears to be no technological reason why such a simple argument could not have been proposed that long ago.
I don't think it would make any difference. Replace the squares with circles, or chiliagons, or any polygon, etc. The hypotenuse would still be 8 units in length. Imagine the board distorted Salvador Dali-style. You could still superimpose a right triangle on it.
"does it even make sense to talk of the quanta being arranged in space, since it's rather the case under the theory that the quanta are space?" That was just the point I made in my response to Jeff and in the penultimate para of my post.
The argument relies on the idea that "length" is best calculated by counting squares. But why? This is just one possible metric among many others. Pythagoras's theorem isn't compatible with it, but that's OK; instead you get Weyl's Theorem, which is that the length of the hypotenuse equals the length of the longest other side; this is just as useful. Or if you don't like that, try choosing a different metric.
(Aside: it might be more telling to try to reconstruct trigonometry in the Weyl space... looks like sin^2 + cos^2 = sin^2 => cos(a)=0 for all a, and the same argument applies to sin(a), which is a tiny bit worrying)
One thing that looks wrong in the Weyl space is that as you rotate the triangle it seems to be deformed. But if the tiles were very tiny we wouldn't notice (you don't notice this on your TV, and that's a Weyl space). I seem to recall someone suggesting that a Penrose tiling would "average out" this effect, but I don't know enough about it and anyway, as I say, the chunks of space may as well be cubes if they're tiny enough.
My general point is that if space's micro-structure is Weyl-like then the maths we commonly use is just an approximation, and we'd need some different maths to work at very small scales. I don't think that's a show-stopper for a theory that denies the existence of a continuum.