Is physical space, the space of the natural world, continuous or discrete? If composed of space atoms, then discrete. The Weyl Tile argument (WTA), however, seems to show that physical space cannot be discrete or 'quantized' and therefore must be continuous. This is relevant to our ongoing debate about potential versus actual infinities. For if physical space cannot be discrete, then it must be at least *compact* (the lowest grade of continuity), where "A series is called *compact* when no two terms are consecutive, but between any two there are others." (Bertrand Russell, *Our Knowledge of the External World*, Norton, 1929, p. 144.) But if between any two points in space there are others, then there are infinitely many others, so that any line segment will be composed of an actual infinity of points.

But before we return to the question of actual infinities we need to get clear about the WTA itself. The *nervus probandi* lies in the following quotation from Hermann Weyl, *Philosophy of Mathematics and Natural Science*, Princeton UP, 1949, p. 43:

If a square is built up of miniature tiles, there are as many tiles along the diagonal as along the side; thus the diagonal should be equal in length to the side.

Take a gander at the chess board below. Consider the right triangle the sides of which are a1-a8 and a1-h1, and the hypotenuse of which is the diagonal a8-h1. The sides and the diagonal are each eight squares long. Count 'em and see. But this flies in the face of the theorem of Pythagoras. If the sides are each eight units in length, then the hypotenuse is equal to the square root of (8^{2} + 8^{2} =128), which is not 8, but the irrational 11.313 . . .

The question this curious fact raises is whether physical space can be quantized, i.e., whether there are space atoms. If so, space is discrete as opposed to continuous. It may help to bear in mind that the above array is a mere model in continuous space of discrete space. So it will do no good to object that if space atoms are squares, then the theorem of Pythagoras hold for them. Space atoms are not squares: they have no shape at all. But I am getting ahead of my story.

We need to define our terms. Space is *discrete* just in case every finite extended spatial region is composed of finitely many *atomic* spatial regions. That amounts to saying that every finite extended region of space is composed of finitely many space atoms, where 'atom,' as its etymology suggests, implies indivisibility. You cannot 'split' a space atom because such atoms are inherently 'unsplittable.' A space atom is thus an individual that has no proper parts: it is a mereological atom. A non-atomic region of space is then a mereological sum of space atoms. Note that every space atom, precisely because it is an atom, is an unextended region of space. It's an itty-bitty unextended bit of space itself, not of something in space. Space atoms are not in space; they compose space.

Now for the argument:

1) The theorem of Pythagoras is not true (or even approximately true) of discrete space.

2) The theorem of Pythagoras is true (or approximately true) of actual space. Therefore:

3) Actual space is not discrete.

To understand this argument, you have to understand that nothing rides on how small the tiles/squares are. Glance back at the chessboard. Consider the small right triangle in the bottom left corner of the board. Opposing sides and hypotenuse all have a length of two units. So it doesn't matter how small the space atoms are. No matter how small the squares, the hypotenuse remains equal in length to the other two sides.

You will be tempted to think of the array of tiles/squares against the backdrop of continuous Euclidean space for which the Pythagorean theorem holds. Thinking in this way, you will imagine that no matter how small you make the tiles, the diagonal will be longer than the sides. You have to resist this temptation to understand the 'Weyl tile' (vile tile?) argument. For if there are space atoms, then they have no shape and hence no different dimensions in different directions. As Wesley C. Salmon puts, "In discrete space, a space atom constitutes one unit, and that is all there is to it. It cannot be regarded as properly having a shape, for we cannot ascribe sizes to parts of it -- it has no parts." (*Space, Time, and Motion*, U of Minnesota Press, 1980, p. 66)

I have found K. McDaniel, "Distance and Discrete Space," *Synthese* (2007) 155: 157-162, very helpful. He has an argument against the WTA which I may discuss in a subsequent post.

What about the observable fact that the chess board has as many tiles along the diagonal (8) as it does along each side (8)?

Posted by: Loquitur Veritatem | Sunday, May 14, 2023 at 04:58 PM

Bill writes: "...every finite extended region of space is composed of finitely many space atoms." Then later he writes: "Note that every space atom, precisely because it is an atom, is an unextended region of space."

I need some clarification here, unless I'm completely confused, these two statements don't seem to me to be consistent. A single atom exists in space, and therefore composes the smallest region of space, under your stipulated definitions, but, analogously, as a single number is finite, then a single atom consists of the minimal finite extended region of space...unless you're saying that this is like talking about sets that are both open and closed.

Posted by: EG | Monday, May 15, 2023 at 07:57 AM

Excellent comment, Erik. I asked myself the same question. How are the two assertions consistent?

Consider a line segment L whose endpoints are A and B. L is a one-dimensional finite extended region of space that extends from A to B. L is composed of points. Each of these points is a space atom. Each such atom has no proper parts. (Each atom has only itself as an improper part.) So each space atom is unextended. To have spatial extension, a space atom would have to have proper parts, which it cannot possibly have given that it is an atom, an indivisible individual.

So it seems to me that the two assertions are logically consistent. It may help to bear in mind what I said, "Space atoms are not in space; they compose space." The idea is that physical space itself is 'grainy' or 'granular.' Space atoms are not like grains of sand on a beach. The latter are material individuals having proper parts. But even if each grain of sand were literally a material atom, the comparison would not hold, and this for the reason that grains of sand fill space. But what we are talking about is space itself. The idea is that empty space is composed of finitely many space atoms. Space atoms are not in space; they constitute space.

Does this make sense? Note that I am not arguing for or against space atoms; I am merely trying to plumb the depths of the Weyl Tile Argument.

Posted by: BV | Monday, May 15, 2023 at 01:09 PM

Erik,

Set theory does not come into this at all. A region of discrete space is not a set of points, but a nonset, an individual. This individual is either a mereological sum/fusion of mereological atoms, or a mereological atom, a space atom.

One reason a sum is not a set is because the parthood relation is not the same as the membership (elementhood) relation. Given my two cats, Max and Manny, there is their sum/fusion F. The parts of F include Max and Manny but also all of their parts, their tails, whiskers, claws, etc. 'all the way down.' The parthood relation is transitive: if X is a part of F, and Y is a part of X, then Y is a part of F. The elementhood relation is not transitive: it is not the case that if x is an element of y and y is an element of z, then x is an element of z. For example, Max is an element of {Max, Manny}, and {Max, Manny} is an element of {{ }, {Max, Manny}, {Max}, {Manny}}, but Max is not an element of the latter, the power set of {Max, Manny}.

Posted by: BV | Monday, May 15, 2023 at 01:51 PM

Bill,

The responses are as usual, excellent and to the point (see what I did there?) It is both the charm and frustrating part of these sorts of arguments and discussions that we have to be very clear about what we mean, how that meaning should be interpreted, relative to the context and argument.

(I also note that "nonset" --> "nonsense" has a certain fittingness that seems appropriate to this discussion.)

I am now agreed with your argument as it has been laid out, and I agree that under the stipulated definition that space (atom(s)) is not itself extended; though, thinking about this, I am now curious about the transition from un-extended to extended--logically this is not so problematic, we can just stipulate it; but physically, this is deeply mysterious.

Posted by: EG | Tuesday, May 16, 2023 at 06:19 AM

Erik,

I did indeed see what you did there, although I confess that, like the tack I sat on, I didn't get the point at first, but I did get it in the end.

Your question could be put like this: How do the space atoms that compose a finite line segment L 'add up to' a finite extended one-dimensional spatial region given that each atom is an indivisible, unextended unit? How do we get from unextended partless parts of a whole to a partite extended whole?

It would be unphilosophical to evade this question by saying, "We just do!"

One might try an emergentist response: L emerges from sufficiently many atoms. Analogy: How is it that a wholly physical/material system can be conscious, given that consciousness is not a physical property? Ans: consciousness is an emergent phenomenon: at a certain level of complexity (the kind that our brains have) consciousness emerges! Admittedly, not a very satisfying answer. But are the alternatives any better? Dennett says that that cs. is an illusion. Now that is surely bullshit if anything is. Danny boy is giving bullshit a bad name!

Posted by: BV | Tuesday, May 16, 2023 at 01:25 PM

Let me now try a different tack.

Suppose space is a continuum. Then our finite line segment L has a nondenumerably infinite number of points. How many? 2-to-the-aleph-nought, which is the cardinality of the continuum. How on Earth -- or in Cantor's Paradise -- do they 'add up to' L? The points are partless and unextended. Supposedly, they add up to an extended line because there are so bloody many of them. How did we get from the partless to the partite, from the unextended to the extended? That's a quantitative-to-qualitative leap, a metabasis eis allo genos. Saltus mirabilis!

But isn't this just as mind-boggling as the notion that finitely many space atoms add up to a line segment?

I'm just shootin' from the hip here, trying out some ideas.

Posted by: BV | Tuesday, May 16, 2023 at 01:45 PM

Hello Everyone,

I do not feel the force of WTA. I am sure this is due to my ignorance, but that is not going to stop me from opining.

Ignoring the parenthetical part of this, it seems easy to raise objections. The Pythagorean Theorem (PT) is a theorem of Euclidean geometry. If space is Euclidean, then space is continuous. However, if space is non-Euclidean, say, hyperbolic or elliptical, then premise (2) is false. There are significant physical theories, like Relativity, that are based on non-Euclidean models where PT does not hold.The second premise of the argument is:

Another objection concerns the ontological status of the kinds of entities where PT applies. Euclidean geometry is simply a model. If right triangles exist at all, then they exist as abstract entities. To go from PT is a true theorem in Euclidean geometry to PT is true of actual space is quite a leap and begs the question concerning continuity.

Lastly, given the chessboard model, there is no such thing as a triangle. If reality is a plane made up of 64 squares, then the shortest path from one corner of the 8 X 8 space to its catty corner is 15 units. There is no diagonal line. There is no right triangle, and there is no length 8√2. Lastly, PT does not hold. These kinds of entities do not exist in this kind of model.

There is a parallel with this last point and say, elliptical geometry. Think of space as being a kind of sphere. What is the shortest distance from one point of this sphere to a point on the other side? With no slight intended towards Jules Verne, there is no journey through the center of the sphere to the other side. The shortest path will always be along the sphere itself. (Note: in this example, space is continuous and PT does not hold.)

Ok, it's now time to be set straight (no pun intended).

Brian

Posted by: Brian Bosse | Tuesday, May 16, 2023 at 04:29 PM

If space is a discrete lattice, like a chessboard, it has preferred directions (the lines connecting the lattice points) and objects with spatial extension can't be rotated to arbitrary orientations - only to the lattice's preferred directions. As in actual space objects can be freely rotated, actual space is not a discrete lattice. It must be, at least, a dense space like the rational numbers, so that between any two points there is always a third point.

For instance, if you pick a contiguous set of squares on a chessboard, and say an object occupies those squares and no others, you can't rotate that object to any angles but the right angle and multiples of it. Any other rotations must distort the object, putting it into a differently shaped set of squares; the set may not even have the same

numberof squares, so such a rotation would actually change the thing's size!Posted by: Michael Brazier | Tuesday, May 16, 2023 at 10:29 PM

Bill, you wonder how continuum-many partless and unextended points can 'add up' to a partite, extended line segment. It's more puzzling still that continuum-many points can also 'add up' to a line segment of twice the length and even an infinite line segment. I suspect the problem is that we are thinking within the metaphor of dividing up a long thin physical object. The puzzle is (partly) dissolved if we think of a point as a place in space, and of the real number that designates it as an 'address' that tells how to arrive at the place. The points 'add up' to the line segment because they amount to all the places along the segment, with no duplication and no place missed out (in the sense that the rational numbers miss out some places). The model for generating 'addresses' is indefinitely repeated binary division. Addresses of points in the first half of the segment begin 0.0..., those in the second begin 0.1... Those in the first quarter begin 0.00.., second 0.01..., third 0.10..., and fourth 0.11... and so on, producing uncountably many binary representations. But no doubt this conception raises further questions!

Posted by: David Brightly | Wednesday, May 17, 2023 at 05:04 AM

Thanks for the comment, David. What you say is helpful and admirably clear. But I plead innocent to the charge of thinking of a line segment as a long, thin physical object. Neither lines nor their constituent points are physical/material things. This may sound vague, but what we are discussing is the structure of physical space itself. not anything in space.

>> think of a point as a place in space, and of the real number that designates it as an 'address' that tells how to arrive at the place.<< I like that formulation. Yes, a point is a place in space. Yes, there is a distinction between a real number and a place in space, in analogy to the distinction between one's physical/postal address and one's domicile. I live in a house, not in a postal designation. Similarly, we ought to distinguish between numbers -- which are broadly 'arithmetical' -- and points, lines, planes, solids, etc. which belong to geometry.

Do you understand and accept that distinction? If you do, then you ought to accept a further distinction between 'broadly arithmetical' continuity and geometrical continuity, and similarly for density (compactness) and discreteness. The reals form a continuum; the rationals, which are a proper subset of the reals, do not form a continuum: they are merely dense. (By the way, am I right to assume that 'dense' and 'compact' have the same sense?)

A further distinction is between geometrical space and actual space, i.e., the space of the actual physical world. A geometry provides the structure of a possible physical space, and then the question arises -- a presumably empirical question -- whether this possible physical space is the actual space of nature. For example, there are Euclidean and non-Euclidean geometries. The still going view, I take it, is that the actual space of nature is non-Euclidean. So one must not confuse a merely possible physical space. as articulated in some geometrical system, with actual space.

Do you buy all these distinctions? You said, >>think of a point as a place in space, and of the real number that designates it as an 'address' that tells how to arrive at the place.<< My only quibble is that we are not told how to arrive at the place, but only the location of the place relative to its 'neighbors' where this world is obviously not to be taken in the 'next-door neighbor' way.

Finally, David, what you say above makes perfect set-theoretical sense. So do you accept set theory, in particular, do you accept that there are actually infinite sets?

Posted by: BV | Wednesday, May 17, 2023 at 12:41 PM

> By the way, am I right to assume that 'dense' and 'compact' have the same sense?

No, topology assigns different senses to those words. A space is compact if every open cover of the space has a finite subcover, where "open covers" are collections of open sets whose union contains the whole space. An implication of this is that any infinite sequence of points in the space must have a subsequence that converges to a point in the space - which is why a closed interval of rational numbers, though dense, isn't compact: the successive decimal approximations to an irrational number are an infinite sequence of rationals that don't converge to a rational.

Intuitively, a compact space is a chunk of some continuum and inherits its continuity from it.

Posted by: Michael Brazier | Wednesday, May 17, 2023 at 04:22 PM

Michael,

Thanks for that clarification. Let's leave compactness out of the discussion. I think we all understand the differences between discrete, dense, and continuous. The nat'l numbers in their usual order are discrete: for each such number there is a uniquely next one. The rational numbers (fractions) in their usual order are dense: between any two fractions there is a fraction. But there are gaps in the rational line. They are filled by the irrationals. The union of the rationals and the irrationals are the reals. To say that the reals are continuous is just to say that they are both dense and gapless.

Posted by: BV | Wednesday, May 17, 2023 at 07:36 PM

Arithmetical and Geometrical Continuums

In view of the completeness axiom in R , we find that there are no gaps in R of the kind Q has. We may, therefore, say that the real numbers form a continuous system. On account of this characteristic, R

is also called the arithmetical continuum.

As provided by the Dedekind-Cantor axiom, we find that the systems of points on a directed straight line also do not posses gaps and, therefore, the system of points on a directed straight line is called the geometrical continuum.

See here: https://www.emathzone.com/tutorials/real-analysis/dedekind-cantor-axiom-of-continuity-o

So we have arithmetical continuity, geometrical continuity, and presumably also (though this is an empirical question) the continuity of actual physical space, the physical space of nature.

Everybody agree? Alles klar?

Posted by: BV | Wednesday, May 17, 2023 at 07:45 PM

Morning Bill,

Perhaps not physically---that's too specific---but at any rate

mereologically, with points composing and constituting lines, being parts of lines, etc.I certainly accept all the distinctions you make between geometric and arithmetic (analytic?) continua. There is of course no unique continuous mapping from R to a line segment L. But having identified the endpoints of L as P0 and P1, say, and having decided to address these as 0.0 and 1.0, the repeated binary division procedure I outlined is purely geometric and yields a sequence of left/right, 0/1 decisions which specify a unique real number in the interval [0,1]. Conversely, given r ∈ [0,1] its binary representation yields a sequence of telescoping subsegments that, in the limit, specify a unique point. So we get a 'nearly natural' way---there is some arbitrariness in it---of coordinatising the segment.

I certainly accept the infinity of natural numbers but I don't understand how the modal terms 'possible' and 'actual' can be applied to them. I am happy to talk about them in the language of sets.

Posted by: David Brightly | Thursday, May 18, 2023 at 04:07 AM

Good day, David.

>>I certainly accept the infinity of natural numbers but I don't understand how the modal terms 'possible' and 'actual' can be applied to them. I am happy to talk about them in the language of sets.<<

Not clear. Do you or do you not accept that there are actually infinite sets? If you accept the infinity of natural numbers, and you talk about them in the language of sets, then you are committed to saying that there are actually infinite sets. Do you agree?

Note also that 'possible' and 'potential' don't have the same sense.

Please explain exactly what you mean by 'the infinity of natural numbers.'

Posted by: BV | Thursday, May 18, 2023 at 04:58 AM

Bill, You ask,

No, I'm afraid not. I don't understand what work 'actually' does here, but let's put that to one side. I say that there are natural numbers. The correspondence 0↔1, 1↔2, 2↔3, ... shows that that 0, 1, 2, ... can be placed in one-to-one correspondence with 1, 2, 3,.... So I say that there are infinitely many of them. This can be said without set-speak. Set-speak is jolly useful for referring to natural numbers plurally. It's a nice way, for example, of clarifying what we mean by the arithmetic of residues modulo n. But number theory can be done, and indeed was done before Cantor without the benefit of set-speak. 'The set of natural numbers is infinite' is just set-speak for what Euclid would have said in his contemporary Greek. So I say there is no set of natural numbers in addition to the naturals themselves or in a 'one-over-many' relation to them.Posted by: David Brightly | Thursday, May 18, 2023 at 06:51 AM

Hello David,

If we are working within a particular set theory to construct the natural numbers, the following axioms would be needed:

- The Empty Set Axiom
- The Axiom of Extensionality
- The Axiom of Pairing
- The Axiom Schema of Specification
- The Axiom of Union
- The Power Set Axiom

From this, we start with the empty set and construct another set. Once we have that set, we then construct a third set, etc…This process of construction is what Aristotle called a potentially infinite process. It is potential in that you can always construct the next set, but it is never completed. It just goes on and on.What is being poorly described above is the construction of the natural numbers. Each constructed set corresponds to a particular natural number. But, yet, within our system, we still cannot construct the set of all of these iterative constructions. What is required is another axiom - the Axiom of Infinity. Without this, you cannot have “the set of Natural numbers.” Informally, this axiom asserts the existence of such an infinite set. This existence assertion is the assertion of the existence of an actual infinity. Anyone who adopts this particular axiom is adopting the existence of the acutal infinite. When you speak of these various sets being in correspondence, it seems you are assuming some kind of set theory requiring the Axiom of Infinity. If so, then you are assuming an actual infinity. If not, and you are speaking colloquially, then what is ‘correspondence’, what do you mean by ‘…’, and what do you mean to say that “there are infinitely many of them”?

Again, ‘correspondence’ has a technical meaning. If you are intending to use this term in its technical sense, then you are assuming some kind of set theory. If so, and if you are applying this to infinite sets, then you are assuming the Axiom of Infinity. If so, then you are assuming an actual infinity.

Brian

Posted by: Brian Bosse | Thursday, May 18, 2023 at 05:42 PM

Brian,

Thanks for your comments and for interacting with David. I'll try to respond to you tomorrow -- there's been a lot of 'incoming.'

By the way, I need to issue a

retractio. When we met over the chessboard, I believe I denied your assertion that the naturals are a proper subset of the integers. If I said that, I was wrong. I had forgotten about the negative integers. There are no negative naturals, but there are negative integers.Posted by: BV | Thursday, May 18, 2023 at 07:05 PM

Hello Brian,

You are referring, of course, to the standard textbook account of the 'construction of the natural numbers'. This name, I think, is a misnomer. The account does not construct the natural numbers. Rather it constructs a set with structural affinities with the natural numbers. This set has something that can play the role of 0 and something that models the successor function. But this is not quite right either. It doesn't construct this set, rather it reveals that this set exists, granted the ZF axioms. What the account constructs is a proof of this claim. Nowhere do we find anything 'incomplete' or 'potential' as opposed to 'actual'.

Contra the textbooks, perhaps, I do not see how we are beholden to any theory of sets, aggregates, pluralities, call them what we will, to understand the infinitude of the natural numbers. Euclid puts it well enough with regard to the conclusion of his theorem on the primes:

This could be taken as a definition of infinite numerousness. Or we might simply say that the primes are uncountable, had this term not already become established with a different sense.Posted by: David Brightly | Friday, May 19, 2023 at 08:00 AM

What was more interesting reading these comments was the interaction itself, and how in a certain sense each interlocutor was defending a specific point, and perhaps excepting Bill were “talking past” one another. And it is perhaps in this sense in which we reach an impasse, but I struggle to pin down where that is.

Posted by: EG | Saturday, May 20, 2023 at 02:22 PM

Bill,

Can I ask you to resummarize but also be as explicit as possible with your definitions, as against the ones that were presented by David, Brian and Mike, so that we can properly distinguish the specific positions in the context of your original argument. I think some of this has slid into sliding into "equivalent" definitions but ones that actually change the context and interpretation of your argument. So to say, a number-theoretic and topological argument are not the same thing, even if they may share some common root definitions or there is some preserving transformation that will move you from one argument to another.

Posted by: EG | Tuesday, May 23, 2023 at 07:54 AM