Maverick Philosopher

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. Think about it! Imagine the tiles (squares) becoming smaller and smaller. No matter how small they become, the number of tiles along the hypotenuse will be equal to the number along either of the other sides. If there are space-atoms, the number of such atoms along the hypotenuse will be equal to the number along the legs.

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. In sum:

1. If a plane in physical space were composed of space-atoms, then the Pythagorean theorem would not be true of it.
2. The theorem in question is true of planes in physical space.
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3. Physical planes are not composed of space-atoms.

One will be tempted to object to (1) by saying that 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.

Weyl's argument, then, strikes this non-mathematician as a good argument against space-atoms as constituents of physical space, the space of our empirical acquaintance.

A reader who came across an earlier post of mine on this topic writes:

I've been trying to wrap my head around this problem lately. And a couple of things still do not make sense. I wondered if you might be able to clarify them. I don't understand how a height/width unit is held to equal a diagonal unit. By geometrical definition, isn't the diagonal of a square longer than either its height or its width? I don't understand, then, how the Pythagorean Theorem can fail to be approximated. If approximation is the problem, then I think this is quite possibly a reflection on the failure of human beings to exactly calculate a hypotenuse and not necessarily on the failure of ultimate space to be exact....

I too cannot understand how the theorem of Pythagoras could fail to be true of space as we experience it. The question, however, is what space must be like if the theorem in question is to be true of it. What Weyl's argument seems to show is that space cannot be composed of space-atoms, discrete bits of space, but must be continuous.

REFERENCES

W. C. Salmon, Space, Time and Motion, pp. 64-66.
Hermann Weyl, Philosophy of Mathematics and Natural Science, p. 43.

By 'sets' I mean mathematical as opposed to commonsense sets. A set in the mathematical sense is a single item 'over and above' its members. If the six shoes in my closet form a mathematical set, and it is not obvious that they do, then that set is a one-in-many: it is one single item despite its having six distinct members. A set is not identical to its members, and so it is distinct from them, though not wholly distinct from them: it is after all composed of them and its very identity and existence depends on them.

One can reasonably deny the existence of mathematical sets as witness our friend 'Ockham.' He is a nihilist about such sets: there are no such entities on his view. But he doesn't deny that there are such commonsense sets as a gaggle of geese or a pride of lions. Suppose I say of my cat Zeno, 'He has a healthy set of teeth.' The truth of that sentence does not commit me to the existence of mathematical sets. I am not using 'set' in the mathematical sense in that sentence. I could just as well have said, 'Zeno has healthy teeth.' I happen to think that there are mathematical sets, and that there is a set consisting of Zeno's teeth. But when I look into his mouth what I see is not a math. set — how could such an entity be literally seen? — but a number of teeth, a mere plurality of teeth, if you will.

The confusion of mathematical and commonsense sets is illustrated by this remark, "It sounds odd to say that a set has a cuisine, but an ethnic group is a set of people that share a common culture and identity, and ethnic groups often have their own cuisines, so therefore some sets have cuisines." No doubt an ethnic group is a set of people in the commonsense acceptation of 'set,' but it doesn't follow that an ethnic group is a mathematical set. One can reasonably deny that there are math. sets but not that there are ethnic groups.

The reason I think line segments are not identical to math. sets is that there is a class of properties that can be ascribed meaningfully to line segments, but cannot be ascribed meaningfully to sets. And so, by the Indiscernibility of Identicals, line segments cannot be sets.

1. One can say of a line segment that it has a length. But there is no clear sense in which sets have length. Whatever has length must have a spatial extension. But no math. set has spatial extension. Therefore, no line segment is (identically) a set. Suppose one line segment is twice as long as another. The corresponding sets of points have the same cardinality, namely, the cardinality of the continuum. Can the metrical predicate, 'twice as long as' be given a purely set-theoretical representation? That is not a rhetorical question.

2. One can say of a ray that it originates in a point. So if lines, rays, and line segments are sets of their constituent points, then one could say of a set that it originates in a point. But no clear sense can be attached to such talk. Math. sets are nonspatial and nontemporal.

3. If a circle is a set of points, then some sets have a diameter, a radius and an area. But no sense can be attached to the claim that a math. set has an area, or that some sets have a larger area than others. Similarly for spheres. Spheres have volume. Do math. sets have volume?

4. Analogous points can be made about temporal intervals. It is tempting to identify temporal intervals with sets of instants. But although it makes sense to say of temporal intervals that they are earlier than, later than, or simultaneous with other temporal intervals, it presumably makes no sense to say the same about sets. Math. sets are nontemporal entities.

Suppose you have a line segment AB. You bisect it. To bisect is to cut into two equal parts. The parts of a bisection are mutually exclusive and jointly exhaustive. If we think of the parts as sets, then they are disjoint and such that their union is equal to the set of points in AB. What I have just said simply unpacks what we mean by 'bisection.'

Assume that lines and line segments are composed of points, and that there are continuum-many points in each line segment. That is to say: there are as many points in each line segment as there are real numbers. Assume these points are all 'out there' in mind-independent reality whether or not we engage in any bisecting. Thus, for any possible bisecting, there is a point of bisection antecedently present in the line: these points are not created by our bisectings.

Let C be the point of bisection of AB. This yields two equal line segments AC and CB. Now we bisect CB and get CD and DB. Bisecting DB yields EB, and so on: FB, GB, HB, IB, JB, etc. each shorter than the preceding one. Endpoint B is a member of each of these segments. For every line segment, unlike a line, has a beginning and an end, and the end of a line segment is a point.

Now if an infinity of such bisectings could be completed, what would be the result? Would there ever come a bisecting that yielded B by itself?

Presumably not; for every segment, no matter how short, contains continuum-many points. Thus every bisecting yields two line segments, never a line segment and an endpoint. But then in what sense is a line composed of points?

Think of it this way. If a line is composed of points, built up out of points, then presumably it can be decomposed into points, broken down into points. One might think that an actual infinity of choppings would chop the line into points if indeed there is sense to lines being composed of points. But no matter how many cuts are made, the result will always be two line segments, never a line segment and an endpoint.

So here is my paradox: A line segment is composed of points, but a line segment cannot be composed of points because the decomposition of a line segment by successive bisectings could never result in anything other than two line segments, never a line segment and a point, or two points. There appears to be a contradiction between the notion of continuity and the notion of composition out of points.

The paradox could be resolved by rejecting continuity or rejecting the notion that lines are composed of points, or both.

But I did make an assumption above that ought to be tested. I assumed that B is a member of each of the shorter and shorter segments. Consider any arbitrarily short segment XB. Since a line segment is a proper part of an infinite line, there has to be points to the right of XB, and we should consider the possibility that the set corresponding to XB has no last element and that B is the first element of the set corresponding to the segment immediately to the right of XB. But even if that is the case, it seems that successive bisecting would never yield points, and if so, then it is hard to understand how a line could be composed of points.

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 8² + 8² = 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.

Let us consider the 'calculus solution' to the paradox. Consider the infinite sequence

1/2, 1/4, 1/8, . . . ,1/2ⁿ, . . .

A sequence is said to be convergent if it has a limit. Roughly, the limit of a sequence is a number L such that the terms of the sequence become and remain arbitrarily close to L as we consider the terms in succession. Thus the limit of the sequence displayed above is 0, since as we run through the sequence, each term gets closer and closer to 0. The larger n becomes, the smaller is the difference between 1/2ⁿ and 0. The sequence converges to 0 as its limit.

We can also speak of the convergence of an infinite series, where an infinite series is the result of adding all the terms in an infinite sequence. But how do we add the terms in an infinite sequence? What meaning can we attach to 'sum of an infinite sequence of terms'? In standard analysis (calculus), the sum of an infinite sequence is the limit of the sequence of partial sums:

1/2, 1/2 + 1/4 = 3/4, 1/2 + 1/4 + 1/8 = 7/8, . . . .

This sequence of partial sums converges to the limit 1. Thus the sum of a convergent infinite series is a number that can be approximated arbitrarily closely by adding up sufficiently many finite terms.

Now how is this supposed to solve the Regressive Dichotomy paradox? Since the sum of an infinite series has been defined as the limit of the sequence of partial sums, and since this sequence converges to a finite limit (1 in our example), it is intelligible how infinitely many terms in a convergent series can have a finite sum. Note, however, that this intelligibility is a matter of stipulative definition: it is just stipulated that 'sum' in the case of infinite series shall have the meaning 'limit of a convergent sequence.' And since this is intelligible, it is supposed also to be intelligible how a runner can traverse the infinity of points lying between A and B.

The rub, however, is in applying the mathematical formalism, which I cannot fault, to the physical world. Our convergent infinite series has a sum, but only in the sense that the sequence of partial sums converges to the limit 1. The series does not have a sum in the sense of some number that is actually computed by someone or some computer. But to get from A to B I must actually complete an infinite sequence of runs. There is an infinite number of acts that I must perform assuming that space is composed of points. It is not enough that I converge upon point B; I must actually arrive there. In the mathematical case, however, it is enough that the infinite series converges upon the limit 1. There is no need that one actually compute up to 1.

Although I haven't expressed this as clearly as I ought to, my sense is that the 'calculus solution,' although fine as mathematics, is no solution to the Dichotomy paradox. One can render intelligible how an infinite series of terms can have a finite sum. But this does not render intelligible how an infinite series of acts can be performed in a finite time.

REFERENCE: Wesley C. Salmon, Space, Time, and Motion, Chapter 2.

The At-At theory is a theory of motion that attempts to view motion as a logical construction out of immobilities. How so? At each instant, the object O is immobile in that O cannot move in an instant, just as Zeno of Elea said long ago. Instants being durationless, nothing can move during an instant. But for Russell and his followers, this is not a problem since there are continuum-many instants: time is a continuum. And so is space. At each of the continuum-many instants in the temporal interval in which O is moving, O is at a different position. Thus motion logically supervenes upon this continuous sequence of immobilities. Although one cannot speak of motion in an instant, one can speak of motion over an interval. O's motion just is O's being at continuum-many different positions at continuum-many different times in the interval in question.

That's neat, and I don't deny that it works mathematically, but it looks to be too much of a concession to Zeno. It seems to expunge from the world the 'dynamism of becoming.' For it amounts to saying that no object is ever such as to possess intrinsically and at an instant a dynamical attribute such as a definite velocity or a definite acceleration, not to mention jerk (the third derivative of position) or jounce (the fourth derivative). To focus the issue, consider what the differences are among:

1. Object O is at rest at time t.
2. O is moving at constant velocity at t.
3. O is accelerating at a constant rate at t.
4. O is increasing its rate of acceleration, and thus jerking, at t.

Intuitively, what we want to say is that these are four quite different intrinsic states for O to be in at a time. An intrinsic state is a state an object is in regardless of what is happening at other times or to other objects. But if time t is a durationless instant, then there would be seem to be no differences among (1)-(4). On Russell's theory, for there to be a difference between (1) and (2) one must consider what is going on at times in the neighborhood of t. If at the nearby times O is in the same position, then O is at rest; if at the nearby times, O is in different positions, then O is in motion. Russell's theory has the consequence that such properties as being at rest, being in motion, being in motion at constant velocity, accelerating, jerking, etc. are not intrinsic properties of objects at times. They depend on what is happening throughout sufficiently small intervals around the time in question. If you focus on one instant, there is no difference between an object's being at rest, moving at constant velocity, accelerating, decelerating, or jerking.

One problem with the At-At theory of motion is that it seems to rule out determinism a priori. In a deterministic system, any state of the system at a time determines all later states. Let the system be a projectile P moving at constant velocity v in a straight line. To keep it simple, we may assume that no forces act upon P as it moves. If determinism is true, and the velocity of P at t is known, then the positions of P at all later times can be calculated. But if the At-At theory is true, it is hard to see how determinism could be true. For on the At-At theory, there is no such property as the velocity of a projectile at a time. Indeed, there is no such property as the being in motion of a projectile at a time.

Determinism may or may not be true; but it doesn't seem credible that it should be ruled out by a theory of motion.

What's worse, the At-At theory seems to rule out even any fixing of probabilities of later states of things given present states of things. Suppose that the present position and velocity of the projectile does not determine its future positions but merely fixes certain probabilities with regard to its future positions. This too presupposes the possession by objects intrinsically and at an instant of such dynamical attributes as velocity.

A third worry is this. On the At-At Theory, to say that an object O is in motion is just to say that it is at different positions at different times. Suppose O moves during interval I. Consider a time t near the beginning of this interval. According to the theory, O is in motion at t because, during I, O is at different positions at different times. This implies that the states of O at times later than t are constitutive of O's being in motion at t. Surely this smacks of absurdity. How can the later states of an object be constitutive of its being in motion or at rest at earlier times?

This argument needs to be presented with greater rigor, but it is time to punch the clock.

REFERENCE: Frank Arntzenius, Are There Really Instantaneous Velocities?

To move is simply to occupy different positions in space at different moments of time. Considering any single point of space and single instant of time, an object simply is at that point of space at that instant of time. There is no distinction between being at motion and being at rest as long as we consider only that one point and that one instant. To get from point A to point B consists merely of being at the intervening points of space at the corresponding moments of time. There is no further question of how the moving arrow [Zeno's arrow] gets from one point to another. (Scientific Explanation, p. 110, emphasis added.)

Spur does not tell us why he thinks it is is impossible "to traverse an infinite." But if his reason is that it involves Zeno's paradox of the arrow, then perhaps the At-At theory is a way of countering Spur's argument. It is not clear to me, however, that the At-At theory is acceptable.

The At-At theory can be seen as a response to Zeno's Paradox of the Arrow. But I will replace the arrow with a ball. The paradox goes like this. A ball is moving from point A to point B. Motion takes time. The interval of time through which the ball moves is composed of instants. Since instants are durationless, the ball cannot move in any instant. At any given instant, the ball is at rest. Not moving at any instant, the ball does not move at all.

Perhaps the most obvious response to the paradox is to say that, while nothing can move during an instant, an object can be in motion at an instant t if at times in the neighborhood of t the object is in different positions. This response amounts to the claim that motion just is occupancy of different positions at different times. Thus X moves iff X is at different positions at different times, and X is at rest iff X is in the same position at different times. There is no more to it than that on the At-At theory.

Note how substantial is the concession that the At-At theory makes to Zeno. The Eleatic held that nothing moves from one place to another. But that is also what the At-At theory says: nothing moves from one place to another since motion just is being at different places at different times. Since motion is a being at different positions at different times, it is not a becoming differently positioned as time passes.

The concession to Zeno is particularly clear when one realizes that the theory implies that there are no instantaneous intrinsic states of motion. Nothing at an instant is intrinsically in motion, and nothing at an instant has intrinsically a velocity.

The At-At theory is thus a static theory of motion. To see what is wrong with it, suppose the ball travels from A to B and then proceeds from B back to A. Suppose it travels in a straight line in both directions and so comes to occupy each point between A and B twice. At any one of these points, what is the difference between the ball's moving from A to B and its moving from B to A? The At-At theory requires that at each of these points, the ball lacks an intrinsic instantaneous velocity; so the At-At theory cannot distinguish between the ball's moving from A to B and its moving from B to A. This implies that the At-At theory cannot explain why the right-ward moving ball continues to move right and the left-ward moving ball continues to move left.

But surely what we want to be able to say is that the the right-ward moving ball is in ever more right-ward positions because at earlier times it was moving right-ward. But this demands that we attribute to the ball an intrinsic instantaneous velocity vector (composed of a magnitude = speed, and a direction). If a moving ball is to the right of where it was a moment ago BECAUSE of where it was a moment ago, then the At-At theory of motion leaves out something essential, namely, the dynamic quality of motion, its instantaneous intrinsic potential to be elsewhere. If memory serves, dynamis is the Greek for L. potentia.

My argument is adapted from this article by Frank Arntzenius.

It is undeniable that nature exhibits regularities. But it is prima facie conceivable that these regularities are all of them temporally or spatially 'local' as opposed to 'global.' Consider the law of thermal expansion. Nonquantitatively expressed, the law is that all materials with the exception of water expand upon being heated. But how do we know that this holds in the vicinity of Alpha Centauri? Perhaps some metals contract upon being heated in that neighborhood. I don't believe this, but I grant that it is prima facie conceivable, and that it is prima facie conceivable that all laws are only locally valid.

Thinking about this a bit more, however, we realize that if this is so, then this is a fact about nature: nature is such that none of its regularities are exceptionless. Nature is uniform in respect of being only locally uniform. No matter where you are in space-time, the regularities you encounter there are such that in some other region of space-time they will not be found. But this is just to say that this fact about nature, if it is a fact, is exceptionless. But then this is a law of nature, whence it follows that there is at least one exceptionless law of nature.

Suppose that in a certain room at a certain time everyone is wearing a wristwatch. (Example from D. M. Armstrong, What is a Law of Nature? p. 46) That amounts to an exceptionless regularity relative to that time and place: every human being is a wristwatch-wearer. But there is nothing law-like about this regularity, exceptionless though it is. Suppose Jones is not in the room at the time in question, and consider the statement, 'If Jones were in the room at the time in question, then he would be wearing a wristwach.' Surely this counterfactual is false. But the statement of the regularity is true. Therefore, the statement of the regularity does not 'support' the counterfactual.

Nothing changes if we consider a regularity that holds for all of time and all of space. Even if, always and everywhere, every F is a G, it doesn't follow that every F's being a G is a law of nature. There is more to a law of nature than exceptionless regularity. The something more is nomic necessity.

Contrast the following case. 'Every water molecule contains two hydrogen atoms.' This supports the counterfactual, 'If there were water in this (empty) glass, then there would be molecules containing hydrogen atoms in this glass.'

Though there is more to a (deterministic) law than exceptionless regularity, every law-statement entails a statement of exceptionless regularity.