Infinite Sequences: Finitist Consequence
Revised draft of my 2003 (Brit. J. Phil. Sci. 54, 591-9)
The following four sections are based upon a draft of my 2003 (here) from which several tedious details have been omitted and to which some more recent thoughts (mostly from 2004) have been added. (Sections 3 and 4 link to separate pages.) My motivation remains the thought that, despite extensive work on set theory during the last century, the distinction between actual and potential infinities remains rather obscure.>
Fletcher’s (2007: 531-51) introduction to that distinction is exceptionally good; but note that while for Fletcher there might conceivably be an endless sequence of physical objects, in some infinite space-time (2007: 565), about which our talk should nonetheless be rigorously constructive, my more platonistic view is that while our talk of reality should always aim to be realistic, even if that necessitates some imprecision, it so happens (see my unpublished) that it is structurally impossible for physical continua to contain endless sequences of objects. So, since for me the relevant distinction lies further from logic than has usually been assumed, by those who (like myself) assert the potential infinitude of the natural numbers (e.g. by Dummett 1993), and lies instead within metaphysics (cf. my 2005), hence I think that the following scenarios might at least help to illuminate that distinction, because of their relatively simple conceptual framework, of classical physics.
1. Indeterminate Summations
References
When we consider the behaviour of À0 (i.e., one per natural number) hypothetical objects, we naturally assume that the combined effect of À0 vectors would be their classical mathematical sum, i.e. the limit of their initial finite sums. Such sums depend upon the ordering of their terms (since that determines the initial finite sums)—in particular the sum of (–1)n·n(–1) for n = 1, 2, 3, … can be any real number or even unbounded, depending upon the order of summation (which need not be the order implicit in the notation, i.e. that of the n).
However, we also naturally assume that such objects would not be intrinsically ordered, but would rather coexist (within their common space) non-hierarchically. For a simple illustration of such coexistence, think of a magnet being brought up to a piece of paper, upon which are some iron filings. However many there were, the filings would be affected simultaneously by the magnet’s approach, each filing being affected independently of the others (cf. my 2005: 104).
For another illustration, consider the following instantiation of the Grandi series, 1 – 1 + 1 – 1 + …, which has no standard mathematical sum because the initial finite sums (first 1, then 0 = 1 – 1, then 1 = 1 – 1 + 1, and so forth) oscillate between 0 and 1 endlessly. Let each ‘1’ of the Grandi series represent the matter of a virtual particle moving leftwards, and each ‘–1’ represent the anti-matter of a virtual anti-particle moving rightwards, so that an w-sequence (a sequence with the order of the natural numbers in their natural ordering) of virtual particle-anti-particle pairs forming spontaneously from a fluctuating vacuum (within some infinite space) might be represented as (1 – 1) + (1 – 1) + …. After a short while the pairs annihilate each other as 1 + (–1 + 1) + (–1 + 1) + …, the net result being the creation of one particle (and while that is not paradoxical, it illustrates how spatial coexistence matters, cf. Moore 1990: 29).
Those two natural assumptions might conflict, were the natural numbers really only a potential infinitude, and so the first part of my 2003 considered such counter-intuitive physical instantiations of (–1)n·n(–1) (for n = 1, 2, 3, …) as an instantaneous collision between À0 classical spheres (for the details, see my 2003), which stick together upon collision, so that kinetic energy is not necessarily conserved, but which do obey the following principle of the conservation of momentum—the sum of the momenta of the interacting particles before the collision equals the sum of their momenta afterwards.
After the collision, the total finite mass will necessarily move with some (possibly zero) component of velocity in some given direction of the collision plane (a disappearing mass being assigned an unbounded velocity). Because momentum is conserved, that is also the value of the sum of the components of the À0 momentum vectors in that direction before the collision. And because the masses have that total momentum before the collision, an ordering corresponding to that value of the sum cannot be imposed upon them by the collision itself. It might not even depend upon there being a collision, in the sense that in a similar set-up (wherein, for example, additional masses interacted to prevent the above collision) there would be the same intrinsic ordering, the same total momentum. (Furthermore, imagine the final mass, with some finite momentum, exploding into the initial pieces which are connected by elastic threads and so collapse back into the same finite mass. There is of course no conservation of energy there, but could there be no conservation of momentum? Could the final momentum, of the finite mass, be different to the initial momentum? Note that a failure of energy conservation is not unrealistic, given our quantum-mechanically fluctuating vacuum, but that momentum is always conserved, e.g. by the virtual particle-pairs that appear and disappear in the vacuum having zero net momentum.)
Since it would be natural to regard the summation of the momentum vectors before the collision as the cause of the resulting velocity of the total mass, we might conjecture that À0 particles would have, behind their transfinite cardinality, a hidden ordinal. Prima facie that may not be an unreasonable conjecture, because transfinite arithmetic is usually based upon ordinal arithmetic. And an obvious ordering would be given by the magnitudes of some physical property, such as mass or momentum. But what about when they have the same values; and anyway, why would any such choice be the correct one? Answers are not easily found, and it is similarly unclear whether or not exactly similar masses would necessarily move with the same velocity, after an exactly similar collision. But what is relatively clear is that if the infinite collection had such an intrinsic ordering, then any finite subset of particles would have one, and that might conflict with our basic notion of cardinality.
For another example of a counter-intuitive physical instantiation of (–1)n·n(–1) for n = 1, 2, 3, …, consider À0 such forces, acting upon some mass, and its resultant acceleration—it would be natural to assume that any acceleration of a unit mass is caused by a net force of the same value because, after all, the possibility of such an explanation is why we postulate the existence of forces in the first place (again, for details see my 2003). And similar vector conjunctions could occur in any physical scenario that gives us À0 directed quantities (with various arbitrary values); e.g., another model for this paradox involves constant fluid flows into and out of a reservoir, which would fill or empty at an indeterminate—and possibly variable—rate. So, since the root of the counter-intuitive behaviour (such as it is) appears to be the infinite summation, rather than its instantiation within any particular kind of physics, it is surely worth considering the possibility that the transfinite À0 is at fault (as I shall do in the rest of this essay).
It is of course also possible that it is our faulty physical intuitions that are the underlying problem. Perhaps there is a plausible treatment of classical indeterminism that would resolve our worries (e.g. could the net momentum before the collision be a range of values, with the collision forcing just one to be selected, so that my statement of the conservation of momentum is what would need revising? The idea of the pre-collision spheres having such a range is certainly an odd one), although there are well-known problems getting realistic theories of indeterminism (see my unpublished). And Furthermore, if the classical mathematics of infinity really is wrong (despite its apparent coherence, and even were alternative resolutions of the scenarios above possible) then we might be warping our physical intuitions in unrealistic ways whenever we try to understand such scenarios in a standard way. So we should also, in order to be as justifiably sure as we can be of our applicable mathematics, consider seriously finitistic resolutions, those that also appear coherent.
So we should occasionally suspend the standard way of judging such scenarios (which is to regard any other explanation that is not self-contradictory as preferable) and look again at all the evidence—in particular, see my unpublished. And although scenarios like the above do not show so much (as my unpublished), they are an example of the kind of counter-intuitive order-dependent behaviour that we could expect from À0 hypothetical objects (given a physics that is prima facie coherent), so they may well give us some insight into the meaning of the standard claim that the natural numbers form an actual infinitude. In the next two sections I shall therefore consider a more paradoxical example of such, i.e. Benardete’s (1964) before-effects.
To keep the physics to a minimum (the underlying mathematics being in question), consider to begin with some non-interacting balls being put into a box one at a time. Repeated endlessly (starting with an empty and possibly infinite box and a plentiful supply of balls) we would have first an empty box, then one ball in the box, then two, three, and so forth. If each successive ball-addition were to take half the time of the previous one then, after a finite time (i.e. twice the time it took to put the first ball in) there would be À0 balls in the box (for an online introduction to such supertasks, see Perez Laraudogoitia 2004). (And incidentally, for the kinematical possibility of putting À0 balls into a box, given only finite accelerations and a maximum speed, see Allis and Koetsier 1995: 245-6.)
So, from that w-sequence of single ball-additions, we obtain the following (w + 1)-sequence for the number of balls in the box: 0, 1, 2, 3… À0. So far, so straightforward—but now consider the task of removing a ball from this box of À0 balls. Removing one would leave À0, and so another ball-removal would also leave À0, and so on. Upon completion of this supertask we might be left with any number of balls in the box, depending upon how the balls were removed, but let us assume that we are left with an empty box (such as we started with). The number of balls in the box would stay the same until we were no longer removing them, when it would suddenly be zero (we may assume).
That is, the (w + 1)-sequence for the number of balls in the box would be À0, À0, À0, À0… 0. Now, a certain counter-intuitiveness arises merely because, had it been a large finite number suddenly becoming zero, then that would have been because they were all removed at that time. And the infinite is of course unlike the finite (which is why extrapolating from finite cases to infinite cases is generally unreliable). Nonetheless, such (w + 1)-sequences are stranger than just that; e.g., even though the balls were only added or removed one by one, the intermediate states of the box—between it containing 0 balls and À0 balls—generated by the single ball-additions are completely different to the intermediate states—between there being À0 balls and 0 balls—generated by the single ball-removals. Now, even if we can see how two such different routes should arise from the transfinite mathematics, the question remains, is such applicable mathematics physically plausible?
(Cf. how every single natural number is in the first no more than 0% of the totality of all and only those same natural numbers in their natural ordering. While that would certainly be a reasonable property were that totality a potential infinitude, it seems implausible that objects actually coexisting within their common space should possess it—yet it would be possessed by any instantiation of the actually infinite À0, even the most plausible.)
Furthermore, although there is a (1 + w*)-sequence (an w*-sequence is the reverse of an w-sequence) from À0 to 0 that does go through all the natural numbers, generated by the time-reverse of the w-sequence of single ball-additions, there is similarly a (1 + w*)-sequence generated by the time-reverse of the single ball-removals, which is a way of putting À0 balls into a box, one by one, so that there would, at any time after starting, already be À0 in the box, and only the final, finite number would remain to be added—the number of balls in the box would be 0, …À0, À0, À0, À0, which has some very counter-intuitive instantiations, e.g. Benardete’s (1964) before-effects.
Suppose that the balls each have unit mass, and that there is a man underneath the box. At any time after starting this sequence of single ball-additions, there would already be À0 unit masses above the man. So although he would presumably have been crushed flat by all those balls, at the instant of his death (assuming that his death is instantaneous) the box would have been empty. Whereas someone’s death might naturally follow (as an after-effect of) his being hit by, say, a falling piano, this man’s death would be a bizarre before-effect of our w*-sequence of single ball-additions.
It is bizarre because he could not be crushed at any time after the balls have begun to be added unless the presence of À0 balls in the box had already failed to crush him, which is why any crushing would have to be a before-effect, but which also means that there might not be any crushing at all—we must ensure that our intuitions for crushing (rather than for, say, his teleportation) as our method of avoiding a contradiction (with the fact that a crushing weight was above him during the whole of that w*-sequence) do not only come from our thinking of similar finite cases (as they might naturally, if unreliably, tend to do). In scenarios involving everyday objects, various alternatives might be excluded via various implicit assumptions (e.g. the need for individuals to be spatio-temporally continuous), but it might be better were more made explicit, as in a classical physical scenario. E.g., Benardete (1964: 234-61) considered the following before-effect of À0 solid boards, which involves everyday objects but also resembles the classical physical scenario described at the end of this section.
A man is walking, from the negative-x-direction, towards an w*-sequence of vertical boards, located at x = 1 (that board having width ½), x = ½ (width ¼), x = ¼, and so forth, limiting to x = 0. All the boards (even the extremely thin ones) are completely rigid and impenetrable. What happens when the man reaches x = 0? He cannot go on, there being no free space into which he could move, but nor could he be stopped by any of the boards, since to get to any of them he would have had to have passed through À0 of them. So, before he reaches any of the boards he is stopped by a before-effect (of their totality) at x = 0.
Or rather, he would be were his halting at x = 0 physically possible, and that depends upon whether or not he needs to be stopped by something more tangible than the open end of that w*-sequence of boards (although it is plausible that he would not, since the space ahead of him is unavailable for him to move into because it is occupied by that w*-sequence); and also were there no other physical possibilities (e.g. teleportation), etc. And incidentally, note that were each of the boards covered with wet paint, for example, which would paint the man were he stopped by that board, nonetheless there would be no need for the man to become, not only halted but also painted (rather mysteriously by paint that came from none of the boards), because there is no reason why such paint should appear—in order to avoid a contradiction, he only has to halt (or teleport, etc.).
Still, it may well be more plausible for the before-effect to be À0 boards ceasing to be rigid and impenetrable; or similarly, for there to be no resolution on the grounds that, with the boards defined to be rigid and impenetrable, the scenario is unrealistically defined. Extremely thin boards would not really be rigid and impenetrable, and so if we think about this scenario realistically we are likely to imagine the man crashing through the first À0 boards. Nonetheless, classical particles are by definition (and rather more plausibly) both rigid and impenetrable (within a physics that is presumably coherent), whilst being of arbitrary size (even point-sized), so consider the following, which is (at least for Angel 2001: 349) an example of a contact interaction within Newtonian mechanics.
A rigid and impenetrable mass approaches (from the negative-x-direction) the open end, at x = 0, of an w*-sequence of masses, at x = 1, x = ½, x = ¼ and so forth, colliding not with any of those masses, but with their accumulation point, x = 0, where it halts or rebounds. Note that although such a collision might seem non-Newtonian (e.g. via Newton’s first law of motion, cf. the end of §3; or because it is intuitively non-contact, there being no contact between the masses, cf. the contradiction of Priest 1999), the effect of excluding such interactions from a Newtonian physics in which À0 objects are allowed would be counter-intuitive, because surely a mass should be able to collide with the other end of that w*-sequence (cf. Perez Laraudogoitia 1996, and see below), and outlawing only certain arrangements of particles would seem to go against classical physical intuitions (although see Fletcher 2007: 575). Nonetheless, including such interactions seems to lead to such conclusions as a Newtonian “space-time weave” (which also seems to go against classical physical intuitions), as follows.
My 2003 grew out of conversations with Peter Fletcher about supertasks (at Keele University, in 2002), while my later thoughts also benefited from contact with Leonard Angel’s and from the discussion that followed my seminar ‘Was Zeno a Zero?’ (at the University of Glasgow, in 2006).
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