KR. Richmond Zeno Lecture 3

Knowledge and Reality: Zeno’s Paradoxes, lecture 3 - Page 1

Knowledge and Reality: Zeno’s Paradoxes (lecture 3). Lecturer: Dr. Alasdair Richmond, Dugald Stewart Building - Room 6.11, Office hours: Thursdays 1 - 3 pm,

Tel. (0131) 650 3656, e-mail: [email protected]

LECTURE 3: FRIDAY MARCH 28TH

2014

ZENO’S PARADOXES: SOME CRITICISMS AND REPLIES

Suggestions for Reading:

Black, Dainton and Salmon readings as per previous lecture.

James Thomson, ‘Tasks and Supertasks’, Analysis, Vol. 15, 1954/5, pp. 1-13, also available on JSTOR.

References:

Besides the Black and Salmon readings, as cited before, this handout also makes references to the following but

please note that the following items are listed for reference purposes only, i.e. you’re not expected to read them:

Paul Benacerraf- ‘Tasks, Super-Tasks and the Modern Eleatics’, as in Salmon [ed.], pp. 103-129.

John Earman - Bangs, Crunches, Whimpers and Shrieks: Singularities and Acausalities in Relativistic

Spacetimes, (Oxford, OUP, 1995), pp. 103-104. (N.B. Earman’s book is exceedingly technical in places.)

A. W. Moore - The Infinite, (London, Routledge, 2nd edition, 2001), pp. 39-40.

Wesley C. Salmon - Introduction to Salmon [ed.], pp. 5-44.

Further reading:

Very useful short introductions to the Achilles, Arrow and Stadium paradoxes can be found in Michael

Clark’s Paradoxes from A to Z, (Routledge, 2002) – see esp. pp. 1-4, 7-8, 159-160, 218.

All the extant Zeno fragments, plus commentary and notes, can be found in Jonathan Barnes (ed.), The Early

Greek Philosophers, (Penguin, Harmondsworth, 1987), pp. 150-158.

An excellent critical discussion of Zeno appears in Jonathan Barnes, The Presocratic Philosophers, (London,

Routledge, 1979), pp. 231-295. (Same pagination in the 1st volume of the original two-volume edition).

Summary of contents:

This lecture look at some replies to the arguments of Zeno’s paradoxes, and considers some problems with the

most popular of these replies. There are three chief sections to today’s lecture:

Part 1: Convergent and Divergent Series.

1. introduces the distinction between those infinite mathematical series with a finite sum (converging

series) and those with an infinite sum (diverging series). (The notion of a convergent series is often

invoked to try to defeat the “Dichotomy” and “Achilles” paradoxes.)

2. introduces the distinction between those infinite mathematical series that have a finite sum (converging

series) and those with an infinite sum (diverging series).

Part 2: Actual and Potential Infinity.

1. two kinds of infinity - one is an infinity which is completed at a time, an actually finished infinite thing,

and the other is an infinity which is formed over time, a process or object which can always have more

added to it. (The first corresponds to actual infinity and the second to potential infinity)

2. a common objection to Zeno’s paradoxes is that Zeno confuses potential and actual senses of infinity.

Part 3: Max Black and Infinity Machines.

1. The coherence of “supertasks”: is it possible to complete an infinite series of tasks in a finite time?

2. Black rejects the “convergent series” reply to Zeno and maintains that supertasks are incoherent.

3. Black imagines some machines for performing supertasks and argues that each machine can’t possibly

exist. (These machines include Black’s series of infinite machines for moving marbles and Thomson’s

lamp, which can be switched off and on an infinite number of times.)

4. do Black and Thomson succeed in showing that the very idea of a supertask means that such tasks are

logically (not merely physically) impossible?

mailto:A.Richmond@%20ed.ac.uk


Part 1: Convergent and Divergent Series.

Wesley Salmon sees Zeno’s problems as like an onion, each layer requiring the available

mathematical resources available at any given time. There may be a fallacy buried in Zeno’s

“Dichotomy” and “Achilles” paradoxes, namely Zeno’s assumption that the sum of an infinite

series of finite terms must always give you an infinite sum as your answer. Some infinite

series add up to infinite amounts, but then again some don’t. As an example of the latter,

imagine adding up parts (or fractions) of apples. You begin with nothing, but after half a

minute, you acquire half an apple. After another quarter of a minute, you gain another quarter

of an apple. After another eighth of a minute has elapsed, you gain another eighth part of an

apple, and so on. The result of this process is that after a full minute has elapsed, you will

have one apple, for all that an infinite number of apple-fractions have been added to your

original half-apple. This is a converging series - the fractions being added get smaller and

smaller, and so the infinite addition draws ever closer to a definite, finite limit.1

The fractions add up thus:

1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 ...

And the sequence of partial sums goes as follows:

1/2, 3/4, 7/8, 15/16, 31/32, 63/64, 127/128, …

This sum draws ever-closer to 1 and 1 thus represents the limit to which the partial sums

tend. To put the same point another way, although at no finite point in the series of sums does

the total reach 1, it can be proven that the total of even an infinite number of such diminishing

fractions will not exceed 1 and that, as the number of fractions rises, the sum draws closer and

closer to 1 without limit.

Obviously if the fractions increase in size at each step, then the sum will be infinite

However, not all sums of continually-diminishing fractions are converging. Suppose that

once again you start out with nothing, then after half a minute you add half an apple, but this

time, after another quarter of a minute you add one-third of an apple, and then after another

eighth of a minute you add a quarter of an apple, and so forth. Thus, the sums go:

1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 ...

This sequence sums to an infinite result, although the fractions are diminishing at each

step. At the end of this minute, you will have an infinite number of apples, for all that in no

single operation at any point in the series did you add one whole apple. In this case, the

fractions diminish at each step but they don’t do so sufficiently quickly so as to allow the

series to converge to a finite sum.

1Consider Rudy Rucker’s (Infinity and the Mind, 1982, Harmondsworth, Penguin, 1997: 73) variant of Hilbert’s

famous Hotel with infinitely many rooms. Rucker’s hotel has a ten-feet-tall first floor and every subsequent

floor is two-thirds as tall as the floor below. This building has infinitely many floors but is only thirty feet tall.


Compare Max Black on the battle between Hercules and the Hydra. In the original myth,

one of the labours of Hercules was to slay the eight-headed serpent, the Hydra. However, as

soon as Hercules cut off one of the Hydra’s heads, another two heads sprang into being in its

place, so Hercules’ efforts will not only fail to diminish the task over time but will actually

make things progressively worse, (see Black, MTBQ, p. 126). If Hercules cuts off all eight of

the original Hydra-heads, he finds himself now facing a sixteen-headed Hydra, and if he cuts

off all these heads, he ends up facing thirty-two replacements, and so on.2

Part 2: Actual and Potential Infinity.

We can distinguish between an infinity which is actual, i.e. the result of a completed

infinite process, and an infinity which is potential, i.e. you could keep on performing the same

operation indefinitely many times without finding the task at an end. The actual / potential

infinite distinction could be defined thus: “The actual infinite is that whose infinitude exists,

or is given, at some point in time. The potential infinite is that whose infinitude exists, or is

given, over time; it is never wholly present. ... Imagine a clock, for example, endlessly

ticking. Its ticking is potentially, but never actually, infinite”, (Moore, p. 40). This

distinction relates to the difference between infinity by addition and infinity by division.

Zeno may be correct in thinking that any finite sum, no matter how small, copied an infinite

number of times, will give you an infinite sum at the end, (that is, an actual infinity).

However, any finite quantity, such as a ruler which is exactly one metre in length, can be

divided up into fractions in indefinitely many ways, (e.g. into one hundred centimetres, a

thousand millimetres, a billion nanometres, and so on downwards), but that doesn’t

demonstrate that there are different physical pieces of the ruler answering to these different

descriptions. The potential for dividing a metre length into infinite pieces does not establish

that there actually are an infinite number of discrete pieces lying waiting in the metre ready to

be uncovered. We must be careful to distinguish the possibility of assigning an unlimited

amount of numbers to the stages of Achilles’ journey with there actually being an infinite

number of distinct distances out there for him to cross. Mathematically sub-dividing the

distance with ever-smaller numbers doesn’t change the size of the physical task: “I do not

think that if each pebble were broken into a million pieces the difficulty of getting over the

road would necessarily have increased; and I don’t see why it should if one of these millions -

or all of them - had been multiplied into an infinity”, (C. S. Peirce, see Black, MTBQ, p. 129,

footnote 5). Any physical quantity can be described an infinite number of different ways.

2Cf. too the “Tristram Shandy” paradox: “Tristram Shandy, in Sterne’s famous novel, on finding that it took him

two years to write an account of the first two days of his life lamented that material for his biography would thus

accumulate faster than he could deal with it, so that he could never come to an end”, G. J. Whitrow, ‘On the

Impossibility of An Infinite Past’, British Journal for the Philosophy of Science, Vol. 28, (1978: 39-45):. 42. But

see Clark, Paradoxes From A to Z: 199.


Staccato vs. legato movement: in the first case, Achilles runs for a short interval, then rests

for a short interval; in the second, the run is smooth and continuous. Staccato running seems

to scupper any chances of seeing the motion as merely potentially infinite, since it seems that

there must be an infinity of distinct tasks performed at each stage of the run, no matter how

short: “The staccato runner takes a rest of finite (non-zero) duration between each of his

infinite succession of runs; hence there can be no question that he performs an infinite

sequence of distinct runs”, (Salmon, MTBQ, p. 143). The problem gets worse if we imagine

the staccato runner planting a flag at each stop. However, even these distinct tasks too can be

made the object of a converging series, so that the rests too get shorter in proportion to the

passing of time.

Part 3: Max Black and “Infinity Machines”.

Max Black objects that the “convergent series” reply doesn’t solve the problem of Achilles

and the tortoise, because there is something logically incoherent in the very idea of Achilles

(or anyone else) being able to perform an infinite series of operations. To illustrate this thesis,

he uses several examples of what he calls “infinity machines”, (Black, MTBQ, p. 125).

Black’s objection to infinity machines is not based on the physical difficulties of constructing

such devices but rather that the performance of an infinite series of operations is logically

self-contradictory and therefore absurd.

The simplest infinity machine is probably the Thomson lamp. This is a desk-lamp with a

switch such that pressing the switch when the lamp is off turns the lamp on and pressing the

switch when the lamp is on turns the lamp off. Suppose that a sequence of switching

operations is performed, so the first pressing takes half a minute, the second takes a quarter of

a minute, the third an eighth of a minute and so forth. After a minute has elapsed the

Thomson lamp will have been switched off and on an infinite number of times. The Thomson

lamp has performed a “supertask”, namely carrying out “an infinite number of operations in a

finite span of time”, (Earman, p. 103). Like Black, Thomson held that the completion of such

tasks is logically impossible, i.e. not merely forbidden by the laws of physics and/or biology

but conceptually incoherent and hence impossible.

Max Black objects that the Thomson lamp’s moving parts must traverse an infinite

distance in a finite time. This objection holds if the Thomson lamp remains the same size

throughout the process. However, if the lamp and all its components shrink in proportion

with the elapsed time (so the lamp and all its components are half the size after 30 seconds

that they were at t0, and half of that size after forty-five seconds, and so on), the distances

traversed in each movement also get shorter and shorter. (Cf. Earman, p. 104.)


The Shrinking Thomson Lamp

ON OFF ON OFF ON OFF ON OFF ON OFF

One minute

... etc.

0 t

1 t

Time

Number of presses

1 t 0 t

Let’s assume that initially at t0, the lamp is switched off. Thomson’s question is then: after

the full minute has elapsed at t1, is the lamp on or off? For every one of infinitely many

operations, whenever the lamp was on it got switched off and whenever it was switched off it

got switched on. Every operation, and its opposite, has been performed an infinite number of

times. So, at the end of 1 minute, the lamp seems to be neither off nor on, nor yet can it be

anything else. However, Thomson’s question is about what state the lamp is in at the limit of

the one-minute period we were considering, i.e. after the period of the infinite task has

elapsed. This question is flawed in its framing - what happens after the end of the period of

the supertask is outside the terms of the supertask and is therefore not defined or determined

by the supertask it succeeds, so either outcome is compatible with the supertask as it is

described. The last element in the series is left undefined by the description of the series and

so a final state either of being off or of being on is equally compatible with the problem, since

it must be specified separately.3

Cf. Salmon’s verdict on the Thomson lamp: “If the function approached a limit at t = 1, it

would be natural to extend the definition of the function by making that limit the value of the

function at that endpoint. But the ‘switching function’ of Thomson’s lamp has no such limit,

so any extension we might choose would seem arbitrary”, (Salmon, 2001, p. 31). Moral:

failures of continuity and determinism aren’t failures of consistency.

Max Black (see Black, MTBQ, pp. 125 ff.) also considers a series of infinity machines

which move marbles from one infinite tray to another infinite tray. To our left, a tray full of

marbles stretches into the indefinite distance. Beside the tray stands a mechanical grab or

scoop. To the right, there’s another but empty tray, which also stretches off into the indefinite

distance. This set-up is the first of Black’s series of infinity machines, “Alpha”. Alpha

3“The only reasons Thomson gives for supposing that his lamp will not be off at t1 are ones which hold only for

times before t1. The explanation is simply that Thomson’s instructions do not cover the state of the lamp at t1”,

(Benacerraf, see Salmon [ed.]: 108).


moves a marble from left to right in one minute and then rests for a minute. It moves the next

marble in half a minute and then rests for half a minute, and so on. After four minutes, Alpha

comes to a stop, having moved all the marbles from one infinitely long tray to the other.

The next step in Black’s argument is to imagine that now we are dealing with two

machines and only one marble. A second machine (“Beta”) returns the marble to the first tray

after every single operation of Alpha. Otherwise, Alpha works as before - first, it moves the

marble in half a minute and then rests for half a minute, then it moves the marble in a quarter

of a minute and rest for a quarter of a minute and so on. While this frenzy of marble-moving

is taking place, Beta is doing the same operation but in reverse - namely, moving the marble

from the second tray back to the first one but taking the same length of time both to move and

to rest as that taken by Alpha, save that Beta works when Alpha is resting and vice-versa.

The problem is deciding where the marble is at the end of the four-minute period - for an

infinite series of operations, once it has appeared in the one tray, it has been transferred to the

other. The operation of switching from tray to tray has been performed successfully an

infinite number of times but at the end of the minute, the marble cannot be in either tray, nor

can it be anywhere else either. The objection used in the Thomson lamp case (i.e. what if the

components shrink in time with each operation) may not work here, because if the

components just get closer together, then the marble ends up exactly mid-way between the

two trays, located in neither of them. However, as before, a failure of continuity need not

imply a breakdown in logical consistency.

Part 4: The Arrow Revisited.

We discussed the notion that space and time are atomistic, i.e. that there are minimal,

indivisible units of both space and time. Such units have sometime been dubbed “hodons”

and “chronons” respectively, (taken from the Greek words hodos for “way” and chronos for

“time”). These units are not to be confused with material atoms but rather are the

fundamental building blocks of space and time themselves. Likewise, it’s perhaps better to


talk of hodons and chronons, rather than quanta of space and time, because it isn’t clear that

modern quantum theory is atomistic in the sense we’re considering. According to the hodon

conception, space has a sort of grid structure and objects/events can only be slotted into this

grid no less than one hodon/chronon apart. Likewise, on the atomistic view, events must be

separated by at least one chronon of time. How then do objects move? Well, they effectively

jump discontinuously from one point in the grid to another. Thus, the Arrow doesn’t ever

pass through the lines on the grid but has an existence which consists of successive

appearances at successive places. Is this discontinuous existence really motion as we

understand it? In the Stadium paradox, the problem was to explain how and when the first

element in row C passed any of the intermediary members of row B. The discontinuous

conception of movement seems to reject this question as ill-formed: at no time does C1 pass

(e.g.) B2; rather, the motion consists in C1 having been to the right of B2 and then, in the next

succeeding time, being to the left of B1.

Bertrand Russell advocated a version of what is known as the “at-at” theory of motion. On

this view, all that is required for an object to truly be said to have moved is that it is found at

different places at different times, (hence “at-at”). However, it does not make sense to ask in

which moment (or moments) the object is moving. The concept of instantaneous motion thus

seems to be incoherent on this account. This has some odd consequences. For example, as

the formulation above suggests, we can truly say of the Arrow that it has (past tense) moved

but apparently not that is (present tense) currently moving.

Coda: ‘The Trojan Fly’.

Imagine the race between Achilles and the Tortoise is accompanied by a fly (“Superfly”)

who buzzes continuously back and forth between the two runners.4 Achilles travels at 8 mph

and the Tortoise at 1 mph. When Achilles draws level with the Tortoise, Superfly starts to

travel back and forth between the runners at 20 mph. One hour later, Achilles has a 7 mile

lead over the Tortoise. Questions:

i) Where is Superfly?

ii) Which way is Superfly facing?

4Apparently first proposed by A. K. Austin, in Mathematics Magazine, 1971. See Ch. 2 of Salmon’s Space,

Time and Motion (California, Dickenson Publishing Co. 1975) and Clark, Paradoxes From A to Z, pp. 200-1.


A0 An-2 An-1 An ...

T0

Tn

Tn-1

Oddly enough, the answers are, respectively: i) anywhere on the course and ii) in either

direction. The set-up as described is compatible with an infinite number of different positions

and with either direction. An oddity of the situation is that Superfly has to change direction

infinitely often to carry out the task.

Documents

KR. Richmond Zeno Lecture 3