ZENO’s PARADOXES

b. 490 BC

Student of Parmenides

Wrote over 40 paradoxes, of which only about a half dozen survive, second-hand, through Aristotle and Simplicius.

Zeno of Elea

Who Was Zeno’s Target?

• We don’t really know, but here are two possibilities.

• Opponents of Parmenides, his teacher. Parmenides argued that there can only be one thing. Others ridiculed this idea. Plato said Zeno wanted to “pay them back in their own kind, with something for good measure.”

• Advocates of the standard Euclidean view of space. Space can be neither infinitely divisible nor atomic, thought Zeno. The Euclidean hypothesis is incoherent for him.

Achilles & the Tortoise Achilles can’t catch Tortoise

The Dichotomy Motion is impossible

The Dichotomy

1. All segments (finite lengths) can be divided into two segments

C1. All segments can be divided into segments without limit.

C1’. All segments are composed of an infinity of segments.

2. All segments have finite length.

3. The length of any segment = sum of lengths of segments that compose it

C2. The length of any segment = an infinite sum of finite lengths.

4. All infinite sums of finite quantities are infinite.

C3. All segments are infinitely long.

The Dichotomy

1. All segments (finite lengths) can be divided into two segments

C1. All segments can be divided into segments without limit.

C1’. All segments are composed of an infinity of segments.

2. All segments have finite length.

3. The length of any segment = sum of lengths of segments that compose it

C2. The length of any segment = an infinite sum of finite lengths.

4. All infinite sums of finite quantities are infinite.

C3. All segments are infinitely long.

Euclidean

Aristotle’s Response

• There is enough time. Time divisible just like space. To get halfway takes half as much time.

• Difference between potential and actual infinities... Can’t cross an actual infinity, but can a potential infinity.

The Dichotomy

1. All segments (finite lengths) can be divided into two segments

C1. All segments can be divided into segments without limit.

C1’. All segments are composed of an infinity of segments.

2. All segments have finite length.

3. The length of any segment = sum of lengths of segments that compose it

C2. The length of any segment = an infinite sum of finite lengths.

4. All infinite sums of finite quantities are infinite.

C3. All segments are infinitely long.

Premise 4

• Finite arithmetic doesn’t tell us what a “sum of an infinite collection of numbers is.” As Huggett says, the operation ‘+’ in finite addition will only ever yield finite sums.

• Until we have a theory of how to add up an infinite series of terms, we can’t conclude that an infinite sum of finite quantities is infinite, i.e., we can’t conclude premise 4.

• Do we have such a theory?

• Hint: 1/2 + 1/4 + 1/8 +... will never sum to a number equal to or excelling 1.

Some History

• Newton and Leibniz independently invented the calculus for mechanics. They needed tangents to a curve and such for defining instantaneous speed, acceleration, and so on. Doing so, they invoked the notorious concept of an infinitesimal.

• Infinitesimal: not zero, but less than any assignable value.

• Worked great! E.g. Principia Mathematica...

But...

• George Berkeley’s, The Analyst, 1734.

• “And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?”

• Berkeley was right.

Later...

• Need for new mathematical concepts: compactness, connectedness, continuity, convergence of an infinite series, derivative, dimension, function, integral, limit, measure, and much more.

• Enter Euler, Bolzano, Cantor, Cauchy, Dedekind, Frege, Hilbert, Lebesque, Peano, Russell, Weierstrass, Whitehead, and others.

Agustin Louis Cauchy

• 1789-1857

• 1820’s: Cauchy gave rigorous definitions of limits, continuity, derivatives, and the definite integral.

• 1817: Bernard Bolzano did too, but work not as well known.

Convergence

• An infinite sequence of increasing positive numbers {n1, n2, ...} converges to a finite limit L iff, for all numbers e>0, there is some number d such that for all (integers) c ≥ d, |L - nc| < e.

• The sum of an infinite series s1 + s2 + ... is the limit of the sequence of sums {s1, s1+s2, s1+s2+s3...}, if such a limit exists.

• lim {1/2, 1/2+1/4, 1/2+1/4+1/8...} = 1

• lim {1, 1+1, 1+1+1...}

www.gresham.ac.uk/

The Dichotomy Premise 4

4. All infinite sums of finite quantities are infinite.

No!

The Paradox of Plurality

• Goal: a contradiction in the idea of an extended object with parts.

1.All finite segments are composed of an infinity of identical points.

2.The points have either zero length or finite length.

3.If zero length, then the total length of the segment is zero.

4.If finite length, then total length of the segment is infinite

5. 2,3,4 imply not-1. how far am I from you?

Limits to the rescue?

• No!

• The points are identical, so they each have length L. Suppose L is, say, 1. Then the infinite sum is 1+ 1+ 1..., and so the sequence is {1, 1+1, 1+1+1,...}, which has no limit. Limits won’t get us out of premise 4.

• And don’t limits commit us to premise 3? After all, the limit of the sequence {0, 0+0, 0+0+0, ...} is indeed zero.

• Uh oh.

Side Excursion: Really Big Numbers

Some infinities are bigger than other infinities.

No way:

{1, 2, 3, 4, 5. 6...} positive integers

{1, 3, 5, 7, 9, 11...} positive odd integers

{2, 4, 6, 8, 10, 12...} positive even integers

The evens and odds can be mapped into the positive integers, one-to-one, without remainder. Therefore they are the same size.

Cantor’s Diagonal Argument

Some infinite sets cannot be put into one-to-one correspondence with each other. Georg Cantor (1845-1918) proved this in general, but we’ll suffice with one example.

The real numbers...

Natural #’s Real #’s

0 --------------0.1212... 1 ------------- 0.3333... 2 ------------- 0.7000... 3 ------------- 0.1412...

: :

How to find a real # not paired with a natural...

Look at the number in the first column and the first row and write down a 0 if that number is a 1 and a 1 if that number is anything other than a 1. Do the same for the next step in the diagnol, the number in the second column, second row, and so on. This procedure will give us a new real number, in this case, 0.0111... .

Is this real paired off with one of the natural numbers? No! By construction, it differs from the first real in the first decimal place, the second real in the second decimal place, the third real in the third decimal place, etc., ad infinitum. It can’t be one of them. And that means that the reals can’t be paired with the natural numbers, which in turn implies that the set of real numbers is bigger than the set of natural numbers.

Countable v Uncountable Infinities

C D

A B

• The sequence {0, 0+0, 0+0+0...} is a countable sequence. But the points on the real number line are uncountable. So Cauchy’s method of infinite summing doesn’t apply.

• In the Dichotomy we tried imposing finite summation on countably infinite sequences -- didn’t work -- and now we’ve tried countable summation on the uncountable -- doesn’t work.

• So, what is the answer? Argument not valid...

• Very deep point: length, and other metrical notions, aren’t intrinsic features of collections of points. Segments don’t inherit their lengths from the points that constitute them. Length is logically independent of points. Points do not have zero length -- they have no length at all.

1. Instants have no parts. 2. If the arrow moves during any instant, then that instant has earlier and later parts.

C1. The arrow doesn’t move during any instant.

C1. The arrow doesn’t move during any instant.3. If the arrow doesn’t move during any instant, it doesn’t move at all.

C2. The arrow doesn’t move at all.

Zeno’s Arrow

“At-at” theory of motion

• William of Occam; Bertrand Russell

• The basic idea here is that motion is a feature of being in different locations at different times (in the right way), and that is that. So it is true that there is no motion during any instant. But motion, says the at-at theory, has nothing at all to do with what happens during instants; it has instead to do with what happens between instants.

• “We must entirely reject the notion of a state of motion…Motion consists merely in the occupation of different places at different times, subject to continuity…” Bertrand Russell, Principles of Mathematics.

“Dynamic Theory of Motion

• An object’s motion at a time is independent of the object’s position at other t imes ; mot ion i s an in t r ins ic p roper ty, a d i s p o s i t i o n t o b e elsewhere than the place it is.

Science and States of Motion

The average velocity v* is

v*=Δx/Δt

where Δx= x(final) –x(initial) and Δt= t(final) –t(initial).

The instantaneous velocity v is

The important thing to note about this is that its value is determined by what is going on at t’s and x’s other than t and x. Velocities are determined by what happens at other locations and times than x and t.

Take- Home Points

• Dichotomy: need notion of a convergent limit of a sequence

• Plurality: metrical notions, e.g., length, logically independent of intrinsic features of points.

• Arrow: need at-at theory of motion

Is it all just definitions?

Supertasks!

http://www.timtim.com/public/images/drawings/large/Weightlifter-2.gif

Impossible... according to Thomson and Black

A B

At t=0, ball at A; at t=1/2, ball moved to B; at t=1/2+1/4, ball moved to A, etc. At t*=1, where is the ball?

“the state of a system at an instant t* is not a logical consequence of which states he has been in before t* “

Benacerraf’s Critique quote from SEP’s “Supertasks” entry

Laraudogoitia 1996 A Physical Supertask