- Assignments due Monday 9/13/04 – web site.
- Note 1st and last assignments worth 10% each.
2. The Dichotomy – 'division into two' – Paradox
Zeno of Elea (in what is now South Italy): ~490BC – 425BC
- Background: nothing exists of Zeno’s work and so we rely mostly on Aristotle, say 100 years later, and Simplicius, another 850 years later, in the first half of the sixth century C.E. – though he seems to have had originals of Zeno’s paradoxes.
- What is the paradox?
- What is 'paradoxical' about the paradox?
Reasonable sounding premises entail something entirely unreasonable! Zeno assumes 'for the sake of argument' that motion occurs, and shows that it follows that an infinite number of distances must be covered in a finite time – something he believes impossible. He concludes that the assumption is false.
- Other 'Paradoxes':
- Olber's: "Space is infinite, and stars evenly distributed on average, so there is a star in every direction, so why isn’t the night sky light?" – a surprising and false conclusion.
- Russell's: "In my home village everyone has a car, and there’s a mechanic who fixes the cars of all and only those who don’t fix their own cars" – if it’s true then it’s false!
- Liar: "The sentence that I am saying now is untrue" – if it’s true it’s false, and if it’s false it’s true!
3. Responses to the paradox
- Diogenes: an experimental argument, but what about Zeno’s reasoning? Should we believe our senses or our reasoning? And if the former we have to say which premise of the argument is false.
- Aristotle: there's an interval of time for every stretch of track.
- But does an infinity of times (or distances) add up to a finite time?
- Analyze the argument:
1. All (finite) intervals of time (or space) can be divided into two.
2. All intervals have finite length.
3. The length of any interval = sum of lengths of intervals of which it is composed.
4. All infinite sums of finite quantities are infinite.
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C1. All intervals are composed of an infinity of intervals. (1)
C2. The length of any interval = an infinite sum of finite lengths. (2, 3, C1)
C3. All intervals are infinitely long. (4, C2)
- We want to keep 1-3, so what justifies 4? The intuition that if you add two (positive) finite numbers together you get a number bigger than either; so, if you add finite numbers without limit, getting ever larger totals, without limit.
- Cauchy: the problem is that finite arithmetic does not apply to infinite sums! We cannot infer from finite arithmetic to infinite -- as we do in the intuition.
Augustin Cauchy – one of the fathers
of the modern calculus
- In other words, an `infinite sum' has no well-defined meaning: e.g,
what is 1 - 1 + 1 - 1 + 1 - …?
- Cauchy's definition:
- Consider the infinite sum, 1/2 + 1/4 + 1/8 + ...
- In finite arithmetic we can total finite parts of this series to obtain the sequence of 'partial sums', 1/2, 1/2+1/4 = 3/4, 1/2+1/4+1/8 = 7/8, ...
- The partial sums grow without end but they don't grow indefinitely large; they approach ever closer to 1 without exceeding it - 1 is the 'limit' of the sequence.
- We define that the sum of any infinite series to be the limit
of the sequence of partial sums -- if it exists.
- What is 1 - 1 + 1 - 1 + 1 - 1 + ... according to Cauchy's definition.
- But this is only a definition – does it give the right answers? Right in what sense?
- The mathematically correct answer? The rule for adding two numbers
does not determine what would happen if an infinity of numbers were added
together. There is no `right’ answer in this sense – another definition might
be given.
- The empirically correct answer? For example, the usual rule for addition works for apples, but not for the volumes of liquids that are mixed! Similarly we can ask whether Cauchy’s definition works for our theory of space: since it makes lines finite and motion possible it seems to work well!