Paradoxes from A to Z Paperback – 20 Sep 2012
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Praise for previous editions:
‘Self-contained courses in paradox are not usually taught as part of a philosophy degree. There is good reason for thinking they should be, and this book would make the ideal text for just such a course.’ – Times Higher Education Supplement
‘Clark’s survey is an entertaining junkshop of mind-troubling problems.’ – The Guardian
‘Paradoxes from A to Z is a clear, well-written and philosophically reliable introduction to a range of paradoxes. It is the perfect reference book for anyone interested in this area of philosophy.’ – Nigel Warburton, author of Philosophy: The Basics
‘An excellent book … Clark’s masterful discussion makes this one of the best general introductions to paradoxes.’ – James Cargile, University of Virginia, USA
‘Very well done … a useful complement to the existing literature.’ – Alan Weir, University of Glasgow, UK
About the Author
Michael Clark is Emeritus Professor of Philosophy at the University of Nottingham, UK. He is editor of the leading journal Analysis, and has published widely in a variety of areas, including philosophical logic and the philosophy of law.
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For the price, it seems too ambitious -- accessible for the 'weekend philosopher,' but only if they have a serious grasp of textbook logic and math.
Contrary to popular belief, the word "paradox" has not one but two basic meanings.
One kind is an irremediable contradiction between two or more things. (An example of this is "What happens when an unstoppable force meets an immovable object?")
The second kind is an apparent contradiction that is in fact not a contradiction when viewed insightfully. (The insight sometimes requires showing that the description of the paradox is not meaningful.) An example of this is the algebra paradox: Suppose a = b = 1. Then a^2 = ab, so a^2 - ab = 0, so a(a-b) = 0. Now dividing both sides by a-b shows that a = 0. But we assumed a = 1. (Resolution of this is left to the reader.)
This book mostly focuses on the second kind, paradoxes that can be explained away with the right point of view. (An example is Zeno's paradox that one can never finish walking to a destination, since first one must reach the halfway point, then reach the 3/4 point, then reach the 7/8 point, ad infinitum.)
Unfortunately, the author does not necessarily get everything right. His most foolish moment is when he grapples with the Numbered Balls paradox (sometimes called the Numbered Marbles paradox). His formulation goes roughly like this: From infinitely many numbered balls, balls '1' and '2' are tossed into a box (of infinite capacity) at 1 minute to noon, but '1' is removed. Then at 1/2 minute to noon, balls '3' and '4' are tossed into the box but '2' is removed. At 1/3 of a minute to noon, balls '5' and '6' are tossed into the box but '3' is removed. Etc. all the way up to noon.
The question is: At noon, how many balls are in the box? The paradox is that, on the one hand, the number of balls in the box increases — infinitely many times all the way up to noon — so there are apparently infinitely many marbles in the box at noon. But, on the other hand, every numbered marble is removed from the box at some point before noon — and never moved again — so there must be no marbles at all in the box at all.
This is one of the paradoxes the author gets terribly wrong. He says: "Although noon is approached asymptotically, each of the numbered balls is thrown out *before* noon, and replaced by two other numbered balls *before* noon. The infinite series does not include noon itself, and therefore the state at the limit, noon, is not determined."
The fallacy of this "argument" is seen by imagining a line that is painted to have a length of 1/2 (unit) at 1 minute to noon, then extended by 1/4 unit at 1/2 minute to noon, by 1/8 of a unit at 1/3 of a minute to noon, and in general by 1/2^n of a unit at 1/n of a minute to noon. How long is it at noon has a perfectly good answer: 1 unit. Likewise, the numbered marbles paradox makes perfectly good sense mathematically and has an unequivocal answer: There are no marbles at all in the box at noon, because the statement beginning with "But, on the other hand" is telling: Each marble has been removed at some point before noon and never moved again.
The author was made aware of controversy surrounding some of his "explanations", including this one, but did not "update" them for this 2012 updated version of the original 2007 book.
* Another large source of paradoxes in one place is the Wikipedia article titled "List of paradoxes".
"So the sequence of partial sums is 1/2, 3/4, 7/8,... It goes on for ever, getting closer and closer ('converging') to 1. In this case 1 is the limit, and so the sum, of the series. Achilles gradually closes in on the tortoise until he reaches it."
What? The "explanation" continues by simply explaining limits. This is inane hand waving. Worse still, Clark cites Salmon's excellent collection of articles on Zeno's paradoxes (of which Achilles and the Tortoise is one). A main theme of many of the articles in Salmon's book is that limits do NOT dissolve the paradox.
In the same entry as Achilles, Clark discusses Thomson's Lamp, where the dominant line taken today is that there is no spatio-temporal continuity through an infinite sequence of tasks. "But the description of the supertask entails nothing about the lamp's state at one minute..."
So be it. But then in "explaining" Achilles, Clark writes, "Why then is Achilles at the limit, 1? ... The answer is that, if he is anywhere, as surely he is- he must be at 1."
The problem is the "as surely he is." This echoes Thomson's own, "Surely the lamp must be on or off." If there is no (spatio-temporal) continuity through an infinite task, as was just explained to dissolve Thomson's Lamp, how is there continuity in the case of Achilles? Put differently, why does a limit process tell us about Achilles but not the Lamp? That Thomson's sequence (0,1,0,1...) i.e. (off, on, off, on...) does not have a limit is precisely his point.
There may be a response involving discrete versus continuous tasks, or some other explanation. But the reader is not told this. In general, my problem with the book is that the treatment is superficial and often presents the paradoxes as though they no longer present problems, when in fact they do. An intelligent reader may also be left bewildered by some of Clark's "explanations."
I wonder for whom this book is intended. I think that this book would make a terrible introduction to paradoxes, but may very well serve as a good reference book for those already acquainted with many of the paradoxes.
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