Thursday, July 9, 2009
Professor: Calman found in practice what Wittgenstein discovered decades before, the series 2, 4, 8 can be followed by 16 but also by 10 or 7004. It's always possible to find a rule or justification that allows a series to continue by any number, it all depends on how complicated the rule is.
The Oxford Murders
I have been pondering over Wittgenstein's Rule-Following paradox since a couple of days, such a splendid thought-inducer it is. Wikipedia's entry is good, but pretty technical. The above quote from the movie The Oxford Murders is an easily understandable version of the paradox. A logical series in a series of numbers governed by a rule. 1, 2, 3... what comes next? 4, you would say. Yes, it could be 4, but if we are to believe Wittgenstein, it could be ANY number at all, because it would be possible to find a logical rule justifying whatever number it would be. [It could be, for instance, be 5 (2+3=5), as well as 8 (2 x 1 + 1= 3, 3 x 2 + 2=8). It could be any number at all, provided you are intelligent enough to think up a justification for that.]
What is an illogical series? An illogical series would be a series of numbers which is following no rule. But, if we are to accept Wittgenstein, it would always be possible to think up of a rule (sufficiently complex, of course) that would be able to explain those numbers in a series which was apparently following no rule. Hence, any illogical series can potentially be converted into a logical one according to this paradox.