Circular References and Logic
Sat, 30 Mar 2002 17:02:42 +0100 (CET)
>#1: Statement #2 is true.
>#2: Statement #1 is false.
>Assume statement #1 is true. Then statement #2 is true. But, statement #2
>These two statements contradict each other, and are neither true nor false.
When mathematicians started to try to formalise mathematics in the
19th century, Cantor described the "set of sets that don't contain
themselves" as an example and noted that the ability to write
apparently legit maths containing such nonsense was... umm... a tad
bothersome. <font size=700>Note: this is a great understatement.</font>
Contradictory statements like the above are complete nonsense.
Discussions of them using naive boolean values for truthhood,
falsehood, provability and so on is just total gobbledy-gook weurh tnm
icue rtcwerg cer grs kgjhk jfds dkhpweotrn viupyiuth nwi ouhtg.
At the beginning of the 20th century, Russell and Whitehead showed how
to solve the problem with hierarchies of kinds of statements to avoid
self-referential garbage and that work was improved later on with
typed lambda-calculus, constructive mathematics etc.
Constructive mathematics is difficult. It makes it much harder to
describe and work with even relatively simple objects. Work is
on-going to build computerised tools which take some of the tedium out
of it (e.g., implementations of the calculus of inductive
constructions by some INRIA colleagues of mine). But the advantage is
that you can't even write down nonsense. False things, sure. But not
Vast areas of maths get by fine without using constructive techniques.
It is understood (by those who care) that in principle they could be
re-worked that way if desired. However some areas really need them to
be done right. Even some ooh-ah profound shit like Godel's work is
guilty in this area.
Note: many gung-ho mathematicians, logicians etc are non-constructivists
and would disagree, but that's OK because they're wrong. Cantor's
problem just has to be taken seriously.
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