Dialectics and maths
wpc at cs.strath.ac.uk
wpc at cs.strath.ac.uk
Mon Feb 27 08:36:17 MST 1995
I think that there may be more of a connection
between a materialist interpretation of maths
and dialectics than is immediately apparent.
Turing's proof of the impossibility of contructing
a program to say if any arbitrary other program will
halt relies upon showing that the existence
of such a program would be inherently paradoxical.
Assuming that such a program h(x)
which returns true if x halts exists, we can
write another program h'(x) of the form :
h'(x) = if h(x) then loop forever
This then gives us a possibility of constructing
a negation of a negation - we apply h to
deterimine if h' will halt when h' is applied
to itself. We then get the result that h
will halt only if it will not halt - an inherent
contradiction.
At one level this is just an application of
reductio ad absurdam, but it is significant
that both this and earlier paradoxes
that caused problems for logic - the liar
paradox or Russell's barber paradox involve negation
of the negation. What Turing does is show that
when interpreted in a materialist fashion - ie
in terms of a machine or mechanism for analysing
formulae, paradoxes involving the negation of
the negation are transformed into non-halting
programs. These involve recursion to arbitrary
depths as the machine oscillates between steps
in which it assumes the proposition to be true
and states in which it assumes it to be false.
In doing this he reveals that in a materialised
logic self negation leads to temporal oscillations.
The logic of the computer on which you are reading
this derives its dynamism from this. It evolves
in time thanks to its circuits being provided
with an oscillating clock signal. The clock
is basically derived from an inverter with
feedback. A= not A may make no sense in
conventional logic, but one can readily put
together a logic circuit of this form. It will
oscillate and the resulting oscillations
can form a clock signal driving all the
rest of a computer's logic.
Hence the old Hegelian association between
the negation of the negation and becomming
is more than an empty phrase.
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