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Known Unknowns 205
spread through the entire body. Think about it. If 1 am allowed to prove
anything either way, of course, my system is complete. It can say anything
it wants, but the proofs I make are worthless.
Let us imagine, for a moment, we created a new system of
mathematics where all the numbers in our new theory behave as we
expect, except for the numbers 5 and 6. You may use them to count, but
they are also equal to each other! This feels bad and it certainly breaks the
Peano axioms. In my new system | plus 5 and 0 plus 5 are the same, so I
can equate 0 to 1. Because 0 and | are the basis of binary arithmetic, all
numbers can be equated. Numbers now have no guaranteed meaning in
my system and, what is worse, since logic uses 1 and 0 to represents true
and false, all of logic falls apart as well. Whenever we allow inconsistency
into mathematics it rapidly brings the whole pack of cards down.
The example I gave was glaring; an inconsistency right in the
middle of the counting numbers! Maybe I was too aggressive and a
subtle and less damaging inconsistency might be tolerable. However,
any inconsistency allows me to make zero equal one somewhere in my
system and, therefore, any theorem based on proof by counterexample
will be suspect.
There might be systems where inconsistency could be a legitimate
part of a mathematical system, but I would always need positive
corroboration for each proof. IfI tried hard enough, I could always prove
something either way. I would need to formulate a new mathematical
rule — something like “I will believe short, sensible-looking proofs to be
right and circuitous proofs to be wrong.” Mathematics would be a bit like
a court of law. You would have to weigh up the evidence from a variety
of sources and the verdict would be a matter of subjective opinion rather
than objective fact. Inconsistency is very bad in mathematics.
The Lucas Argument
J.R. Lucas of Oxford University believes Gédel’s theorem says something
fundamental about the nature of the human mind. In 1959, he wrote a
paper, Minds, Machines and Gédel, where he argued humans must be able
to think outside a fixed set of formal rules. The paper has been causing
arguments ever since. Strong AI proponents have a visceral reaction to
it. Forty years later, in 1989 Roger Penrose picked up the baton and put
the Lucas argument on a stronger theoretical footing. The Lucas-Penrose
argument is this:
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