Mikael Johansson wrote:
> I wouldn't place myself in any of these slots. It would be one thing if he
> came down unto us and spake of our misunderstanding -- i.e. "Hey, you
> humans! Your logic is _incompatible_with_the_universe_ you live in, but
> again, that is understandable, you were built that way..."; that's a
> statement of the relationship between the mathematics and the real world,
> but it still wouldn't influence the validity of the pure mathematics --
> axiomatic systems still would be valid, they just wouldn't say anything
> useful about the world around us.
This is very helpful as it helps isolate the source of disagreement. That
mathematics might be full of sound and fury but signifying nothing is of
course not particularly comforting. Someone like Goedel, who thought we had
special intuitionistic access to the logical/mathematical structure of the
universe, I am sure is rolling over in his grave at your suggestion that
this might be the case.
> If we 'solve' the threefoil by embedding it in 4-space, we step out of the
> original system. In 3-space, the problem of finding such an isomorphy is
> definitio unsolvable.
> 'Violations of our most cherished logical laws' cannot be understood
> their axiomatic system -- but these violations form _ANOTHER_ axiomatic
> system, with its own validity, its own pecularities and its own results.
> study of it will with any luck even be rather interesting; but it doesn't
> have any impact whatsoever on the axiomatic systems already in study,
> they build on different axioms.
In response let me quote from the great fideist Descartes, (who often goes
around in rationalistic drag):
"I would dare not even dare to say that God cannot arrange that a mountain
should exist without a valley, or that one and two should not make three;
but I only say that He has given me a mind of such a nature that I cannot
conceive a mountain without a valley or a sum of one and two which would not
be three, and so on, and that such things imply contradictions in my
conception." (Letter to Arnauld, 29 July 1648).
With respect to your example, Descartes would have to say he cannot conceive
of finding such an isomorphy in 3-space because God has given him such a
mind, but he dare not say that god could not find such a solution. Of course
you will say that this would be to change the axiom system. Descartes will
respond that this may not be so, to assume that it must be the case is to
assume that your understanding extends as far as God's will along this
vector. Of course what the transhumanist fideist must do here is substitute
a created God III here for God.
You didn't warm to my last analogy but let me try another. Children
sometimes have a difficult time grasping the concept of volume, the usual
mistake is to think that a squat beaker which holds say 300 ml is smaller
than a tall slender beaker of equal volume. So if you ask a child what will
happen if you pour the contents of the full tall beaker into the empty squat
beaker they say that it will overflow. Even when one demonstrates that the
two beakers are of equal volume, by pouring the liquid from one to the
other, some still think that the taller beaker holds more liquid. They
sometimes think that there is some slight of hand going on. On this analogy,
then, what looks impossible to us, the 3-space solution may be like the
child thinking that the two beakers are equivalent in volume. To say that
any solution requires changing the axioms is like the child saying there
must be some slight of hand--what the child believes is impossible is indeed
entirely possible. The fideist response then is whether you might believe
the magnificently powerful being who claims to be a God III and says that
there is a 3-space solution within the axioms given, it is just that humans
are too stupid to understand it. Could you imagine your trust going this
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