From: Marc Geddes (m_j_geddes@yahoo.com.au)
Date: Fri Jan 27 2006 - 02:54:29 MST
Note:
People may wonder why I'm talking about what seems
like metaphysics again. But I'm discussing the nature
of mathematics here and since AI work is basically
pure mathematics, I'd say understanding the nature of
mathematics could be critical to AGI.
From: Marc, who is aiming to quickly 'close out' the
Singularity game in a spectacular blaze of glory.
O.K, let me now start to bring this game to a swift
and shocking end...
Ben Heaton said >>>
>Just to check: is this related to the idea that, for
example, Fermat's Last Theorem was not true until a
proof for it was discovered?
No
>If your idea is something more meaningful than that,
then can you give an example of a mathematical claim
that you think has even a slight metaphysical
possibility of being time-dependent? I'm not asking
for an example of one that definitely is, just one
where it wouldn't be obviously ridiculous to believe
that it might be. All the examples of mathematical
theorems I can think of lack any indexical references
to times, but if you know of something I'm
overlooking, I'd like to learn about it.
Deductive mathematical proofs cannot by definition be
time dependent. So if there are any time indexed
mathematical properties they must be inductive (in the
sense of probability theory).
If one subscribes to mathematical Platonism (the idea
that mathematical entities have objective existence)
then there may be a way to define the 'changes in
state' of these mathematical entities.
If you imagine a Platonically existing proof that no
mathematician has yet discovered, then imagine a proof
which every one knows about, one could say that the
mathematical entities which are the 'theorems'
(proofs) under go 'changes of state' when they change
from existing only in Platonic reality, to existing as
representations in the physical world. The
mathematical entity is not the same thing as a
particular instantiation of that entity in the
physical world. The mathematical entity can exist
purely Platonically, or it can 'project down' into the
physical world and be instantiated in physical matter.
When the entity 'projects into' the physical world,
it can be defined as having 'changed its state'. For
instance, a mathematical theorem which is not known by
any one is in a 'different state' to a mathematical
theorem which also exists in the minds of
mathematicians.
The time indexed mathematical theorems then, are
likely the one's expressing the relationship between
platonic properties, physical properties and mental
(volitional) properties (see the example I gave above
- a mathematical entity can be said to have 'changed
state' when it goes from existing only in the Platonic
realm , to becoming manifest in the physical and
mental realms).
Now these ideas may not help much in identifying
exactly what the time indexed mathematical properties
are, but they do, I hope, show you that platonic
mathematical entities are not necessarily the static,
timeless things many philosophers think they are.
Again, you may all ask, how is all this metaphysics
relevant to AGI work? Well, consciousness appears to
be intimately connected to the integration of
information. See:
http://www.sci-con.org/tiki-read_article.php?articleId=188
The integration of information is in turn intimately
connected to the discovery of mathematical truth. As
I suggested above, if platonic mathematical entities
can undergo 'changes of state' when they go from
existing only platonically to existing in someone's
mind and if the mathematical truths representing this
process are time indexed (or time dependent) then
consciousness itself (and information integration)
would be precisely equivalent to the 'movements' of
platonic mathematical entities or the sub-set of
mathematical truths that were time dependent...
"Till shade is gone, till water is gone, into the shadow with teeth bared, screaming defiance with the last breath, to spit in Sightblinder’s eye on the last day”
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