Brent, I intend to reply more directly to your post soon, as I think
there's a lot to be said in response.
But in the meantime:
So I just finished reading David Deutsch's "The Fabric of Reality",
and I'm curious what you (Brent, Bruno, and anyone else) make of the
following passage at the end of chapter 10, The Nature of Mathematics.
The first paragraph is at least partly applicable to Brent's recent
post, and the second seems relevant to Bruno's last response. It
makes one wonder what other darkly esoteric abstractions may stalk the
abyssal depths of Platonia???
The passage:
"Mathematical entities are part of the fabric of reality because they
are complex and autonomous. The sort of reality they form is in some
ways like the realm of abstractions envisaged by Plato or Penrose:
although they are by definition intangible, they exist objectively and
have properties that are independent of the laws of physics. However,
it is physics that allows us to gain knowledge of this realm. And it
imposes stringent constraints. Whereas everything in the physical
reality is comprehensible, the comprehensible mathematical truths are
precisely the infinitesimal minority which happen to correspond
exactly to some physical truth - like the fact that if certain symbols
made of ink on paper are manipulated in certain ways, certain other
symbols appear. That is, they are the truths that can be rendered in
virtual reality. We have no choice but to assume that the
incomprehensible mathematical entities are real too, because they
appear inextricably in our explanations of the comprehensible ones.
There are physical objects - such as fingers, computers and brains -
whose behaviour can model that of certain abstract objects. In this
way the fabric of physical reality provides us with a window on the
world of abstractions. It is a very narrow window and gives us only a
limited range of perspectives. Some of the structures that we see out
there, such as the natural numbers or the rules of inference of
classical logic, seem to be important or 'fundamental' to the abstract
world, in the same way as deep laws of nature are fundamental to the
physical world. But that could be a misleading appearance. For what
we are really seeing is only that some abstract structures are
fundamental to our understanding of abstractions. We have no reason
to suppose that those structures are objectively significant in the
abstract world. It is merely that some abstract entities are nearer
and more easily visible from our window than others."
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Received on Tue Jul 21 2009 - 01:22:36 PDT