Rex Allen wrote:
> 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,
I find that dubious. Even if it were true, I don't think we could ever
*know* it was true.
> the comprehensible mathematical truths are
> precisely the infinitesimal minority which happen to correspond
> exactly to some physical truth
There seem to be many mathematical truths that do not correspond to
physical facts. In any case the correspondence is what needs explanation.
> - 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.
I don't think Godel sentences "appear intextricably in our explanations
(proofs?) of other theorems." They are entailed by the same axioms and
rules of inference, but that seems different to me. They come from
infinites, which I regard as convenient approximations of "very big".
> There are physical objects - such as fingers, computers and brains -
> whose behaviour can model that of certain abstract objects.
A very Platonic way of putting it.
> 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."
What would it mean for a structure in the abstract world (of
mathematics?) to be insignificant?
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