On 8/16/2014 4:57 PM, James Lindsay wrote:
Hi Brent,
Thanks for the note. I like the thought about mathematics as a
refinement
of language. I also think of it as a specialization of philosophy,
or
even a highly distilled variant upon it with limited scope. Indeed,
I
frequently conceive of mathematics as a branch of philosophy where
we
(mostly) agree upon the axioms and (mostly) know we're talking about
abstract ideas, to be distinguished from what I feel like I get
from many
philosophers.
I am not familiar with Bruno Marchal,
Here's his paper that describes his TOE. It rests on two points
for which
he gives arguments: (1) If consciousness is instantiated by certain
computational processes which could be realized in different media
(so
there's nothing "magici" about them being done in brains) then they
can
exist the way arithmetic exist (i.e. in "platonia"). And in
platonia
there is a universal dovetailer, UD, that computes everything
computable
(and more), so it instantiates all possible conscious thoughts
including
those that cause us to infer the existence of an external physical world.
The problem with his theory, which he recognizes, is that this apparently
instantiates too much. But as physicist like Max Tegmark, Vilenkin,
and
Krause talk about eternal inflation and infinitely many universes
in which
all possible physics is realized, maybe the UD doesn't produce too
much.
He thinks he can show that what it produces is like quantum
mechanics
except for a measure zero. But I'm not convinced his measure is
more than
wishful thinking.
He's a nice fellow though and not a crank. So if you'd like to
engage him
on any of this you can join the discussion list
[email protected]
<mailto:[email protected]>.
and I am not expert in theories of anything, much less everything,
based
upon computation or even computation theories. I remain a bit
skeptical
of them, and overall, I would suggest that such things are likely
to be
/theories/ of everything, which is to say still on the map side of
the
map/terrain divide.
I agree. But some people assume that there must be some ultimate
ontology
of ur-stuff that exists necessarily - and mathematical objects are
their
favorite candidates (if they're not religious). I don't think this
is a
compelling argument since I regard numbers as inventions (not
necessarily
human - likely evolution invented them). I think of ontologies as
the
stuff that is in our theories. Since theories are invented to
explain
things they may ultimately be circular, sort of like: mathematics->
physics-> chemistry->biology-> intelligence-> mathematics. So you
can
start with whatever you think you understand. If this circle of
explanation is big enough to include everything, then I claim it's
"virtuously" circular.
Brent
"What is there? Everything! So what isn't there? Nothing!"
--- Norm Levitt, after Quine
Regarding your note about my Chapter 2, that's an interesting point
that
he raises, and interestingly, I don't wholly disagree with him that
it is
an integral feature of arithmetic that it is axiomatically
incomplete
(though maybe I thought differently when I wrote the book).
Particularly,
I don't think of it as a "bug," but I don't necessarily think of it
as a
"feature" either. I'm pretty neutral to it, and I feel like I was
trying
to express the idea in my book that it reveals mostly how
theoretical, as
opposed to real, mathematics is. I'm not sure about this "more than
a
map" thing yet, as by "map" I just mean abstract way to work with
reality
instead of reality itself and hadn't read more into my own
statement than
that.
I would disagree with him, however, that it is related to the hard
problem of consciousness, I think, or perhaps it's better to say
that I'm
very skeptical of such a claim. Brains are, however "immensely"
complex,
finite things, and as such, I do not think that the lack of a
complete
axiomatization of arithmetic is likely to be integrally related to
the
hard problem of consciousness. Maybe I just don't understand what
he's
getting at, though. Who knows?
I also tend to agree with you--in some senses--about the
ultrafinitists
probably being right. My distinction is that I'm fine with infinity
as a
kind of fiction that we play with or use to make calculus/analysis
more
accessible. I certainly agree with you that infinity probably
shouldn't
be taken too seriously, particularly once they start getting weird
and
(relatively) huge.
There's something interesting to think about, though, when it comes
to
the ideas of some infinities being larger than others. I was
thinking a
bit about it the other day, in fact. That seems to be a necessary
consequence of little more than certain definitions on certain
kinds of
sets (with "infinite" perhaps not even necessary here, using the
finitists' "indefinite" instead) and one-to-one correspondences.
Anyway, thanks again for the note.
Kindly,
James
On Sat, Aug 16, 2014 at 1:14 AM, meekerdb <[email protected]
<mailto:[email protected]>> wrote:
After seeing your posts on Vic's avoid-L list, I ordered your book.
I'm generally inclined to see mathematics as a refinement of language
- or in your terms a "map", not to be confused with the thing
mapped.
However I often argue with Bruno Marchal, a logician and
neo-platonist, who has a TOE based on computation
(Church-Turing) or
number theory. I thought you book might help me. But I think
Bruno
would rightly object to your Chapter 2. He considers it an
important
feature of arithmetic that it is axiomatically incomplete, i.e.
per
Godel's theorem it is bigger than what can be proven from the
axioms.
He takes this as a feature, not a bug, to explain that if
conscious
thought is a computation this is why it cannot fully explain
itself;
and that is why "the hard problem" of consciousness is hard. I
think
there are simpler, evolutionary explanations for why
consciousness
does not include perception of brain functions, but I think
Bruno has
a point that arithmetic is bigger than what follows from Peano's
axioms and so it is more than a map.
I'm inclined to say Peano's axioms already "prove too much" and
the
ultrafinitists are right. Infinity is just a convenience to
avoid
saying how big, and shouldn't be taken too seriously.
Brent Meeker
