On 2/14/2011 4:12 PM, Jason Resch wrote:
On Mon, Feb 14, 2011 at 11:23 AM, Brent Meeker
<[email protected] <mailto:[email protected]>> wrote:
On 2/13/2011 11:24 PM, Jason Resch wrote:
On Mon, Feb 14, 2011 at 12:52 AM, Brent Meeker
<[email protected] <mailto:[email protected]>> wrote:
On 2/13/2011 10:13 PM, Jason Resch wrote:
On Sun, Feb 13, 2011 at 10:46 AM, Brent Meeker
<[email protected] <mailto:[email protected]>>
wrote:
On 2/13/2011 5:21 AM, 1Z wrote:
On Feb 12, 3:18 am, Brent
Meeker<[email protected]
<mailto:[email protected]>> wrote:
What do you think the chances are
that any random object in
Plato's heaven, or any random Turing
machine will support intelligent life?
1 in 10, 1 in 1000, 1 in a billion?
Zero.
Does that allow us to argue:
1) A universe selected from an uncountably infinite
number of
possibilities has measure
0
2) Our universe exists so it has measure>0
3) Our universe is not selected from uncountably
infinite
possibilities
4) MUH indicates any universe must be selected from
uncountable
infinite possibilities (since all
of maths includes the real line, etc)
5) MUH is false.
Hmmm. I think we argue that objects in Plato's heaven
and Turing machines are not the right kind of things to
support life.
I am very puzzled by this statement. You could help me
understand by answering the following questions:
Why couldn't there be an accurate simulation of life on a
Turing machine?
Because a Turing machine is an abstraction. If you mean a
realization of a Turing machine, then I suppose there could
be a simulation of life on it.
How can entities within a universe that exists in Plato's
heaven distinguish it from a universe that does not?
I doubt that Plato's heaven exists. So no universes would
exist in it.
Brent
Exists is a funny word. It seems to embody knowledge and opinion
from one observer's viewpoint based on their own limited
experiences and interactions within their local portion of reality.
Indeed. I'm not sure it's unqualified use is meaningful.
If Plato's heaven is such a thing that contains all possible
structures, does the fact that it contains all possible
structures hold true whether or not it exists?
All possible brick structures? Please explain as precisely as
possible what Platonia is.
If there are universes existing abstractly inside Plato's heaven,
and some of those universes contain conscious observers, does
ascribing the property of non-existence to Plato's heaven or to
those universes make those observers not conscious, or is the
abstraction enough?
What does "abstractly existing" mean.? How is it different from
just exsiting?
What properties can something which is non-existent have?
It seems there are two choices: 1. Things which are non-existent
can have other properties besides non-existence.
Sure. Sherlock Holmes is non-existent and has the property of
being a detective.
E.g., a non-existent universe has atoms, stars, worlds, and
people on some of those worlds. Or 2. Non-existent things cannot
have any other properties besides non-existence. It sounds like
you belong to this second camp.
However, this seems to lead immediately to mathematical realism.
As there are objects with definite objectively explorable
properties in math. 7's primality and parity are properties of
7. But how can 7 have properties if it does not exist? If
non-existent things can have properties, why can't consciousness
be one of those properties? What is the difference between a
non-existent brain experiencing a sunset and an existent brain
experiencing a sunset?
Only one of them exists.
Please explain as precisely as possible what it means for
something to not exist.
If I can kick it and it kicks back it exists.
Brent
What do you think about this passage from Fabric of Reality, where
David Deutsch argues numbers do "kick back":
"/Do/ abstract, non-physical entities exist? Are they part of the
fabric of reality? I am not interested here in issues of mere word
usage. It is obvious that numbers, the laws of physics, and so on do
‘exist’ in some senses and not in others. The substantive question is
this: how are we to understand such entities? Which of them are merely
convenient forms of words, referring ultimately only to ordinary,
physical reality? Which are merely ephemeral features of our culture?
Which are arbitrary, like the rules of a trivial game that we need
only look up? And which, if any, can be explained only in a way that
attributes an independent existence to them? Things of this last type
/must/ be part of the fabric of reality as {222} defined in this book,
because one would have to understand them in order to understand
everything that is understood.
This suggests that we ought to apply Dr Johnson's criterion again. If
we want to know whether a given abstraction really exists, we should
ask whether it ‘kicks back’ in a complex, autonomous way. For example,
mathematicians characterize the ‘natural numbers’ i, 2, 3,... in the
first instance through a precise definition such as:
1 is a natural number.
Each natural number has precisely one successor, which is also a
natural number.
1 is not the successor of any natural number.
Two natural numbers with the same successor are the same.
Such definitions are attempts to express abstractly the intuitive
/physical/ notion of successive amounts of a discrete quantity. (More
precisely, as I explained in the previous chapter, that notion is
really quantum-mechanical.) The operations of arithmetic, such as
multiplication and addition, and further concepts such as that of 1
prime number, are then defined with reference to the ‘natural
numbers’. But having created abstract ‘natural numbers’ through that
definition, and having understood them through that intuition, we find
that there is a lot more that we still do not understand about them.
The definition of a prime number fixes once and for ill which numbers
are primes and which are not. But the /understanding/ of which numbers
are prime — for instance, how prime numbers are distributed on very
large scales, how clumped they are, how ‘random’ they are, and why —
involves a wealth of new insights and new explanations. Indeed, it
turns out that number theory is a whole world (the term is often used)
in itself. To understand numbers more fully we have to define many new
classes of abstract entities, and postulate many new structures and
connections among those structures. We find that some of these
abstract structures are related to other intuitions that we already
had but {223} which, on the face of it, had nothing to do with numbers
— such as /symmetry, rotation/, the /continuum, sets, infinity/, and
many more. Thus, abstract mathematical entities we think we are
familiar with can nevertheless surprise or disappoint us. They can pop
up unexpectedly in new guises, or disguises. They can be inexplicable,
and then later conform to a new explanation. So they are complex and
autonomous, and therefore by Dr Johnson's criterion we must conclude
that they are real. Since we cannot understand them either as being
part of ourselves or as being part of something else that we already
understand, but we /can/ understand them as independent entities, we
must conclude that they /are/ real, independent entities.
I don't think it follows. We can understand numbers because we have
invented them. We infer their properties from the axioms we adopt. How
would we know that there wasn't a biggest number (as children sometimes
suppose) unless we adopted the axiom of infinity? The fact that is
difficult to forsee all the implications of our axioms, and so we
discover them, doesn't imply anything about their reality. Bruno points
to Godel's incompleteness theorem, which shows that there are true
arithmetical statements not provable from Peano's axioms, but the axioms
are intended to capture our intuition about counting. The reflective
statements Godel constructs don't have much to do with that intuition.
Nevertheless, abstract entities are intangible. They do not kick back
physically in the sense that a stone does, so experiment and
observation cannot play quite the same role in mathematics as they do
in science. In mathematics, /proof/ plays that role. Dr Johnson's
stone kicked back by making his foot rebound. Prime numbers kick back
when we prove something unexpected about them especially if we can go
on to explain it too. In the traditional view, the crucial difference
between proof and experiment is that a proof makes no reference to the
physical world. We can perform a proof in the privacy of our own
minds, or we can perform a proof trapped inside a virtual-reality
generator rendering the wrong physics. Provided only that we follow
the rules of mathematical inference, we should come up with the same
answer as anyone else. And again, the prevailing view is that, apart
from the possibility of making blunders, when we have proved something
we know with /absolute certainty/ that it is true.
No, we only know it follows from the axioms, given the rules of
inference. You can change either of them with impunity. They won't
kick back.
Mathematicians are rather proud of this absolute certainty, and
scientists tend to be a little envious of it. For in science there is
no way of being certain of any proposition. However well one's
theories explain existing observations, at any moment someone may make
a new, inexplicable observation that casts doubt on the whole of the
current explanatory structure. Worse, someone may reach a better
understanding that explains not only all existing observations but
also why the previous explanations seemed to work but are nevertheless
quite wrong. Galileo, for instance, found {224} new explanation of the
age-old observation that the ground beneath our feet is at rest, an
explanation that involved the ground actually moving. Virtual reality
— which can make one environment seem to be another — underlines the
fact that when observation is the ultimate arbiter between theories,
there can never be any certainty that an existing explanation, however
obvious, is even remotely true. But when proof is the arbiter, it is
supposed, there can be certainty."
All the more reason not to suppose that mathematical things exist. But
as you observed "exist" is a funny word - it seems to act like an
indicial, dependent on a context.
Brent
Now it is precisely in cleaning up intuitive ideas for mathematics that
one is likely to throw out the baby with the bathwater.
--- J.S. Bell ; ‘La nouvelle cuisine’, 1990.
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