On 2/14/2011 4:12 PM, Jason Resch wrote:

On Mon, Feb 14, 2011 at 11:23 AM, Brent Meeker <meeke...@dslextreme.com <mailto:meeke...@dslextreme.com>> wrote:

    On 2/13/2011 11:24 PM, Jason Resch wrote:

    On Mon, Feb 14, 2011 at 12:52 AM, Brent Meeker
    <meeke...@dslextreme.com <mailto:meeke...@dslextreme.com>> wrote:

        On 2/13/2011 10:13 PM, Jason Resch wrote:

        On Sun, Feb 13, 2011 at 10:46 AM, Brent Meeker
        <meeke...@dslextreme.com <mailto:meeke...@dslextreme.com>>

            On 2/13/2011 5:21 AM, 1Z wrote:

                On Feb 12, 3:18 am, Brent
                <mailto:meeke...@dslextreme.com>>  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?


                Does that allow us to argue:

                1) A universe selected from an uncountably infinite
                number of
                possibilities has measure
                2) Our universe exists so it has measure>0
                3) Our universe is not selected from uncountably
                4) MUH indicates any universe must be selected from
                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.


    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.


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.

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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