On Tue, May 22, 2012 at 11:08 AM, Stephen P. King <stephe...@charter.net>wrote:

>  On 5/22/2012 10:56 AM, Joseph Knight wrote:
>
>
>
> On Tue, May 22, 2012 at 7:36 AM, Stephen P. King <stephe...@charter.net>wrote:
>
>>  On 5/21/2012 6:26 PM, Russell Standish wrote:
>>
>> snip
>>
>>  Hi Russell,
>>
>>     I once thought that consistency, in mathematics, was the indication
>> of existence but situations like this make that idea a point of
>> contention... CH and AoC <http://en.wikipedia.org/wiki/Axiom_of_choice>are 
>> two axioms associated with ZF set theory that have lead some people
>> (including me) to consider a wider interpretation of mathematics. What if
>> all possible consistent mathematical theories must somehow exist?
>>
>
>  Joel David Hamkins introduced the "set-theoretic multiverse" idea 
> (link<http://arxiv.org/abs/1108.4223>).
> The abstract reads:
>
>  "The multiverse view in set theory, introduced and argued for in this
> article, is the view that there are many distinct concepts of set, each
> instantiated in a corresponding set-theoretic universe. The universe view,
> in contrast, asserts that there is an absolute background set concept, with
> a corresponding absolute set-theoretic universe in which every
> set-theoretic question has a definite answer. The multiverse position, I
> argue, explains our experience with the enormous diversity of set-theoretic
> possibilities, a phenomenon that challenges the universe view. In
> particular, I argue that the continuum hypothesis is settled on the
> multiverse view by our extensive knowledge about how it behaves in the
> multiverse, and as a result it can no longer be settled in the manner
> formerly hoped for."
>
>
>  Hi Joseph,
>
>     Thank you for this comment and link! Do you think that there is a
> possibility of an "invariance theory", like Special relativity but for
> mathematics, at the end of this chain of reasoning?
>

I am doubtful, simply because, for example, the Continuum Hypothesis and
its negation are both consistent with ZF set theory. Ditto for the axiom of
choice, of course.

I find it fascinating that, at this level of the foundations of
mathematics, mathematics becomes almost an intuitive science. Questions are
asked such as: *Ought *the axiom of choice be true? Are its consequences in
line with how we intuit sets to behave? This is the intersection of minds
and mathematics.


> My thinking is that any form of consciousness or theory of knowledge has
> to assume that there is something meaningful to the idea that knowledge
> implies agency <http://en.wikipedia.org/wiki/Agency_%28philosophy%29> and
> intention <http://plato.stanford.edu/entries/intention/>...
>
>
>
>
>>
>>
>>  Its one reason why Bruno would like to restrict ontology to machines,
>> or at most integers - echoing Kronecker's quotable "God made the
>> integers, all else is the work of man".
>>
>>
>>
>>
>>      I understand that, but this choice to restrict makes Bruno's
>> Idealism even more perplexing to me; how is it that the Integers are given
>> such special status, especially when we cast aside all possibility (within
>> our ontology) of the "reality" of the physical world? Without the physical
>> world to act as a "selection" mechanism for what is "Real", why the bias
>> for integers? This has been a question that I have tried to get answered to
>> no avail.
>>
>
>  I think Bruno gives such high status to the natural numbers because they
> are perhaps the least-doubt-able mathematical entities there are. The very
> fact that talks of a "set-theoretic multiverse" exist makes one ask, how
> real are sets? Do set theories tell us more about our minds than they do
> about the mathematical world? (Obviously, as David Lewis pointed out, you
> need something like a set theory in order to do mathematics at all, and as
> Russell says, for the average mathematician it really doesn't matter.)
>
>
>     My skeptisism centers on the ambiguity of the metric that defines "the
> least-doubt-able mathematical entities there are".
>

I understand. At the end of the day, it may be up to the individual to
decide what is doubt-able and what is not.


> We operate as if there is a clear domain of meaning to this phrase and yet
> are free to range outside it at will without self-contradiction. Set
> theory, whether implicit of explicitly acknowledged seems to be a
> requirement for communication of the 1st person content. Is it necessary
> for consciousness itself? Might consciousness, boiled down to its essence,
> be the act of making a distinction itself?
>

This is an extremely interesting line of thought. Sets do seem to be
necessary for the communication of mathematical ideas, maybe even the
communication of ideas period. I will have to give this more thought.


>
>
>
>  Also: *No one here has questioned the reality of the physical world. *Should
> I append this statement to every email until you stop countering it?
>
>
>     I frankly have to explicitly mention this because the "reality of the
> physical world" is, in fact, being questioned by many posters on this list.
>

Only its status as fundamental is being questioned, to my knowledge. There
are a couple of posters whose messages I ignore, however, so I could be
missing something.


> That you would write this remark is puzzling to me. I think that I can
> safely assume that you have read Bruno's papers... Maybe the problem is
> that I fail to see how reducing the physical world to the epiphenomena of
> numbers does not also remove its "reality".
>

It's "real" because I see it, I interact with it. It's not fake, whatever
that could possibly mean. It's just made epiphenomenal by COMP.


>
>

>
>
>
>>
>>
>>
>>   This is the origin of Bruno's claim that COMP entails that physics is
>> not computable, a corrolory of which is that Digital Physics is
>> refuted (since DP=>COMP).
>>
>>
>>      Does the symbol "=>" mean "implies"? I get confused ...
>>
>>
>>  Yes, that is the usual meaning. It can also be written (DP or not COMP).
>>
>>
>>      "=>" = "or not"]
>>
>
>  Actually "a implies b" is defined as "not a or b".
>
>
>     Thank you for this clarification! Would you care to elaborate on this
> definition?
>

We understand A to imply B. If A is true, then B should be. If B is false,
then A better be too. If A is false, then we don't really care about B.

This is the standard definition of "implies" throughout mathematics -- as a
definition, in terms of "not" and "or".


>
> --
> Onward!
>
> Stephen
>
> "Nature, to be commanded, must be obeyed."
> ~ Francis Bacon
>
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-- 
Joseph Knight

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