Many thanks! I'll give my current attitudes to your hints:
Bruno:
You mentioned the ASSA. Yesterday, motivdated by your hint, I have
read about the ASSA/RSSA debate that is said to have divided the list
into two camps. Since I have trouble with the reasoning I read, I will
probably send a new message hoping for leaving the misunderstanding
behind.
Searching for the Universal Dovetailer Argument, I found a quite
formal demonstration that you wrote in the list, and an even more
formal demonstration that you published in the original work. I do see
the advantage to have such a formal demonstration when it comes to
detailed discussions, but sometimes I'd prefer a simplified outline to
get the basic idea and the main conclusions before going into detail.
If you have written such an outline (in English or in French as well)
I would be thankful to get the link. Otherwise I'll read one of the
formal versions in the future.
Hal (and partially Russell):
I still like your approach to the Everything ensemble using a
countable set P of 'properties'. In fact, if we describe any object or
world by a sequence of properties, the objects form a set equivalent
to {0,1}^P (e.g. we assign 0 if the object does not have the property
and 1 if it has the property) which is the power set of P
(equivalently we could have formed subsets of P). Since P is
countable, we can work with the Everything ensemble {0,1}^IN of
infinite bitstrings. As you have mentioned, this set is uncountable.
So far, there isn't any mathematical problem. In contrast to Marc, I
do also agree identifying objects with the corresponding subset of P.
In this picture, "states and behaviours" as Marc calls it, must also
lie in the properties. Thus, the term 'property' is used in a more
comprehensive sense than in programming.
But now, we come to much more serious criticism. Russell noticed that
regarding the ensemble of infinite bitstrings to be based on
properties jumbles the ensemble (a simple mathematical entity) with
interpretations by the observer. His separation between "syntactic"
and "semantic" space is essential. I agree with Russell, but I do also
see the necessity to interpret (not in an exact sense) mathematical
entities in our theories within our "everyday theory"; because this is
what makes a mathematical theory a (meta)physical theory as I have
pointed out. Russell also uses such an interpretation, but on a more
implicit level: An observer reads bits of the world's description. In
order to make this a (meta)physical theory, we must be able to find
ourselves within the theory, namely as observers. So, we must know
what the process of reading bits of the word's description is meaning
for us. And I'd say that it means measuring 'properties' of the world.
To give a concise explanation: Properties should not be a fundamental
ingredient to the mathematical theory. The mathematical theory uses
"syntactic" space. Though, in order to understand the mathematical
theory by means of the everyday theory (and thus to link the
mathematical theory to "concrete reality"), we need (at some point of
our theories) a translation. This translation can possibly be done by
interpreting the ensemble via 'properties'. Conversely, we can
motivate the ensemble of infinite bitstrings (ant thus "syntactic"
space) starting from a countable set of 'properties'.
Maybe it would be the best for your theories, Hal, to interrupt after
having motivated the ensemble of infinite bitstrings. Then, the
infinite bitstrings are considered to be fundamental (and no longer
the properties themselves). Russell (and surely others, too) has
provided a good framework to work with this ensemble and the role of
observers. Perhaps, you can try to translate some of your ideas to
Russell's more strict and formal language. Then, it will be easier for
us to follow your thinking.
Marc:
Thank you very much for the definitions. I did not know how this was
commonly called.
Brent:
I do still defend extensional definitions even for infinite sets.
Mathematics shows how useful this is. I come back to the example of a
real function f that maps every real number to another real number. In
mathematics, this function is defined by the infinite set {(x,f(x)); x
being a real number}. And the space of all these functions has very
nice mathematical properties, we can work with it and prove theorems.
Of course, in practice I will not have the set but merely a formula
defining f. For example f(x)=x+1. But this does not disprove the
possibilty of working with the sets on an abstract level. Mathematics
indeed proves that it is possible.
Your second point, Russell's ("Bertie's") paradox, is much more
striking. In fact, if we allow every property the English (or the
German, following Cantor) language can express, we will end up with
contradictions. This is why the set of properties is somehow
restricted. We need, as I wrote, "a set of distinct and independent
properties". I don't really know if such a postulate makes sense.
Youness Ayaita
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Received on Tue Sep 18 2007 - 06:42:58 PDT