Le 15-août-06, à 20:32, David Nyman a écrit :


> But don't we just 'derive' natural numbers by establishing a semantic
> equivalence between '6' and the collection of faces on a cube?


But what is a cube?



> And
> their additive and multiplicative structures likewise by analogy and
> generalisation? Must it not be the case that all we can know of the
> number realm is in practice wholly instantiated in indexical
> 1st-persons as information, and that ideas about its further extent,
> while possibly justified as theory, are not, empirically, instantiated
> *anywhere* to our knowledge?


Not if we assume comp, by UDA. Concept like "information", "cube", etc. 
are all more complex than numbers. Recall we search a theory  (even a 
testable theory).
Note also I have not yet seen physical theory which does not assume 
numbers.


> As far as I can see, the only alternative
> to this is the belief that we have 'direct contact' with this realm, as
> Penrose claims, which is surely equivalent to claiming knowledge of God
> by 'direct revelation'.

I disagree. I am just now doing math with six years old children. I 
have not the feeling that they have any direct contact with the number 
realm. They have contact with empirical reality (the one which I don't 
take for granted, i.e. the one I want to explain (without eliminating 
the person)). Only through some contact with that apparent reality can 
children (and adults) learn to discern the abstract math pattern. But 
this does not mean the math pattern would be senseless in absence of 
those empirical evidences. Accepting I am myself, at least partially 
empirical, only the knowledge of numbers would disappear in absence of 
knower, not the number and their relations themselves. Now Arithmetical 
realism is just part of the comp hypothesis, which I want to show is 
testable.
Empirical physics assumes numbers too, in any assumed theories. I could 
prove too you that no axiomatization of physics can bypassed some 
belief in number. Actually string theory cannot even bypass deep and 
strange arithmetical truth like the famous Ramanujan formula which says 
that:

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ...   =   -1/12    (!)    (it is 
zeta(-1) extended on the complex plane)


>  In this case we're merely substituting 'numbers
> made us', for 'God made us'.

This makes sense (as I see you have seen in your more recent post) if 
you take "God" as the 0-person notion, i.e. arithmetical truth.



> While any such belief may be *true*, it
> isn't logical or necessary truth. So what precisely is 'essential'
> about the number realm, in the sense of making it the basis of
> 'indexical David' - whom I claim and assert to be necessarily real?
>
>> Of course, we have access to numbers only via our first person view. 
>> But this
>> fact does not logically entails that numbers themselves are a 
>> necessarily
>> personal or an indexical construction per se.
>
> Despite your claim that they are the basis both of the personal and
> indexical? I ask you again, for them to play such a profound role, what
> status, beyond that of an idealised notion, are you giving them?


I give the status of a-temporal a-spatial truth, independent of me (and 
you),  to the elementary propositions in arithmetics. I believe in the 
independence of propositions like "3 * 7 = 21", where 3, 7, and 21 have 
their usual meaning (on the planet TETRA, where people have only four 
fingers at each hand, they write "3 * 7 = 25", but it really means the 
same ("25" means 2*8 + 5 = 21).



>
> Having remonstrated with you thus, might I suggest that I could
> understand your meaning better thus:
>
> "Let's proceed *as if* the number realm were the sole 'primitive', and
> everything else we observe could be derived from it.


I propose a reasoning (UDA). I am just saying that in case we can 
survive with an artificial digital brain, then, even if that artificial 
brain looks material, we HAVE TO derive persons and appearances of 
matter from numbers, and nothing else. UDA is made simple by being not 
too much constructive. Then I show that computer science is already 
powerful enough to begin (at least) the concrete and constructive 
derivation of the comp physical laws---and then we can compared it to 
the empirical physics and so we can perhaps refute comp or confirm it. 
Until now I got only a confirmation through the appearance of an 
"arithmetical quantization" or of an arithmetical interpretation of 
quantum logic.
My hope is to be able to show it is enough for explaining why any 
lobian machine looking to its neighborhood and trying to get its "more 
probable computational continuations" will discover that "its" bottom 
is described by a quantum computer (actually a topological QC).
I believe that Everett, Hartle, Graham, Deutsch .... are mainly correct 
when they explain how Bits appears from Qubit. But looking closely, 
Everett ... need comp. And with comp there is a way to reverse the 
arrow and we got an explanation how Qubits arise from Bits. Qubits are 
eventually "just" description of bits from an indeterminate first 
person plural view, where the indeterminateness comes exclusively from 
the comp 1-indeterminacy.




> If we succeed in
> this venture, we will have gained much in the way of insight. No doubt,
> there will still remain further questions as to the nature and true
> origins of the 'reality' so conjured into existence, possibly
> unanswerable.

What is cute about natural numbers is that, although we cannot justify 
their existence without assuming them, we can at least justify why any 
entity cannot justify them without assuming them.
This justification works for all self-referentially correct entities 
(not just digital machines).


> But since the question - why am I in this situation at
> all in which I am able to be surprised that I am in this situation at
> all? - regresses inevitably to a point beyond reason, perhaps it
> doesn't put us in a worse position in this regard than any other
> assumption."
>
> Does this work for you?

Perhaps. Numbers can explain our discourse, including the part " ... 
but numbers cannot explain qualia". But numbers can explain why 
"numbers cannot explain qualia" in any third person way, and this (it 
is the key point) without eliminating them.
"Why am I Bruno and not David" can then be explained by relative 
indeterminacy, and is akin to the question "why am I the one 
reconstituted in Moscow" after a self-duplication Washington/Moscow, 
for example. The duplication is an easy way to explain why we cannot 
explain something.

Note that your "indexical" approach fits nicely with the idea that 
"everything" (everything observable or measurable + qualia 
(perceptible)) comes ultimately from the self-referential intellect (G, 
G*).
Perhaps some misunderstanding grew up between us, because the 
self-referential intellect looks like a "first person" (because of the 
self-reference), but I have to put it in the third person (because it 
corresponds to a discourse about some 3-person description of 
yourself). I have been stuck during years on that "subtlety" .

Bruno


http://iridia.ulb.ac.be/~marchal/


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