On 09 Jul 2011, at 14:26, Bruno Marchal wrote:


On 09 Jul 2011, at 09:10, Russell Standish wrote:

On Fri, Jul 08, 2011 at 11:04:56PM +0200, Bruno Marchal wrote:

On 08 Jul 2011, at 03:39, B Soroud wrote:

I mean if you went back to classical greece... or classical
india.... could it have been predicted or shown to deduced?

Excellent question. China was close. Reading the treatise "number"
by Plotinus, and having a bit study Diophantus, I am not sure that
in the world were Plato academia lasted longer they could have find
it.  Nature found it before (quantum vaccum, DNA, Brain, humans,
Human thought, computers, ...).
It is the little God. The one you can named (Like FORTRAN, Java,
c++, LISP, game of life, etc.) but when you name it, its names
multiplies.

David Deutsch has an interesting discussion about this in his
"Beginning of Infinity". He actually introduces several notions of
universality, one of which is universality of the numbering
system. Our numbering system is universal,

Well, carefull. It is unidversal in some sense, but is not Turing universal.




since the discovery of the
zero, but ancient Greek & Roman systems were not.

But they are universal in some other sense.



Archimedes came
close to a universal numbering system in the "Sand Reckoner", but
mysteriously shied away from true universality (his system included
some rather arbitrary restrictions preventing it from true
universality).

But they were way far from Turing universality.



Similarly, Babbage and Lovelace came very close to the Turing
universality concept, but again mysteriously shied away from
it.

Here I disagree. I have made research, and I am convinced that babbage has been aware of the Turing universality, of, its notation system to describe its machine. He said that this was his real big discovery, but none understand it.

Then Emil Post is the second one, but nobody will listen (nor will Post really insist). Only with Church and Turing will the notion be admitted by the many. But still very badly understood, despite the concrete computers, which when programmed, hides their universality.



Deutsch remarks that we as a species seem to have a reluctance to
making systems universal, which is quite curious.

So in answer to this question, even if Plato's academy had continued,
it probably still would not have discovered Turing universality.

I think it would have taken some more centuries. They might have discovered it in the 12 or 13th century. They would not have been able to miss it, especially with the development of math and calculus, which they would have developed much faster than Newton and Leibniz. OK, that is just my current opinion. We can't change history.


Let me tell you why I think so. Basically it is the text by Plotinus on Numbers which makes me think like that. This is not numerology, which annoyed Plotinus, but a deep reflection on the infinite, and the question if there is a number of numbers, and the dialectic between the ONE and the Many. This prefigures Cantor. But mathematicians have inherit Aristotle fears on the infinite, and it is only by a reflection on the infinite which can lead, through the diagonalization technic, to the discovery that the "numberable" (enumerable) functions (computable functions) own a universal function, and thus an (infinity) of universal numbers. Plotinus was the first to see the infinite as a possible positive attribute of "God", and not as a defect like for the ealier platonicians. When you "see" the quality of Hypatia research and teaching on Plotinus and Diophantus, I think that if Platonism would have been able to continue a little longer, may be we would be farer in the field, although the Platonists would have perhaps consider the building of such machine a sort of blasphemy too, so it is impossible to know. The notion of universal machine is, with Church thesis, a precise discovery in math, and it is changing the world, more than the zero, more than the logarithm (which initiated the computation revolution), more than calculus (which initiated the industrial revolution). To be sure, I have few doubt that without the fire of Alexandra library, we would also be far away. And without some asteroïds hurting our planet ... "we" might still be dinosaurs.

Bruno







Bruno


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



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