Quentin Anciaux writes:
> Le Lundi 01 Août 2005 05:32, Hal Finney a écrit :
> > I am generally of the school that considers that calculations can be
> > treated as abstract or formal objects, that they can exist without a
> > physical computer existing to run them.
> I completely agree with that... but I have problem with the word 
> "instantiating" in front of an abstract calculation, because if the 
> calculation is abtract that means the calculation just is, no need of 
> instantiation.

I agree, and if I used that terminology then it was probably a
mistake.  Looking back at the message you replied to, I did not talk of
instantiating an abstract calculation.  I did mention the question of
whether a given calculation instantiated a given OM.  Maybe "instantiate"
is not the right word there.  I meant to consider the question of whether
the first calculation added to the measure of the information structure
corresponding to the OM.  If you can find any other place where I used
the word confusingly, let me know.

> On the other hand I have still problem with abstract 
> calculation... take for example a mathematic demonstration written on a sheet 
> of paper, it doesn't mean anything if there is no observer to read it and 
> understand it (thereby "instantiating" the calculus in his own mind), what do 
> you think of that ?

I can interpret your question in two ways.  One is, does a mathematical
proof written on paper has an intrinsic meaning, or is the meaning
in the mind of the reader?  And the other is, do mathematical proofs
have abstract, logical/mathematical existence, in the same way that,
say, numbers or geometric figures might be said to abstractly exist?

As far as the first question, I would analyze it by asking whether someone
who did not know the language it was written in, not even recognizing
the symbols, would be able to deduce what the proof was.  I believe the
answer is yes, for reasonably long proofs.  There would be no ambiguity.

As a concrete way to understand this, suppose we want to ask the question,
does this string of symbols represent proof X, where X is some valid
mathematical proof.  We could write a translation program which, given the
symbols, would output proof X.  If the string of symbols is reasonably
long and actually does match proof X, the translation program will be
short, much shorter than the proof itself.  However if the string of
symbols is not a proof of X, then the translation program will have to
be long.  By the same type of argument I have used repeatedly, this gives
us a tool for evaluating whether a string has a given "meaning".  If the
translation program is short, then the meaning is in the string.  If the
translation program is long, then the meaning is in the translation.
I believe that this shows that it is in fact reasonably to suppose that a
complex proof written on paper does in fact have intrinsic meaning and it
is not just a matter of how it is interpreted in the mind of the reader.

In terms of the other question, whether proofs have abstract mathematical
existence just as (we suppose) integers and triangles do, again I think
that the answer is yes.  Proofs are merely more complex.  They have
relationships amongs their parts.  They depend on an axiom system.
The implicit "causality" and "time ordering" among steps of the proof
could be represented graphically, by colored arrows leading from one
step to another.  I could imagine a representation where valid proof
steps would be as apparent and obvious as the question of whether a set
of lines in a geometric figure all meet at a common point.

In short, I do think that proofs, and for that matter computations,
can be sensibly thought of as having abstract existence just like other
complex mathematical objects.  Some of the constructions of set theory
are far more complex than any humanly understandable proof, yet it is
reasonable to say that sets exist in the abstract.  The fact that a
proof has many parts and has complex relationships between the parts is
no obstacle to its having abstract mathematical existence.

Hal Finney

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