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
