I’d like to inform you of something. You might be aware that I’ve been studying the applications of PCT to logic and pure mathematics. By studying Godel’s work (which ultimately led to the work of Alan Turing and to the invention of the universal computer), we see that Godel invented a special language to express propositions about mathematical systems which could only be viewed as true by observing the internal content of these systems from the outside. I understand Bill had something to say about this, but I think there is more to be said.
Godel’s language was very much like a “functional” programming language. Most programming languages, such as C, C++, FORTRAN, etc, are usually described as being “imperative”, because the successive lines of programs written in these languages can be thought of as commands to be executed by the computer. In the “functional” programming languages, such as LISP or PROLOG, the lines of a program are definitions of operations. That is, rather than telling the computer what to do, they define what it is that the computer is to provide.
This seems linked to Bertrand Russell's concept of "propositional functions" where a proposition only become true after its variables have been verified to be of the exact type necessary.
For instance the proposition "cats have four legs" only become true when you put an instance of an actual cat that fit the generic cat definition into the proposition.
The proposition can only be true or false for actual instance of the variable and not "generally" as in "for all 'cats' ".
This is related to the superset of formal logic called non-Aristotelian logic (developed by Alfred Korzybski, 1933) and to epistemology.
For instance, the actual instance of a cat to be fitted into the propositional function "cats have four legs" has to be provided from outside the functional system (for instance out nervous system), probably as what we could call a "type" of disturbance that has to be managed either with probability (<1) or more careful observation (or both).
Nick
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Le 2015-02-26 à 18:51, PHILIP JERAIR YERANOSIAN (pyeranos@ucla.edu via csgnet Mailing List) <csgnet@lists.illinois.edu> a écrit :
[philip 2/26.15:30]
Hi everyone,
I'd like to inform you of something. You might be aware that I've been studying the applications of PCT to logic and pure mathematics. By studying Godel's work (which ultimately led to the work of Alan Turing and to the invention of the universal computer), we see that Godel invented a special language to express propositions about mathematical systems which could only be viewed as true by observing the internal content of these systems from the outside. I understand Bill had something to say about this, but I think there is more to be said.
Godel's language was very much like a "functional" programming language. Most programming languages, such as C, C++, FORTRAN, etc, are usually described as being "imperative", because the successive lines of programs written in these languages can be thought of as commands to be executed by the computer. In the "functional" programming languages, such as LISP or PROLOG, the lines of a program are definitions of operations. That is, rather than telling the computer what to do, they define what it is that the computer is to provide.
Thank you for the response. What is most relevant to what you are describing, I believe, is the notion of variable binding and substitution. We must refer to the lambda calculus and the Church-Turing hypothesis for further insight.