John, (01)
> Gian-Carlo Rota, mathematician:
>
> Shocking as it may be to a conservative logician, the day will come
> when currently vague concepts such as motivation and purpose will be
> made formal and accepted as constituents of a revamped logic, where they
> will at last be alloted the equal status they deserve, side-by-side with
> axioms and theorems. (02)
This sounds very good, but it is also very hard to imagine how motivation
and purpose could be made as formal as mathematical axioms, theorems, and
rules of inference. However, motivation and purpose can be given
axiomatically e.g. as a statement of a guideline that is held in some book,
for instance. These motivation-purpose axioms just have to be separated from
the axioms of some specific system in the book. (03)
Avril (04)
Lainaus "John F. Sowa" <sowa@xxxxxxxxxxx>: (05)
> Rick,
>
> Thanks for the reference. Soloman Feferman is a very good logician,
> who edited Gödel's _Collected Papers_, and his views are well taken.
>
> But the question about the potential of human minds vs. computer
> programs is extremely complex, and I'd like to mention one commonly
> claimed, but erroneous argument:
>
> 1. Gödel proved that there are infinitely many true theorems that
> cannot be proved by a machine that enumerates all proofs.
>
> 2. But humans can prove some theorems that are outside some list
> of enumerated proofs.
>
> 3. Therefore, humans can do things that no machine ever could.
>
> The point to note is that humans can't enumerate all proofs any
> better (in fact much worse) than a computer. What they have done
> to "cherry pick" the low-hanging fruit. To be specific, they have
> found some examples of true statements that are outside a certain
> list of theorems.
>
> But it's also possible for a computer to "cherry pick" simple
> theorems that happen to lie outside a given enumeration. So
> that argument and many others, which may be disguised by side
> issues about neurons and quantum mechanics, prove nothing.
>
> In any case, it is quite obvious that people don't think like
> computers. The neatly formalized proofs in mathematical
> journals have no resemblance to the way the authors actually
> discovered them. See the quotations below.
>
> > Seems to me that efforts like Common Logic and LBase would either
> > have to a) be defined within this type of an open ended system,
> > let's say as the natural language description of the constraints
> > to which the axioms that make up the theory of such a system
> > would adhere; or b) evolve into an open ended system that
> > exhibits characteristics of transformation across languages,
> > logics, models and theories.
>
> I strongly agree that we have to keep our systems open ended.
> But Common Logic and the LBase subset are neutral with respect
> to the methodologies for designing and using open-ended tools
> and ontologies.
>
> Common Logic is a generalization of a wide range of logic-based
> tools and systems that have been developed over the past 30
> years, and it can support many more systems and methodologies.
>
> There have also been some requests for more features, which
> have been included in the IKL proposal. There are still many
> questions about which extensions should be included in a future
> version of CL, but they can be considered as time goes by.
>
> John
> ___________________________________________________________________
>
> Paul Halmos, mathematician:
>
> Mathematics -- this may surprise or shock some -- is never deductive
>
> in its creation. The mathematician at work makes vague guesses,
> visualizes broad generalizations, and jumps to unwarranted conclusions.
> He arranges and rearranges his ideas, and becomes convinced of their
> truth long before he can write down a logical proof... the deductive
> stage, writing the results down, and writing its rigorous proof are
> relatively trivial once the real insight arrives; it is more the
> draftsman's work not the architect's.
>
> Richard Feynman, physicist:
>
> We have a habit in writing articles published in scientific
> journals to make the work as finished as possible, to cover up all the
> tracks, to not worry about the blind alleys or describe how you had the
> wrong idea first, and so on. So there isn't any place to publish, in a
> dignified manner, what you actually did in order to get to do the work.
>
> Gian-Carlo Rota, mathematician:
>
> Shocking as it may be to a conservative logician, the day will come
> when currently vague concepts such as motivation and purpose will be
> made formal and accepted as constituents of a revamped logic, where they
> will at last be alloted the equal status they deserve, side-by-side with
> axioms and theorems.
>
>
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> (06)
--
Always forward towards the supreme maxim of scientific philosophizing (07)
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