A comment from an interested onlooker

Gentlemen, I am delighted to find someone discussing my book! I hope it
will not seem presumptuous of me if I venture an opinion in one topic
glancingly touched on.

Mr. Novak takes me to task for a violation of the Goedelian
Incompleteness Theorem. I must say he does me great honor by holding me
to so exacting a standard. I cannot recall the last pulp space opera I
read where the author was called upon to get his meta-mathematical
ontology right.

For the purposes of drama, it is assumed that the reasoning process of
artificial superintelligences, called a sophotechs, is beyond human
comprehension. It is said to be more like intuition (such as when you
look at a diagram in a geometry book and grasp at a glance that
vertical angles are equal), and less like linear thinking (such as
going through the formal proof of Euclid 1.14). We do not currently
have a formal, logical system for expressing intuitions, or for
reducing matters of judgment and opinion to measurable Eudoxian units,
and so it is premature to make dogmatic assertions about what the
limitations of such a system, if it is ever invented, if it ever could
be invented, must be.

In any case, Mr. Novak's objection, if I understand it (I am sure he
will be gracious enough to fill in the gaps in my understanding)
concerns a scene where one sophotech, is attempting to reprogram or
persuade another about in error in its moral reasoning. Mr. Novak
objects that (1) both sophotects must be reasoning about morality using
a formal, closed system, similar to the Principia of Newton; but (2)
that Goedel proves that no reasoning within a formal closed system can
be complete, that is, prove all its statements; ergo (3) it is not
possible that one sophotech could correct an error in moral reasoning
afflicting another sophotech.

Mr. Novak is free to interpret the scene he read in any way he wishes:
an author cannot complain about that. I merely note that nowhere in the
text does it say whether reasoning about morality among sophotechs
takes place within and only within a formal, closed system. Of course,
the text does not expressly deny this either. The matter never comes
up; the book leaves it open to the opinion of the reader.

But Mr. Novak is not free to interpret Goedel any way he wishes. If a
man miscalculates the ballistic fall of an apple, Goedel does not
prevent Newton from correcting the mistake. Goedel merely says that at
least one problem (like the three body problem) or at least one axiom
(like the equality of action to reaction) cannot be proved by a closed
formal system from within that same system. You cannot use Newton to
prove Newton.

Goedel does not say anything one way or the other about self-evident
metaphysical axioms, does not say anything about the axioms in one
system resting on conclusions proved in another and more powerful
system, and it certainly does not say that no self-contradiction can
ever be corrected.

Goedel says rather that for any closed formal theory, it is possible to
construct a statement that is true but not provable in the theory. That
is, each and every statement in a closed formal system cannot be
proved. I should hasten to add this does not apply to all formal
systems. There are first-order axiomatizations of Euclidean geometry
that fall outside Goedel, for example. Systems with recursive axioms
(i.e. non-closed), for another example, also fall outside Goedel.

In this case, my humble book never says the one sophotech or the other,
in their high-speed debate of morality, were called upon to prove each
and every statement within a given closed, formal system. It is not
clear that what is being described is a closed, formal system in the
Goedelian sense. The sophotechs came from a common cultural background;
nothing in the text implies that they would not share certain axioms
and procedures in common. Any axiom which both sophotechs took for a
self-evident first principle would not come into question. Even if the
moral reasoning were confined to a formal system, if the system were
below a certain level of expressiveness or if it contained an infinite
number of axioms, Goedel's limit would not necessarily apply; or if the
system did not use itself to prove itself.

Goedel says that everything cannot be proved: I certainly hope Mr.
Novak does not take this to mean nothing can be proved.

Here is the paper in question:
http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf.
Here is a scaled-down explanation, more suited to bewildered amateurs
like me: http://www.apronus.com/math/goedel.htm

Now, on the other hand, if one wanted to make the argument that the
premise of THE GOLDEN AGE was absurd, that is an easier argument to
make. The premise of the book is that the human mind in its entirely,
personality and memory and all, can be recorded and redacted, rewired
and reworked to any specification. This might imply that a closed
formal system of symbols exists to depict the human mind in all its
complexity; and an argument can be made that such a system would
violate Goedelian incompleteness. Ah! There you have me.

But if the reader is willing to swallow the camel that the human mind
can be recorded and reduced to a system, it is comical to choke on the
gnat that the moral sentiments can be reduced to a system. Are not the
moral sentiments one part of the human mind? If you record a mind, why
not record a conscience? If you can record a conscience, why not play
back that conscience like a virus into an enemy mind, to bring him over
to your side?

This is just what the characters attempt; but their attempt is based on
a faulty assumption, and so the attempt fails. Or perhaps it succeeds,
but in a way they did not expect. From the author's point of view,
Phaethon does not exactly cover himself with glory by the last act of
the scene, but he should be given points for his stubbornness.

John C. Wright
http://www.sff.net/people/john-c-wright/

.

Relevant Pages

  • Re: Goedel - interesting problem?
    ... Let's refer to the "Masterpiece of explanatory text" aka "Effect of the ... Goedel Theorem" as "the article" for convenience. ... >say exactly what particular axioms of arithmetic one has in mind. ... distinction changes lots of editing criteria for the article. ...
    (sci.logic)
  • Re: Help writing a paper on Godels Incompleteness Thorem
    ... Are you refuting Godel or not? ... That's not necessarily about the most axiom-like or examplar of axioms, ... Gregory Chaitin has taken the methodogy of Goedel even ... Boucher posits a maximal element of sorts of the natural integers. ...
    (sci.logic)
  • Re: Robot Evolution
    ... In the _same terms in which Goedel worked_, ... demesne of accomplishing that proof as Goedel ... well be provable under a stronger set of axioms, ... he then showed that the stronger set of axioms would ...
    (sci.bio.evolution)
  • Null-Axiom Set Theory
    ... The idea of having no axioms is for several reasons. ... empty, or an ur-element or minimal element, exist, that from anything ... Goedel sentence G_0 and the infinite chain of ghost axioms that each ... and applied mathematics. ...
    (sci.logic)
  • Re: GCH vs. Axiom of Choice.
    ... |familiar that somone has proven ... Only "finitistic" reasoning is actually required. ... That someone was Goedel. ... nonsequitur. ...
    (sci.math)