Shaking Up Foundations Of Math: Roger Penrose On Kurt Gödel's Groundbreaking Work
Roger Penrose
Best known for his Incompleteness Theorem, Kurt Gödel
(1906-1978) is considered one of the most important
mathematicians and logicians of the 20th century. By showing that
the establishment of a set of axioms encompassing all of
mathematics would never succeed, he revolutionized the world of
mathematics, logic and philosophy.
Sir Roger Penrose is known worldwide for his work in
mathematics and mathematical physics, in particular general
relativity and cosmology. Currently Emeritus Rouse Ball Professor
of Mathematics at the University of Oxford and Emeritus Fellow of
Wadham College, he has been awarded numerous honors for his
scientific contributions. His many books include, The
Emperor’s New Mind: Concerning Computers, Minds, and the Laws
of Physics; Shadows of the Mind: A Search for the Missing
Science of Consciousness; The Nature of Space and Time
(with Stephen Hawking) and most recently The Road to Reality:
A Complete Guide to the Laws of the Universe.
Q: Kurt Gödel’s 1931 Incompleteness Theorem
disrupted German mathematician David Hilbert’s agenda for the
20th century mathematical research and rocked the very
foundations of mathematics in general. What was this pivotal
insight that turned the foundations of mathematics on its
head?
A: Hilbert was hoping to be able to formalize mathematics in a
completely clear way, so that the issue of whether a result was
to be considered to be “proved” could be made completely
unambiguous. This desire had been prompted by the appearance of
“paradoxes,” such as Bertrand Russell’s "set of all sets
that are not members of themselves." If some area of
mathematics could be formulated in such a way that the proof
procedures are completely unambiguous and clear cut (in a sense
that I shall come to below), one should be able to make sure that
contradictions, such as Russell’s paradox, didn’t occur (i.e.
were not part of the accepted proof procedures), then that area
of mathematics would be put on a sound basis.
Gödel’s Incompleteness Theorems showed that Hilbert’s
program was unachievable — at least for sufficiently broad areas
of mathematics (such as the ordinary number theory of the
integers). Gödel showed that for such an area of mathematics, for
any proposed formal system F (a “formalization,” in the above
Hilbertian sense) which intended to describe it would always fail
to be able to establish some result (that could be explicitly
constructed in terms of the rules of F)—-let us call this result
G(F)—-even though G(F) could be seen to be necessarily true, by
methods outside the rules of F, provided that the rules of F
could themselves be trusted as yielding only true results. In the
form of Gödel’s result that is most commonly referred to is his
“second” Incompleteness Theorem, in which G(F) effectively
asserts that F is consistent, so the argument tells us that the
consistency of F cannot be proved within the rules of F
itself.
In my view, this has the appearance of somewhat downgrading
the significance of Gödel’s theorem, because it gives it
perhaps a somewhat circular appearance, “consistency” being a
somewhat internal matter of concern.
Q: How would you explain Kurt Gödel’s
Incompleteness theorem to a layman?
A: I prefer to state Gödel’s result in a more direct way,
using Turing’s notion of computation. The point about a formal
system F is that it provides a proposed method of “proof” which
has the character that the correctness of any such “proof”,
according to the rules of F, is computationally checkable. That
is to say, there is a computer program P[F] which when applied to
any such “proof” will always say “OK” or “NOT OK” after a finite
time. What Gödel tells us is that if we are presented with F
which we “believe in” in the sense that we are prepared to accept
as actually true any mathematical statement that P[F] says “OK”
to a proposed proof of, then there is a specifically
constructable mathematical statement G(F) which we must also
accept as actually true, but which there is no “proof” within the
rules of F which P[F] will say “OK” to.
To put this another way, if we accept F as giving us a sound
set of procedures of mathematical proof, then we are able (via
Gödel’s ingenious argument) to transcend the methods of F to
see the truth of results that are beyond the scope of F. Thus, if
we trust F, then we can transcend F.
Q: In your book “The Emperor’s New Mind” you made
clever use of Gödel’s proof to advance the view that
artificial intelligence is impossible, or that machines cannot
think. Can you briefly explain the main thrust of your
argument?
A: Basically the thrust of my argument is that the quality of
“understanding” is something outside the capabilities of a
computer. It is through understanding that we can use the Gödel
argument to extend our belief in the trustworthiness of some F to
the belief in the truth of G(F), even though G(F) is unobtainable
by means of the rules of F. The generality of Gödel’s argument
simply illustrates how powerful conscious reasoning (through
understanding) can be. Just following rules (which is what
computers do—-albeit extraordinarily well) is something very
different from understanding. (This is something that
educationalists know very well!) I argue that understanding
(whatever it is) requires “consciousness” (whatever “that”
is!).
To take the argument further, I take the view that the quality
of consciousness is something that is potentially out there in
the physical world, and is not necessarily something unique to
human beings. But I regard the Gödel argument as showing that
conscious understanding is something that cannot be properly
imitated by a computer. So I argue that if consciousness is part
of physics—-describable by the “true” laws of physics—-then the
true laws of physics must be non-computable. It is known (using
Gödel-Turing-type arguments) that there are many areas of
mathematics which are actually non-computable, so I am claiming
that the true laws of physics (not yet fully known to us) must
also be non-computable. But the known laws of physics are
(more-or-less) computable, so we must look outside the known
laws. I argue, further, that the only plausible loophole in the
laws that we know lies in the issue of quantum measurement, and
that the “measurement paradox” (basically “Schrödinger’s cat”)
points to where we need to make further progress in our
understanding of the laws of physics in order to uncover what is
actually non-computable in the true laws).
Q: It’s been 20 years since the publication of “The
Emperor’s New Mind.” How has your viewpoint held up?
A: In my book “Shadows of the Mind” I developed these ideas
quite considerably, mainly in three directions (1) strengthening
the Gödelian argument (making it more rigorous) (2) improving my
criterion for the onset of new physics, in relation to the
“measurement paradox” (3) learning from Stuart Hameroff about
microtubules, and taking the view that it must be at the level of
neuronal microtubules, basically, (rather than neurons) that the
required coherent quantum processes (and “non-computable
beyond-quantum-mechanics” processes) must manifest
themselves.
How has it held up? Of course, many people have remained
skeptical. But despite the many (often aggressive) arguments from
others, my arguments seem to me to have stood up well enough (and
are described in the soon-to-be published proceedings of the
Vienna conference honoring Gödel’s centenary, with the
approval of some of my sternest critics from the community of
logicians). On the biological side, there are some recent very
striking results concerning microtubules, but these are not
published as yet. On the quantum physics side, there are some
theoretical developments, but the (extremely difficult)
experiments are still being developed.
Q: Philosopher J.R. Lucas advanced a similar argument
in a paper entitled “Minds, Machines and Gödel” in the journal
Philosophy in 1961. Are you familiar with his paper?
A: Yes, Lucas put forward a similar type of argument to my own
before I did (and Nagel and Newman before Lucas, and Gödel before
them), although I believe that my own argument has rather more
mathematical rigour than Lucas’s one did. Of course, Lucas was
arguing from the point of view of a philosopher, and I from the
point of view of a mathematical physicist.
Q: How did Gödel’s proof influence Alan Turing’s
work?
Quite a lot. Turing was very impressed by Gödel’s argument,
and he developed that argument further, phrasing it in terms of
non-computability, more-or-less in the way that I have done
(following Turing) above. Turing’s philosophical standpoint (at
least later in his life) was different from Gödel’s, however.
Gödel seemed to think that human minds must transcend physics,
whereas my view is that conscious minds must transcend the
presently known physics, but that physics is too limited. Turing
seemed to base his later views on a computer model of minds in
which the way around the Gödel theorems lies in the fact that
conscious humans make mistakes. I try to argue in my books that
this is an implausible let-out.
Q: Around the same time that Gödel was working on his
Incompleteness Theorems, another logician by the name of Alfred
Tarski was working on similar results. Why do you suppose
Tarski’s work hasn’t garnered as much attention as
Gödel’s?
A: I don’t know the history well enough. There were several
other logicians who were on to the same sort of issues that Gödel
was. My guess is that Tarski’s results weren’t so developed as
Gödel’s at the time, but I don’t really know the details. I had
the impression that Gödel’s results were a bit of a bomb-shell,
though taking a bit of time to be appreciated fully.
Q: Another fundamental result that Gödel worked on was
his proof of the consistency of two problematic hypotheses with
the axioms of set theory in 1939. Can you briefly explain
this?
A: I think this must refer to the Gödel work that was carried
on by Paul Cohen. They showed that Cantor’s continuum
hypothesis and the axiom of choice (two famous assertons in
mathematics) cannot be proved or disproved within one of the
standard formal systems for mathematics (known as the
Zermelo-Frankel system). This is very interesting, of course, but
as we already know from the Gödel Incompleteness Theorems, proof
within a specific formal system is not the same as being able to
see that something in mathematics is true or false by general
mathematical argument.
Q: Gödel also dabbled outside of his field of
expertise by proving that time travel to the past was possible
under Einstein’s equations. Should we give any credence to his
proof?
A: He only showed that such time travel was possible within
his specific cosmology. This, of course, is fascinating, but we
don’t now believe that this particular cosmological model
actually holds for our own universe. Yet Gödel’s arguments were
ahead of their time and certainly influential in the development
of relativity theory.
Q: Like Gödel, you are a Platonist who views
mathematical truth as "absolute, external, and eternal, and
not based on man-made criteria." Is there any proof one can
evince to support such a standpoint? Or is it just a
belief?
A: I think there are many possible different understandings of
what "Platonism" means. Some "Platonists"
like Gödel were very "strong Platonists" in the sense
that they would believe that all mathematical statements must
have an absolute truth value---so the truth is in a sense
"out there," and not the product of our minds, having
some subjective aspect to them. In my own case, I do not feel so
strongly as Gödel seemed to that all mathematical truth is
objective, but I would probably go much of the way with him.
There is a separate issue having to do with the basis of physical
reality. Is physical reality based on a deeper mathematical
reality? I think that my own picture is best expressed in my
"Three-worlds" picture (in "Shadows of the
Mind" and "The Road to Reality"), in which I
indicate how the physical world, the mental world of conscious
experience, and the Platonic world of mathematical forms
inter-relate to one another via three "mysteries". This
is not really a belief system, however, but rather a clarifying
picture, in my view.
