If you were to be marooned on a deserted island, and could bring
three books, which ones would you choose? This familiar question,
suitable to get a dinner table conversation going, or to serve as
an icebreaker among strangers, deserves careful consideration. Who
knows, one day you might be forced to leave your home or your country
in a hurry, with just a minute to decide what to bring...
As for me, I would make sure to choose some books which would last
for a long time, and which I had anyway saved up for a more careful
reading once I retired and had more time for reading. (Alas, now
that I have been retired for four years, it turns out that I do
not have the time and patience needed to fulfil my youthful resolve
to broaden my education, which also included learning Chinese and
Arabic.) Upon careful consideration, here is my selection:
 "The Complete Works of William Shakespeare"
(Abbey Library, 1100 pages in small print)
 A chess book. I have not been able to decide between my old
favorite "Schachgenie
Aljechin" by H. Müller and Pawelczak, and
"Fundamental Chess Endings" by K. Müller
and Lamprecht, or perhaps "My 60 Memorable Games"
by Bobby Fischer or "My Great Predecessors"
by Kasparov (I have not read the two lastmentioned).
 "Gödel, Escher, Bach" by Hofstadter
(Penguin Books, 777 pages).
None of these books is what you might call a pageturner. But
they all offer a good deal of resistance, and so invite revisiting
and rereading.
"Gödel, Escher, Bach" struck me as a
bombshell in 1980. It addressed many of the subjects that had
fascinated me for a long time: the foundation of mathematics,
symbolic logic, the nature of language and meaning, artificial
intelligence, and the origin of life. And it raised as many
questions as it answered.
The book spans over many fields, from Zen Buddhism to the art of
Escher and Magritte and to DNA molecules. The main text is interspersed
with playful dialogues between "Achilles" and "the
Tortoise", offering an offbeat, whimsical perspective of the
more serious discussion. The latter is presented in accordance with
Einstein's dictum: "Make everything as simple as possible,
but not simpler." A mathematical background is not required,
although it may make the reading a little easier.

A selfreferential Escher
drawing.

The book sparkles with ideas and associations. It seeks to tie
together the works of Gödel, Escher and Bach, but it is clear
that it deals mainly with Gödel. Escher and Bach are introduced
chiefly to offer examples of the unifying concept of selfreference,
and to lighten up the text with some interesting discussions and
illustrations which provide variations of the main theme in a different
context
Out of the great achievements of 20th century science, that of
Gödel is probably the least known and understood by the scientific
community, relative to its fundamental importance, to say nothing
of the general public. You might say that Gödel's main work,
published in 1931, completed the great earthquake that happened
in science with the establishment of the Special and General Theories
of Relativity and of Quantum Theory. It overturned the foundations,
not only of physics, but of mathematics itself.
Ever since Euclid, it had been accepted that mathematics dealt
with deriving the logical consequences of a few simple postulates:
axioms, or "selfevident truths". Euclid's own work "Elementa"
was exactly that: a collection of truths about geometry derived
from a few axioms. Much later, in the 17th century, the Calculus
was developed by Leibniz and Newton. It involved the calculation
of areas and volumes through successive approximations, leading
to infinite series. The methods worked, but caused a great deal
of confusion due to the underlying concept of adding an infinite
number of "infinitely small" increments to arrive at the
result. During the 19th century, a more solid foundation was laid
by mathematicians such as Cauchy and Weierstrass.
A low point in my own education in mathematics
came in high school, when I had missed a few lessons due to
illness. I was unable to grasp, or at least to apply, Weierstrass'
famous definition of continuity (and I failed miserably in a
math test for the first and last time): "f(x) is continuous
at ξ if for every positive number ε, no matter how
small, there can be determined another positive number δ
= δ(ε) such that f(x)  f(ξ) < ε
for all points x for which x  ξ < δ." Today
this seems pretty obvious to me, but I can see why this would
be a stumbling stone for a high school student!
A further attempt was made to anchor arithmetic in basic axioms
of logic. This was a rather esoteric exercise. "If a
implies b, and b implies c, can we be certain
that a implies c, or do we need a further axiom
If a implies b and b implies c, then a implies c"?
Are proofs involving induction valid? Many similar issues, some
perhaps more contentious than these, had to be settled. This
work culminated in Russell's and Whitehead's Principia Mathematica
in 19101913, where in part II, page 362, we find:
From this proposition it will follow, when arithmetical addition
has been defined, that 1 + 1 = 2.
In the course of his investigations, Russell discovered this paradox:
"Consider the set of all sets that are not members of themselves.
If such a set exists, it will be a member of itself if, and only
if, it is not a member of itself."  This paradox echoes
the famous paradox of the Cretan Epimenides: "All Cretans
are liars" (to be understood that all Cretans lie all the
time) and points forward to Gödel's results.
One may ask why mathematicians are so obsessed with paradoxes,
which may seem contrived, involving selfreferences at different
levels of abstraction: "The following sentence is true.
The previous sentence is false." etc. The answer, pointed
out again and again in Hofstadter's book, is that selfreference
is a key feature of intelligence, and of life itself.

Kurt Gödel 1906 
1978.

Gödel's incompleteness theorem tells us that all consistent
axiomatic formulations of number theory (including Russell's
Principia Mathematica) include undecidable propositions,
i. e. propositions whose truth or falsehood cannot be decided within
the theory. In other words, given any consistent set of arithmetical
axioms, there are true mathematical statements that cannot be derived
from the set.  I should point out that this does not prove that
there are any "unprovable truths", just that there are
truths which cannot be derived within the axiomatic formulation
of number theory.
Gödel's proof rests on the observation that any sentence consisting
of a string of letters or symbols can be assigned a unique natural
number by using prime numbers. The Gödel number thus generated
allows us to retrieve a sentence through factorisation. There is
a onetoone correspondence between a sentence and its Gödel
number.
Now, as each sentence within the axiomatic system corresponds to
a natural number, the rules of arithmetic can be applied to show
that within the system a sentence referring to itself can be formed,
the truth of which cannot be determined within the system. Moreover,
it can be shown that the consistency of the axiomatic system cannot
be demonstrated within the system itself, so any axiomatic system
on which number theory is based will necessarily be incomplete,
and it will be impossible to prove its consistency within the system
itself.
So, what does it all mean?
The crucial point is that Gödel's discoveries concern selfreferential
systems, and such systems are much more than a mathematical curiosity.
For instance the DNA molecule contains a coded message with instructions
on how to produce descendants of itself in a very roundabout way
(using proteins and enzymes to generate cells and organs and bodies
and gametes). Likewise, the Holy Grail of Artificial Intelligence
is the development of computer programs that modify themselves in
response to their environment. And the human brain itself is extremely
selfreferential, on several levels. Just think of what goes on
in your brain, when you think of yourself, or when you think about
thinking of yourself! Hofstadter believes that this process underlies
the experience of consciousness, meaning and self; and that complex
systems built upon simple mechanistic components, such as neurons
and transistors, can evolve awareness and fundamentally unpredictable
behavior.

Douglas Hofstadter (b.
1945)

This raises any number of interesting questions. For instance:
The human brain contains a hundred to a thousand billion neurons.
This is a number only moderately larger than the number of humans
on this planet, some 7 billion. Suppose we ran a simulation of a
brain, using one person per neuron, who would be in touch with perhaps
a thousand other "neurons" and send signals to other "neurons"
in response to inputs according to mechanistic rules  of course
under the assumption that we know what those rules are, and that
the structure and detailed wiring of the brain, at a given moment,
had been mapped. (Yes, I realize that that is quite an assumption!)
Such an assembly should in principle be capable of simulating a
brain, even if at much lower speeds. Could such an entity "experience"
the world? Could it become conscious?  I suspect that one would
have to add sensory impressions, or one might be stuck with a catatonic
being, a "cabbage case". But does the substrate matter
of the brain itself matter: neurons, transistors, or persons simulating
neurons?
In "The Emperor's New Mind", R. Penrose argues
that it does. Computers are restricted to algorithmic processing,
while humans are able to achieve insights through nonverbal thinking
and an intuitive sense of Truth and Beauty. Even a child can see
that the conscious mind cannot work like a computer, no matter how
complex. (Hence the title of his book.)  Personally, I find Penrose's
arguments unpersuasive, but of course that does not mean much ...
The debate rages on with undiminished intensity today, 25 years
later.
The fascinating question of what a mind is, is further explored
in "The Mind's I" by Hofstadter and Dennett.
Further reading
 A concise essay
on Gödel's findings.
 A discussion of some common misconceptions
about Gödel's results. (A collection of links.)
 A Gödel biography,
with some technical information on his work. (Stanford Plato archive.)
 "The Man Who Would Teach Machines To Think", article
in The Atlantic by James Somers 2013.
"Books"
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