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[Cross-posted from goodreads]

If Godel proved that no sufficiently complex system, i.e one that is capable of arithmetic, can prove its own consistency or if you assume the system is consistent there will always exist (infinitely many) true statements that cannot be deduced from its axioms, in what system did he prove it in? Is that system consistent? In what sense is the Godel statement true if not by proof? You’ll have hundreds of questions popping in your mind every few minutes, and this short book does a very good job of tackling most of them.

Godel numbering is a way to map all the expressions generated by the successive application of axioms back onto numbers, which are themselves instantiated as a “model” of the axioms. The hard part of it is to do this by avoiding the “circular hell”. Russell in Principia Mathematica tried hard to avoid the kind of paradoxes like “Set of all elements which do not belong to the set”. Godel’s proof tries hard to avoid more complicated paradoxes like this :

Let p = “Is a sum of two primes” be a property some numbers might possess. This property can be stated precisely using axioms, and symbols can be mapped to numbers. ( for e.g open a text file, write down the statement and look at its ASCII representation ). The let n(p) be the number corresponding to p. If n(p) satisfies p, then we say n(p) is Richardian, else not. Being Richardian itself is a meta-mathematical property r = “A number which satisfies the property described by its reverse ASCII representation”. Note that it is a proper statement represented by the symbols that make up your axioms. Now, you ask if n(r) is Richardian, and the usual problem emerges : n(r) is Richardian iff it is not Richardian. This apparent conundrum, as the authors say, is a hoax. We wanted to represent arithmetical statements as numbers, but switched over to representing meta-mathematical statements as numbers. Godel’s proof avoid cheating like this by carefully mirroring all meta-mathematical statements within the arithmetic, and not just conflating the two. Four parts to it.

1. Construct a meta-mathemtical formula G that represents “The formula G is not demonstratable”. ( Like Richardian )
2. G is demonstrable if and only if ~G is demonstrable ( Like Richardian)
3. Though G is not demonstrable, G is true in the sense that it asserts a certain arithmetical property which can be exactly defined. ( Unlike Richardian ).
4. Finally, Godel showed that the meta-mathematical statement “if ‘Arithmetic is consistent’ then G follows” is demonstrable. Then he showed that “Arithmetic is consistent” is not demonstrable.

It took me a while to pour over the details, back and forth between pages. I’m still not at the level where I can explain the proof to anyone clearly, but I intend to get there eventually. Iterating is the key.

When I first came across Godel’s theorem, I was horrified, dismayed, disillusioned and above all confounded – how can successively applying axioms over and over not fill up the space of all theorems? Now, I’m slowly recuperating. One non-mathematical, intuitive, consoling thought that keeps popping into my mind is : If the axioms to describe arithmetic ( or something of a higher, but finite complexity ) were consistent and complete, then why those axioms? Who ordained them? Why not something else? If it turned out that way, then the question of which is more fundamental : physics or logic would be resolved. I would be shocked if it were possible to decouple the two and rank them – one as more fundamental than the other. I’m very slowly beginning to understand why Godel’s discovery was a shock to me.

You see, I’m good at rolling with it while I’m working away, but deep down, I don’t believe in Mathematical platonism, or logicism, or formalism or any philosophical ideal that tries to universally quantify.

My primary reason for reading Origin is its historical significance. It bothers me that humans, until as recently as Darwin, did not seriously ask about the origin of species despite interacting with its members every day. That this revolutionary idea should have occurred to one person as opposed to many in an incremental fashion, is far from obvious – and extremely interesting retrospectively. Are there any other such revolutionary ideas that’ll later be trivial to everyone?

Darwin supported his argument meticulously by observations, made by him and others, and their sheer number is so mind boggling that I fell asleep more times reading this book than any other. It amazes me how he summoned up so much enthusiasm to study and compare the most boring habits of some of the dullest creatures.

There is no reason to read this book to understand natural selection – our knowledge now is a far more superior and complete. I had already read The Selfish Gene, had already experienced the profound “OMFG! That’s Brilliant!” reaction that rapidly morphed into a “Duh! Isn’t that obvious?” . Yet, Origin was an enthralling read for the most part, with many opportunities to pause, wonder, daydream, extrapolate, apply the theory to modern humans and computers, and so on.

This is a fairly long book about a seemingly tautological argument. I highlighted a large number of lines and took down a few notes on my kindle. A short summary of my observations, and a few quotes to give you the taste of what most of the book is like, follows.

1. Variation – Darwin appreciated that variability exists in nature,
but he did not seem to explore the causes or consequences – which is understandable as genetics was ahead of his time.

“It may seem fanciful, but I suspect that a similar parallelism
extends to an allied yet very different class of facts. It is an old
and almost universal belief, founded, I think, on a considerable body
of evidence, that slight changes in the conditions of life are
beneficial to all living things.”

“Again, both with plants and animals, there is abundant evidence, that
a cross between very distinct individuals of the same species, that is
between members of different strains or sub-breeds, gives vigour and
fertility to the offspring.”

“Dominant species belonging to the larger groups tend to give birth to
new and dominant forms; so that each large group tends to become still
larger, and at the same time more divergent in character.”

“Widely ranging species vary most, (…) and varieties are often at
first local,–both causes rendering the discovery of intermediate
links less likely. (…) And if there be any variability under nature,
it would be an unaccountable fact if natural selection had not come
into play.”

2. Cooperation – Darwin appreciated that environment in the form of
sunshine, water, temperature ( non living stuff ), wasn’t the primary
reason for success of life in its complexity, and that inter-species
interaction is a large cause. However, he talks about competition,
much like Dawkins, but not much about cooperation, like Lynn Margulis. I
think cooperation is much less apparent when you just look at the
phenotype. It seems to me that it is more widespread on the smaller
gene level with microbes and viruses exchanging chemicals frequently, and
hence completely understandable why Darwin was unaware of it.

“This long appeared to me a great difficulty: but it arises in chief
part from the deeply-seated error of considering the physical
conditions of a country as the most important for its inhabitants;
whereas it cannot, I think, be disputed that the nature of the other
inhabitants, with which each has to compete, is at least as important,
and generally a far more important element of success.”

“Bearing in mind that the mutual relations of organism to organism are
of the highest importance, we can see why two areas having nearly the
same physical conditions should often be inhabited by very different
forms of life; for according to the length of time which has elapsed
since new inhabitants entered one region; according to the nature of
the communication which allowed certain forms and not others to enter,
either in greater or lesser numbers; according or not, as those which
entered happened to come in more or less direct competition with each
other and with the aborigines; and according as the immigrants were
capable of varying more or less rapidly, there would ensue in
different regions, independently of their physical conditions,
infinitely diversified conditions of life,–there would be an almost
endless amount of organic action and reaction,–and we should find, as
we do find, some groups of beings greatly, and some only slightly
modified,–some developed in great force, some existing in scanty
numbers–in the different great geographical provinces of the world.”

3. Ontology of god – Darwin did not comment much on Man, but he did
make strong rational arguments against creation, and understood the philosophy of science.

“Why should all the parts and organs of many independent beings, each
supposed to have been separately created for its proper place in
nature, be so invariably linked together by graduated steps? Why
should not Nature have taken a leap from structure to structure? On
the theory of natural selection, we can clearly understand why she
should not; for natural selection can act only by taking advantage of
slight successive variations; she can never take a leap, but must
advance by the shortest and slowest steps.”

“But many naturalists think that something more is meant by the
Natural System; they believe that it reveals the plan of the Creator;
but unless it be specified whether order in time or space, or what
else is meant by the plan of the Creator, it seems to me that nothing
is thus added to our knowledge.”

4. Altruism – Darwin identified and seemed to have understood the
conundrum posed by “selfless” individuals of the ant of the bee
communities. Since his argument is at the species level, it was easier
for him to reconcile the selfless acts of a few “slave” members as
beneficial to the species. Dawkins’ must have had a tougher time as
his arguments were at the genetic level, but even that was resolved
when it was shown that slave members had a different gene composition
that made it advantageous for their genes to die in the service of the
queen’s genes.

“Finally, it may not be a logical deduction, but to my imagination it
is far more satisfactory to look at such instincts as the young cuckoo
ejecting its foster-brothers,–ants making slaves,–the larvae of
ichneumonidae feeding within the live bodies of caterpillars,–not as
specially endowed or created instincts, but as small consequences of
one general law, leading to the advancement of all organic beings,
namely, multiply, vary, let the strongest live and the weakest die.”

“for if on the whole the power of stinging be useful to the community,
it will fulfil all the requirements of natural selection, though it
may cause the death of some few members.”

5. Falsification – I think it is important to note that when a
groundbreaking theory is introduced, it should make some substantial claims
that are against common knowledge. Although Darwin did not tackle all
the implications of his theory in this book, he did make remarks
which, at that time, would perhaps have been considered quite bold.

“hence there seems to me to be no great difficulty in believing that
natural selection has actually converted a swimbladder into a lung, or
organ used exclusively for respiration.”