Godel, Escher, Bach – Douglas Hofstadter


“Relying on words to lead you to the truth is like relying on an incomplete formal system to lead you to the truth. A formal system will give you some truths, but as we shall soon see, a formal system—no matter how powerful—cannot lead to all truths. The dilemma of mathematicians is: what else is there to rely on, but formal systems? And the dilemma of zen people is: what else is there to rely on, but words?” — Douglas Hofstadter, in Godel, Escher, Bach.

Two Types, Two Types.

There are two types of systems, formal and informal. The former is the type of system which is completely contained within a specific set of rules. Think of Euclidian geometry for example, in his first book of The Elements Euclid states the definitions, postulates, and common notions that are used to deduce all of the theorems in his work. Informal systems—or “informal systems”, as Hofstadter calls them— are “those which obey so many rules of such complexity that we do not yet have the vocabulary to think about it” or that as far as we know, could be not contained within just one set of rules. There is no strict rule on how to strictly differentiate one from the other, all systems could be seen as formal systems of different complexities really.

Another way to think of types of systems would be as open and closed, have in mind that this is classification is not precisely isomorphic to the one mentioned before. Closed systems would be those which allow no inputs, and cannot produce outputs that have effects outside of its system. Euclidian geometry would be an example of this once again, the universe is (arguably) another example. Open systems are the opposite, in the sense that they do allow inputs and produce outputs that affect the way they operate. Think of our minds, outputs, such as opinions presented to us, have an effect in what we think; or the free market.

Johann Sebastian and Maurits Cornelis

Bach’s music, especially the contrapuntal compositions Crab Canon and Neverending Canon, and Escher’s Drawing Hands and many other drawings are an exploration into self-reference and recursion. When objects, ideas, art, or systems for that matter, speak about themselves reality starts to fade away and we enter a conceptual dreamland, the strange-loop created by self-reference leads us into a labyrinth of contradictions.


Once-in-a-big-bang mastermind mathematician Kurt Gödel had a couple of things to say about systems. Gödel’s incompleteness theorems prove that there is no system of mathematical logic that could be both complete and consistent, meaning that no system can prove all truths about the relationships between natural numbers and not fall victim of internal contradictions. There will always be statements that are unprovable within the system; The second part of the incompleteness theorem proves that a system cannot demonstrate its own consistency, it needs a system of equal or more strength.

“If Gödel was right, belief in mathematics also requires a leap of faith. All significant mathematical systems are open and incomplete. Even in mathematics, truth goes beyond our ability to prove that it is true.” Kitty Ferguson, in Fire in the Equations

The Mind

So, what type of system would the brain be? The brain is the most complex object known in the universe, that is, to the brain itself. What results the most intriguing about this is that on the lower levels the brain is a combination of chemicals and energy working together, under rules that are not too complex. When we go deeper into the question however—Or should I say, when we look at it from a farther perspective—we notice the emergence of the epiphenomenon of our minds. Thoughts don’t seem to obey many strict rules, you could say they seem to obey a very vast amount of ever-changing rules, they are spontaneous, sometimes coherent and more often not. It seems as if a very informal system emerges from the not too informal structure of the brain.

“If our psychology is entirely explainable in terms of physical processes, as we said above it might be, then any meaning you or I attach to events might be similarly explained by science. For instance, the birth of my child may have meaning for me beyond the physical event because of my psychology and chemistry. Suffering, beauty, evil might all be reduced to physics, chemistry, and the way we have evolved to feel, think, and react. Perhaps there is no meaning in any of this beyond the ability of science to explain.” Kitty Ferguson, in Fire in the Equations

Painting the brain in this way is definitely an over simplification. The brain, with it’s more than 10 billion neurons connecting all over the place is definitely not our definition of a simple, clean, formal system. But this is not the point in discussion right now.

In order to understand our brains we need to understand the higher levels, the thinking processes, the very complex and very informal part of our brains. We need to understand where intelligence comes from.

Artificial Intelligence

Defining intelligence is very complicated and I have no intentions of reaching a definite answer, however, I will give a very brief explanation of a few of the elements of intelligence and the difference between ‘mechanical processes’ and ‘intelligent processes’.

Mechanical processes are those done by calculators or computer programs. They are very tightly defined formal systems. Their functions cannot exceed those that are contained within the rules of the program, in other words, they cannot ‘pop out’ of the system. They don’t have room for inconsistencies and contradictions.

Intelligent processes on the other hand can change levels of systems, understand why something isn’t working, change the rules and try again. They can reference themselves, they can understand what they’re doing. They can accept and work with contradictions. Humans are intelligent in big part due to their subjunctive function, they can imagine the outcomes of different scenarios without actually living them. They can find and recognize patterns.

There is no clear line drawn of where mechanic processes end and intelligence begins. In fact, there’s a joke that goes like this, I’m paraphrasing here: every new advancement in AI shows us what intelligence is not.

What is tricky about this is that just as with formal and ‘informal’ systems, what is seen as an ‘informal’ system is no more than a system that is too complex for us to understand right now, intelligence could be a lot of mechanic processes that are too complex for us, and someday we might have the vocabulary and the capacity to understand and even create it.


To understand the world that surrounds us, we first must look within us. Everything that we experience is perceived through our senses, thought of through the lenses of our thoughts, past experiences, and assumptions about how life should be.

We try to understand our minds by the isomorphisms to the world that surrounds us, we try to find ourselves in the universe. Without them we would be left wandering in an endless sea of irrational thoughts. This comparison, this search for the similitudes, is what the search for intelligence is all about.