Douglas Hofstadters ‘Gödel-Escher-Bach’
In the 1980s, I read Douglas Hofstadter’s cult book ‘Gödel-Escher-Bach’ with fascination. Central to it is Gödel’s incompleteness theorem. This theorem shows the limit for classical mathematical logic. Gödel proved this limit in 1931 in conjunction with the fact that it is, in principle, insurmountable for all classical mathematical systems.
This is quite astonishing! Is mathematics imperfect? As inheritors of the Age of Enlightenment and convinced disciples of rationality, we consider nothing to be more stable and certain than mathematics.
Hofstadter’s book impressed me. However, at certain points, e.g. on the subject of the ‘coding’ of information, I had the impression that certain aspects were greatly simplified by the author. In my opinion, the way in which information is incorporated into an interpreting system plays a major role in the recognition process in which information is picked up. The integrating system is itself active and participates in the decision-making process. Information is not exactly the same before and after integration. Does the interpreter, i.e. the receiving (coding) system, have no influence here? And if it does, what influence does it have?
In addition, the aspect of ‘time’ did not seem to me to be sufficiently taken into account. In the real world, information processing always takes place within a certain period of time. There is a before and an after. A receiving system is also changed by this. In my opinion, time and information are inextricably linked. Hofstadter seemed to miss something here.
Strong AI
My reception of Hofstadter was further challenged by Hofstadter’s positioning as a representative of ‘strong AI’. The ‘strong AI’ hypothesis states that human thinking, indeed human consciousness, can be simulated by computers on the basis of mathematical logic, a hypothesis that seemed – and still seems – rather daring to me.
Roger Penrose is said to have been provoked to write his book ‘Emperor’s New Mind’ by a BBC programme in which Hofstadter, Dennett and others enthusiastically advocated the strong AI thesis, which Penrose obviously does not share. As I said, neither do I.
But of course, front lines are never that simple. Although I am certainly not on the side of strong AI, Hofstadter’s presentation of Gödel’s incompleteness theorem as a central insight of 20th century science remains unforgettable to me. With enthusiasm, I also read the interview with Hofstadter that appeared in Spiegel this spring (DER SPIEGEL 18/2014: ‘Language is everything’). In it, he postulates, among other things, that analogies are decisive in the thinking of scientists and he differentiates his interests from those of the profit-oriented IT industry. These are thoughts that one might very well endorse.
Selfreference and incompleteness
But let’s go back to Gödel. What – in layman’s terms – is the trick in Gödel’s incompleteness theorem?
The trick is the same as in the barbar paradox and all other real paradoxes. The trick is to make a sentence, a logical statement and …
1. To refer it to itself (Selfreference)
2. and then to deny it. (Negation)
That’s the whole trick. With this combination, any classic formal system can be broken.
I’m afraid I need to explain this in more detail …
→ „Self-reference 2“ (will be translated to English soon)
Summary
Self-referentiality causes classical logical systems such as FOL or Boolean algebra to crash.
More on the topic of logic → Overview page Logic
German original (2015): Selbstreferenz
Translation; Juan Utzinger