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“

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Summary

Self-referentiality causes classical logical systems such as FOL or Boolean algebra to crash.

More on the topic of logic → Overview page Logic

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German original (2015): Selbstreferenz

Translation; Juan Utzinger

LDC (Logodynamic cards)

LDC is the new project of my team. It combines a digital slip box (Zettelkasten) for your personal collection of ideas, notes and internet-links with a safe exchangebility of personally selected zettels (cards).

As a surplus we have added a novel deliniation of logic, especially convenient for the dynamics in reflections and discussions.

LDC is freeware and publicly available at ld-cards.com. Thank you for your interest.

Scales are mathematical patterns

When you hear a melody, it is based on a musical scale. The scale consist of the small number of tones which are allowed and may occur in the melody. In a linear sequence, these tones constitute the musical scale. Most melodies that can be heard in our cultural area can be traced back to one single scale, the Ionian or major scale, which is made up of seven notes in very specific scale steps.

Thousands of scales

However, there are thousands of different scales. Presumably you are familiar with the minor as well as with major scale and may have heard something about pentatonic scales or about whole tone scales, Lydian and Phrygian scales, Indian ragas, Japanese and African scales. All these scales differ from each other.

As we will see, however, they have some astounding similarities. Why should people all over the world, in all cultures and with all their differences, comply voluntarily and strictly with these similarities? The reasons for this are easy to explain if we don’t merely look at one world (‘world’ in the sense of Penrose), but at the interaction of all three worlds.

In which of the three worlds do the scales exist?

Scales are part of our reality, no matter how we define reality – unless we define reality as that which we call matter. In that case, the scales are not part of matter. They may manifest themselves in the physical world, for instance if a human being sings or plays them, but they have an identity which is independent of the individual way they are performed. In this sense, scales are non-local, as is typically the case with entities of the Platonic world. Between the scale and its performance, there is thus the relationship of an abstract, i.e. Platonic pattern with its material instance. This is always a 1/n relationship, for the pattern is unique but can be the source of any number of instances.

As a pattern, scales are part of the Platonic world, even though they manifest themselves in the material world. Mathematics, in particular, has much to do with the form of scales, which can be demonstrated easily, yet you don’t need to know anything at all about this kind of mathematics in order to recognise the scales correctly or to sing them. Your mental world in which you experience these scales has no need of figures and formulae.

Scales thus exist in all three worlds:

Platonic world: here, a scale exists as an entity, i.e. as a unity and as a whole. Here, every scale exists only once.

Physical world: here, a scale exists as any number of occurrences – whenever melodies are produced on the basis of it.

Mental world: here, i.e. in your head, you recognise the melodies and the scales.

Of course, each world is organised in its very own way. Now, how do the three worlds interact?

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This is a text in the series about the theory of the three worlds.

Translation: Tony Häfliger and Vivien Blandford