Gödel, Escher, Bach

An Eternal Golden Braid

by Douglas R. Hofstadter

Basic Books, 1979; Penguin Books, 1980

Gödel, p 18-19

Gödel had the insight that a statement of number theory could be about a statement of number theory (possibly even itself), if only numbers could somehow stand for statements.

[...]

The grand conclusion? That the system of Principia Mathematica is "incomplete" —there are true statements of number theory which its methods of proof are too weak to demonstrate.

[...]

Gödel proof pertained to any axiomatic system which purported to achieve the aims which Whitehead and Russell had set to themselves.

[...]

Modern readers may not be as nonplussed by this as readers of 1931 were, since in the interim our culture has absorbed Gödel's Theorem, along with the conceptual revolutions of relativity and quantum mechanics, and their philosophically disorienting messages have reached the public [...] There is a general mood of expectation, these days, of "limitative" results —but back in 1931, this came as a bolt from the blue.

Meaning, information

Isomorphisms Induce Meaning, p 50

The perception of an isomorphism between two known structures is a significant advance in knowledge —and I claim that it is such perceptions of isomorphism which create meanings in the minds of people.

Recursively Enumerable Sets vs. Recursive Sets, p 72

I was quite convinced that not only the primes, but any set of numbers which could be represented negatively, could also be represented positively [...] How could a figure and its ground not carry exactly the same information?

[...]

There exists formal systems whose negative space (set of non-theorems) is not the positive set (set of theorems) of any formal system. [...]

There exists recursively enumerable sets which are not recursive.

Implicit and Explicit Meaning, p 82

We see the meaning without seeing the isomorphism. The most blatant example is human language, where people often attribute meaning to words in themselves, without being in the slightest aware of the very complex "isomorphism" that imbues them with meanings. This is an easy enough error to make. It attributes all the meaning to the object (the word), rather than to the link between that object and the real world.

Definition, Peano

Undefined Terms, p 93-94

Undefined terms [...] do get defined in a sense: implicitly —by the totality of all propositions in which they occur, rather than explicitly, in a definition.

[...]

A full formalization of geometry would take the drastic step of making every term undefined [...]

[...] symbols automatically pick passive meanings in accordance with the theorems they occur in.

Is Number Theory the Same in All Conceivable Worlds? p 100

It seems to be the consensus of most modern mathematicians and philosophers that there is [...] a core number theory, which ought to be included, along with logic, in what we consider to be "conceivable worlds". The core of number theory, the counterpart to absolute geometry, is called Peano arithmetic [...]

Genotype

Genotype and Phenotype, p 159

A molecule of DNA —a genotype— is converted into a physical organism —a phenotype— by a very complex process, involving the manufacture of proteins, the replication of the DNA, the replication of cells, the gradual differentiation of cell types, and so on. Incidentally, this unrolling of phenotype from genotype —epigenesis‐ is the most tangled of tangled recursions [...]

The Five Peano Postulates, p 216

1. Genie is a djinn.
2. Every djinn has a meta (which is also a djinn).
3. Genie is not the meta of any djinn.
4. Different djinns have different metas.
5. If Genie has X, and each djinn relays X to its meta, then all djinns get X.

Chunking and Chess Skill, p 286

[The chess master] thinks on a different level from the novice [...] The trick is that his mode of perceiving the board is like a filter: he literally does not see bad moves when he looks at a chess situation —no more than chess amateurs see illegal moves [...] This might be called implicit pruning [...]

Thought, classes, diagonal method

New Perspectives on Thought, p 337

It was only with the advent of computers that people actually tried to create "thinking" machines, and witnessed bizarre variations on the theme of thought. [...] As a result, we have acquired, in the last twenty years or so, a new kind of perspective on what thought is, and what it is not.

Classes and Instances, p 351

There is a general distinction concerning thinking: that between categories and individuals, or classes and instances. [...] It might seem at first sight that a given symbol would inherently be either a symbol for a class or a symbol for an instance —but that is an oversimplification. Actually most symbols may play either role, depending on the context of their activation.

What Does a Diagonal Argument Prove? p 423

Cantor's proof uses a diagonal in the literal sense of the word. Other "diagonal" proofs are based on a more general notion, which is abstracted from the geometric sense of the word. The essence of the diagonal method is the fact of using one integer in two different ways —or, one could say, using one integer on two different levels [...]

Mathematics

The Two Ideas of the "Oyster", p 438

[The idea of Gödel numbering] is an idea which goes far beyond the confines of mathematical logic, whereas the Cantor trick, rich though it is in mathematical consequences, has little if any relation to issues in real life.

Supernatural Numbers, p 454

[R]ecall that there is another number whose square is also minus one: -i. Now i and -i are not the same number. They just have a property in common. The only trouble is that it is the property which defines them! We have to choose one of them —it doesn't matter which one— and call it "i". In fact there's no way to tell them apart. So for all we know we could have been calling the wrong one "i" for all these centuries and it would have made no difference.

[Supernaturals Are Useful] ...But Are They Real? p 456

By taking the step of formalization, we were committing ourselves to accepting whatever passive meanings these terms might take on.

Bifurcations in Number Theory, and Bankers, p 457

Mathematics only tells you answers to questions in the real world after you have taken the one vital step of choosing which kind of mathematics to apply.

Genetics, locality, AI

Reductionistic Explanation of Protein Function, p 522

But for better or for worse, this is a general phenomenon which arises in the explanations of complex systems. In order to acquire an intuitive and manageable understanding of how parts interact —in short, in order to proceed— one often has to sacrifice the exactness yielded by a microscope, context-free picture, simply because of its unmanageability. But one does not sacrifice at that time the faith that such an explanation exists in principle!

Two Types of Form, p 582

[...] "semantic" properties are connected to open-ended searches because, in an important sense, an object's meaning is not localized within the object itself.

The Crux of AI: Representation of Knowledge, p 616

A large amount of work in AI has [...] gone into systems in which the bulk of the knowledge is stored in specific places —that is, declaratively.

The Concept Network, p 653

Science and the World of Bongard Problems, p 660

The Kuhnian theory that certain rare events called "paradigm shifts" mark the distinction between "normal" science and "conceptual revolutions" does not seem to work, for we can see paradigm shifts happening all throughout the system, all the time. The fluidity of descriptions insures that paradigm shifts will take place on all scales.

Soul, death and western science

Can Machines Possess Originality? p 686

Unless you are a soulist, you'll probably say that [your sense of having a will] comes from your brain —a piece of hardware which you didn't design or choose. And yet that doesn't diminish your sense that you want certain things, and not others. You are a "self-programmed object" [...]

Can We Understand Our Own Minds or Brains? p 697

[...] might there not be some vaguely Gödelian loop which limits the depth to which any individual can penetrate into his own psyche? Just as we cannot see our faces with our own eyes, is it not reasonable to expect that we cannot mirror our complete mental structures in the symbols which carry them out?

All the limitative Theorems of metamathematics and the theory of computation suggest that once the ability to represent your own structure has reached a certain critical point, that is the kiss of death: it guarantees that you cannot represent yourself totally.

Gödel's Theorem and Personal Nonexistence, p 698

Perhaps the greatest contradiction in our lives, the hardest to handle, is the knowledge "There was a time when I was not alive, and there will come a time when I am not alive".

Science and Dualism, p 699

Step by step, inexorably, "Western" science has moved towards investigation of the human mind —which is to say, of the observer.

Ism once again, p 704

Cage had led a movement to break the boundaries between art and nature. In music, the theme is that all sounds are equal [...] Leonard B. Meyer, in his book Music, the Arts, and Ideas, has called this movement in music "transcendentalism" [...]

Undecidability Is Inseparable from a High-Level Viewpoint, p 708

G's nontheoremhood is, so to speak, an intrinsically high-level fact. It is my suspicion that this is the case for all undecidable propositions; that is to say: every undecidable proposition is actually a Gödel sentence, asserting its own nontheoremhood in some system via some code.

Strange Loops as the Crux of Consciousness, p 709

My belief is that the explanations of "emergent" phenomena in our brains —for instance, ideas, hopes, images, analogies, and finally consciousness and free will— are based on a kind of Strange Loop, an interaction between levels in which the top level reaches back down towards the bottom level and influences it, while at the same time being itself determined by by the bottom level.