BOOKS: What I learned from “Godel, Escher, Bach” – Part II

Continued from Part I.

In this post we will look more closely at “symbols”, “levels of meaning” and “isomorphism”.

Symbols – the carriers of meaning

Strange loops arise in systems that are powerful enough to capture meaning. Meaning is achieved when networks of signals stand for one particular concept in the world.  Thus each letter of the alphabet signals a particular sound. These signals in turn get arranged into a pattern – a word – that refers to one particular object or idea from the external world.

In a similar vein, the firing of a single neuron in response to a specific stimulus is a signal. For instance, studies on the visual cortex have shown that different specific neurons are stimulated upon presentation of and variation in very specific features of the visual stimuli such as length and orientation.

When we sense (see, hear, or touch) a new object, all the neurons responding to its various features are activated in the pertinent area of our brain are activated and form a neural circuit. Circuits in different regions of the brain are themselves interconnected: As you talk about an apple you are using i) the muscles of your respiratory system to create the right sounds, ii) the corresponding visual circuit to visualize the apple, and iii) circuits in the language areas representing the word apple as well as the corresponding phonetic pattern needed to pronounce the word correctly.

The above described network caters to one meaningful concept in the world. It is to such networks that Hofstadter applies the term symbol. The world is full of such information-preserving  symbols and below are some examples:

Meaning is not inherent in the symbol

By now it is clear that the power of a symbol does not reside in the signals that it is made up of; rather it is the correspondence with a specific concept from the outer world. Every word will be a meaningless pattern of sound if it was not associated with something we are familiar with. Words such as ‘mother’, ‘money’ and ‘love’ evoke strong personal reactions in most of us, not because there is something in the special arrangement of those particular sounds, but because of what each of these words refers to.

Thus the fact that each constituent of the symbol stands for a particular sound is explicit (i.e. apparent). On the other hand, the fact that the pattern as a whole stands for something else altogether is implicit (i.e. hidden) –  the meaning is not readily apparent to anybody who’s not  well-practiced in the use of these symbols (e.g. a child, a person not familiar with the English language).

Meaning is thus independent of any rules for combining signals to produce patterns. That is how, even though the formal system in Principia Mathematica was especially designed to shun explicit self-reference, it is by association with a different (a higher, and less readily apparent) level of meaning that self-reference is achieved.

Isomorphism – reading meaning into patterns of signals

The key is that the transition from the explicit to the implicit level is  information-preserving. In math, such a case wherein elements of two sets correspond with each other in an information-preserving fashion (math theorems and Godel numbers in the above example) is called an isomorphism.

Thus the word “table” is isomorphic to that piece of furniture on which my PC sits. So is the neural circuit that gets activated in my mind when I think of a table. The symbol ‘=’ is isomorphic to the concept of “is equal to”.  Genes are isomorphic to the protein synthesized from them. A code is isomorphic to the text of the message it hides. The camera film is isomorphic to the color photograph printed from it.

In short, human thinking and culture is fraught with isomorphisms of various kinds.

As far as we cannot detect and read the isomorphism, the structural similarity between two different sets of elements, we will be oblivious to the fact that one is a message encoded by the other. In Hofstadter’s terms, meaning is induced in the explicit lower level matter (or components of a pattern) by identifying its isomorphism with a real world concept at a more abstract level.

We recently have had an eerie reminder of this fact, when a scientist Craig Hogan realized that he may have hit upon some observations supporting the holographic principle. It seems that the totality of information on all the particles in our 3D universe may be contained on the 2D cosmic horizon…

Isomorphisms and mental life

The swirls of neuoronal activity back and forth across the brain are isomorphic to mental activity. In other words: consciousness of our inner life is attained because we can read off the explicit neurological processes at a much more higher and implicit level.

When you are looking at a TV screen, the data you are receiving is nothing but a fluctuating pattern of pixels. But you are not conscious of this ‘lower level’ of the message. You can simply read off the higher level meanings coded for by those pixels – feat we call as perception.

This shows how its totally unnecessary to be conscious of the lower level in order to read the implicit. Fluent readers are rarely conscious of the exact letter sequences making up the words they are reading. A practiced reader in a book describing highly visual scenes will simply see the scene by scene depiction of the story on the pages. The feel of reading is completely replaced by the sense of watching in such instances…

In sum, both intelligence and consciousness may be redefined as our capacity to perceive the meaningful isomorphisms in the world and within ourselves.

Tangled hierarchies and strange loops

Many a times a clear differentiation of ‘lower’ and ‘higher’ levels is possible when dealing with two isomorphic sets. A majority of the examples of such sets given above illustrate this differentiation. There are times however when such a clear differentiation is not possible, since levels keep leading back to each other.

It is the entangledness of our concepts that leads to recursion in human thinking. All our concepts are interrelated, are constantly activated by each other, and this constant exchange among themselves and with information from the outside leads either to modification or reinforcement of every concept.

We define our preferences and loves in relation to our own selves. We reflect upon the outer world and interacting with it obtain further information from ourselves thus re-affirming, enhancing or adjusting our self-concepts. All this modification is indeed not just at the abstract, conceptual changes. The changes are reflected in the underlying patterns of connections across networks of neurons. This is what happens in any level-crossing feedback loop. The system does not just mirrors meaning, it has the capacity to  change in response to changing information.

That is also why human intelligence is definitely superior to machine intelligence. Where a computer will get hanged, the human will leave the level on which it was working (for e.g. some office task) and work on other levels to solve the problem (for e.g. confronting the supervisor who didn’t explain the task fully, confronting and modifying one’s own level of knowledge and skill required to achieve the task, etc.).

The beauty of Hofstadter’s ideas is that they apply equally well to human intelligence and human consciousness. Fluid Concepts and Creative Analogies is the result of his research into the intricacies of human cognition, whereas I am a Strange Loop presents more fully Hofstadter’s ideas of the emergence of the human ‘I’.

An example of the all-tangled up semantic network underlying GEB (Click on the image to see in full size)

After I have completed reading I am a Strange Loop, I’ll find some excuse to post about it as well, InshaAllah!

BOOKS: What I Learned from ‘GODEL, ESCHER, BACH’ – Part I

In keeping with UNESCO’s World Book Day which came to pass yesterday, I would love to revise my concepts from Gödel, Escher, Bach: An Eternal Golden Braid. They book first published in 1979, won for its author, Douglas Hofstadter, the Pulitzer Prize (1980) in non-fiction.  GEB may give the appearance of being a fun book, about music, art, computers, riddles and puns, but it is not just that. Using observations  from the diverse fields of math, language, and genetics, the book actually introduces very innovative concepts in an attempt to explain the nature of human thinking and intelligence.  The art of Escher and the compositions of Bach are actually referred to as analogies to help illustrate the highly abstract ideas presented. The inventiveness and originality of the author makes  the book a joy and a treasure in many ways. It is surely a book one would repeatedly return to.

So as a tribute to i) the love of books, ii) Gödel, Escher, Bach: An Eternal Golden Braid, and iii) Douglas Hofstadter, here is my attempt at summarizing what I learned from GEB.

Here we go…

In order to understand intelligence the GEB way, we first need to get certain key concepts straight. These concepts are “self-reference”, “paradox”, “recursion”, “isomorphism”, and “symbol”. As we whirl between and across these concepts, the view of intelligence a la Hofstadter will gradually emerge.

Self-reference and paradox

Sentences that talk about their own selves and people reflecting upon themselves are examples of self-reference. A typical example will be

This sentence is false.

Now if this sentence is really false, then it means the sentence’s claim is true. So it turns out that the sentence is actually true. But how can it be true when  it is false….?

Now here we are caught up in a looping paradox! If it’s false, it means it is true; but if its true it means it’s false!

Lesson? When a system that represents  meaning starts  referring to itself (and it can and it will) it will lock itself into a loop.  Each ‘level’ of  the loop is constantly leading up to the other and there’s no way out of it … an infinite regress! This is what Hofstadter calls a Strange Loop – a  concept he brings out more fully in his latest book I am a Strange Loop. (See Here for an interesting anthology of paradoxes.)

Paradoxical Mathematics…!

What!!#!@!!

Math…..paradoxical? Well… Kurt Gödel, a young Austrian mathematician, proved that.

Gödel used Bertrand Russell and Alfred Whitehead‘s system of formal mathematics (from their book called Principia Mathematica) to show that it contains irresolvable inconsistencies in the spirit of the paradox stated above.

Any formal system of mathematics attempts to show that i) all mathematical truths (or theorems) are derivable through strict rules of logic from previous truths (initial truths from where the derivations are build up are taken as true a priori and are called as axioms) [completeness] and that ii) no mutually contradicting statements are derivable in the system [consistency].

On the other hand, Gödel proved how sufficiently strong and expressive formal systems cannot prove their own consistency (provided that they are in fact consistent).

How did he do it?

First of all, he devised a system of numbers to represent statements in formalized mathematics. Next, he assigned these “Godel numbers” to steps in derivation of the theorems of the math system. He found that these strings of Gödel numbers can themselves be derived through the rules of math just like the actual derivations of theorems. For every logical rule applied in the derivation, a corresponding mathematical rule will yield the very Gödel number that represents the next step of the derivation. Thus Gödel discovered a heretofore undiscovered level of meaning in the formal system.

But where is the paradox?

Gödel showed how this process of “Gödelization” can be used to construct the following paradoxical statement in terms of the formal system.

This theorem cannot be proved in this system.

Now if the concerned system of formal math is REALLY complete (i.e. able to provide logical proofs to all its theorems) then a proof for the above statement will automatically prove the system’s inconsistency (as the system now proves a contradictory statement). On the other hand, the inability to prove the statement (indeed that is the case) is evidence  of the INCOMPLETENESS of mathematical logic!

Herein lurks another paradoxical self-reference – a Strange Loop!.

Essential incompleteness and Recursion

Paradoxical statements like these defy the neat categorization of ‘truth’ and ‘false’. In fact, truth and falsehood seem to be constantly leading to each other in a continuous fashion. This is the strangeness of the loop and it involves a process of recursion.

Recursion is a procedure which entails repeating its own steps indefinitely. The shampoo instructions

1. Lather; 2. Rinse; and, 3. Repeat.

are recursive. If acted upon literally, there will be a continuous repetition of the procedure making your stay in the washroom infinite! Hence, recursion is a self-reference with a potentially infinite continuity.

A common and visual example of recursion is when you are standing between two large mirrors….  “endless corridors” (a term used by Hofstadter in I am a Strange Loop) on both sides!

Some other visual examples of recursion are as follows:

A picture embedded inside itself inside itself inside itself…

Example of a fractal pattern made using a recursive math equation

Waterfall - Lithograph by Escher (1961)

Example of a pattern resulting from a video feedback loop, from the cover of “I am a Strange Loop”

The last example most closely illustrates the concept of a Strange Loop. When successive iterations of the recursive process are designed to have a slight variation, novel and complicated patterns emerge from the  starting and much simpler ingredients. The  patterns could never be predicted by looking at the starting levels alone.

Hofstadter presents an illustration of recursive cognitive processes by referring to the chess situation. Before making the next move in a game of chess, you would like to see with your mind’s eye how will the opponent respond to the move you are now considering. You may also try to “look ahead” further into the game by trying to determine a possible move against the opponent’s response to your next move and then again reflect on how the opponent will respond to that…. This process of repeatedly generating solutions belonging to a particular class (in math, members of a particular set such as the Fibonacci numbers) is called as recursive enumeration.

Hofstadter seems to suggest that a recursively defined set of procedures entails an increasing level of complexity of behavior. Furthermore, where a machine (for e.g., a computerized chess program) must be given a decision procedure to settle on a solution rather than engage in an infinite “looking ahead”, humans have the capability to “jump out of the system” at their will. They can put a break on the recursion, observe the workings of their  own reasoning process, draw some conclusions, and look for alternatives. Hofstadter illustrates his point beautifully by engaging his readers in a task involving recursive enumeration in the first chapter of his book. (For a go at the task, check out this site.)

The human ability to stand back and observe the results of their own thinking guarantees an unpredictability of ideas and behavior that the machine lacks. Ultimately, it is these strange loops of thought that are also responsible for the general self-consciousness that defines the human condition. Akin to the beautiful, totally unpredictable and unique pattern that emerges out of a video feedback loop.

There is so much more still left to delve into…

I’ll complete this summary of my GEB concepts in the next post InshaAllah.