3. Looking into the mirror

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When I was young, visiting the barber shop was a fascinating experience because of the mirrors. In most barber shops in India, there are two rows of mirrors on opposite parallel walls (i.e. the mirrors face each other) and when I was put atop that fancy barber's chair; I would see an infinite number of images of my head in the mirrors. Naturally; as a child, I used to endeavour to see where the series of images all ended and could never fathom why even after me shifting my head around, I could never quite catch the ending of the series. It only resulted in me getting a bad haircut, and the barber getting annoyed. And that has more or less remained a trade mark with me (in fact, both the haircut and the annoying of people :-) )

Talking about barbers, there was this question that Bertrand Russel had asked (which eventually ended up with Godel's incompleteness theorem ) : In a town, if the rule was that the only barber in town will shave those who will not shave their own heads, who then shaves the barber's head?

The above paradox can be traced to the ancient Greeks and the liar's paradox which can be stated quite elegantly: What would you make of if someone walks upto you and said "I am lying"? If that statement would have been right that would mean he was lying which meant the statement would be false, which means he would be telling the truth and that contradicted what he had said (he said he was lying) and so on and so forth ad infinitum….notice the multitude of images forming in the mirror?

I remember this being discussed in my college where some of the geeks (from the non-architecture departments) used to go around the campus with a fat book called Godel, Escher, Bach: An Eternal Golden Braid by a then young author Douglas Hofstadter; telling people wherever they went that "they were lying".

They had a bewildered look in their eyes as if the book had first fallen on their heads before they read it. Now, most of the reactions to such a statement were distinctly non-mathematical, and much of us architects were not particularly impressed by such displays of obvious geekiness. We thought all this stuff cannot apply to us. After all, designing architecture is a rarified experience and we need not get into details of who is lying or not and/or reading that book for the answer. We obviously knew! … and we dont need anyone to tell us anything!

I did get around to reading the book in bits and parts. The sheer reading of it itself is a fascinating experience. Parts of the book uses the Indian literary concept of the suthradhar* and clever devices used by the suthradhar reveals concepts to you in layers. Much misunderstanding can arise from the book. I would dare say that what is stated in that book (and here) could be, just maybe, used by some to bring in further hogwash than what already exists in the name of architectural philosophy. Hofstadter was trying to explain something deeper using references from music (Bach), mathematics (Godel) and art (Escher)) so let us give this the intense thought it deserves. Do I make myself clear?

Anyway, so what on earth did Hofstadter say?

His book was about the seeking of "the self". Just like the boy trying to look into the mirrors, and seeing an infinite amount of reflections. Hofstadter was pointing out that recursion is unavoidable when we try to seek out the notion of the self. Godel supplied the mathematical proof.

What Godel stated was that in any mathematical system, not all the theorems in the system can be proved using the tenets of the system themselves. What was fascinating is that he effectively said something like this: "You chose what you think constitute mathematics and I would still find some statements which cannot be determined to be true"

Let me quote from a nice summary I found about Godel Incompleteness theorem from the web ( See references at the end of this chapter)

Begin quote:

The proof begins with Godel defining a simple symbolic system. He has the concept of a variables, the concept of a statement, and the format of a proof as a series of statements, reducing the formula that is being proven back to a postulate by legal manipulations. Godel only need define a system complex enough to do arithmetic for his proof to hold.

Godel then points out that the following statement is a part of the system: a statement P which states "there is no proof of P". If P is true, there is no proof of it. If P is false, there is a proof that P is true, which is a contradiction. Therefore it cannot be determined within the system whether P is true.

As I see it, this is essentially the "Liar's Paradox" generalized for all symbolic systems…

Godel's proof is designed to emphasize that the statement P is *necessarily* a part of the system, not something arbitrary that someone dreamed up. Godel actually numbers all possible proofs and statements in the system by listing them lexigraphically. After showing the existence of that first "Godel" statement, Godel goes on to prove that there are an infinite number of Godel statements in the system, and that even if these were enumerated very carefully and added to the postulates of the system, more Godel statements would arise. This goes on infinitely, showing that there is no way to get around Godel-format statements: all symbolic systems will contain them.

:End quote

I believe that in architectural designing, what an architect does is to seek the self within the architecture he is about to materialize. That process is carried out using a model. A model is a mirror of the real world, held parallely to it. For if we didnt hold it parallely, we would be distorting parts of it and who would want such a model?. This approach is okay if we can distinguish the mirror from what is being modeled. But what if the reality contains mirrors also that needs to be modeled? That is when the series of infinite images comes in. We can then never be sure about some aspects of the model.

The constructing of a model is exactly same as the construction of a mathematical system. In fact, a model is nothing but a mathematical system. This may come as a bit of a surprise to those who rejected the "modeling" approach to architecture. I've often heard youngsters state things like "I do not want to explain my work of architecture. It has to be experienced" Of course, architecture ought to be experienced. That is the end of the excercise and undoubtedly we need to be focused on that.

Stephen Covey says in his book "Seven Habits of Highly Effective People":

Begin Quote:

"Begin with the end in mind" is based on the principle that all things are created twice. There's a mental or first creation, and physical or second creation to all things.

:End Quote

He goes on further to explain that if we do not get the mental image (the model) right, things can go real wrong indeed.

In the earlier web reference, I find this:

Begin Quote:

Could we make a complete system? The only way I can see to do that would be to include an infinite number of axioms, which deterministicly describe all happenings in the past, present and future. This would only work in a deterministic universe, and it would be difficult to draw a distinction between the data of this 'complete' system and reality itself.

Thinking of the data required is perhaps the right direction to move in: it is the reason the symbolic system is incomplete. The symbolic systems we use to describe the universe are not separate from the universe: they are a part of the universe just as we are a part of the universe. Since we are within the system, our small understandings are 'the system modeling itself' (system meaning reality in this case). Completion of the model can never happen because of the basic self-referential paradox: the model is within the universe, so in effect the universe would have to be larger than itself. Or you can view it iteratively: the model models the universe. The universe includes the model. The model must model itself. The model must model the model of itself.. ad absurdum.

:End Quote

So, we were talking about how architecture materializes in the first place, and even if you did not want to explain it. We use axioms (starting assumptions) and we then apply rules on those axioms (to tie the entire model into a tautology).

So we may say things like: "In this project, it is obvious that the privacy of the family needs to be preserved " (axiom) and then go on "therefore, I assume that the bedroom should be placed here" (logical operation within the tautology). Now a few of us may not be able to verbalize or articulate such models and still produce good architecture. However, that is not the point.

There will always be models. What was fascinating about Godel's theorem was that he did not make any explicit assumption about a mathematical system. A bare minimum set of axioms and logical operations would do, and one can still find Godel statements within such a framework.

Godel's incompleteness theorem has had far reaching implications. Alan Turing's explanation of the "halting problem" in computers can be traced to the Godel's theorem. Turing had stated that in certain situations a computer will never halt and therefore never produce a result. This is a death knell to those who think that a computer can supply all the answers. No, it cannot.

In some round about manner, Godel's theorem explained the need for the intuitive in any modeling. But that is one aspect which can be grossly misunderstood and exploited by architects. I can imagine a student shrugging her shoulders at her professor indicating that even Godel understood the need for intuition and therefore what she has done is actually intuitive and therefore she should be given good grades. What she may miss is that the moment she says that "I've done something intuitive", she has actually contradicted herself (liar's paradox once again) because if it was truly intuitive, there is nothing that she need to have said. In fact, she has only constructed an axiom by stating that she had done something intuitive.

I have often experienced another variant of this argument: "If I look at this objectively, then this a,b,c is what I think about it …. but if one looks at it subjectively, then …x, y, z…" And I will invariably interject (annoy the barber!) by pointing out that if it was indeed subjective then why do you want to explain it? The subjective will always be in existance, and trying to articulate it would do you no good!

There is a philosophical implication of Godel's theorem which is best explored in the Philosophy section of our website, but I can give it a mention here: I believe that much of western logic can be traced to the Aristotelean system of logic. (Godel is supposed to be one of the three important logicians that civilization has ever seen. Aristotle being the first and Ferge being the second.) Godel pointed out the roots of some of the problems that we face due to such a logic. There are noticeable chinks in the armour that are leading society to sometimes misbehave in an unacceptable manner (like crashing planes into buildings), and a more Eastern way of looking at things may now be required.

Having said all this about Godel et al, what advice do I have for architectural designing? Let me express my answer in terms of what we learnt here itself. Let me say that my advice (a model) is a set of axioms and I would then propose a set of operations using those axioms to form a tautology. Well, within that tautology there would always be inconsistencies! So, my advice would have shortfalls. There are no magic solutions for modeling.

Unfortunately (for us) when a model has such inconsistencies, we can never be sure that the thingamajig that was not provable was a simple thing which didn't have consequences for the eventual architecture. If so, we could have got away by chucking that aspect of the model altogether. But what if it really was important? Who will dare to bell the cat?

There are ways that uses concepts from Eastern philosophy but they have to be brought in logically and carefully lest it is thought as some quick-fix Eastern mysticism.


  1. http://www.myrkul.org/recent/godel.htm
  2. http://www.forum2.org/tal/books/geb.html
  3. http://plus.maths.org/issue20/xfile
  4. * A suthradhar is a narrator, who stands away from the story being told and makes running commentaries on it.

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