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August 2016

Gödel, Escher, Bach

GodelDouglas Hofstadter’s Gödel, Escher, Bach is a fascinating exploration of how consciousness can arise from inanimate matter. It's an intellectual tour-de-force, covering a fantastically diverse range of subjects, including mathematics, art, music, molecular biology, neuroscience, Zen Buddhism, extraterrestrial life, computer science, and artificial intelligence.

At the core of Hofstadter’s beliefs about consciousness lies the idea of a “Strange Loop”:

“The "Strange Loop" phenomenon occurs whenever, by moving upwards (or downwards) through the levels of some hierarchical system, we unexpectedly find ourselves right back where we started.”

Hofstadter cites the Epimenedes Paradox - the statement “this statement is false” – as an example of a one-step Strange Loop. He spends a large part of the book discussing Gödel’s Incompleteness Theorem, which can be thought of as the translation of the Epimenedes Paradox into mathematical terms.

“Gödel says that no sufficiently powerful formal system can be perfect, in the sense of reproducing every single true statement as a theorem... The fact that truth transcends theoremhood, in any given formal system, is called "incompleteness" of that system.”

A “formal system” is a system that has a set of axioms and can generate statements by following a set of rules. A “theorem” is just a statement made by the system (including the axioms). “Sufficiently powerful” means a system that has the ability to make statements about itself.

While Gödel’s Theorem is about mathematical systems, an analogy can be drawn with the English language. This can be thought of as a system with a set of axioms (words) and a set of rules (grammar) for combining those words into sentences. Sentences can be constructed that are true (“the sky is blue”) or false (“ice is hot”). Gödel’s Incompleteness Theorem is analogous to saying it is impossible to create a book that contains every true statement that could be made in English, regardless of how large that book is.

The gist of the proof is as follows. Consider the sentence, “This sentence is not in this book”. Would that sentence be found in our hypothetical book, or not?

Suppose it was in the book. In that case, the sentence is untrue. But since we have said that the book only contains true statements, this is impossible. Therefore, the sentence must not be in the book. But in that case, the book must be incomplete, since we know the sentence is true, yet is not contained within it!

Hofstadter also notes that whether a statement is true or false is not inherent in the statement itself – it depends entirely on the interpretation we choose for the symbols. For example, consider, “the sky is blue”. This is only true if we interpret “the sky” as “the sky on planet Earth”. Were we to interpret “the sky” as “the sky on Mars”, the statement would be false. This mapping between the symbol “sky” and the concept of the thing above our heads is called an “isomorphism”. Indeed, the only thing that gives the inherently meaningless squiggles S-K-Y meaning is our recognition that they refer to the thing above our heads. As Hofstadter puts it, “Meaning is an automatic by-product of our recognition of any isomorphism”.

Whether a system is internally consistent also depends on the interpretation chosen for it. For example, consider a system with “1p1q2” as an axiom. We might choose to interpret p as “plus” and q as “equals”. Now suppose we decided to create a new system by adding the axiom “1p1q4”. Isn’t this new system inherently inconsistent? We might think so, since not only is 1+1 not equal to 4, we now seem to have two axioms that disagree with each other. However, we only have a problem because we have retained the same interpretation for the symbols p and q. Reinterpret the symbols appropriately (for example, by reinterpreting q as “less than or equal to”), and our system is consistent and meaningful once more.

Later in the book, Hofstadter offers a great visualisation of a multi-level Strange Loop:

“Think of chess. Clearly the rules stay the same, just the board position changes on each move. But let's invent a variation in which, on your turn, you can either make a move or change the rules [according to some constraints]...

Now we have two layers of rules: those which tell you how to move pieces, and those which tell how to change the rules... You could even express rules and metarules as positions on auxiliary chess boards...

Now we can have any number of adjacent chess boards: one for the game, one for rules, one for metarules, one for metametarules, and so on, as far as you care to carry it. On your turn, you may make a move on any one of the chess boards except the top-level one, using the rules which apply (they come from the next chess board up in the hierarchy). Undoubtedly both players would get quite disoriented by the fact that almost anything - though not everything! - can change...

Now it is possible to go considerably further in removing the pillars by which orientation is achieved. One step at a time... We begin by collapsing the whole array of boards into a single board. What is meant by this? There will be two ways of interpreting the board: (1) as pieces to be moved; (2) as rules for moving the pieces. On your turn, you move pieces - and perforce, you change rules!... The distinction between game, rules, metarules, metametarules, has been lost. What was once a nice clean hierarchical setup has become a Strange Loop... There are still different levels, but the distinction between "lower" and "higher" has been wiped out.”

Hofstadter argues that consciousness arises from a similar tangling of different levels in the brain:

“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 the bottom level. In other words, a self-reinforcing "resonance" between different levels... The self comes into being at the moment it has the power to reflect itself.”

Mark Vernon on Friendship, Love and Sex

The Meaning of Friendship

“A friendship which includes a sexual element is the best sort of relationship that many people hope to have,” says Mark Vernon in The Meaning of Friendship. “The downside is that sex can clearly imperil friendships by its possessiveness or its inappropriateness.”

Sex presents a number of problems to friendship. First, relationships based on physical attraction can form quickly, with little need for personal disclosure. In this respect, they are similar to relationships based on utility or a shared interest. As a result, the relationship may fall apart if the couple later realise there is no real basis for any friendship between them.

Conversely, friends who sense a sexual undercurrent developing between them may hesitate to pursue those feelings lest they put the friendship at risk.

However, even just the possibility of such feelings developing can cause problems:

“Imagine a man and woman becoming friends at work - good friends - and deciding to go out for dinner together as an apparently natural extension of the friendship. Then, as they're sat across the table from each other - starched linen, candles and a rose between them - they start to feel awkward. Unwittingly, they have been drawn into uncharted waters as dinner for two is the sort of thing that lovers do, not friends. The evening is one of embarrassment, and the friendship flounders. What's happened is that cultural assumptions about the activities associated with a sexual relationship have imperilled a friendship quite as effectively as any actual erotic attraction itself…

The complication that comes from such an unresolved sexual frisson is the suspense. Indeed, suspense is as much a cause of erotic frisson as any actual sexual attraction might be: people do not even need to fancy each other, just be conscious that they might. In Evelyn Waugh's phrase, even 'a thin bat's squeak of sexuality' can frighten people off or distract them from becoming friends.”

However, sex doesn’t just imperil the friendship between lovers, it can jeopardise their other friendships too:

“In today's world, there is a myth of romantic love based upon the idea that two lovers become one flesh, a totalisation of life in the other, supremely enacted in sexual ecstasy which is symbolic of that union. The myth or ideal tends to exclude others, not because lovers do not want friends, but because it tells them that their friends are incidental - pleasant but non-essential adornments to the lover's life together. Although few people in real life believe the myth in its entirety, it is difficult to ignore it entirely too...

Think of the estrangement that can come about between friends after one of them marries another... Or recall just how hard it can be to sustain a friendship when your friend started a new sexual relationship..."

Vernon suggests this may be because the partiality of friendship may be seen as a threat to the supposed primacy of the sexual relationship.

“Is there not a steely strand in the ethic of modern marriage which repels anything that compromises the unconditional commitment of husband and wife - 'forsaking all others', as the service says? Close friendship can count as infidelity quite as much as a fling or affair.”

To my mind, this is one of the great tragedies of modern marriage. The idea that one person can satisfy all of our sexual, emotional and intellectual needs is infantile. Our notions of exclusivity, indoctrinated in us by our culture, all too often cut us off from a richer, albeit more complex, tapestry of relationships that have the capacity to profoundly improve our lives.

At the root of the problem is our tendency to value sexual relationships above mere friendships. We often talk about how a friendship can blossom into “something more”.  Yet by doing so, we have created an incentive to misunderstand our feelings:

“In a culture where sexual consummation is seen as the highest expression of love that two people can hope for, a fascination for someone is easy to mistake for falling in love, even when it is simultaneously obvious that a sexual relationship would be inappropriate, unsustainable and possibly ruinous of the friendship.”

Aristotle valued things differently:

“[Aristotle] assigns close friendship top place in the hierarchy of human relationships, regarding it as a key ingredient in any flourishing life. There's a place for 'friendly lovers' too, if lower down: they can hope for some contentment. They belong in his second category of friendship, the kind that form because of some mutual shared pleasure, in this case that being sex...”

Vernon seems to agree:

“Perhaps friendship should assert itself more strongly in our romance obsessed world... For contra the myth, there is a love that does not desire to possess. It is called friendship. It loves the other, and wants them both to be free.”

Mark Vernon on the Three Kinds of Friendship

The Meaning of FriendshipWhat is friendship? Why are we friends with some people but not others? Why do some friendships last while others wither away? These are some of the questions considered by Mark Vernon in The Meaning of Friendship.

According to Aristotle, there are three reasons people may be friends: because they are useful to each other, because they enjoy some shared interest, or because they value each other for who they are.

Most work friendships fall into the first group. Consider how we typically feel when a colleague leaves:

“Why is it that you can have known a colleague for years, enjoyed their company day after day, worked with them, even helped them when personal matters spilt into the workplace, and yet, when they left, it was, overnight, almost as if you had never known them? You might miss them for a day, perhaps a week, and hope their new job is going well. But, in truth, most of the people with whom we were once friendly at work disappear from our lives with little more than a toast in the canteen, or best wishes on a card. It is very odd, when you consider all the time you spend with these people, and the genuine exchange of good feeling. And yet, it is entirely understandable when you realise that the relationship was, at heart, one of utility, based mostly on what was done together. Take that shared activity away, which is what happens when people leave work, and the friendship withers like a cut flower. It is not that they were not liked or had nothing in common with you. It is that the thing held in common - the work - is gone; without doing that together the relationship ceases to have reason or purpose.”

The second group is similar to the first, except that the shared interest is pleasurable rather than utilitarian. It may be a common love of board games or birdwatching, shopping or sex. Like the first group, however, the friendship only thrives for as long as the shared pleasure continues to exist.

Vernon argues that it is friendships of the third kind – those who enjoy being together because of who they are in themselves – that are the most durable. However, while this may be a necessary component of the best friendships, it may not be sufficient:

“In [philosopher Friedrich] Nietzsche's book, longevity is not the determining measure of friendship. A short-lived friendship may nonetheless be the most important of your life. It's not that there is anything wrong with long-lived friends per se; it is rather that time can suck the authenticity out of friendship... Such friends do not really share that much, beyond their association, and so wind away the hours talking about this and that, conspiring in indecision and perhaps in all honesty becoming nuisances to one another...

[Nietzsche] also suspects that such relationships are untrustworthy because when the dynamism disappears from a friendship, but the individuals concerned cannot quite bring it to an end, they constantly strive to re-establish their intimacy with each other - by dwelling on the 'old times' or college days; the past, not the future. This is a sign that habit has become a substitute for any real affection or closeness.

Nietzsche is not saying that a shared past is not important to close friends. Rather, he's arguing it's not enough. His observation about the future orientation of the best friendships is an arresting one.”

To Nietzsche the true joy to be found in friendship is not the collapse of boundaries between the individuals, nor a sense of like-mindedness. Rather, like the Missing Piece and the Big O, it’s the realisation that they are moving in the same direction.

Most of our friendships don't reach such heights. Yet for Nietzsche there was value in more superficial friendships too:

"Just as in order to walk beside an abyss or cross a deep stream by a plank one needs a railing, not so as to hold on to it - for it would at once collapse if one did that - but to give the eye a feeling of security, so as a youth one has need of people who without knowing it perform for us the service of a railing. It is true that, if we were really in great danger, they would not help us if we sought to rely on them, but they give us the quieting sensation that there is protection close at hand."

Savour the Mundane

“Perfect happiness comes in tiny, incremental units only, perhaps no more than five minutes at a time,” writes Alain de Botton in The Course of Love. “This is what one has to take with both hands and cherish.”

Yet these moments need not be rare. In Peace is Every Step, Thich Nhat Hanh argues that they can be found in even the most mundane of tasks, if we take the time to savour them:

“To my mind, the idea that doing dishes is unpleasant can occur only when you aren't doing them. Once you are standing in front of the sink with your sleeves rolled up and your hands in the warm water, it is really quite pleasant. I enjoy taking my time with each dish, being fully aware of the dish, the water, and each movement of my hands. I know that if I hurry in order to eat dessert sooner, the time of washing dishes will be unpleasant and not worth living. That would be a pity, for each minute, each second of life is a miracle.”