Seventeenth Night

APOSTOLOS DOXIADIS

The extract that follows is part of a transcription of a long dialogue that took place between the Greek writer, mathematician and film and theatre director, Apostolos Doxiadis and his friend, G.E. It forms the second part of a book entitled “From Insanity to Algorithms”, published in Athens in 2006 by Ikaros Publishing. The first part of the book is a play fictionalising the last days of the great logician Kurt Gödel. The entire journey into the relation between logic, mathematicians and insanity was inspired when, few years back, Doxiadis read in an introduction to an article about the logician Alonzo Church by Gian-Carlo Rota, that five of the main actors of the creation of modern logic spent a part of their life in psychiatric asylums.

Trying to disentangle the apparently inconsistent connection between the very epitome of logic, algorithms, and its diametric opposite, insanity, Doxiadis visits the personalities of some of the greatest minds of modern mathematics, the principles that govern logical reasoning and the potentially necessary trade-off between absolute proffesional order and logic and personal psyhological malady and deviance.

[...] Barry Mazur has written somewhere that the Dutch have the most accurate word for mathematics: ‘wiskunde’. It’s made up of two words, ‘wis’, which means certainty and ‘kunde’, which means ‘area of knowledge’. Mathematics is, then, referred to as sureology, namely, the science of absolute certainty – a definition most mathematicians would endorse.

Sureology… I like that!

Now look: whenever, in this world of sureology, you wish to insert a truth so original and, as it was later revealed, so precarious as the one Cantor introduced, a truth that has exceptional powers and produces almost unbelievable proofs -theorems, in other words, that even their author finds it hard to believe- you need to be equipped with a certain mental apparatus, you need to be ready to deal with great emotional strain. Think for a minute: it can’t be easy having Poincaré referring to your theory as a “disease” from which mathematics will, eventually, be cured.

Without, however, calling it ‘wrong’.

Yes, because Poincaré does not judge Cantor’s proofs, which were themselves impeccable, but the introduction of a new construction based on completely novel foundations. In such a situation, the terms ‘right’ and ‘wrong’ become meaningless, in the same way that between Euclidean, Hyperbolic and Elliptic geometries, you cannot speak of one being ‘more correct’ than the others. Notice, again, Poincaré’s description: disease. It’s not at all common for mathematicians to hear such cruel words for their work.

And you’re saying that Cantor had the kind of resistance needed to withstand the criticism…

My claim is even more specific: that the creation/inspiration of set theory was facilitated by the psychological courage of its creator. Look… on the one hand we know that insanity is an ability to see causal links where they don’t exist. The insane person can interpret simple glances as signs of being followed, detect in others’ whisperings innuendos against himself, see encrypted messages aimed against him in otherwise innocuous newspaper articles. We call these readings insane because we, ‘sane’ people, consider them unfounded. Nonetheless, they are not in themselves, necessarily, devoid of logical coherency. Every insane reading forms a theory, a model that explains reality in a particular way. And these models are, by and large, very logical – sometimes, even more logical than other, less insane readings, because they formalize, they ‘clean up’ and simplify reality. They explain it in a very organized way.

But, nonetheless, wrong.

Definitely as wrong -in its exaggeration- as it is logically neat. It’s what we often hear, that ‘this great artist found refuge in insanity’. It’s an escape from inner chaos to an absolutely organized, logical representation of reality.

It is, in a way, the same certainty that some people find in political ideologies.

Yes, especially when the ideologies are Manichean or when they are fanatically embraced. This kind of addiction is, for example, typically found in the extreme form of worshiping of football teams or similar concepts. In any case, what is important here is that the shortcomings of insane models do not reside in their internal logic, which is flawless, but in their inability to comprehensively describe the world out there.

And does this happen because of false axioms?

You could say that. The problem is not the internal logic, but the starting points, the points on which theory relates to reality. But -and this I want to stress- this method of fabricating insane explanatory models of the universe can, sometimes, actually get it right!

It’s like an American joke from the sixties: “If you think people are after you, it’s usually because they are”.

Indeed. Or, to put it differently, being insanely paranoid and actually being persecuted, at the same time, is not impossible; likewise a hypochondriac can, in fact, be sick. Here’s a story about the great American scientist Archibald Wheeler, who has a very eccentric form of genius. Someone said about him that “during his lifetime he formulated three hundred entirely crazy theories, two of which happened to be true!”

And it is to these two that he owes his eminence.

Precisely. In other words, no one is especially bothered about the two hundred and ninety eight wrong ones -of course, the numbers are only random- and Wheeler is regarded as a great physicist thanks to the two right ones. The rest were, in a way, the price he had to pay for an imagination so vivid and creative, so extraordinarily productive. This mode of creative imagination is something we know well from artistic creation. In some historical periods, certain painters, composers or poets create within the frames of the ruling lege artis -same as the scientists in periods of normal science, to use Kuhn’s term. At other times -much more rarely- some daring individuals lead the way to something entirely new, breaking all links with the past. They pave new roads and, at the same time -these two things usually go together- burn the bridges to the past. This is what the first painters who thought ‘let’s do away with representation’ did, or the first composers who decided to defy the rules of tonality.

Like Jackson Polock who, suddenly, puts his canvases on the floor and drips paint on them…

…And he calls this painting. This last bit is crucial: many others before him had done something similar, but they called this ‘doodling’. But beware. While for an artist, to try something new, unusual, weird, and be successful -which, by the way, means to be recognised- seems to us nowadays, at a time when originality is a value in itself, entirely normal, this should not make us blind to the fact that a similar thing happens also in science. That is why I mentioned Wheeler’s example. The gradual acceptance of a scientist’s work, the regulating effect of the constraints of empirical verificationism, and the subsequent implementation of the scientist’s work in the canonical science makes us forget the innovative element -or, many a time, the seeming insanity- of new ideas. This kind of insanity is noticeable only if we examine them in the context of the time they appeared. Which is why I said that historical perspective is so important.

Is this what happened with Cantor? He said something ’crazy’ that ended up ‘working out’?

Not entirely. Because, mind you, in mathematics we don’t speak in terms of experimental verification, we have proof -so, it’s not possible for anyone to say anything without facing up to the consequences. The mental resilience of Cantor, like I prefer to call it, has to do with his persistence, in spite of difficulties; with his sincerity in exclaiming about one of his proofs ‘I see it but I don’t believe it’ and with his determination to go straight ahead, subjecting his theory, along with himself, to grave consequences and continuing his career attempting to prove the so-called ‘Continuum Hypothesis’ on infinite sets.

Someone else in his place would have stopped?

Quite possibly. In the history of ideas, apart from genealogy, preparation or environment and fact, to which we have already referred, there is also another substantial factor: the ‘carrier’ of an idea, the particular individual who receives the effects of genealogy and preparation, who lives the event and, finally, contributes his or her own part and brings out to the open the final idea. We could refer to him or her as a good gardener, who makes the plant blossom. So, at some point in our conversation we had better leave aside mathematics and talk about the ‘carrier’ of mathematical ideas, mathematicians themselves. Because who these people were, played an important role in the final form that their ideas took. Do you know the joke about an astronomer, a physicist and a mathematician sitting in a train?

No.

Well, the three of them are sitting in a train car and the train crosses the border between two countries, say from Germany to France. And, in the first pasture they cross on their way, they see a brown cow. So, the astronomer says: “Aha! In France cows are brown”. The physicist corrects him: “No, that’s not right. In France, some cows are brown”. And then the mathematician puts things in place: “No, my friends. All we know is that in France there is at least one pasture in which at least one side of at least one cow is brown”. This describes brilliantly the psychological type of a mathematician. This is the stance taken by an entirely logical person, who does not unwarrantedly generalize, does not exaggerate, does not claim anything more than what he can logically establish. We know well, however, that in life there is also the other side of a coin; that for everything you gain, you lose something else. And here, in this type of extremely ‘mathematical’ view on the world that this anecdote shows us, the price is not but a compromised ability to handle complicated situations.

What do you mean by that?

I mean that, if someone wishes to be precise to such an extent, like the mathematician in the story, if he wants, in other words, to always be right, there is no other way to achieve this but by reducing, confining, oversimplifying reality. The world is infinitely complex and so, to handle it absolutely logically you must, in each case, reduce it, simplify it. But, as the simplification process continues, you gradually cease to refer to the real world and you start moving on to another world which is, in a sense, imaginary, namely not the real one but a simplified version of it. And, as per the common saying, half a truth is, usually, a great lie.

Are you referring here to the world as defined by mathematical axioms?

Yes. But, of course, in mathematics, this is acceptable. Even more, oversimplification, namely absolute axiomatization, is mathematics’ greatest advantage. The moment, however, that you attempt to apply these models outside mathematical discourse, you begin to flirt with insanity, especially if you go to extreme simplifications. Nonetheless, let me get to my main subject. I said earlier that in the history of mathematics we rarely find cases of serious mental disease. In fact, if we compare mathematicians to poets or composers, the occurrences are extremely rare. We will have a very hard time to find in the biographies of great mathematicians instances of real psychosis. We will, of course, encounter more or less attractively weird people -some who forget their appointments, some who are incapable to walk to their local newsagent without getting lost- people, in other words, who exhibit eccentricities or absentmindedness, both being the side effects of an increased capacity for concentration. But we are not likely to find serious psychological deviance…

What you are describing is the behaviour of the ‘crazy scientist’ type we see in cartoons, the person who is so dedicated to a particular field that he is completely out of touch with reality.

The history of mathematics is full of types like that who are, to a certain degree, detached from reality, lost in their own theories. What we frequently find in mathematicians is, in other words, a psychological ground that can support a kind of mental functioning similar to that of the mathematician from our joke: one that allows a degree of detachment from reality that can allow the formulation of logical certainties. But this obsession, this detachment is, usually, not pathological and it rarely extends beyond mathematics. You know, of course, the adage ‘history is written by the winners’.

I do.

I find another variation even more interesting: ‘history is written for the winners’. Think about it in terms of our discourse: as a rule we write, narrate, read and hear the biographies of successful scientists, like those of successful artists, politicians, business people etc… Their success is, of course, given, since we are reading their stories in retrospect, i.e. after they have achieved fame. And, therefore, even if the lives of some successful individuals were, as it usually happens, rich in personal failures and tragic moments, we tend to read the biographies looking for the reasons that lead only to their success. This is because we are intelligent people who wish to understand the world logically: we seek everywhere and always the answer to ‘why’, the causal link. Narration itself as a cognitive function is, to a large degree, a mechanism of detecting causal relations in the flow of time, a tool to establish links that take us from a to b to c with causal efficacy.

How does that relate to our topic?

When, for example, we read in the biography of a successful scientist -remember, we said we are referring to success stories- that he or she discovered a new theory, the natural tendency is to attribute their accomplishment to their great intellect, persistence, their moral or spiritual vigour, or anything really that is good and positive.

But it’s not as if these elements don’t exist in the life of a creator, is it?

Sure, but what we tend to ignore in this way of looking at things, is that other scientists, who may have equal or greater intellect, persistence, moral vigour etc., failed, by proposing theories that were proven to be wrong. And what fundamentally differentiates the former from the latter is that the former happened to be right.

“Happened”? You mean they were just luckier?

Writing the history of science, we, as a rule, omit the factor of chance and the role it plays in its progress. In other words, we ignore the possibility of the evolution of scientific ideas as a more general mechanism of production and control of new ideas that is reminiscent of the Darwinian notion of natural selection. If we leave out for a while the central, glorified image of the scientist-researcher and look at the history of science more macroscopically, investigating the history of a branch in its wider context, we realize that every time there is a big open problem, a number of alternative solutions are formulated, from which a number of them are confirmed, whereas the majority are eventually rejected and forgotten. The scientist who develops the right solution is very likely to be endowed with the psycho-mental characteristics of persistence, laboriousness and moral vehemence – characteristics he may very well share with those who failed. That is why I put it so; that for someone to come up with a good idea, he or she also needs to, besides any other virtues, luck. In other words, for the right idea to be found, it is necessary that many wrong ones are expressed and, very often, the question of which scientist has which idea is, partly, a matter of chance. And, often, the ideas that do not survive are equally, if not more, intelligent, creative and interesting as the ‘right’ ones that survive this conceptual survival struggle. Those who express wrong ideas can be equally good scientists -or even better- as the others.

I think that this notion is very nicely illustrated in the autobiographical text “The Double Helix”, where James Watson elaborates how he came to the discovery, along with Francis Crick, of the structure of DNA.

This is indeed a very interesting book. And it should certainly be read by anyone caring to see this side of science that, I believe, Watson describes beautifully, even to the extent of presenting himself in an unfavourable light. What I mean by this is that he does not try to gloss over the fact that he and Crick were helped by luck and that they happened to be at the right place at the right time, amongst the right people who came up with the right ideas, with which they could work and, ultimately, arrive first at their conclusion. And there’s something else, which is particularly interesting for our discussion: the psychological reasons that lead a scientist to the correct solution of a problem have occasionally less to do with his capabilities than with his weaknesses. Namely, if we accept that, to a certain extent, the right idea chooses the person through which it will be expressed -by ‘choosing’ I of course don’t allude to some external wilful act of sensible choice-, then, the choice of the carrier, namely of the person most capable to express the idea, may depend, on his or her neuroses or, even, psychoses. For an idea to be expressed properly, then, the appropriate emotional makeup must be in place. Which can, at times, be pathological.

Something tells me we are getting closer to our main subject.

Precisely, we are reaching the point of the observation I made at the beginning, that the history of the quest for the foundations of mathematics is full of severely pathological cases. From Frege onwards -himself included, as we shall see- the magnificent history of the creation of modern logic is, largely, a story of psychopathology. We already saw – this was Rota’s remark with which we began that most of the people who attempted to build mathematics their on foundations of absolute logical certainly, exhibit in their private lives serious deviations from logicality. The brilliant construction that leads, through the last in the chain, Alan Turing, to the invention of electronic computers is, for the most part, deeply rooted on a ground of psychological malady.

  • About

    Apostolos Doxiadis is a Greek writer, mathematician and theatre and film director. His international best-seller "Uncle Petros and the Goldblach Conjecture" helped start the ‘mathematical fiction’ trend.


    His article 'Seventeenth Night' appears in Bedeutung Magazine Issue 1/ Nature & Culture, available here for purchase.