The Art of Conjecturing

Reading Katy Börner’s Atlas of Forecasts: Modeling and mapping desirable futures for New Scientist, 18 August 2021

My leafy, fairly affluent corner of south London has a traffic congestion problem, and to solve it, there’s a plan to close certain roads. You can imagine the furore: the trunk of every kerbside tree sports a protest sign. How can shutting off roads improve traffic flows?

The German mathematician Dietrich Braess answered this one back in 1968, with a graph that kept track of travel times and densities for each road link, and distinguished between flows that are optimal for all cars, and flows optimised for each individual car.

On a Paradox of Traffic Planning is a fine example of how a mathematical model predicts and resolves a real-world problem.

This and over 1,300 other models, maps and forecasts feature in the references to Katy Börner’s latest atlas, which is the third to be derived from Indiana University’s traveling exhibit Places & Spaces: Mapping Science.

Atlas of Science: Visualizing What We Know (2010) revealed the power of maps in science; Atlas of Knowledge: Anyone Can Map (2015), focused on visualisation. In her third and final foray, Börner is out to show how models, maps and forecasts inform decision-making in education, science, technology, and policymaking. It’s a well-structured, heavyweight argument, supported by descriptions of over 300 model applications.

Some entries, like Bernard H. Porter’s Map of Physics of 1939, earn their place thanks purely to their beauty and for the insights they offer. Mostly, though, Börner chooses models that were applied in practice and made a positive difference.

Her historical range is impressive. We begin at equations (did you know Newton’s law of universal gravitation has been applied to human migration patterns and international trade?) and move through the centuries, tipping a wink to Jacob Bernoulli’s “The Art of Conjecturing” of 1713 (which introduced probability theory) and James Clerk Maxwell’s 1868 paper “On Governors” (an early gesture at cybernetics) until we arrive at our current era of massive computation and ever-more complex model building.

It’s here that interesting questions start to surface. To forecast the behaviour of complex systems, especially those which contain a human component, many current researchers reach for something called “agent-based modeling” (ABM) in which discrete autonomous agents interact with each other and with their common (digitally modelled) environment.

Heady stuff, no doubt. But, says Börner, “ABMs in general have very few analytical tools by which they can be studied, and often no backward sensitivity analysis can be performed because of the large number of parameters and dynamical rules involved.”

In other words, an ABM model offers the researcher an exquisitely detailed forecast, but no clear way of knowing why the model has drawn the conclusions it has — a risky state of affairs, given that all its data is ultimately provided by eccentric, foible-ridden human beings.

Börner’s sumptuous, detailed book tackles issues of error and bias head-on, but she left me tugging at a still bigger problem, represented by those irate protest signs smothering my neighbourhood.

If, over 50 years since the maths was published, reasonably wealthy, mostly well-educated people in comfortable surroundings have remained ignorant of how traffic flows work, what are the chances that the rest of us, industrious and preoccupied as we are, will ever really understand, or trust, all the many other models which increasingly dictate our civic life?

Borner argues that modelling data can counteract misinformation, tribalism, authoritarianism, demonization, and magical thinking.

I can’t for the life of me see how. Albert Einstein said, “Everything should be made as simple as possible, but no simpler.” What happens when a model reaches such complexity, only an expert can really understand it, or when even the expert can’t be entirely sure why the forecast is saying what it’s saying?

We have enough difficulty understanding climate forecasts, let alone explaining them. To apply these technologies to the civic realm begs a host of problems that are nothing to do with the technology, and everything to do with whether anyone will be listening.

How many holes has a straw?

Reading Jordan Ellenberg’s Shape for the Telegraph, 7 July 2021

“One can’t help feeling that, in those opening years of the 1900s, something was in the air,” writes mathematician Jordan Ellenburg.

It’s page 90, and he’s launching into the second act of his dramatic, complex history of geometry (think “History of the World in 100 Shapes”, some of them very screwy indeed).
For page after reassuring page, we’ve been introduced to symmetry, to topology, and to the kinds of notation that make sense of knotty-sounding questions like “how many holes has a straw”?

Now, though, the gloves are off, as Ellenburg records the fin de siecle’s “painful recognition of some unavoidable bubbling randomness at the very bottom of things.”
Normally when sentiments of this sort are trotted out, they’re there to introduce readers to the wild world of quantum mechanics (and, incidentally, we can expect a lot of that sort of thing in the next few years: there’s a centenary looming). Quantum’s got such a grip on our imagination, we tend to forget that it was the johnny-come-lately icing on an already fairly indigestible cake.

A good twenty years before physical reality was shown to be unreliable at small scales, mathematicians were pretzeling our very ideas of space. They had no choice: at the Louisiana Purchase Exposition in 1904, Henri Poincarre, by then the world’s most famous geometer, described how he was trying to keep reality stuck together in light of Maxwell’s famous equations of electromagnetism (Maxwell’s work absolutely refused to play nicely with space). In that talk, he came startlingly close to gazumping Einstein to a theory of relativity.
Also at the same exposition was Sir Ronald Ross, who had discovered that malaria was carried by the bite of the anopheles mosquito. He baffled and disappointed many with his presentation of an entirely mathematical model of disease transmission — the one we use today to predict, well, just about everything, from pandemics to political elections.
It’s hard to imagine two mathematical talks less alike than those of Poincarre and Ross. And yet they had something vital in common: both shook their audiences out of mere three-dimensional thinking.

And thank goodness for it: Ellenburg takes time to explain just how restrictive Euclidean thinking is. For Euclid, the first geometer, living in the 4th century BC, everything was geometry. When he multiplied two numbers, he thought of the result as the area of a rectangle. When he multiplied three numbers, he called the result a “solid’. Euclid’s geometric imagination gave us number theory; but tying mathematical values to physical experience locked him out of more or less everything else. Multiplying four numbers? Now how are you supposed to imagine that in three-dimensional space?

For the longest time, geometry seemed exhausted: a mental gym; sometimes a branch of rhetoric. (There’s a reason Lincoln’s Gettysburg Address characterises the United States as “dedicated to the proposition that all men are created equal”. A proposition is a Euclidean term, meaning a fact that follows logically from self-evident axioms.)

The more dimensions you add, however, the more capable and surprising geometry becomes. And this, thanks to runaway advances in our calculating ability, is why geometry has become our go-to manner of explanation for, well, everything. For games, for example: and extrapolating from games, for the sorts of algorithmical processes we saddle with that profoundly unhelpful label “artificial intelligence” (“artificial alternatives to intelligence” would be better).

All game-playing machines (from the chess player on my phone to DeepMind’s AlphaGo) share the same ghost, the “Markov chain”, formulated by Andrei Markov to map the probabilistic landscape generated by sequences of likely choices. An atheist before the Russian revolution, and treated with predictable shoddiness after it, Markov used his eponymous chain, rhetorically, to strangle religiose notions of free will in their cradle.

From isosceles triangles to free will is quite a leap, and by now you will surely have gathered that Shape is anything but a straight story. That’s the thing about mathematics: it does not advance; it proliferates. It’s the intellectual equivalent of Stephen Leacock’s Lord Ronald, who “flung himself upon his horse and rode madly off in all directions”.

Containing multitudes as he must, Ellenberg’s eyes grow wider and wider, his prose more and more energetic, as he moves from what geometry means to what geometry does in the modern world.

I mean no complaint (quite the contrary, actually) when I say that, by about two-thirds the way in, Ellenberg comes to resemble his friend John Horton Conway. Of this game-playing, toy-building celebrity of the maths world, who died from COVID last year, Ellenburg writes, “He wasn’t being wilfully difficult; it was just the way his mind worked, more associative than deductive. You asked him something and he told you what your question reminded him of.”
This is why Ellenberg took the trouble to draw out a mind map at the start of his book. This and the index offer the interested reader (and who could possibly be left indifferent?) a whole new way (“more associative than deductive”) of re-reading the book. And believe me, you will want to. Writing with passion for a nonmathematical audience, Ellenberg is a popular educator at the top of his game.

Stanisław Lem: The man with the future inside him


From the 1950s, science fiction writer Stanisław Lem began firing out prescient explorations of our present and far beyond. His vision is proving unparalleled.
For New Scientist, 16 November 2016

“POSTED everywhere on street corners, the idiot irresponsibles twitter supersonic approval, repeating slogans, giggling, dancing…” So it goes in William Burroughs’s novel The Soft Machine (1961). Did he predict social media? If so, he joins a large and mostly deplorable crowd of lucky guessers. Did you know that in Robert Heinlein’s 1948 story Space Cadet, he invented microwave food? Do you care?

There’s more to futurology than guesswork, of course, and not all predictions are facile. Writing in the 1950s, Ray Bradbury predicted earbud headphones and elevator muzak, and foresaw the creeping eeriness of today’s media-saturated shopping mall culture. But even Bradbury’s guesses – almost everyone’s guesses, in fact – tended to exaggerate the contemporary moment. More TV! More suburbia! Videophones and cars with no need of roads. The powerful, topical visions of writers like Frederik Pohl and Arthur C. Clarke are visions of what the world would be like if the 1950s (the 1960s, the 1970s…) went on forever.

And that is why Stanisław Lem, the Polish satirist, essayist, science fiction writer and futurologist, had no time for them. “Meaningful prediction,” he wrote, “does not lie in serving up the present larded with startling improvements or revelations in lieu of the future.” He wanted more: to grasp the human adventure in all its promise, tragedy and grandeur. He devised whole new chapters to the human story, not happy endings.

And, as far as I can tell, Lem got everything – everything – right. Less than a year before Russia and the US played their game of nuclear chicken over Cuba, he nailed the rational madness of cold-war policy in his book Memoirs Found in a Bathtub (1961). And while his contemporaries were churning out dystopias in the Orwellian mould, supposing that information would be tightly controlled in the future, Lem was conjuring with the internet (which did not then exist), and imagining futures in which important facts are carried away on a flood of falsehoods, and our civic freedoms along with them. Twenty years before the term “virtual reality” appeared, Lem was already writing about its likely educational and cultural effects. He also coined a better name for it: “phantomatics”. The books on genetic engineering passing my desk for review this year have, at best, simply reframed ethical questions Lem set out in Summa Technologiae back in 1964 (though, shockingly, the book was not translated into English until 2013). He dreamed up all the usual nanotechnological fantasies, from spider silk space-elevator cables to catastrophic “grey goo”, decades before they entered the public consciousness. He wrote about the technological singularity – the idea that artificial superintelligence would spark runaway technological growth – before Gordon Moore had even had the chance to cook up his “law” about the exponential growth of computing power. Not every prediction was serious. Lem coined the phrase “Theory of Everything”, but only so he could point at it and laugh.

He was born on 12 September 1921 in Lwów, Poland (now Lviv in Ukraine). His abiding concern was the way people use reason as a white stick as they steer blindly through a world dominated by chance and accident. This perspective was acquired early, while he was being pressed up against a wall by the muzzle of a Nazi machine gun – just one of several narrow escapes. “The difference between life and death depended upon… whether one went to visit a friend at 1 o’clock or 20 minutes later,” he recalled.

Though a keen engineer and inventor – in school he dreamed up the differential gear and was disappointed to find it already existed – Lem’s true gift lay in understanding systems. His finest childhood invention was a complete state bureaucracy, with internal passports and an impenetrable central office.

He found the world he had been born into absurd enough to power his first novel (Hospital of the Transfiguration, 1955), and might never have turned to science fiction had he not needed to leap heavily into metaphor to evade the attentions of Stalin’s literary censors. He did not become really productive until 1956, when Poland enjoyed a post-Stalinist thaw, and in the 12 years following he wrote 17 books, among them Solaris (1961), the work for which he is best known by English speakers.

Solaris is the story of a team of distraught experts in orbit around an inscrutable and apparently sentient planet, trying to come to terms with its cruel gift-giving (it insists on “resurrecting” their dead). Solaris reflects Lem’s pessimistic attitude to the search for extraterrestrial intelligence. It’s not that alien intelligences aren’t out there, Lem says, because they almost certainly are. But they won’t be our sort of intelligences. In the struggle for control over their environment they may as easily have chosen to ignore communication as respond to it; they might have decided to live in a fantastical simulation rather than take their chances any longer in the physical realm; they may have solved the problems of their existence to the point at which they can dispense with intelligence entirely; they may be stoned out of their heads. And so on ad infinitum. Because the universe is so much bigger than all of us, no matter how rigorously we test our vaunted gift of reason against it, that reason is still something we made – an artefact, a crutch. As Lem made explicit in one of his last novels, Fiasco (1986), extraterrestrial versions of reason and reasonableness may look very different to our own.

Lem understood the importance of history as no other futurologist ever has. What has been learned cannot be unlearned; certain paths, once taken, cannot be retraced. Working in the chill of the cold war, Lem feared that our violent and genocidal impulses are historically constant, while our technical capacity for destruction will only grow.

Should we find a way to survive our own urge to destruction, the challenge will be to handle our success. The more complex the social machine, the more prone it will be to malfunction. In his hard-boiled postmodern detective story The Chain of Chance (1975), Lem imagines a very near future that is crossing the brink of complexity, beyond which forms of government begin to look increasingly impotent (and yes, if we’re still counting, it’s here that he makes yet another on-the-money prediction by describing the marriage of instantly accessible media and global terrorism).

Say we make it. Say we become the masters of the universe, able to shape the material world at will: what then? Eventually, our technology will take over completely from slow-moving natural selection, allowing us to re-engineer our planet and our bodies. We will no longer need to borrow from nature, and will no longer feel any need to copy it.

At the extreme limit of his futurological vision, Lem imagines us abandoning the attempt to understand our current reality in favour of building an entirely new one. Yet even then we will live in thrall to the contingencies of history and accident. In Lem’s “review” of the fictitious Professor Dobb’s book Non Serviam, Dobb, the creator, may be forced to destroy the artificial universe he has created – one full of life, beauty and intelligence – because his university can no longer afford the electricity bills. Let’s hope we’re not living in such a simulation.

Most futurologists are secret utopians: they want history to end. They want time to come to a stop; to author a happy ending. Lem was better than that. He wanted to see what was next, and what would come after that, and after that, a thousand, ten thousand years into the future. Having felt its sharp end, he knew that history was real, that the cause of problems is solutions, and that there is no perfect world, neither in our past nor in our future, assuming that we have one.

By the time he died in 2006, this acerbic, difficult, impatient writer who gave no quarter to anyone – least of all his readers – had sold close to 40 million books in more than 40 languages, and earned praise from futurologists such as Alvin Toffler of Future Shock fame, scientists from Carl Sagan to Douglas Hofstadter, and philosophers from Daniel Dennett to Nicholas Rescher.

“Our situation, I would say,” Lem once wrote, “is analogous to that of a savage who, having discovered the catapult, thought that he was already close to space travel.” Be realistic, is what this most fantastical of writers advises us. Be patient. Be as smart as you can possibly be. It’s a big world out there, and you have barely begun.