The number-crunchers. How we learned to stop worrying and love to code.

My grandmother was a calculator, and I don’t mean to say I’m the newest from Texas Instruments. I mean my grandmother did calculations for a living, with pencil on paper, using a slide rule and logarithmic tables. She calculated the positions of stars on the night sky, for five minute intervals, day by day, digit by digit.

Today you can download one of a dozen free apps to display the night sky for any position on Earth, any time, any day. Not that you actually need to know stellar constellations to find True North. Using satellite signals, your phones can now tell your position to within a few meters, and so can 2 million hackers in Russia.

My daughters meanwhile are thoroughly confused as to what a phone is, since we use the phone to take photos but make calls on the computer. For my four-year old a “phone” is pretty much anything that beeps, including the microwave, which for all I know by next year might start taking photos of dinner and upload them to facebook. And the landline. Now that you say it. Somebody called in March and left a voicemail.

Jack Myers dubbed us the “gap generation,” the last generation to remember the time before the internet. Myers is a self-described “media ecologist” which makes you think he’d have heard of search engine optimization. Unfortunately, when queried “gap generation” it takes Google 0.31 seconds to helpfully bring up 268,000,000 hits for “generation gap.” But it’s okay. I too recall life without Google, when “viral” meant getting a thermometer stuffed between your lips rather than being on everybody’s lips.

I wrote my first email in 1995 with a shell script called “mail” when the internet was chats and animated gifs. Back then, searching a journal article meant finding a ladder and blowing dust off thick volumes with yellowish pages. There were no keyword tags or trackbacks; I looked for articles by randomly browsing through journals. If I had an integral to calculate, there were Gradshteyn and Ryzhik’s tables, or Abramovitz and Stegun's Handbook of Special Functions, and else, good luck.

Our first computer software for mathematical calculations, one of the early Maple versions, left me skeptical. It had an infamous error in one of the binomial equations that didn’t exactly instill trust. The program was slow and stalled the machine for which everybody hated me because my desk computer was also the institute’s main server (which I didn’t know until I turned it off, but then learned very quickly). I taught myself fortran and perl and java script and later some c++, and complained it wasn’t how I had imagined being a theoretical physicist. I had envisioned myself thinking deep thoughts about the fundamental structure of reality, not chasing after missing parentheses.

It turned out much of my masters thesis came down to calculating a nasty integral that wasn’t tractable numerically, by computer software, because it was divergent. And while I was juggling generalized hypergeometric functions and Hermite polynomials, I became increasingly philosophic about what exactly it meant to “solve an integral.”

We say an integral is solved if we can write it down as a composition of known functions. But this selection of functions, even the polynomials, are arbitrary choices. Why not take the supposedly unsolvable integral, use it to define a function and be done with it? Why are some functions solutions and others aren’t? We prefer particular functions because their behaviors are well understood. But that again is a matter of how much they are used and studied. Isn’t it in the end all a matter of habit and convention?

After two years I managed to renormalize the damned integral and was left with an expression containing incomplete Gamma functions, which are themselves defined by yet other integrals. The best thing I knew to do with this was to derive some asymptotic limits and then plot the full expression. Had there been any way to do this calculation numerically all along, I’d happily have done it, saved two years of time, and gotten the exact same result and insight. Or would I? I doubt the paper would even have gotten published.

Twenty years ago, I like most physicists considered numerical results inferior to analytical, pen-on-paper, derivations. But this attitude has changed, changed so slowly I almost didn’t notice it changing. Today numerical studies are still often considered suspicious, fighting a prejudice of undocumented error. But it has become accepted practice to publish results merely in forms of graphs, figures, and tables, videos even, for (systems of) differential equations that aren’t analytically tractable. Especially in General Relativity, where differential equations tend to be coupled, non-linear, and with coordinate-dependent coefficients – ie as nasty as it gets – analytic solutions are the exception not the norm.

Numerical results are still less convincing, but not so much because of a romantic yearning for deep insights. They are less convincing primarily because we lack shared standards for coding, whereas we all know the standards of analytical calculation. We use the same functions and the same symbols (well, mostly), whereas deciphering somebody else’s code requires as much psychoanalysis as patience. For now. But imagine you could check code with the same ease you check algebraic manipulation. Would you ditch analytical calculations over purely numerical ones, given the broader applicability of the latter? How would insights obtained by one method be of any less value than those obtained by the other?

The increase of computing power has generated entirely new fields of physics by allowing calculations that previously just weren’t feasible. Turbulence in plasma, supernovae explosion, heavy ion collisions, neutron star mergers, or lattice qcd to study the strong nuclear interaction, these are all examples of investigations that have flourished only with the increase in processing speed and memory. Such disciplines tend to develop their own, unique and very specialized nomenclature and procedures that are difficult if not impossible to evaluate for outsiders.

Lattice QCD. Artist’s impression.

Then there is big data that needs to be handled. May that be LHC collisions or temperature fluctuations in the CMB or global fits of neutrino experiments, this isn’t data any human can deal with by pen on paper. In these areas too, subdisciplines have sprung up, dedicated to data manipulation and -handling. Postdocs specialized in numerical methods are high in demand. But even though essential to physics, they face the prejudice of somehow being “merely calculators.”

The maybe best example is miniscule corrections to probabilities of scattering events, like those taking place at the LHC. Calculating these next-to-next-to-next-to-leading-order contributions is an art as much as a science; it is a small subfield of high energy physics that requires summing up thousands or millions of Feynman diagrams. While there are many software packages available, few physicists know all the intricacies and command all the techniques; those who do often develop software along with their research. They are perfecting calculations, aiming for the tiniest increase in precision much like pole jumpers perfect their every motion aiming after the tiniest increase in height. It is a highly specialized skill, presently at the edge of theoretical physics. But while we admire the relentless perfection of professional athletes, we disregard the single-minded focus of the number-crunchers. What can we learn from it? What insight can be gained from moving the bar an inch higher?

What insight do you gain from calculating the positions of stars on the night sky, you could have asked my grandmother. She was the youngest of seven siblings, her father died in the first world war. Her only brother and husband were drafted for the second world war and feeding the family was left to the sisters. To avoid manufacturing weapons for a regime she detested, she took on a position in an observatory, calculating the positions of stars. This appointment came to a sudden stop when her husband was badly injured and she was called to his side at the war front to watch him die, or so she assumed. Against all expectations, my grandfather recovered from his skull fractures. He didn’t have to return to the front and my grandma didn’t return to her job. It was only when the war was over that she learned her calculations were to help the soldiers target bombs, knowledge that would haunt her still 60 years later.

What insight do we gain from this? Precision is the hallmark of science, and for much of our society science is an end for other means. But can mere calculation ever lead to true progress? Surely not with the computer codes we use today, which execute operations but do not look for simplified underlying principles, which is necessary to advance understanding. It is this lacking search for new theories that leaves physicists cynical about the value of computation. And yet some time in the future we might have computer programs doing exactly this, looking for underlying mathematical laws better suited than existing ones to match observation. Will physicists one day be replaced by software? Can natural law be extracted by computers from data? If you handed all the LHC output to an artificial intelligence, could it spit out the standard model?

In an influential 2008 essay “The End of Theory,” Chris Anderson argued that indeed computers will one day make human contributions to theoretical explanations unnecessary:

“The reason physics has drifted into theoretical speculation about n-dimensional grand unified models over the past few decades (the "beautiful story" phase of a discipline starved of data) is that we don't know how to run the experiments that would falsify the hypotheses — the energies are too high, the accelerators too expensive, and so on[...]

The new availability of huge amounts of data, along with the statistical tools to crunch these numbers, offers a whole new way of understanding the world. Correlation supersedes causation, and science can advance even without coherent models, unified theories, or really any mechanistic explanation at all.”

His future vision was widely criticized by physicists, me included, but I’ve had a change of mind. Much of the criticism Anderson took was due to vanity. We like to believe the world will fall into pieces without our genius and don’t want to be replaced by software. But I don’t think there’s anything special about the human brain that an artificial intelligence couldn’t do, in principle. And I don’t care very much who or what delivers insights, as long as they come by. In the end it comes down to trust.

If a computer came up with just the right string theory vacuum to explain the standard model and offered you the explanation that the world is made of strings to within an exactly quantified precision, what difference would it make whether the headline was made by a machine rather than a human? Wouldn’t you gain the exact same insight? Yes, we would still need humans to initiate the search, someone to write a code that will deliver to our purposes. And chances are we would celebrate the human, rather than the machine. But the rest is overcoming prejudice against “number crunching,” which has to be addressed by setting up reliable procedures that ensure a computer’s results are sound science. I’ll be happy if your AI delivers a theory of quantum gravity; bring it on.

My grandmother outlived her husband who died after multiple strokes. Well in her 90s she still had a habit of checking all the numbers on her receipts, bills, and account statements. Despite my conviction that artificial intelligences could replace physicists, I don’t think it’s likely to happen. The human brain is remarkable not so much for its sheer computing power, but for its efficiency, resilience, and durability. You show me any man-made machine that will still run after 90 years in permanent use.