A Fellow of the Royal Society, Simonyi Professor for the Public Understanding of Science and Professor of Mathematics at the University of Oxford, author of seven books and a play, mathematician Marcus du Sautoy says his latest book Thinking Better: The Art of the Shortcut was inspired by the twin realizations that a certain laziness is at the heart of human creativity and that mathematics is “the art of the shortcut” allowing humans to get by doing as little as possible. Multiplying the number of mixed metaphors at play in response to interviewer Roger Highfield’s query if the book proved laziness was a sort of virtue, du Sautoy reported that nature writer Robert MacFarlane called mathematicians rock climbers as opposed to walkers whose modus operandi involves slow but constant progress along the path of least resistance. Rock climbers, by contrast, spend a lot of time stationary, waiting to sight their next foothold and then there’s a burst of activity. du Sautoy cavils this isn’t quite right, and the seeming stasis of mathematicians is the recce work of a ravening lion completely consumed in its hunger for a solution. Highfield is appropriately amused with this cheeky recasting of the slothful seeming mathematician, lazy by du Sautoy’s own admission, as a lion poised for attack.
du Sautoy’s “mathematical journey” began when at age 12 a chemistry teacher told his class the story of how as a schoolboy Carl Friedrich Gauss surprised his teacher with a quick answer to the problem of giving the sum of the numbers from one to hundred. The trick for adding numbers from 1 to n, he’d figured out, was multiplying (n/2) with (n +1); and the answer in this case was (50)(101) = 5,050. After regaling his students with this tale, du Sautoy’s teacher dubbed mathematics “the art of shortcuts”; tricks like this one which allowed one to tunnel through to a problem’s solution in a way that made it irrelevant how big the mountain was. du Sautoy was definitely sold on the branding of mathematics as the art of shortcuts even if it took him many years to produce a book with that title.
Highfield points out that Gauss looms large in the book, and du Sautoy acknowledges this happened by accident. Every time he found a nifty example of solving a big problem with a clever hack Gauss had been there before: “he has his name on many things” and “his biography” provides one way to tour the shortcuts revealed in this book. What does du Sautoy mean by saying nature has found these short cuts before, prods Highfield. As one can guess, du Sautoy responds it is a common adage in science that “nature too is lazy”, that it “likes to find the low energy solutions, the fastest way to get to something.” Light for instance takes the shortest path to its destination, and so underwater it refracts because it wants to get to the medium where it travels the fastest, namely air. The more interesting question, du Sautoy says, isn’t how nature finds these short cuts but how humans find them at all. “Calculus is our amazing shortcut to match nature” in the case of figuring out how light finds the fastest solution, the low energy solution, to travelling to its destination. Citing his own research Highfield concurs that nature does seems to seek shortcuts as you see when bouncing neutrons off soap bubbles it appears that “they seek a minimal energy surface if you put in a frame of a certain structural size.” Highfield, following claims by Max Tegmark, seems to conclude that this implies the world is not just mathematical, or mathematically tractable, but it is mathematics. du Sautoy wants to believe this being a mathematician he says, but a lot of mathematical objects seem to have no physical counterparts: like quarks in the theory of quantum mechanics. However, while physicists seems to lose interest in a proposed mathematical structure when they find it fails to describe an intended explanandum as a mathematician du Sautoy remains interested in the explanans as theoretical objects exhibiting certain properties. He is cautious in spite of his own enthusiasm, and calls mathematicians to be mindful of the tension between “beauty and messiness.” Many of our leading theories seem to be motivated by the desire to find beauty in the universe, but there isn’t any prima facie reason to think physics isn’t like biology and that there are no good reasons why there are “cats and not unicorns.” The beauty of mathematics “may be distracting us from messier ways of looking at the world which will have to do” he warns.
Speaking of biology as a possible model for physics, Highfield wryly notes researchers in biology are drowning in data and are at sea for want of a theory; there needs to be theory even as data are not ignored. This is, du Sautoy notes, another place mathematical shortcuts might be in order. Deciding just how much data one needs to look at before coming up with a satisfactory theoretical explanation with practical consequences is a question about sampling methods. For instance, you need to check the food preferences of only 250 cats to reach good conclusions about the preferences of 7 million cats. “An amazingly small number” says Highfield, “yeah” concurs du Sautoy. “19 out of 20 times is 5% away from what the true value in 7 million is: 250.” Yet this type of shortcut hunting isn’t about cutting corners, it is about getting results that work while expending the least amount of effort. It isn’t enough to get the preferences of 250 cats as data; the cats must be chosen at random for the trick to work, for instance.
To get a handle on the many types of licit shortcuts that might be imagined du Sautoy spoke to economists, naturalists, and psychologists about shortcuts they might have noticed in their disciplines. He discovered Robert MacFarlane who really loves walking didn’t care to cut his meanderings short, but entrepreneur and founder of LastMinute.Com Brent Hoberman had a favourite: getting as close to the boundaries of the law, pushing it, without breaking it. This has an analogue in mathematics, says du Sautoy: the concept of the square root of -1. Since its square is positive it can be used to do a lot of useful mathematics. √-1 was vital for handling calculations that allow aeroplanes to land safely, and you can’t do quantum physics without complex numbers. But there are other things which don’t allow for shortcuts: say, learning to play the cello. You can use patterns on the fingerboard, you can read the notation like it were forming words instead of individual notes, but ultimately your body is undergoing changes as it familiarizes itself with the instrument and this cannot be sped up. Another place you can’t really use shortcuts is psychology says du Sautoy drawing on the insights of his psychologist wife. You can’t process trauma or overcome ingrained patterns of thought using some shortcut.
Finally, Highfield touches on the topic of intuition as a sort of shortcut in mathematics. The case of the Indian savant Srinavasa Ramanujan illustrates how the subconscious can glimpse peaks which can only be conquered later by deliberate excursions. The mathematical terrain intuited by Ramanujan is still being worked out by mathematicians, yet these ideas came to him in flashes of inspiration. du Sautoy is alive to the powers of the subconscious and acknowledges it can sometimes achieve what grunt work often cannot. Nevertheless, it would be unfair to think intuition an art because it is all caprice and no method. It comes and goes as it pleases, yet the tools of mathematics are reliable even if boring in their stolidity.
JLF Toronto (2021). “Marcus du Sautoy in conversation with Roger Highfield on Thinking Better: The Art of the Shortcut.” YouTube. Updated: 7 October, 2021. Retrieved: 22 October 2021. URL:<https://youtu.be/HYTPGsCkVX4>.