Consider, for example, the oft-repeated legend of the Grand Vizier in Persia who invented chess. The King, so the legend goes, was delighted with the new game, and invited the Vizier to name his own reward. The Vizier replied that, being a modest man, he desired only one grain of wheat on the first square of a chessboard, two grains on the second, four on the third, and so on, with twice as many grains on each square as on the last. The innumerate King agreed, not realizing that the total number of grains on all 64 squares would be 2⁶⁴-1, or 18.6 quintillion—equivalent to the world’s present wheat production for 150 years.

Fittingly, this same exponential growth is what makes chess itself so difficult. There are only about 35 legal choices for each chess move, but the choices multiply exponentially to yield something like 10⁵⁰ possible board positions—too many for even a computer to search exhaustively. That’s why it took until 1997 for a computer, Deep Blue, to defeat the human world chess champion. And in Go, which has a 19-by-19 board and over 10¹⁵⁰ possible positions, even an amateur human can still rout the world’s top-ranked computer programs. Exponential growth plagues computers in other guises as well. The traveling salesman problem asks for the shortest route connecting a set of cities, given the distances between each pair of cities. The rub is that the number of possible routes grows exponentially with the number of cities. When there are, say, a hundred cities, there are about 10¹⁵⁸ possible routes, and, although various shortcuts are possible, no known computer algorithm is fundamentally better than checking each route one by one. The traveling salesman problem belongs to a class called NP-complete, which includes hundreds of other problems of practical interest. (NP stands for the technical term ‘Nondeterministic Polynomial-Time.’) It’s known that if there’s an efficient algorithm for any NP-complete problem, then there are efficient algorithms for all of them. Here ‘efficient’ means using an amount of time proportional to at most the problem size raised to some fixed power—for example, the number of cities cubed. It’s conjectured, however, that no efficient algorithm for NP-complete problems exists. Proving this conjecture, called P¹ NP, has been a great unsolved problem of computer science for thirty years.

Although computers will probably never solve NP-complete problems efficiently, there’s more hope for another grail of computer science: replicating human intelligence. The human brain has roughly a hundred billion neurons linked by a hundred trillion synapses. And though the function of an individual neuron is only partially understood, it’s thought that each neuron fires electrical impulses according to relatively simple rules up to a thousand times each second. So what we have is a highly interconnected computer capable of maybe 10¹⁴ operations per second; by comparison, the world’s fastest parallel supercomputer, the 9200-Pentium Pro teraflops machine at Sandia National Labs, can perform 10¹² operations per second. Contrary to popular belief, gray mush is not only hard-wired for intelligence: it surpasses silicon even in raw computational power. But this is unlikely to remain true for long. The reason is Moore’s Law, which, in its 1990’s formulation, states that the amount of information storable on a silicon chip grows exponentially, doubling roughly once every two years. Moore’s Law will eventually play out, as microchip components reach the atomic scale and conventional lithography falters. But radical new technologies, such as optical computers, DNA computers, or even quantum computers, could conceivably usurp silicon’s place. Exponential growth in computing power can’t continue forever, but it may continue long enough for computers—at least in processing power—to surpass human brains.

To prognosticators of artificial intelligence, Moore’s Law is a glorious herald of exponential growth. But exponentials have a drearier side as well. The human population recently passed six billion and is doubling about once every forty years. At this exponential rate, if an average person weighs seventy kilograms, then by the year 3750 the entire Earth will be composed of human flesh. But before you invest in deodorant, realize that the population will stop increasing long before this—either because of famine, epidemic disease, global warming, mass species extinctions, unbreathable air, or, entering the speculative realm, birth control. It’s not hard to fathom why physicist Albert Bartlett asserted "the greatest shortcoming of the human race" to be "our inability to understand the exponential function." Or why Carl Sagan advised us to "never underestimate an exponential." In his book Billions & Billions, Sagan gave some other depressing consequences of exponential growth. At an inflation rate of five percent a year, a dollar is worth only thirty-seven cents after twenty years. If a uranium nucleus emits two neutrons, both of which collide with other uranium nuclei, causing them to emit two neutrons, and so forth—well, did I mention nuclear holocaust as a possible end to population growth?

— FROM SCOTT AARONSON, “Who Can Name the Bigger Number?” Retrieved from https://www.scottaaronson.com/writings/bignumbers.html