Randomness, often viewed as the opposite of order, plays a crucial role in science and technology, according to mathematician Avi Wigderson. The 2021 Abel Prize laureate discussed this topic during his lecture titled “Randomness” at Stony Brook University on November 12. The event was part of the Simons Center for Geometry and Physics’ Della Pietra Lecture Series.
Wigderson, who is the Herbert H. Maass Professor at the Institute for Advanced Study in Princeton, explained how randomness has shaped fields such as computer science, mathematics, game theory, and cryptography. Addressing an audience in the Della Pietra Family Auditorium, he described how unpredictability is not only present in nature but also serves as a tool for understanding complex systems.
“Randomness has fascinated humanity for millennia,” Wigderson said. “It’s been used to settle disputes, in gambling, in statistics, in science. The question is not only whether the universe is inherently deterministic or probabilistic, but whether we can tell the difference.”
Using examples like coin tosses and randomized algorithms, Wigderson demonstrated how random processes contribute to order and efficiency. He noted that even though people perceive certain patterns as more likely when flipping coins repeatedly, mathematically all sequences have equal probability if each flip is fair.
“If I could flip a coin perfectly, heads has a 50 percent probability and tails has a 50 percent probability,” he said. “Suppose I flip a coin 20 times. What sequence is more likely? The answer, of course, is that any sequence is equally likely, and has a probability of two to the power of 20, whether the sequence is 100 percent heads or 50 percent heads and 50 percent tails. This is what we call the perfect randomness or uniform probability on sequences.”
He highlighted how randomization enables computers to perform tasks quickly that would otherwise be impossible using only deterministic methods. For example, estimating configurations of molecules or securing digital communications relies on randomized techniques.
“With randomness,” Wigderson said, “we can make fast, approximate calculations where exact computation would take forever.”
The lecture also addressed challenges associated with generating true randomness and introduced pseudorandomness—deterministic processes designed to appear random even under close scrutiny by powerful computers.
“Randomness,” Wigderson said, “is in the eye of the beholder. What looks unpredictable to one observer might be predictable to another with more computational power. That means randomness isn’t an absolute property of nature — it’s relative to our ability to compute.”
He illustrated this point with an example involving coin tossing: while most people cannot predict outcomes by sight alone, someone equipped with advanced technology could calculate results based on physical factors.
“The experiment doesn’t change,” he said, “but the observer does. What’s random to us is deterministic to someone with greater computational resources.”
This perspective has influenced developments in error-correcting codes and data security by showing that apparent randomness can be harnessed even when underlying processes are not truly random.
Wigderson further discussed how probabilistic algorithms can sometimes be replaced by deterministic ones under certain assumptions about computational complexity—a concept known as “hardness versus randomness.”
“Every efficient randomized algorithm can, in theory, be simulated deterministically,” he said,“if just one sufficiently hard problem exists.”
“When we talk about pseudorandomness,” he added,“we’re really talking about creating order that looks like chaos. It’s one of the few areas where mathematics allows us to imitate nature.”
He concluded by emphasizing that randomness aids understanding rather than hindering it:
“Randomness lets us explore, approximate,and predict in ways that deterministic thinking alone cannot.Even in a world that may be deterministic at its core,the appearance of chance gives us new ways to make sense of it.”
Wigderson recommended his book “Mathematics and Computation: A Theory Revolutionizing Technology and Science” for those interested in learning more about these ideas; it can be downloaded free from his website.
