John von Neumann, who created game theory, advanced many branches of mathematics and physics, did more than anyone else to design modern computers, was someone who attracted some mythology.
One story about Von Neumann has a colleague who sets him a fun puzzle. In this puzzle, two trains 20 miles apart slowly roar at each other on one track. The fly starts in front of one train, zips towards the other train, then quickly turns around and zips again. They fly back and forth until the trains collide with each other and crush each other between them.
The fly travels at 15 mph, with each train puffing at 10 mph. How far does the fly move before it flies?
“15 miles,” von Neumann said quickly.
“Yeah, you found a trick,” the colleague dismayed.
“What kind of trick? I’ve summed up a converging infinite series,” replies von Neumann.
Author of Von Neumann’s biographer Ananyo Bhattacharya – A man from the future – For mathematicians like Von Neumann who summarise a blurred, endless series, it’s not a feat more than we imagine, so we suspect that the narrative of a vaguely quick calculation shows a kind of thing and is exaggerated. But there is an easier way. Note that the train crashed after an hour and the fly traveled at 15mph.
Because if so, if so, suggests that even the great von Neumann is engaged in the strange practice of “mindless mathematics.”
For example, consider this question. How much dirt does the empty hole have in the empty hole, 3 feet x 3 feet x 3 feet? Congratulations if you answered “27 cubic feet.” That’s the correct number for hole capacity. Unfortunately, that’s the wrong answer. The hole is empty.
Another example of business school researchers Asher Lawson, Richard Larick and Jack Sol is: “Joey went to the store and bought a pack of chips. A bottle of water costs $3, a pack of chips costs $1, and a pack of gum costs $2.
In their survey, a quarter of respondents answered $6. These people were probably too busy looking at numbers and the word “sum” and calculating the answer to read the questions.
“The presence of numbers in a problem seduces people to perform mathematical operations,” Lawson and his colleagues wrote, “even if the correct answer does not require mathematics.”
Mindless Mathematics is an example of what psychologists Abraham and Edith Routines called the Einstelng effect. This German “setting” or “attitude” can be best translated here as “setup”. Luchinses gave experimental participants a set of problems to solve a set of problems in which a particular approach always provides a solution. After that setup they were given further problems with a simpler solution, but instead cranked through established methods and without those simple shortcuts. In a parochial way, they continued to reach for a reliable hammer, even when the situation demanded a driver.
So one lesson is to stop and think. It’s not surprising that people who tend to fall into trick puzzles also tend to fall due to disinformation. The good news is that finding fake news doesn’t require enormous expertise. The habit of slowing down and calming down is usually sufficient.
But this isn’t just about finding fake news. Mindless mathematics is closely related to what I called “premature enumeration.” This sad pain often affects people who are used to numbers, jumping to calculate and analyze numerical claims seen in the news before asking some basic questions.
Such questions include, “What is the cause of this claim?” and “What definition do they use?”
The source is important. Some research is a vast and expensive scientific endeavor. Others are clickbaits designed to sell ice cream. Some claims are made by independent analysts, while others are made by partisans.
Definition is also important. Whether you are pondering statistics about gun violence and drinking, teenage mental health, or extreme weather events, there is no equation to solve, and no graph to plot.
So, once more: stop and think about it. Students taking exams around the world are encouraged to take their time to read their questions properly. That’s healthy advice. But even for us, whose final exam was decades ago, it will help us to look back at the arguments and questions in front of us for a moment. It is difficult to reach the right destination without establishing a starting point first.
For Von Neumann and The Fly, it’s the same with Bhattacharya. Von Neumann wouldn’t have fallen into it. Part of his brilliance was his relentless eyes to the simplest solutions to any problem.
In the early 1950s, scientists at Rand Think-Tank approached Von Neumann with advice on how to design cutting-edge computers to tackle the scary parts of mathematical analysis. Von Neumann asked for details on the issue and was treated to a two-hour presentation of all complexity, with his colleagues scribbled blackboards to pull out tables and charts.
With his head in hand, von Neumann stared vaguely for a moment. He seemed to “his heart slid his face out of gear,” one witness recalled.
Then von Neumann spoke. “Gentleman, you don’t need a computer. I have the answer.”
Written for and first published Financial Times March 21st, 2025.
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