Brain Snacks #139: Human Computers, Euclid’s Mistake and Hofstadter’s 641
How Computers Used to Be Humans, How GPS Needed To Give Up Euclidean Geometry and A Paradox to Help Think About How We Think
Welcome to this week's edition of Brain Snacks. I share three short, exciting stories every Sunday to inspire your week.
I started the week with a great conversation with Helena (my spouse) about getting old. We are both close to turn 40, and we were confortable with our age, and actually no wishes of being younger.
It feels like we are now completely comfortable with who we are. We know ourselves fully and can enjoy life more freely. This realization came after we discussed how raw our 4-year-old is in being completely present and unapologetically his full self at all times.
It feels as we get older we start getting more self-conscious and start losing that feeling, and for us it feels hitting 40s that we are recovering a a bit of that feeling again.
I hope you enjoy this week’s snacks.
Brain Snack #1: Human Computers
If you ask an average person when they think the word computer start being part of our language, they would probably guess the 20th century. But long before “computer” meant a glowing screen in your desk, it was meant for a person who does calculations. The Oxford English Dictionary records the earliest use in 1613, in Richard Braithwait’s book The Yong Mans Gleanings, where he described a man as a “good Computer.”
Back in the 17th century, the rise of astronomy, navigation, and ballistics demanded ever more accurate tables of numbers. Ships needed precise star charts, gunners needed firing tables, and engineers needed logarithmic tables. Teams of human computers were hired to calculate, check, and re-check the figures by hand, and therefore the job “computer” was born.
By the 19th century, whole institutions were organized around this work. The British Nautical Almanac Office employed dozens of clerks to churn out astronomical tables. At Harvard College Observatory, women were hired in large numbers to examine photographic plates of the night sky, painstakingly classifying stars by brightness and spectrum. These “computers” built the raw data sets that underpinned scientific revolutions in astronomy and physics.
The reliance on human computers reached its peak in the world wars, when armies of mostly female clerks calculated ballistics trajectories, weather predictions, and cryptographic tables. Only in the mid-20th century did the word begin to slip away from them, transferred first to electromechanical engines and then to electronic machines, forever tying the word computer to a machine.
Go deeper: Computers Were Originally Humans - Herbert Bruderer (Communications of the ACM blog
Brain Snack #2: Euclid’s Mistake
School geometry gives us a very simple and practical rule: parallel lines never meet and triangles add up to 180°. Handy, and almost true. The “almost” is Euclid’s small mistake. He treated physical space as perfectly flat. That works on a desk, a street, even a bridge. But stretch those lines across oceans, or send timing signals between satellites and phones, and the world’s not flat enough.
On a sphere the rules bend. “Straight lines” become great circles, like the Equator or a meridian, and any triangle you draw, say, two meridians meeting at the poles plus the Equator, has angles summing to more than 180°. That’s why long-haul flights arc over the map: they’re following geodesics on a curved Earth. Satellite navigation has to speak this spherical language (more precisely, an ellipsoidal one), or positions drift.
Einstein pushed the idea further: space and time themselves are curved by mass and motion. GPS only works because its engineers correct for this non-Euclidean reality. A satellite’s clock ticks differently from one on Earth, faster due to weaker gravity, slower because it’s moving, with a net offset of about +38 microseconds per day. Leave that “tiny” mismatch unpatched and location errors balloon by kilometers. So GPS (and Galileo/BeiDou) pre-bias satellite clocks and apply relativistic formulas continuously as signals race along bent paths in curved spacetime.
Still, Euclid isn’t obsolete, he’s still used everyday. For building a house, laying a football pitch, or sketching a neighborhood map, flat-space geometry is perfect: simple, fast, and accurate enough. The bigger lesson is methodological: useful models are often deliberate reductions. They trade universal truth for tractable insight, then flag when to switch gears.
Go deeper: Maths in a minute: Not always 180 - Maths.org
Brain Snack #3: Hofstadter’s 641
Figuring out how our brain works has been the pursuit of many scientists throughout history, but most of it still remains a mystery. One possible explanation for why we have not been able to make as much progress comes from cognitive scientist Douglas Hofstadter, as he poses his “641 argument.”
The 641 argument sets up the following scenario: imagine a chain of dominoes. If 641 is identified as prime, then a particular stretch of dominoes in the “results” section of the chain will remain standing. Now imagine someone who is not aware of the computations and tries to explain why some dominoes are still standing. One explanation could be: “because the previous domino didn’t hit it.” True, but it doesn’t tell the whole story, which is “because 641 is prime.”
Hofstadter notes it is perhaps even the only satisfactory answer on the level that truly explains the phenomenon. This distinction is the heart of Hofstadter’s lesson. Some explanations only exist at a higher level of abstraction and can never be uncovered by staring harder at the low-level mechanics. You could track every vibration of every domino, every molecule of wood and air, and still never discover the concept of “prime.”
For Hofstadter, this is a clue to why the mind itself is so hard to pin down. Brains are made of neurons firing, ions flowing, and chemicals binding, but describing that activity alone doesn’t explain why you feel joy, remember a childhood song, or suddenly realize a joke is funny. Just as primality is invisible in the physics of dominoes, consciousness may be invisible in the biochemistry of neurons.
Go deeper: Do Neurons Push Thoughts Around? Or Do Thoughts Push Neurons Around? - Philosophy Break