Richard Feynman and the Isle of Maths (Plural)
It was 1952, still only a few years after the War, when Dr. Richard Feynman, esteemed professor of theoretical physics at Caltech, boarded a small passenger plane in Brazil, ultimately bound for the United States. His penultimate destination was some patch of ocean in the Gulf of Mexico, where the plane crashed.
The good doctor woke up on a beach. He found his clothes in tatters, and himself somewhat dehydrated (he could use a stiff drink), but he was otherwise no worse for wear. He took off what was left of his shirt, revealing his upper body, toned from years of intense physical calculations. (Unfortunately, his trousers were in one piece.)
Taking immediate stock of the situation, he decided to start exploring the island (which he knew it to be from the title). After walking along the beach for less than 15 minutes (judging by the sun), he almost tripped over a bottle of vodka, conveniently chilled by some damp, shaded sand. Picking it up, his spirit was further lifted when he spotted a grove of coconut trees, bearing countless ripe coconuts. A mixed drink with coconut water would make exploration of this island quite pleasant indeed.
Approaching the grove, he rolled up his pant legs and quickly scaled a coconut tree, like any skilled professor would. He easily dislodged two large coconuts. Looking down to see where they had landed, he noticed three coconuts in a triangle (for no other shape was possible) on the ground. He went about harvesting two more, and again checked his work. This time he was surprised to see six coconuts on the ground (in what he knew technically as a jumble).
He climbed down and examined the pile of coconuts. Taking two from the pile of six, he noticed that there were only three left. Putting them back, there were six total. Dividing the pile in half, there were two piles of three. However, picking up a coconut from one of the piles, the pile only had one coconut left. In other words, by his best professional assessment of these coconuts, it appeared that 1 + 1 = 3.
Leaving aside the coconuts for a moment, he downed a shot of vodka. (However, trust me, your author, when I insist this did not impair his impeccable judgment in the slightest.)
He decided he would come back to the coconuts later. Suddenly he spotted a stream emerging from the forest, further down the beach, and running into the ocean. Upon reaching it, he saw that there was a clear path along it into the woods, and wondered if he might find other inhabitants of this island.
Following the stream for about half an hour (judging by the number of paces at his standard walking speed, because the sun was hidden behind the trees), he eventually came to a large, round pool. His keen sense of vision as a theoretical physicist told him the pool was perfectly round and precisely 100 meters across. He also noticed that the surface of the pool was as smooth as glass (atomically).
Thinking nothing of it, he decided to semi-circumnavigate the pool and continue up the stream, which resumed on the exact opposite side. He walked precisely along the shore, with one foot in the water (because it was a hot day). However, upon reaching the other end of the 1-dimensional stream, he was surprised to notice that his path along half the pool’s perimeter, which should have been 157.08 meters long (approximately), had actually been 150.00 meters long (precisely).
To confirm his hypothesis, he walked around the other half of the pool, and firmly established that its total circumference was precisely 300 meters. In other words, it seemed that at least when pertaining to this pool, the value of pi was exactly 3.
He downed another shot from the bottle of vodka he was still carrying. It was starting to get dark, so he decided to head back to the beach and try to start a fire to signal for help. On the way back to the beach, he found some logs conveniently cut for that purpose.
Dropping about 100 kilograms of logs in a suitable spot on the beach, he set about building a fire. First, he took some dry logs and arranged them in an optimal conflagration configuration. Then he added some dry leaves as kindling. Then he pulled some flint out of his pocket (a physicist is always prepared) and struck it with a rock (from another pocket). Sparks flew onto the dry leaves but nothing happened.
Dismayed, he tried again. More and bigger sparks flew onto the leaves, but disappeared without leaving the slightest burns. He struck the flint again and again, shooting giant balls of fire at the leaves and logs. But there was no hint of ignition.
After hundreds or even thousands of attempts, he finally messed up once. The rock weakly glanced off the flint, making no sparks. Then, slowly, the leaves began to burn. He stepped back as the fire grew.
Surprised at this occurrence, and mindful of the previous events in his short time on this curious isle, he devised some simple experiments. He quickly concluded that there were three basic conditions to start a fire: dry logs, dry kindling, and striking the flint to produce sparks. However, the fire would only start when either the logs were wet, the kindling was wet, or no sparks were produced. In other words, true and true and true was false, but true and true and false was true.
Richard Feynman swallowed a third shot of vodka and cast the empty bottle into the water. The thought of rescue faded from his mind. Sitting down on the beach, he made his plans for the next day. He would investigate the Arithmetic of the Coconuts, the Geometry of the Pool, and the Logic of the Fire, and get to the bottom of the mysteries of this island.
The End (because I’m not smart enough to write the rest)
I’ve long been curious about the idea of whether math and logic could vary. If we can’t rely on logic, then Descartes’ “cogito ergo sum” is wrong. Without logic, we can’t conclude “cogito,” much less “cogito ergo” anything.
It’s easy to imagine alternate universes with different physics, for example a different strength of gravity, or different subatomic particles. However it’s hard to imagine a universe with different math. Already in this story, there are inconsistencies: two groups of three coconuts should add to become three groups of three coconuts; in reality, the conditions to start a fire are not firmly defined, making it hard to know when exactly one of them is false.
My interpretation is that unlike physics, math as we know it is true in every possible universe. The evidence for this is four-fold:
- The difficulty in working out a counter-factual but internally consistent form of math
- The ease with which such a counter-factual math might let you solve hard problems (imagine a universe where the halting problem is trivial to decide in constant time)
- The difficulty of building computer simulations within our universe that don’t inherit our math (despite the ease of not inheriting our physics)
- Not all mathematical operations are local (if you own a house, and buy a vacation home in the Bahamas, do you suddenly get a third house? where?)
My conclusion is that physics isn’t physics, math is physics. Beyond that I’m at a loss.
Thank you for reading!
Keywords: fiction, math