Tweel is one of the all-time great science fiction characters, the hero of Stanley G. Weinbaum’s wonderful 1934 story, A Martian Odyssey. The story is set on Mars in the 21st century and begins with astronaut Dick Jarvis crashing his mini-rocket. Jarvis then happens upon the ostrich-like Tweel being attacked by a tentacled monster. Jarvis saves Tweel, they become friends and Tweel accompanies Jarvis on his long journey back to camp and safety, the two meeting all manner of exotic Martians along the way.
A Martian Odyssey is great fun, fantastically inventive pulp science fiction, but the weird, endearing and strangely intelligent Tweel raises the story to another level. Tweel and Jarvis attempt to communicate, and Tweel learns a few English words while Jarvis can make no sense of Tweel’s sounds, is simply unable to figure out how Tweel thinks. However, Jarvis gets an idea:
“After a while I gave up the language business, and tried mathematics. I scratched two plus two equals four on the ground, and demonstrated it with pebbles. Again Tweel caught the idea, and informed me that three plus three equals six.”
That gave them a minimal form of communication and Tweel turns out to be very resourceful with the little mathematics they share. Coming across a weird rock creature, Tweel describes the creature as
“No one-one-two. No two-two-four”.
Later Tweel describes some crazy barrel creatures:
“One-one-two yes! Two-two-four no!”
A Martian Odyssey works so well because Weinbaum simply describes the craziness that Jarvis encounters, with no attempt to explain it. Tweel is just sufficiently familar – a few words, a little arithmetic and a sense of loyalty – to make the craziness seem meaningful if still not comprehensible.
But now, here’s the puzzle. The communication between Jarvis and Tweel depends upon the universality of mathematics, that all intelligent creatures will understand and agree that 1 + 1 = 2 and 2 + 2 = 4, and so forth.
But why? Why is 1 + 1 = 2? Why is 2 + 2 = 4?
The answers are perhaps not so obvious. First, however, you should go read Weinbaum’s awesome story (and the sequel). Then ponder the puzzle.
Thanks to those who have posted so far. Everyone is circling with the right ideas, but perhaps people are searching for something deeper than intended. Anyway, for this first update (to which people are free to object in the comments), here is our suggested, simplest answer to why 1 + 1 = 2:
“1 + 1 = 2” is true by definition.
To take a step back, what does 2 mean? It depends slightly on how you think of the natural numbers being given, but there are really only (ahem) two, similar choices. If you accept that addition is around then 2/two is simply a new symbol/name that stands for 1 + 1.
Or, more fundamentally, we can follow Number 8 and go Peano-ish, in which case 2 is defined as S(1), as the “successor” of 1. But then we have to define addition, and the first(ish) step for that is to define n + 1 = S(n); that is, 1 + 1 is defined to be S(1), which we have decided to call 2. There’s a good discussion of it all here.
With 1 + 1 = 2 done (modulo objections), why now is 2 + 2 = 4?
It’s probably close enough to round this one off. To clearly state why 2 + 2 = 4, we first have to clearly state what 2 and 4 and + are. So, as discussed above, 1 + 1 = 2 by definition (more or less). And, similarly, we define 3 = 2 + 1 and 4 = 3 + 1. So, the question of why 2 + 2 = 4 comes down to understanding why
2 + (1 + 1) = (2 + 1) + 1
So, our question amounts to a simple instance of the associative law of addition. And, how do we know the associative law is true? Naively, we can accept that’s the way numbers work. Or, we can go Peano-ish again, and the above example of associativity becomes part of the definition of addition.
In summary, to know that 1 + 1 = 2 all we need is the notion of natural numbers, of counting. To know that 2 + 2 = 4, however, requires the notion of addition.