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.

If we assume the Peano axioms, doesn’t it just drop out using induction?

Maybe. What exactly are you assuming and what drops out?

I’m assuming 1 is a natural number and that there exists a function S(n) such that if n is a natural number then S(n) is a natural number.

Well, maybe, but it looks like you’re trying to write a Cambridge text. It shouldn’t be that hard. (I’m not saying it isn’t, but it shouldn’t be.)

1+1=1+S(0)=S(1+0)=S(1)=S(S(0))=2

Then 2+2=2+S(S(0))=S(S(0+2)=S(S(2)=4

Making all sorts of assumptions here of course (at least five)

Nah. Way too Cambridge. A primary kid knows that 1 + 1 = 2 and 2 + 2 = 4. They may not know why they know, but they know. So, at it’s heart, how does a kid come to understand that 1 + 1 = 2 and that 2 + 2 = 4? How can we characterise these truths as simply as possible?

OK. A bit of clarification. I always *thought* I knew these facts but the more I began to think about the assumptions I was making, the more I came to realise I didn’t fundamentally understand the assumptions I was making. Geometry was fine, Euclid’s 5 postulates (really the 5th is the vital one), but numbers really did my head in. Things like why a negative times a negative was positive – it took me a while thinking about the distributive law to get to the bottom of this one. Same with the basic question of natural numbers.

Maybe it is too complicated for school (Primary or otherwise) but to my current state of mind it works perfectly.

Doesn’t answer your question I know.

Although, if you re-work my previous answer – you have one “thing” and you count it. One. You have another “thing” and you count it. One. You put them together and count. Two. The process of putting them together you call addition (in this case) and so 1 put with 1 is 2.

The Peano axioms just formalise the assumptions behind this a bit more.

So, concentrating on your second last paragraph, why, in the simplest possible language, does 1 + 1 equal 2?

Because addition groups “things” and then we count. Counting 1 thing and then 1 thing is the same as counting 2 things.

Is it got to do with our experience inside an apparently consistent universe? A far as we know, the laws of physics hold true and don’t change their behaviour at a whim. If Jarvis has a pebble in each hand and compares it to the two pebbles Tweel has next to him, he’ll see that he can match the pebbles together. So there is the same amount. No other pebbles will randomly appear next to Tweel, as the laws of physics are (as far as well can tell) consistent.

Ah, Potii, perhaps, but I was thinking along directly mathematical lines: why do WE know that 1 + 1 = 2, and 2 + 2 = 4? What do these equations mean for us?

I’d say they mean equivalence; the same amount on either side.

Well, more precisely, equality. But why are the two sides equal?

Interesting side note – I learned only recently of the history of the equality symbol. Scotland’s greatest contribution to Mathematics perhaps? One does wonder how we came to survive without it for so long, though…

But back to the topic, there is something quite beautiful about the lack of direction in this symbol (as is relevant here) 1+1=2 and 2=1+1 are considered mathematically equivalent (I hope!) but the former speaks much more to our way of considering the world than does the latter.

Combining different collections, a collection of 1 pebble and collection of another pebble, has the same number of total pebbles as a collection already containing 2 pebbles.

I agree with Potii to an extent.

It’s interesting to consider why the author has selected three duplication operations (1+1=2, 2+2=4, 3+3=6). Should we focus on the duplicative process? Commutativity? In any case I think it’s reasonable to consider first what addition is.

I doubt we learn the first ten numbers as {1, (1+1), (1+1+1),…}. I would rather hypothesize that a basic sequence, perhaps {1,…,10}, is imprinted in our minds during early cognitive development.* Assuming that we internalize this sequence as a ruler with notches in our minds, counting is the one-to-one matching of items in a set to a notch position on this metaphorical ruler.**

Addition is a related process: move along the metaphorical ruler until you reach the first addend’s position, then make N more jumps along the metaphorical ruler, where N is the second addend. The mental leap is to match the result of these jumps (the RHS) to an addition expression (the LHS). This cognitive act of pattern matching justifies the meaning of equality. This is a crucial step – with it the protagonist can understand that his and Tweel’s intuitive understandings of addition are the same (meta-equality!).

For example:

One plus one

= “jump once then jump once”

= two

Three plus three

= jump thrice, then jump thrice again

= six

Thinking back to my prep year, we were first shown addition with marbles. We had to match the additive operation (the action of adding one set of marbles to another) to its result (the final quantity of the collection). Only after this link had been made could we translate this into the abstract written form of the equation. It’s a shame we don’t play with marbles anymore.

*This is an unsubstantiated guess. My linguist friends point out instances of language groups in the Amazon and Australia that do not verbally express numerals. The Pirahã for example can perceive sets of size greater than one, but cannot specify the set’s size [1]. In any rate, my hypothesis of a numerical ruler can be adjusted to accommodate this phenomenon.

As an aside, this ruler implicitly starts at zero. Everyone is aware of the concept of nothingness, but an explicit numeral zero is a conceptual leap that not all human civilizations made. Notably the North African and Iberian Muslim civilizations played a crucial role in introducing it to Western Europe). The Wikipedia article summarizes the history of its independent discoveries and diffusion [2]. This is an innovative cultural meme that requires a conscious awareness of nothingness for its definition. I do wonder, would Tweel understand and agree on 1 – 1 = 0?

**This is briefly described in the Wikipedia article on Counting under the subsection ‘Education and Development’, and more formally presented in the subsection ‘Counting in Mathematics’ [3].

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Further reading:

[1] http://www.spiegel.de/international/spiegel/brazil-s-piraha-tribe-living-without-numbers-or-time-a-414291.html

[2] https://en.wikipedia.org/wiki/0#History

[3] https://en.wikipedia.org/wiki/Counting