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.
Update
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?
Second Update
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.
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].
_____________
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
OK. I’ve been pondering this for quite a while (trying to not think about VCAA amongst other things) and have come to a few possible explanations:
1. 1+1=2 and 2=1+1 because both sides correspond to the same *idea* in the natural world but approached in different ways.
2. 1+1=2 because it is convenient to think this way and there is nothing in the natural world to contradict this, as far as we can tell.
3. 1+1=2 and 2+1=3 etc because there is a counting instinct in us as humans. The equalities 1+2=3 and 2=1+1 were then formalised later.
Am I getting close?
Yes. Forget about nature.
Going off Number 8’s first point:
2 and 1+1 relate to the same value, whether it be physical quantity (pebbles) or abstract (number line). 2 and 1+1 also have the same properties, in that 2 can be split into 1 and 1 and that 1 and 1 can be combined back into 2. Again this can be though of physically (splitting, combining and matching pebbles) and abstractly (a value added by 2 is the same as adding 1 then 1 again). So there is a symmetry between 2 and 1+1 as both hold true for their value and properties being the same. This means that Tweel and Jervis can agree on the equality 1+1=2; unlike a sound they both could make due to the sound meaning something else (value) or structurally being used differently in communication (properties).
I think this is what you were getting at previously about what the equality 1+1=2 means to us and thinking about it more mathematically. I hope so, as I’m struggling to break this down anymore simply.
Yep, that’s pretty much it. I’ll reply in some detail in the next day or so.
If 1+1=2 then 1+1+1+1=(1+1)+(1+1)=2+2.
Since we have defined 1+1 to be “two”, we can define 1+1+1 to be “three” and 1+1+1+1 to be “four”.
The rest is then a bit trivial?
That’s about it, just a little more care with the brackets. What do 1 + 1 + 1 and 1 + 1 + 1 + 1 mean? Why does 1 + 1 + 1 + 1 equal (1 + 1) + (1 + 1)?
OK. To me the associative law is quite common sense, but I do agree that it must be an agreed definition before it can be used in an argument.
Essentially what I am saying is that for FINITE collections of objects, if you want to count them, the order of the counting does not matter, and the result of the count gives you the size of the collection, hence, by agreed definitions, the two are equal.
This isn’t Methods. You don’t get to wave your hands.