The stereotypical or canonical very simple arithmetic problem is “two plus two is four”, I think.

But why isn’t it “one plus one is two”? Is it that that’s just counting, and you have to get into twos to really have what feels like arithmetic?

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# Two plus two is four

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5 thoughts on “Two plus two is four”

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A mathematician blogs.

The stereotypical or canonical very simple arithmetic problem is “two plus two is four”, I think.

But why isn’t it “one plus one is two”? Is it that that’s just counting, and you have to get into twos to really have what feels like arithmetic?

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Yes, I think it’s the simplest arithmetic problem that doesn’t feel like a definition.

Nice question. Maybe it’s because children learn to count before they learn to add, so when first memorizing an addition table it’s natural to skip the “1+x” row and the “x+1” column as already known.

Well, in my country, we always say “one plus one is two”, so culture may matter

I was surprised by this observation, because – like in Khang Vo’s country – in Dutch we also often say “één en één is twee” (one plus one is two). So I did a search on Google and found that “one plus one is two” beats “two plus two is four”, but the difference isn’t big. (Moreover, “one and two is three” is more prevalent than “one and one is two”, which is in turn more prevalent than “two and two is four”.)

A quick Google NGram test does confirm your observation for English (unfortunately, there are no data for Dutch): http://books.google.com/ngrams/graph?content=one+and+one+is+two%2Ctwo+and+two+is+four%2Cone+plus+one+is+two%2Ctwo+plus+two+is+four&year_start=1800&year_end=2008&corpus=15&smoothing=1&share=

Here’s a long-winded explanation for why this situation is reasonable, though it is not causally the reason:

The most common version of the Peano axioms as a first-order axiomatization of the natural numbers has the constant zero (0), a successor function (S), and two binary operations addition and multiplication (+ and *).

The most direct translation of “one plus one is two” into this system is “S0 + S0 = SS0”, which requires one application of the inductive axiom for addition (“a + Sb = S(a+b)”) and one use of the base case for addition (“a + 0 = a”). The proof roughly reads “S0 + S0 = S(S0 + 0) = SS0”. This is in fact a slightly nontrivial theorem.

However, a common alternative axiomatization begins with the constant one (1) instead of zero. In this system, the most direct translation is “1 + 1 = S1”, which is a direct application of the base case for addition (“a + 1 = Sa”). In other words, in this system, two things come together: the most natural definition for “two” is “the successor of one”, and successor axiomatically means “adding one”.

In both systems, the statement “two plus two is four” translates into something that requires multiple applications of two axioms and is not a definition:

In the first, “SS0 + SS0 = S(SS0 + S0) = SS(SS0 + 0) = SSSS0”. In the second, “S1 + S1 = S(S1 + 1) = SSS1.”

The key point is that although some people would like to define “four” as “two plus two”, making the theorem a definition, “four” is fairly reasonably “the number after three”. In any case, the real content of the theorem is that two signifiers of a number, “the successor of three” and “two plus two”, coincide. The other theorem, “one plus one is two” doesn’t necessarily have content.