2+2=4 is Not a Necessary Truth Kevin Karn (2015) The statement "2+2=4" (or the multiplication variant "2x2=4") is often cited as a necessary, a priori truth. The cliché is so familiar that even great novelists have used it as a motif. For instance, the unnamed protagonist of Dostoyevsky's Notes from Underground regards 2x2=4 as an insult to his dignity as a human being: "But 2 × 2 = 4 is nevertheless an intolerable thing. Twice two is four is, in my opinion, nothing but impudence. 'Two and two make four' is like a cocky young devil standing across your path with arms akimbo and a defiant air. I agree that two and two make four is an excellent thing; but to give everything its due, two and two make five is also a very fine thing." Orwell takes the opposite tack, treating 2+2=4 as the last sanctuary of sanity in a totalitarian world of lies. In 1984, the hero Winston Smith keeps a secret diary in defiance of Big Brother. He writes: "Freedom is the freedom to say that two plus two make four. If that is granted, all else follows." But of course, Winston is eventually tortured by the state, and ends up cheerfully scribbling in a notebook: "TWO AND TWO MAKE FIVE." It's a distressing moment; 2+2=5 has become a symbol of total degradation and slavery. ___________ Literary allusions aside, this paper aims to answer the question: Is "2+2=4" a necessary truth? The answer will be: No. This will be shown constructively, by exhibiting a system of arithmetic in which "2+2=4" is false and "2+2=5" is true. For convenience, I will be calling this new system "System A", and the default system we all learned in school will be called the "ordinary system."