Before we start, I'd like to thank listener D. Zemke, who posted another nice review on iTunes. Thanks D!
Now, on to today's topic. Recently I heard someone quote a clever metaphor in a casual conversation, "Life is when nature takes 2 and 2 to make 5." It's a nice statement of how living creatures are more than the sum of their parts. If you took all the chemical compounds in my body and dumped them on the ground in the right proportions, all you would get is a mess. Yet somehow I am here, and at least sentient enough to record math podcasts. I went online to try to find the source of this quotation, and was surprised to see the number of references to this seemingly silly nonsense equation, 2+2=5.
Most of us are probably familiar with the equation from George Orwell's classic novel 1984. As you probably recall, in the book, people are told that if the government says that 2+2=5, it is the duty of all citizens to believe it-- not just say it, but actually come to believe that it is true. Surprisingly, Orwell did not come up with this out of thin air: a real-life totalitarian government, the Soviet Union, actually did use 2+2=5 as part of its propaganda, in a poster with the title ""2+2=5: Arithmetic of a counter-plan plus the enthusiasm of the workers." It wasn't quite as blatantly absurd as in 1984, but the Soviet propaganda poster used it as a metaphor: supposedly a 5-year plan could be completed in 4 years, because the entuhsiasm of the workers provided an invisible additive factor. Sadly, most of this "enthusiasm" was mainly due to fear of being sent to the Gulag prison camps, which resulted in many managers doctoring the statistics to match the results that the government wanted-- on paper only. It's also reported that Nazi Herman Goering actually used this metaphor in real life, once saying "“If the Führer wants it, two and two makes five!”
The phrase 2+2=5 actually existed in the arts dating from the early 19th century. According to Wikipedia, the phrase was first coined in a letter from Lord Byron, where he wrote ""I know that two and two make four—& should be glad to prove it too if I could—though I must say if by any sort of process I could convert 2 & 2 into five it would give me much greater pleasure." He may have been making an indirect reference to Rene Descartes' Meditations, where the famous philosopher discussed whether equations such as 2+3=5 exist outside the human mind, and whether they can be doubted: "And further, as I sometimes think that others are in error respecting matters of which they believe themselves to possess a perfect knowledge, how do I know that I am not also deceived each time I add together two and three, or number the sides of a square, or form some judgment still more simple, if more simple indeed can be imagined?"
Later Victor Hugo used this concept in a critique of tthe mob rule that had led to Napolean, foreshadowing Orwell's later political metaphor: ""Now, get seven million five hundred thousand votes to declare that two and two make five, that the straight line is the longest road, that the whole is less than its part; get it declared by eight millions, by ten millions, by a hundred millions of votes, you will not have advanced a step." Russian authors Ivan Turgenev, Leo Tolstoy, and Fyodor Dostoyevsky also made use of this metaphor. Turgenev used it to symbolize divine intervention: "Whatever a man prays for, he prays for a miracle. Every prayer reduces itself to this: Great God, grant that twice two be not four." In the 20th century, there are many instances of authors following Orwell's lead and again using this metaphor for the struggle against totalitarianism, including Albert Camus and Ayn Rand.
An intriguing question is whether there are cases when it is actually valid to say that 2+2=5. A well-known mathematicians' joke is that "2+2=5, for particularly large values of 2." This may refer to issues with rounding: if you start, for example, with the obviously correct equation "2.4 + 2.4 = 4.8", and ask someone to round all the numbers to the nearest integer, you do indeed derive "2+2=5" from this true equation. It also might be a case of playing with the definitions of symbols: perhaps you can define the symbol that we normally write as "2" to actually be an algebraic variable representing the value 2.5. A trickier example is a "proof" that 2+2=5 that is circulating the web, where many lines of complex algebra are used. These many lines are artificially complex in order to misdirect you from one invalid step, where a term t is replaced with the square root of t squared. Remember that you can only do such a replacement if t is positive, a fact glossed over in the so-called proof. I won't bore you by trying to cite all the lines of equations in an audio podcast, but you can find them linked online in the show notes if you're curious.
An amusing spoof article online points out some real-life situations where 2 and 2 might really make 5. Ancient Incas used knotted ropes to track business transactions, and if you tie together two ropes that each have two knots, the resulting rope will have 5 knots, including the one used to tie them together. Another example is if you put 2 male and 2 female rabbits in a cage-- pretty soon you will see numbers way larger than 5. I'm pretty sure that most people who experience these situations in real life can make the distinction between the messiness of reality and the related arithmetic though.
But that last example brings us back around to the quote that started this whole thing. Ironically, my web searching did not succeed in uncovering the source of the clever comparison between life and making two plus two equal five. Most likely I didn't remember the phrasing exactly right, or else someone was just coining this on the fly and it didn't really come from a famous quote. If you have heard it before and know its origin, please send me an email!
And this has been your math mutation for today,