Oh, God, why is two plus two equal to four?
—Alexander Pope
Luo Ji: I hear a new bookstore has opened on the 5th floor?
Bei Lun: Yes.
Luo: Then I must go have a look.
Bei: Fine, we’re on the 2nd floor now. Go up three more escalators and we’ll be there……
Luo: Why?
Bei: ……Why what?
Luo: I’m asking why going up three more escalators gets us there?
Bei: ……Because we’re on the 2nd floor now.
Luo: I’m asking why going up three floors from the 2nd floor makes the 5th floor?
Bei: This…… because…… because 2+3=5! Come on, let’s go, hurry up—I really regret indulging you just now until you were stuffed……
Luo: Really? I don’t care anymore; I’ll just follow you.
Bei: One floor…… two floors…… huh!
Luo: Look, sure enough it’s a big bookstore, but there’s nothing good here. Let’s go back.
Bei: Uh…… I remember we just came up from the 3rd floor…… this is the 5th floor……
Luo: Mm, simple. That means what you said before was wrong. Perhaps 2+2=5 is the right answer.
Bei: Ah, I know! This is just a matter of taboo, because the pronunciation of 4 is unlucky, so the floor above the 3rd floor is the 5th floor.
Luo: So in any case it’s 2+2=5, right?
Bei: No, no!
Luo: Then could it be 3+1=5?
Bei: I already said it’s because there’s no 4th floor here out of avoidance of an unlucky number! The numbers on the floors are just labels! Even donkeys know 2+2=4!
Luo: Why exactly is 2+2=4? Just because that’s the donkeys’ opinion?
Bei: Good heavens…… because all psychologically normal people think so!
Luo: I’ve heard that at one time, nearly all psychologically normal people thought the earth was motionless……
Bei: That was certainly wrong! Because science is advancing!
Luo: So 2+2……
Bei: That’s not the same!!
Luo: Don’t get excited, I’m listening.
Bei: Whew…… It’s true that some theories are relative truths, but there are also some things that are so obvious……
Luo: The development of science often overturns those obvious common-sense notions, doesn’t it?
Bei: I know, but 2+2=4 is mathematics……
Luo: But mathematics also progresses. I’ve heard it goes from positive integers to rational numbers, irrational numbers, negative numbers, complex numbers……
Bei: Fine, today I’ll stay with you to the end. Uh, first of all, 1, 2, 3, 4 are “natural numbers”……
Luo: Mm, you’ve finally started to get serious. What is a “natural number”?
Bei: Natural numbers are 1, 2, 3, 4, ……
Luo: Ahem……
Bei: Fine, fine, I know I can’t get away with that: natural numbers are the numbers produced when people count; numbers used to indicate the quantity of objects are called natural numbers,
Luo: Count?
Bei: For example, here there is 1 book, 2 books,……
Luo: Mm, this one is Zhenbo Chen’s Logic and Philosophy, pretty good; this one is Studies in Paradox, and after flipping through a few pages I got dizzy……
Bei: I just picked up two books at random, nothing special by it.
Luo: What I mean is: this one and that one are different things, so how can you lump them together?
Bei:……
Luo: For instance, here is a left-hand glove, here is a right-hand glove, and then here is a pair of gloves……
Bei: Ha! Isn’t a pair just two gloves?
Luo: Is it? Then one cup of water, one spoonful of sugar……
Bei: You’re going to say one cup of sugar water, right? These word games don’t count. When we say 1+1, we’re not saying one something plus another something. Numbers are abstract, transcending concrete things……
Luo: Sorry, could you put that in plain language? You know my Chinese is poor; I don’t really understand the word “abstract.”
Bei: I mean I extract the common feature from things! Mathematics is the highest abstraction from things. When primitive humans began to realize the shared property among four fingers, four beasts, four days and nights, four people, and so on, mathematics was born!
Luo: That sounds really exciting, but you still haven’t answered how one can count different things together.
Bei: When we count some things together, we don’t regard them as different things; we only look at their common features.
Luo: So you regard these two books as having no difference?
Bei: It’s ignoring their differences.
Luo: But if things have no difference, aren’t they the same thing? Here is one you, and here is another you, and added together it’s still one you……
Bei: I only said to ignore the differences! Things that are completely identical, of course, cannot be added together……
Luo: Then does that mean that when counting, one must first deal with different things, but then regard them as the same?
Bei: More or less.
Luo: “More or less”? Then where exactly does it differ?
Bei: Well…… yes, that’s right, that’s how counting works.
Luo: But how do you identify the “common feature” you’re talking about?
Bei: For example, these two books are different, but they are both “books.”
Luo: What I don’t understand is precisely how the notion that “they are both books” is established?
Bei: Books…… something like this is a book: look, it has a cover, it has text, a stack of paper bound together…… that’s called one book.
Luo: I still don’t quite understand, but that’s intuitive enough…… Hey! Don’t tear the book apart!
Bei: As for addition, it means that we can first count two piles of objects separately, then add the abstracted quantities together, which is equivalent to directly counting the things in those two piles.
Luo: Sounds really nice. For example, here there are “○△ ○□”, and over there “◇◇ □△”; put them together……
Bei: That depends on how you count! It can be understood as 1+1, 2+2, 4+4……
Luo: Mm, it depends on how you count…… Look, if I count it like this, on this side ○△□—how many kinds of shapes are there again?
Bei: Three kinds!
Luo: Oh, mm, three kinds of shapes. On that side, ◇□△ is also three kinds, right?
Bei: Right.
Luo: 3+3=?
Bei: 6!
Luo: Fine, then……
Bei: Stop! I know your scheme now! You can’t add it like that!
Luo: I counted that “○△ ○□” contains three kinds of shapes, that’s right, isn’t it?
Bei: Of course…… but when you combine the two groups of symbols, you should count again!
Luo: Numbers are abstracted from concrete things, aren’t they?
Bei: Yes, I said that.
Luo: Then I abstracted “3” from “○△ ○□”, and also abstracted “3” from “◇◇ □△”, and I did it in exactly the same way. Now you say we still can’t perform addition. So what use is this abstract addition, exactly?
Bei: You’re quibbling. Calm down, let me think……
Luo: Real things are much more complicated than “○△ ○□”!
Bei: I’ve figured it out! Actually, I said it earlier: identical things cannot be added together; only things regarded as the same can be added. Here △ and □, by your method of counting, are the same, but when you merge them you count them twice……
Luo: You mean that in order to avoid counting the same thing twice, we first have to see how much overlap there is among all the things?
Bei: Yes, to avoid double-counting.
Luo: In other words, wouldn’t it be better to just count all the things directly?
Bei: That’s not it…… when we divide a pile of things into parts and count and sum them, we must avoid overlapping divisions in advance. The parts being added are independent of one another. For example, this book costs 2 yuan, that book costs 3 yuan; buying them together adds up to 5 yuan.
Luo: Oh, so addition is actually quite useful.
Bei: Of course!
Luo: Seeing it that way, I really need to figure out why 2+3=5, otherwise I’d be wasting money!
Bei: Do you really doubt that 2+3=5?
Luo: I also think 2+3=5, but who knows? Perhaps in fact 2+2=5, but there is a devil who always makes everyone have the wrong impression……
Bei: That’s Descartes’ sophistry…… never mind, even if there were a devil, it would still have to be logical!
Luo: Fine, let’s not talk about a devil. Aren’t our sensations limited by the structure of our body and brain? We can distinguish colors, but most animals can’t; though some insects can see ultraviolet light……
Translated from the Chinese original with AI assistance. The original text is authoritative.
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