Last week, in Number Sense 043, Awkward Goat and Billy Goat Gruff thought they proved that clock arithmetic was commutative and associative. But there is trouble in paradise...
“I have some bad news and some good news,” said Awkward Goat. “Which do you want to hear first?”
“The bad news, of course!” replied Billy Goat Gruff. “We must be prepared! Is it an invasion of Sheep? Mounting Lions? Trolls?”
“Ahhh, no, none of those.”
“What, then?”
“Well,” began Awkward Goat, “you remember the Great Horned Owl that lives in the old oak tree by the parade ground?”
“You mean old Lost in Space?” asked Billy Goat Gruff.
“Flies Through Space,” said Awkward Goat. “Anyway, he was listening to our discussion last week, and says we made a mistake in proving our clock addition was associative.”
“You mean he says I made a mistake,” grumbled Billy Goat Gruff. “I thought up that proof! What's the good news?”
“Well, the mistake has a name, so we can avoid it in the future,” replied Awkward Goat.
“You have a strange idea of good news,” said Billy Goat Gruff. “Where did I mess up? I suppose avoiding future mistakes is a good thing after all.”
“We began with three colors, red, yellow and green. We had an operation, addition, which combines two of these colors to give us a third color.”
“Right, I remember that part.”
“Then we tried to prove addition was associative by adding three colors together.”
“Of course,” said Billy Goat Gruff. “If there are only two colors, you don't need association to figure out which to add first.”
“But our addition table only gives us sums for two numbers, not three,” said Awkward Goat. “And I don't think we made up a table for our color arithmetic.”
“Then let's do that first,” suggested Billy Goat Gruff. “I'll understand it better if I have something to look at.”
“We'll start like this,” said Awkward Goat. “And fill it in. Do we want an identity element?”
“That's the one that doesn't change anything, right?” asked Billy Goat Gruff.
“Correct,” said Awkward Goat. “Shall we use Yellow?”
“Somehow that doesn't look right,” said Billy Goat Gruff.
“What seems to be wrong?” asked Awkward Goat. “When we add Yellow to something, we get something with no changes. That's how identity elements work.”
“Yes, but, well,” said Billy Goat Gruff, “I can understand adding Yellow and Yellow to get Yellow, and I can sort of understand adding Yellow and Green to get a kind of Green, but adding Yellow to Red is Orange.”
“So this doesn't work with how you understand adding colors,” suggested Awkward Goat.
“Exactly!” agreed Billy Goat Gruff.
“All right,” said Awkward Goat. “I can't imagine any way of getting you to change your understanding of addition, so let's make it about some other operation.”
“Some other operation?” wondered Billy Goat Gruff. “Can we just make up some other operation?”
Of course we can,” answered Awkward Goat. “We are defining the operation with our table. So we can make it work however we want, and we can call it whatever we want. What do you want to call it?”
“But,” stammered Billy Goat Gruff, “but, but...”
“All right, then,” said Awkward Goat, “we'll call it Butt. Yellow butts Yellow gives Yellow, Yellow butts Green gives Green, and Yellow butts Red gives Red. Yellow must not butt them very hard, then, since they don't change. We can abbreviate it with a capital B”
“Now, do we want Butting to be commutative?” asked Awkward Goat.
“We get to decide that, too?”
“Sure,” said Awkward Goat. “We're making this operation up from scratch. If it's commutative, then Green butts Yellow will be Yellow, and Red butts Yellow will be Red, but if it's not commutative, then those results can be... something else.”
“We just changed addition to butting,” said Billy Goat Gruff. “Let's keep other changes to a minimum. Make it commutative.”
“All right, since Yellow butts Green is Green, Green butts Yellow is also Green. Same for Yellow butts Red. I suppose to keep changes to a minimum, we want this operation to be closed? Nothing in the table but Red, Yellow and Green?”
“Right.”
“So, what shall we make Green butts Green? We can choose Green, or Red, or Yellow.”
“Red”
“And how about Green butts Red?”
“Make that Yellow. I want to use all the colors in each row.”
“We can finish filling the table in, then, since butting is commutative, Red butts Green is also Yellow, and the last spot is filled in with Green to get all three colors in the last row.”
“Now, last week I thought I proved that Red plus Yellow plus Green was the same as Yellow plus Green plus Red.”
“Right,” said Awkward Goat. “Now, Flies Through Space told me that, since our operation only works with two things at a time, we should have put in parentheses to show what order we combine things. So, if butting is associative it would look like this.”
“We could put Yellow butts Green first, followed by butts Red” said Billy Goat Gruff, “so it would look like the addition picture.”
“We could,” said Awkward Goat, “but switching two terms is just commutativity, and we know that is a property already.”
“All right. Now, the first thing I did,” said Billy Goat Gruff, “was use commutativity to switch Red and Yellow.”
“And that's fine,” said Awkward Goat. “That's commutativity.”
“Then I used commutativity a second time to switch Red and Green,” said Billy Goat Gruff.
“But you can only do that if Red and Green are grouped together,” said Awkward Goat. “That's what Flies Through Space pointed out. Before you can use commutativity a second time, you have to assume you can make that second group.”
“And what's wrong with that?” asked Billy Goat Gruff. “It makes sense, doesn't it?”
“It makes sense,” replied Awkward Goat, “but only if associativity is true for butting. And that's what we are trying to prove. We don't get to use associativity to prove associativity. That's called 'Begging the Question' or petitio principii.”
“Petitio principii?”
“Flies Through Space knows Latin. It means 'assuming the initial point.' It's a logical fallacy.”
“How about that?” said Billy Goat Gruff, “I made a mistake with a Latin name.”
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