Visiting the Infinite Pig Farm
(** Note: This article is an old piece from several years ago, an imaginative visualization by the author of the mathematical concept of dual space. Since the next article involves the concept of dual space, it is published here (however, it has no causal relationship with other articles, and this article has no logic whatsoever - just read it for fun. Only at the end are some correspondences between concepts given) **)
(Mr. Tengse was invited to visit the Infinite Pig Farm. The staff member is responsible for introducing him to the management model here.)
Staff member: Welcome to the Infinite Pig Farm. This is an infinitely large pig farm where we raise countless pigs, have countless feeders, and our management model is very special. Each feeder numbers the pigs for easy identification, but each feeder has a strong personality, and their pig numbering systems are all completely different.
Mr. Tengse: Won’t your lack of unified numbering lead to management chaos?
Staff member: No, the personality diversity of feeders in the Infinite Pig Farm is also infinite. We must fully respect the feeders’ personalities, and we have our own solutions for this.
Mr. Tengse: Do you number the feeders?
Staff member: Actually, the pigs have already done something similar. Whenever a feeder comes to feed the pigs, the pig hears the feeder calling its number. Different feeders generally give it different numbers, so the pig simply uses these numbers that were originally used to call it to number the feeders. Feeders give different numbers to different pigs, so different pigs also number the same feeder differently. And numbering feeders is exactly what one of these infinite pigs does every day.
Mr. Tengse: I don’t want to do what a pig does. I think I understand - you mean you don’t number the feeders, but let the feeders and pigs identify each other, right?
Staff member: Right. But that doesn’t mean the management staff has nothing to do. We still have many numbers to assign. For example, we number all combinations of every two pigs.
Mr. Tengse: Number all combinations of every two pigs? I can’t imagine what use that could possibly have.
Staff member: No, doing this is actually very important for our management, because the purpose of this numbering is to assign a feeder to each pig.
Mr. Tengse: How can that be?
Staff member: Look, we assign the number b to the combination of Pig A and Pig B, number c to the combination of Pig A and Pig C, number d to the combination of Pig A and Pig D, and so on. As long as we have a given Pig A and another pig, this number is determined. So given a Pig A, there’s a corresponding rule for numbering other pigs. Remember, we have countless feeders, so you can always find a feeder whose pig numbering rule is the same as that rule for numbering other pigs. We then have this feeder feed Pig A.
Mr. Tengse: It sounds a bit confusing. Your feeders really have strong personalities. You management staff must have it tough.
Staff member: No, this is actually nothing, because we management staff also have very strong personalities.
Mr. Tengse: You don’t also have countless staff members, do you? If that’s true, it’s terrifying.
Staff member: You guessed it. Otherwise, how could a finite number of us manage an infinite pig farm? The staff’s personality is reflected in each staff member having their own scheme for numbering pig pairs. That is, each staff member assigns feeders to pigs differently. Each staff member’s scheme is executed in rotation, thus fully respecting each feeder’s personality.
Mr. Tengse: I still have one doubt. How do feeders distinguish infinite pigs? Can their brains hold information about infinite pigs?
Staff member: That’s a good question. To be honest, no matter how strong the feeders’ personalities are, we need to have standards, otherwise the infinite amount of information would have driven us crazy long ago. Whenever a pig is born, we attach an electronic display board to its ear with a string of numbers. Feeders all have unique algorithms to map the numbers on the board to their preferred numbering.
Mr. Tengse: These feeders really don’t mind the trouble!
Staff member: That’s not even the real trouble. The real trouble is that later the feeders objected to the numbers on the electronic displays, saying it showed favoritism to certain numbering systems and was disrespectful to other feeders. Their personalities are very strong - they almost went on strike. Finally, we reached an agreement: every morning we modify the computer program, using an algorithm to change the numbers displayed on each pig’s board to different numbers.
Mr. Tengse: With the numbers changing every day, how can these feeders still distinguish infinite pigs?
Staff member: We have our ways! The collective wisdom of infinite staff is infinite - no problem is too difficult for us. Every day we announce to the feeders the algorithm that the computer program uses to change the numbers on each pig’s board from the previous day to that day. When feeders see a pig’s board, they use this algorithm in reverse to deduce yesterday’s number, and the problem is solved. In other words, when the pig’s numbers change, the feeders’ algorithms change accordingly, and in the opposite direction. For example, if one day all pig numbers are doubled and then one is added, the feeders must subtract one and then divide by two to continue calculating the numbers they want in their original way.
Mr. Tengse: Running an infinite pig farm really isn’t easy! Do your staff members’ supervisors number the staff?
Staff member: No need. We number ourselves. Specifically, when I meet a staff member, I know he assigns feeders to pigs. Unless I encounter an oddball feeder who assigns the same feeder to several (or even infinite) pigs, generally pigs and feeders correspond one-to-one, meaning they’re also assigning pigs to feeders, i.e., numbering pairs of feeders. And I number pairs of pigs. You randomly find two feeders and two pigs, multiply that staff member’s number for the two feeders by my number for the two pigs to get a number, then find N such combinations of two feeders and two pigs. As long as these N combinations don’t have special relationships, adding these numbers together always gives a fixed number, regardless of which combinations you found. This is the number I give when I meet a staff member.
Mr. Tengse: What do you mean by these N combinations not having special relationships?
Staff member: As long as the feeders’ methods of numbering pigs in these N combinations are all independent and haven’t copied other staff members’ approaches, there’s no special relationship.
Mr. Tengse: Are you sure this always gives a fixed number? And you have to identify whether their pig numbering methods are independent each time - this operation needs to be done N times. How many times exactly is N?
Staff member: Well, this method is indeed hard to understand, but I’m sorry, these are trade secrets of our pig farm relating to the entire farm’s structure. I cannot reveal too many details. Our supervisors don’t number us because this is an internal staff matter. Supervisors are divided into infinite levels with hierarchical management, but each supervisor does something very similar: numbering any arrangement of M pigs and N feeders. Doing this is equivalent to assigning any combination of pigs, feeders, or supervisors to another combination of pigs, feeders, or supervisors. In other words, we can handle all management tasks.
Mr. Tengse: Alright, I was already confused by your management system, and since it involves your trade secrets, I won’t ask further. Let’s go visit the pig pens over there.
(Subsequently, the staff member led Mr. Tengse to visit some pig pens and employee dormitories. Mr. Tengse spent a pleasant day at the pig farm.)
Explanation: The story’s protagonist Tengse (Chinese pinyin for 藤瑟) is a homophone for “tensor.” Pigs correspond to vectors (column vectors, contravariant vectors), feeders correspond to dual vectors (row vectors, covariant vectors). Staff members correspond to inner products in vector spaces (metrics, or linear mappings from vectors to dual vectors). The electronic displays on pigs’ ears correspond to vector coordinates. Oddball feeders correspond to degenerate metrics, and N is the dimension of the vector space.