For an introduction to this series of posts, see here.
Imagine Joe going up to the drive-through, to order for himself and his room mates. Their orders are all combined on a piece of paper. He reads it at the order window: “2 hamburgers, 1 fries, 1 coke, 1 cheeseburger, 1 fries, 3 hamburgers, 2 cokes.” At the pick up window, he pays and gets the stuff. Before driving away, he checks to make sure everything is there. He counts 3 cokes, 1 cheeseburger, 2 fries, 4 hamburgers. Does this match his order?
There is nothing particularly complicated about this scenario, and I trust you can quickly determine the answer. What’s interesting in this scenario is all the things we tend to take for granted when we think about this situation.
Let’s compare the original shopping list with Joe’s count after receiving the order:
Joe’s original Count
In the shopping list, hamburgers come first, whereas in Joe’s count, cokes come first. The shopping list has seven entries, and Joe’s count has only four. Yet we don’t fault Joe’s count on those grounds, but rather that one hamburger is missing. If it wasn’t for the one hamburger, we’d think of Joe’s count as equivalent to the shopping list. The fact that the shopping list has two separate entries with fries doesn’t seem to be critical; what’s critical to Joe is that the totals match. What does that mean? It means that there is the right number of hamburgers, as well as the right number of fries, as well as the right number of everything else. For Joe, extra fries wouldn’t necessarily compensate for missing hamburgers, and cheeseburgers wouldn’t work as substitutes for missing cokes. What Joe expects is that the count shows the right stuff, and in the right amount for each of the stuff.
After Joe gets an additional hamburger, he is ready to head home. His count is updated as follows:
Joe’s updated Count
If you are thinking that his updated count should really look like this:
then we are partially in agreement. Yes, your count shows the same totals as the original shopping list. Yes, your version of the count seems particularly clean and lean: there is only one line that pertains to hamburgers, and only one line that pertains to cokes, etc. In contrast, the following count seems to lose vital information:
Counting Gone Overboard
There is something that the original Shopping List, Joe’s updated Count and Your Count have in common, which is not shared by Joe’s Original Count nor the Counting Gone Overboard. What those three have in common is that they are really all talking about the same order. In contrast, Joe’s Original Count is missing a hamburger, and the Counting Gone Overboard doesn’t distinguish between hamburgers and cokes (it could refer to an order of 11 hamburgers just as easily).
To constitute a clear order, our piece of paper must have numbers on it, but numbers alone are insufficient: it must be clear for each number exactly what is being counted: it isn’t just “3”, but “3 cokes”. So, more precisely, the order isn’t just a bunch of numbers, but rather a bunch of entries, where each entry combines a number with the thing that is being counted. In this bunch of entries, the order of entries is not important, and we don’t particularly mind if there are multiple entries for hamburgers, or multiple entries for coke. What we do care about is that we end up with the right total amount of coke, and the right total amount of hamburgers, and the right total amount of fries, etc.
In this series of posts, I’m going to use the name vector for these kinds of shopping lists. What is important at this point is that you can look at the shopping list as a single thing, as a whole, and not merely as a collection of parts (entries). Just as the shopping lists, a vector can be written in a number of equivalent ways. As long as Joe comes home with the right stuff, it doesn’t particularly matter how the order was written.
In the next post in this series, I will play with this notion so you can see its usefulness, and introduce a useful notation to go with it.