Key Math Ideas Not Taught In School – Transactions
Things take time. As the saying goes, time is God’s way to keep everything from happening all at once. There is a sequence to solving a problem, and even the most rote-based solutions tend to involve multiple steps. You may know how much is by memory, without having to think about it. For most of us, arithmetic like that involves multiple steps. With multiple steps comes the challenge of keeping track of things.
I introduced one specific aspect of this general notion of keeping track, in the posts on invariants. A related, but distinct, concept is the notion of a transaction. Though the word transaction has a number of informal meanings, I’m using the word in a fairly precise meaning, which I will lay out. As such, I’m using it consistently with the way it is used in the field of data bases, though my first example will be the process of buying a house, where it is more commonly referred to as escrow, at least in the USA.
Typically, when I buy a house, the act involves many parties. There is not only the buyer and the seller, and their agents, but also mortgage lenders, the title company, and government. When the sale goes through, the seller, the seller’s agent, and the various lien holders on the property (this would include the holder of the seller’s mortgage for the remaining principal, and the state for some pro-rated amount of real estate taxes, and perhaps others) get their shares of the proceeds. Government has records of who owns what parcel of land, and who has liens (claims) against it, and these records are changed when the sales goes through. The title company’s major role is in orchestrating the whole affair, and then issuing something called title insurance to the new owner, guaranteeing that the new owner does in fact hold title (i.e. is the real owner) of the parcel of real estate. The proceeds that the title company disburses to the various claim holders typically come from funds released by the mortgage company of the buyer.
As you can readily imagine, there is a challenge here of keeping track. But there is more involved than keeping track of all the funds. There is a process with a duration, and there is a critical moment in time, with a ‘before’ period and an ‘after’ period. What characterizes the ‘before’ period is that everything that’s done can still be rolled back in an orderly fashion. Though the parties may have deposited funds and signed papers ‘in escrow’, they do so knowing that it is ‘safe’ to do so, knowing that they will get it all back if something goes wrong along the way, e.g. when one of the parties backs out. At the critical moment, if all parties have not delivered their necessary part at that point, then the whole process will be rolled back. But if at that moment all parties have done what is needed to proceed, then the sale will go through. What this means is that at this crucial moment the whole transaction is committed, there is no longer any concern for being able to roll anything back, we’re now firmly in the process of rolling forward. In the ‘after’ period, every player can rely on the sale having succeeded, and that both the legal and financial implications will proceed accordingly.
If we step back a bit from the specifics of the real estate sale, we can see that there are multiple pieces that are being fashioned into a whole. This whole is the real estate sale, which either happens or doesn’t happen: it is committed or it is rolled back. The whole requires all the pieces, but the pieces don’t have to worry about being stuck if other pieces back out. E.g. when the mortgage lender backs out, the buyer doesn’t get his or her house, but won’t be left liable to the seller for hundreds of thousands of dollars. And the seller knows that he or she won’t end up in the situation where title to the house is lost but without proceeds being distributed. There is an integrity to the transaction: it is either wholly committed, or wholly rolled back.
When you get $60 cash out of the ATM, you expect cash in your hands, and your bank balance to go down in the amount of $60. You might be disappointed if a particular ATM indicated it was out of order, and you couldn’t get your cash. But you would consider it completely unacceptable if the ATM first reduced your bank balance by $60, and then flashed an out of order message on the screen. If cash doesn’t come out, you expect your bank balance to be unaltered or to be restored to what it was before. Conversely, the bank wants to make sure it doesn’t hand you cash without being able to deduct the amount from your balance.
When you walk down the aisle with your prospective spouse, and you have said your “I do!”, you fully expect the other person to also say “I do!” However, it is still possible for the other person to back out. Without an “I do!” from the other person, there is no marriage. The officiating minister or judge will not say “I now pronounce you husband and wife” or “I now pronounce you married.” If this happens, you are likely disappointed and upset. Yet you would consider it completely unacceptable if you were considered to be married while the other person is not: e.g. if you were arrested for bigamy when marrying somebody else years later. Getting married is organized as a transaction, it is either committed or rolled back in its (legal) entirety. The moment of commitment is clear: it is when the officiant says “I now pronounce you …”. If you want to back out after that moment, it requires a divorce proceeding – something else entirely.
Once you start looking for them, you see examples of transactions in lots of places in life. Some are simpler than others. As consumers, we are involved in many of them, but the burden of worrying about the ability of roll back is with the other party. When we buy an airplane ticket, there is a period in which we haven’t committed yet. During this period, other consumers may be looking at buying a ticket for the same flight. We leave it up to the airline to keep track of the number of seats on the flight that are still open. If it sells too many seats on the flight, we consider that the airline’s problem.
Having introduced the notion of transactions, it behooves me to connect it more clearly to mathematical operations and mathematical ideas. But this can wait till a future post.