Addendum to Too Many Thank Yous
Since writing the essay on the proliferation of the salutation Thank you, it occurs to me that a mathematical basis for using the salutatory couplet Thank you/You're welcome, can be given.' I admit much of what follows is not meant to be interpreted in a serious vein, but it gives me a chance to have a little fun with the idea.
If we consider the exchange of Thank you/You're welcome to be a simple arithmetic expression that sums to 0, we have a solid basis for why these phrases should be used as they normally are.
Let Thank you = 1 and let You're welcome = -1.' If this is true then, the exchange Thank you/You're welcome can be rewritten as 1+-1 = 0.' This expression can be interpreted to mean no further salutatory statement is needed from conversing parties.' Now, let's consider when two (or more) persons engaging in salutatory exchanges say Thank you/Thank you.' This amounts to 1+1 = 2. This means they have double thanked each other, and to balance the expression, two You're welcome's are needed. Of course, this is only true if we assume that salutatory exchange is a group under the operation of addition. If we consider it under the operation of multiplication, a different outcome results. Let's look at that.
Here we have Thank you/You're welcome as 1*0 = 0 or -1*0= 0. Thank you = 1 and You're welcome = 0. I know that the 0 seems to imply that one party contributes nothing but that doesn't have to be the interpretation. And 1 or -1 doesn't matter here since the output is 0 in either case. What happens if the parties say Thank you/Thank you? Oh, I'm so glad you asked! We have -1*-1 = 1 or 1*1= 1. In either case we have one too many thank yous. And yes, you guessed it, somebody has to say to one You're welcome to balance the exchange. It wouldn't matter which side of the exchange did it either. Because a You're welcome approached from right or left would balance the exchange. So, the next time somebody thanks you for thanking them, add a you're welcome and tell them politely, please don't thank me again. And you'll know in a mathematical sense why you're saying that.
I know this is something anybody reading this would chuckle over, but I just had to take it to this extent. Hmmmm, I wonder how we could capture a complex number in all of this.....oh well..
P.S. It may not be surprising to find that two conversing parties can be mutually thankful to each other. While these same two parties would not be able to be welcomeful to each other. Thus, we can have Thank you/Thank you, but never You're welcome/You're Welcome. And still, to be thankful is at best redundant grammatically, and at worst oh well I won't say.