4th letter of the English alphabet
With all the fun we had meeting forgotten friends from middle school I thought of going to kindergarten. My memory is little misty, but my guess is that we all had to answer questions such as the one in the title to graduate. Foremost sightings first:
1, 2, 3…
and
A, B, C… Z
What we have in the case of numbers is the following scenario:
2 after 1
3 after 2
.
.
.
Our impression of reading aloud alphabet is not that different:
B after A
C after B
.
.
.
Z after Y
You don’t get to go home after getting to Z even if you yelled (in a good way) each and every one of the letters in between A and Z, and in order. After Z comes (maybe a candy, but definitely) A, and the cycle of recitation:
A -> B -> C -> … -> Z -> A
repeats if you went to a catholic school like the one I went to (Ok, St. Ann’s Convent was not that bad ;-) Sister Caroline told us that the first letter of the alphabet is A and then asked for the fourth, which gave me the impression that she is, as usual, doing the easy part and is asking me to do the hard part—a recurring theme of schooling, which I didn’t bring to her attention simply because I knew, by now, how she’s going to answer: you, naughty boy! Here’s what we are given:
f(1) = A
Here’s what we are asked to figure out:
f(4) = ?
What we are looking at is Sister Caroline walking with a stack of hats looking like the ones magicians pull rabbits from, and then she places the hats—one on top of each one of the letters lying (in a circular formation as if I need to be reminded of repeating after reaching Z) on the floor. By the time she was done I forgot which letter went under which hat. She went through the exercise so fast though she looked like she knew what she was doing. With her characteristic ‘Listen’ she, then went back and, lifted the hat labeled '1' to reveal the letter 'A' underneath. Fearing what might follow, I began to wonder:
What’s under the hat labeled ‘2’?
Here’s what I remember:
Sister C enters with hats.
One hat on top of each one of the letters lying in a circle.
Seeing the numbers pasted on the hats forming a circle:
1 -> 2 -> 3 -> … -> 26 -> 1
f(1) = A
Sister C got plan.
f(2) = ?
Assuming that the circular structure α:
1 -> 2 -> … -> 26 -> 1
of numbers on top of hats is preserved in the letters under the hats i.e. in the circular formation β:
A -> B -> … Z -> A
we find that
f(2) = f(number after 1) = letter after f(1) = letter after A = B
f(2) = f(α(1)) = fα(1) = βf(1) = β(f(1)) = β(A) = B
The answer-key we used to find the 2nd letter is
which is a statement of respecting structure. Now I hope you sympathize with me when I find kindergarten kids (telling me that B is the 2nd letter of the English alphabet) respectful. This is all the motivation I need to study:
Objectification of properties in dynamical systems
---Conceptual Mathematics, page 175
P.S. Now you know why I find myself at the receiving end: grow up, get a life ;-)