Friday, August 10, 2007

abstract algebra sometimes drives me nuts

The problem with learning new abstractions is that, from then on, the abstractions have the infuriating tendency to crop up everywhere. All someone must do is take a concrete idea, throw away some information, and recognize what's left. Applied geometry has some trivial examples. To compute a geometrical property of a ball, throw out such facts as: what color it is, the store it came from, its texture. Recognize the shape that's left, a sphere, and apply the geometrical formulas for spheres. There's nothing exotic about this process.

However, because shapes in the real world are so complex, geometrical abstractions don't break into my thoughts too often. Abstract algebra is worse, perhaps because its abstractions have minimal assumptions or requirements: sets, mappings, operations, groups, rings. Mathematicians and scientific theorists explicitly apply these concepts all the time, if only for classification.

But I don't want to keep thinking along those lines in everyday life. Specifically, I was thinking about changing the days of the week I water a plant, so the same number of days passes in-between. Right now I water on Monday and Thursday, leaving one gap of two days and another gap of three days. (Yes, I realize this problem is most likely not worth the mental expenditure I'm describing--but what are hobbies for?). Mentally starting at Wednesday, I counted in three day increments until I reached Wednesday again...

...and after I was done I went on a long tangent looking up information about the cyclic group Z7, merely one in this list at wikipedia, but also an important group because it is a simple group. By associating each day of the week with an integer in the range 0-6 (like 0 := Sunday and then assigning progressively greater numbers in chronological order), and setting the group "multiplication" operation to be plain addition followed by modulo 7, the days of the week match that group perfectly. Although people might be a bit bewildered if you mention multiplying Monday by Friday (or Friday by Monday, abelian, baby!) to get Saturday [(1 + 5) modulo 7 = 6].

This group has no nontrivial subgroups (the trivial subgroup is just 0 alone, which is useless because I must assume that time is passing and I must water the plant more than once!). The lack of subgroups implies that no matter what interval of days in the range 1-6 I try, I'll end up having to include every day of the week in my watering schedule!

I mentioned before that I tried 3, which is Wednesday according to the mapping (isomorphism, whatever). Three days past Wednesday is Saturday [(3 + 3) modulo 7 = 6]. Three days past Saturday is Tuesday [(6 + 3) modulo 7 = 2]. Three days past Tuesday is Friday [(2 + 3) modulo 7 = 5]. Three days past Friday is Monday [(5 + 3) modulo 7 = 1]. Three days past Monday is Thursday [(1 +3) modulo 7 = 4]. Three days past Thursday is Sunday [(4 + 3) modulo 7 = 0]. Finally, three days past Sunday is Wednesday [(0 + 3) modulo 7 = 3]. The cycle is complete, but all I have is the same group I started with--the entire week!

I'm not sure what's worse: that all this pondering about abstract algebra didn't lead to any useful insights, or that I got caught up in doing it. I don't know how I manage to get anything done. Ghaah, I can be so nerdy even I can't stand it...