Monday, March 10, 2014

Mastery Against Generality

LtU pointed me to Design Principles Behind Smalltalk. I'll quote the first design principle named, because it illustrates what I think is a mistake in reasoning common to those of a mathematical bent:

Personal Mastery: If a system is to serve the creative spirit, it must be entirely comprehensible to a single individual.

The point here is that the human potential manifests itself in individuals. To realize this potential, we must provide a medium that can be mastered by a single individual. Any barrier that exists between the user and some part of the system will eventually be a barrier to creative expression. Any part of the system that cannot be changed or that is not sufficiently general is a likely source of impediment. If one part of the system works differently from all the rest, that part will require additional effort to control. Such an added burden may detract from the final result and will inhibit future endeavors in that area. We can thus infer a general principle of design:

Good Design: A system should be built with a minimum set of unchangeable parts; those parts should be as general as possible; and all parts of the system should be held in a uniform framework.

I strongly sympathize with the point of view outlined here. If one can master simple, general principles, then that reduces the burden for understanding some set of more specific ideas, and can potentially greatly increase the number and scope of the ideas one can understand and use at any given time -- it can improve one's intellect.

That said, it is considerably more difficult to impart understanding of general ideas than of specific ones. If this isn't immediately obvious to you, consider the order in which you learned some mathematical concepts. Take the following problems:

  1. If joey has three apples, and gives two away, how many does he now have?
  2. Solve for x: x = 3 - 2.
  3. Prove that the addition operation under the set of integers modulo some constant forms a group.
  4. Give an example of a non-Abelian group.
  5. What the hell is a left-adjoint functor?

I think you can be expected to gain mastery of each of these problems in the same order in which the problems are listed. Each problem is more abstract than the previous, and each successive problem is, in a sense, simpler and more general than the previous. But each successive problem is also, to my mind, more difficult than the previous: We needed to understand the more specific ideas before we could be expected to generalize. The mechanism for "good design" quoted above can probably be considered to be in some tension with the stated goal of maximizing "personal mastery" of the system.

The implications of this tension are, I think, important. In particular, under the assumption that a more abstract understanding of a problem domain can make the problem more tractable, it's usually in any given author's interest to move "up" the "abstraction ladder", in order to better solve a problem herself. To the extent that this means she happens upon a good solution faster than others, this is all to the good. But to the extent that this means she happens upon much different solutions than others would, then however much more "elegant" her own solution is will be weighed against the cost in comprehension for others she works with.

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