Lambda the Ultimate

To clarify: I did not mean that the paper is old fashioned because of its date of publication. I said the paper sounds old-fashioned to me, and in fact meant to emphasize that this is remarkable since 1987 was not that long ago...

I said the paper sounds old-fashioned to me, and in fact meant to emphasize that this is remarkable since 1987 was not that long ago...

I think I know what it is you are detecting, Ehud.

It is the feel of the days when people still believed that AI was on the verge of success...

:-)

In fact I think it is something more subtle. I expected the AI vibe to be more worrying than it turned out to be. I guess my feeling has more to do with the broad perspective and intellectual courage that is often lost when disciplines mature...

Topological information can be carried by using algebra with a textural representation.

If I understand Larkin and Simon properly, they don't claim that diagrams necessarily carry any more information than sentential representations, just that they're a more convenient representation for searching, pattern matching, and drawing the resulting inferences — somewhat like the case of indexed tables in databases, which in theory are isomorphic to unindexed tables, but in practice may be much quicker.

In the representations we call diagrammatic, information is organized by location, and often much of the information needed to make an inference is present and explicit at a single location. In addition, cues to the next logical step in the problem may be present at an adjacent location. Therefore problem solving can proceed through a smooth traversal of the diagram, and may require very little search or computation of elements that had been implicit.

The authors do a good job of extolling the virtues of diagrams and of explaining the importance of going back and forth between the diagram and the text to get a good representation of the problem. They then try to separate the visual (topology) and the text (algebra) and try to show that visual is better. This is a categorical mistake. Topology and algebra are joined at the hip, they are points of view about the same thing. You can't have it both ways.

"Joined at the hip" doesn't make any difference to the human brain and its unconscious inference systems. We work with the wetware we have.

Also, I think it's helpful to remember that text itself is just another diagram - a more complex one with more edges and corners - until its parts are interpreted as algebraic ideals. For example, "h" is the height of the ball, but (*intuitive leap*) it's also just a real number, so I can do this and this and this with it...

That example illustrates why I don't think it's appropriate to say that diagrams are universally better than text. "Better" suggests a utility judgment. An AI professor of mine pounded into my head to always ask "For what?" when making those judgments.

I think a diagram is most useful at the beginning, to make something concrete. From there, it can serve as a representative of an "equivalence class" of similar problems, and from that you can make abstractions. But diagrams are terrible at abstraction (besides, say, lossy abstractions of real systems). Because of their concrete nature, you can only do it by leaving the varying part off to be put in later. As soon as you replace that "varying part" with a symbol - a box, an arrow, or whatever - it's a variable, you're halfway to algebra, and you might as well go all the way. For a lot of useful abstractions, you can retain the labeled diagram to leverage our brains' affinities for the concrete.