Gelernter's program, likewise, was not focussed on creativity as such. (It was not even intended as a model of human psychology.) Rather, it was an early exercise in automatic problem-solving, in the domain of Euclidean geometry. However, it is well known that the program was capable of generating a highly elegant proof (that the base-angles of an isosceles triangle are equal), whose H-creator was the fourth-century mathematician Pappus.
Or rather, it is widely believed that Gelernter's program could do this. The ambiguity, not to say the mistake, arises because the program's proof is indeed the same as Pappus' proof, when both are written down on paper in the style of a geometry text-book. But the (creative) mental processes by which Pappus did this, and by which the modern geometer is able to appreciate the proof, were very different from those in Gelernter's program -- which were not creative at all.
Consider (or draw) an isosceles triangle ABC, with A at the apex. You are required to prove that the base-angles are equal. The usual method of proving this, which the program was expected to employ, is to construct a line bisecting angle BAC, running from A to D (a point on the baseline, BC). Then, the proof goes as follows:
Consider triangles ABD and ACD.
AB = AC (given)
AD = DA (common)
Angle BAD = angle DAC (by construction)
Therefore the two triangles are congruent (two sides and included angle equal)
Therefore angle ABD = angle ACD.
Q.E.D
By contrast, the Gelernter proof involved no construction, and went as follows:
Consider triangles ABC and ACB.
Angle BAC = angle CAB (common)
AB = AC (given)
AC = AB (given)
Therefore the two triangles are congruent (two sides and included angle equal)
Therefore angle ABC = angle ACB.
Q.E.D.
And, written down on paper, this is the outward form of Pappus' proof, too.
The point, here, is that Pappus' own notes (as well as the reader's geometrical intuitions) show that in order to produce or understand this proof, a human being considers one and the same triangle rotated (as Pappus put it, lifted up and replaced in the trace left behind by itself). There were thus two creative aspects of this proof. First, when "congruence" is in question, the geometer normally thinks of two entirely separate triangles (or, sometimes, two distinct triangles having one side in common). Second, Euclidean geometry deals only with points, lines, and planes -- so one would expect any proof to be restricted to two spatial dimensions. But Pappus (and you, when you thought about this proof) imagined lifting and rotating the triangle in the third dimension. He was, if you like, cheating. However, to transform a rule (an aspect of some conceptual space) is to change it: in effect, to cheat. In that sense, transformational creativity always involves cheating.
Gelernter's geometry-program did not cheat -- not merely because it was too rigid to cheat in any way, but also because it could not have cheated in this way.
It knew nothing of the third dimension. Indeed, it had no visual, analogical, representation of triangles at all. It represented a triangle not as a two-dimensional spatial form, but as a list of three letters (e.g. ABC) naming points in an abstract coordinate space. Similarly, it represented an angle as a list of three letters naming the vertex and one of the points on each of the two rays. Being unable to inspect triangles visually, it even had to prove that every different letter-name for what we can see to be the same angle was equivalent. So it had to prove (for instance) that angle XYZ is the same as angle ZYX, and angle BAC the same as angle CAB. Consequently, this program was incapable not only of coming up with Pappus' proof in the way he did, but even of representing such a proof -- or of appreciating its elegance and originality. Its mental maps simply did not allow for the lifting and replacement of triangles in space (and it had no heuristics enabling it to transform those maps).
How did it come up with its pseudo-Pappus proof, then? Treating the "ABC's" as (spatially uninterpreted) abstract vectors, it did a massive brute-search to find the proof. Since this brute search succeeded, it did not bother to construct any extra lines.
This example shows how careful one must be in ascribing creativity to a person, and in answering the second Lovelace question about a program. We have to consider not only the resulting idea, but also the mental processes which gave rise to it. Brute force search is even less creative than associative (improbabilist) thinking, and problem-dimensions which can be mapped by some systems may not be representable by others. (Analogously, a three-year old not showing flexible imagination in drawing a funny man: rather, she is showing incompetence in drawing an ordinary man.)
It should not be assumed from the example of Pappus (or Kekule) that visual imagery is always useful in mapping and transforming one's ideas. An example is given of a problem for which a visual representation is almost always constructed, but which hinders solution. Where mental maps are concerned, visual maps are not always best.
24.feb.1999
Glosario de Carlos von der Becke