The material I recently posted as Metamorphosis, Identity, and Metaphysics in Dumbo and Fantasia started out as a piece on topology in Pink Elephants on Parade. And then it went somewhere else. In that context the topology material didn’t quite hold together. But it has some interest in itself, not so much for the topology but for Piaget’s remarks on children’s acquisition of intuitive topology and geometry. So I’m posting that material as a separate piece.
The mathematics of space comes in several varieties, of which ordinary Euclidean geometry is the most familiar. Topology is a more recent mathematics of space, one that is considerably more general than geometry. Robert Bruner at Wayne State observes that “the thing that distinguishes different kinds of geometry from each other ... is in the kinds of transformations that are allowed before you really consider something changed,” a view he attributes to Felix Kline, who is most popularly known for the Klein Bottle, which manages to be both inside and outside itself at the same time.
As Bruner explains, geometry is quite restrictive with respect to identity-conserving transformations while topology is quite liberal:
In ordinary Euclidean geometry, you can move things around and flip them over, but you can't stretch or bend them. This is called "congruence" in geometry class. Two things are congruent if you can lay one on top of the other in such a way that they exactly match.In projective geometry, invented during the Renaissance to understand perspective drawing, two things are considered the same if they are both views of the same object. For example, look at a plate on a table from directly above the table, and the plate looks round, like a circle. But walk away a few feet and look at it, and it looks much wider than long, like an ellipse, because of the angle you're at. The ellipse and circle are projectively equivalent.
And it wasn’t only Renaissance mathematicians who were interested in projective geometry. As Bruner’s remarks imply, artists were also interested in it, and used it in their painting. Animators, obviously, must be intuitive masters of projective geometry, for they have draw many carefully calibrated views of complex objects, such as people and animals, moving in space.
In topology, any continuous change which can be continuously undone is allowed. So a circle is the same as a triangle or a square, because you just `pull on' parts of the circle to make corners and then straighten the sides, to change a circle into a square. Then you just `smooth it out' to turn it back into a circle. These two processes are continuous in the sense that during each of them, nearby points at the start are still nearby at the end.
This is the crucial point, all points have the same neighbors when a circle is transformed into a square.
The circle isn't the same as a figure 8, because although you can squash the middle of a circle together to make it into a figure 8 continuously, when you try to undo it, you have to break the connection in the middle and this is discontinuous: points that are all near the center of the eight end up split into two batches, on opposite sides of the circle, far apart.
Donuts and teacups are the same from a topological point of view, but not a geometrical point of view. The hole in the teacup handle corresponds to the hole through the middle of the donut. Teapots are more complex. There’s the hole through the handle and the hole through the spout. Put another hole through your donut—anywhere (think about it, visualize it)—and it becomes topologically equivalent to a teapot.
[You can find illustrations and discussion of children’s spatial and geometric thinking at this site.]
In a slender volume, Genetic Epistemology (Colulmbia UP 1970) has some remarks on topology, noting that, in children’s active understanding “the first intuitions are topological. The first operations, too, are those of dividing space, of ordering in space, which are more similar to topological operations than to Euclidean or metric ones” (p. 31). That is to say, the developmental order is opposite from that in the history of mathematics, where Euclidean geometry came long before topology.
Of course, we are talking about very different things here. Mathematics involves explicit assertions involving language and mathematical symbols. Those things are taught more or less in the order in which they arose in the history of mathematics. One learns plane geometry in middle or high school. Topology is reserved for college if it is learned at all.
Piaget is talking about what children draw and what they point to. Here’s what he says (pp. 31-32):
Pre-operational children can of course distinguish various Euclidean shapes—circles from rectangles, from triangles, etc.—as Binet has shown. They can do this at about 4 years of age, according to his norms. But let us look at what they do before this age. If we show them a circle and ask them to copy it in their own drawing, they will draw a more or less circular closed form. Once again, if we show them a triangle, they will draw just about the same thing. Their drawings of these shapes are virtually indistinguishable. But if, on the other hand, we ask them to draw a cross, to copy a cross, they will draw something totally different from their drawings of the closed figures. They will draw an open figure, two lines that more or less come to a cross or touch each other. In general, then, in these drawings we see that the children have not maintained the Euclidean distinctions in terms of different Euclidean shapes, but that they have maintained the topological distinctions. Closed shapes have been drawn as closed, an open shapes have been drawn as open.
I note further that Piaget’s discussion is not limited to geometry and topology. Following the Bourbaki mathematicians he also discusses the algebraic group (which Lévi-Strauss employed as a rhetorical device in Mythologies) and “the order structure. This structure applies to relationships, whereas the algebraic structure applies essentially to classes and numbers” (Genetic Epistemology, p. 25).
Finally, I want to quote a passage which has to do with identity, which IS relevant to the problem of metamorphosis (pp. 56-57):
The last experiment that I should like to mention is carried out by Boyat on plant growth. He started by experimenting with the growth of a bean plant, but that took too long, so instead he uses a chemical in a solution, which grows in a few minutes into an arborescent shape looking something loike a seawee. Periodically, as a child watches this plant grow, he is aked to draw it, and then with his drawings as reminders he is asked whether, at the various points in its growth, it si still the same plant. We refer to the plant by the same term the child uses for it—a plant, seaweed, macaroni—whatever he happens to use. Then we ask him to draw himself when he was a baby, and himself a little bigger, and still a little bigger, and as he is now. And we ask the same questions as to whether all these drawings are drawings of the same person, whether the person is always he. At a relatively young age, a child will deny that the same plant is represented in his various drawings. He will say that this is a little plant, and that is a big plant—it is not the same plant. In referring to drawings of himself, however, he will be likely to say that all show the same person. Then, if we go back to the drawings of the plant, some children will be influenced by their thoughts about their drawings of themselves and will say now that they realize that it is the same plant in all the drawings, but others will continue to deny this, maintaining that the plant has changed too much, that it is a different plant now