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Fortunately, later research provided very interesting support for Kant, and showed that incongruency is very probably caused by outer relations to space. With this help, Kant’s argument has been rehabilitated and now stands as one of the most provocative arguments for absolute space. The central point is simple. The letters b and d are incongruent only when confined to the surface of this page. No horizontal sliding will allow one into the place occupied by the other. But if we are allowed to lift the letters off the page and rotate them around, it is easy to show they are congruent because they exactly fit into the same place, and therefore have the same size and shape. Thus two objects that are incongruent in two dimensions are in fact congruent in three dimensions. Whether or not objects are congruent depends on the dimensions of the space they inhabit.

The same is true of left and right hands. In a strange space that had four spatial dimensions in addition to time, a left hand could be “rotated” into a right hand. A left hand could thus be fitted into a right-hand glove simply by flipping it around.

Another intriguing example is provided by a long, thin, rectangular plastic strip whose end is twisted halfway around (by 180 degrees) and smoothly glued to its other end. This loop with a twist is called a Möbius strip after its inventor, the mathematician A. F. Möbius. Surprisingly, a letter b that slides around this strip and makes a complete journey around the loop will return as a letter d (when viewed from the same direction, as if the strip were transparent and the letter were in the surface). That is, the letters b and d are incongruent in an ordinary, flat two-dimensional space, but not in a two-dimensional space with a twist! The incongruency of the letters depends on the overall shape of the space!


Suppose, likewise, that the entire universe had some sort of peculiar twist in it like the Möbius strip. Astronauts travelling in one direction would then find themselves back at their starting point. In this case, a left-hand glove could be converted into a right-hand glove just by sending it along with the astronauts through the twist. In fact, the astronauts too would return mirror-reversed: their hearts would be on their right sides!

Mathematicians say that a space with a twist in it has a different topology from ordinary, flat space. The word “topology” just means the study of place, and is the name of an important branch of mathematics today. The topology of a space is the way its points are connected to each other, and this stays the same if the distances between points are shrunk or expanded. Analogously, a balloon’s shape and size change as it is blown up, but its topology doesn’t change: the bonds between molecules stretch but do not break.  

The fact that incongruence depends on dimensionality and topology very strongly suggests that Kant’s strong assumption (K) was correct. The letters b and d are incongruent because of outer relations, but not outer relations to other objects. Since altering the space affects whether or not the letters are incongruent, their shape must depend on the surrounding space, and not on the objects contained in it. Research continues on this subject, but many philosophers think Kant’s argument is good evidence for some form of spatial structure over and above the bodies they contain. 

After Kant’s investigations of the peculiarities of incongruent counterparts, they played an extraordinary role in chemistry and physics. Two molecules that are incongruent counterparts of each other are called “isomers” in chemistry (from the Greek: “iso” is “same” and “mer” is “parts”). Many medicines and industrial chemicals depend on the remarkably different properties of isomeric molecules. 

There was tremendous surprise in 1956 when two physicists, Tsung-Dao Lee and Chen Ning Yang, showed that incongruent counterparts play a role in fundamental physics. They studied very fragile subatomic particles, which can be produced by physicists but quickly decay and fall apart. Some of these particles come in pairs of incongruent counterparts; that is, pairs of particles that have the same properties except that their shapes are mirror images of each other (like hands). In a series of dramatic experiments, Madam Wu (ChienShiung Wu), a physicist in New York, showed that the lifetime of certain particles depends on whether they were left-handed or righthanded! That is, even the most fundamental physical laws are sensitive to handedness. The excitement about this discovery was so great that Lee and Yang were given the Nobel prize in record time.

Even if we live in a three-dimensional space without twists and thus left and right hands must remain incongruent counterparts, Kant’s argument is strengthened by the possibility that more dimensions would render them congruent.

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