"All right," said the Cat;
and this time it vanished quite slowly,
beginning with the end of the tail,
and ending with the grin, which
remained some time after the rest
of it had gone.
"Well! I've often seen a cat
without a grin," thought Alice;
"but a grin without a cat!
It's the most curious thing
I ever saw in all my life!"
The phrase "grin without a cat" is not a bad description of pure mathematics. Although mathematical theorems often can be usefully applied to the structure of the external world, the theorems themselves are abstractions that belong in another realm "remote from human passions," as Bertrand Russell once put in a memorable passage, "remote even from the pitiful facts of Nature . . . an ordered cosmos, where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape from the dreary exile of the actual world."
Martin Gardner: The Annotated Alice. New York, London 2000, page 68, footnote 11
The process that Alice is experiencing here, is precisely what we are trying to teach children in school. We know what two apples are and we know what the two pears are. But what is 'two' without anything attached? Does the number two exist in reality, or does it exist only in my head? That is exactly the same philosophical question that Alice is wondering about. The cat's grin - in what sense does it exist independently of the cat?
Bjarne Toft, cited by Lise Brix: Eksempler på matematikken i Alice i Eventyrland, Videnskap.dk, Valby 2013
But the Cheshire Cat that is seperated from its grin is not only a mathematical idea. It is a physical reality, too. Recently, scientists observed the existens of a 'quantum Cheshire Cat': if a quantum system is subject to a certain pre- and postselection, it can behave as if a particle and its property are spatially separated. The scientists report an experiment in which they send neutrons through a perfect silicon crystal interferometer and perform weak measurements to probe the location of the particle and its magnetic moment. The experimental results suggest that the system behaves as if the neutrons go through one beam path, while their magnetic moment travels along the other.
Inside the interferometer, the Cat goes through the upper beam path, while its grin travels along the lower beam path.
Figure courtesy of Leon Filter.