  If you are not skilled in colloquial astronomy, and I tell you that the morning star is the evening star, I have given you information—your knowledge has changed. If I tell you the morning star is the morning star, you might feel I was wasting your time. Yet in both cases I have told you the planet Venus was self-identical. There must be more to it than this. Naively, we might say the morning star and the evening star are the same in one way, and not the same in another. The two phrases, “morning star” and “evening star” may designate the same object, but they do not have the same meaning. Meanings, in this sense, are often called intensions, and things designated, extensions. Contexts in which extension is all that matters are, naturally, called extensional, while contexts in which extension is not enough are intensional. Mathematics is typically extensional throughout—we happily write “\(1+4=2+3\)” even though the two terms involved may differ in meaning (more about this later). “It is known that…” is a typical intensional context—“it is known that \(1+4 = 2 + 3\)” may not be correct when the knowledge of small children is involved. Thus mathematical pedagogy differs from mathematics proper. Other examples of intensional contexts are “it is believed that…”, “it is necessary that…”, “it is informative that…”, “it is said that…”, “it is astonishing that…”, and so on. Typically a context that is intensional can be recognized by a failure of the substitutivity of equality when naively applied. Thus, the morning star equals the evening star; you know the morning star equals the morning star; then on substituting equals for equals, you know the morning star equals the evening star. Note that this knowledge arises from purely logical reasoning, and does not involve any investigation of the sky, which should arouse some suspicion. Substitution of co-referring terms in a knowledge context is the problematic move—such a context is intensional, after all. Admittedly this is somewhat circular. We should not make use of equality of extensions in an intensional context, and an intensional context is one in which such substitutivity does not work.