> In conclusion, I want to confess what my real goals are in teaching this material. In a society in which information is passed in 60 second sound bites and reasoning limited to monosyllabic simple sentences, careful, analytic thinking is in danger of extinction. And this is a grave danger in a democratic society beset by a host of very complex moral and social problems.
Maybe something like constructive category theory would give us a better way to formally put ideas in ...
It is an established convention in mathematics to write definitions in the form "X is Y if P(X)". For example: "A metric space M is complete if every Caychy sequence in M converges in M".
One may question whether this is a good convention, but it is a convention that most mathematicians tend to follow.
"X is a rectangle if it's a quadrilateral with all right angles."
Okay, so you can use this statement to look at an object and then check if it's a quadrilateral with all right angles, and then conclude that, by definition, it's a rectangle.
But for the other direction, if I tell you it's a rectangle, it's implicit that you can conclude all right angles. Contrapositively, it's also implicit that if I say it doesn't have all right angles, you can conclude that it's not a rectangle, i.e. that this condition has to be satisfied for anything worthy of the "rectangle" name.