Learning the Language of Mathematics (2000) [pdf] 103 points by mutor on May 22, 2017 | hide | past | web | favorite | 6 comments

 This was written in May 2000, but the following bit sounds like it was written May 2017.> 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.
 "This is unacceptable because mathematics is written as English is written in complete, grammatical sentences." Unfortunately natural languages have many flaws and will easily falls in contradiction like Russell's paradox. Even formal mathematical axiomatic systems encountered GĂ¶del's incompleteness..Maybe something like constructive category theory would give us a better way to formally put ideas in ...
 The article says: 'A definition MUST be an "if and only if" statement.'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.
 That's because the other direction is implicit."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.
 I wish we had such a paper for programming.
 What's the difference? Discrete mathematical concepts are applicable to programming as is without modification.

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