Re: [Haskell-cafe] Haskell is a declarative language? Let's see how easy it is to declare types of things.

(Folks, let's rescue this increasingly tendentious thread.) Some points to ponder: (1) "Any" can be often be clarified to mean "all", depending on how polymorphic functions are exegeted. In a homotopy-flavored explanation of natural transformation, its components (read "parametric instances") exist /all at once/ like the collinear rays of a rainbow. So given this:

f :: Bool -> t f True = 'x' f False = 'y'

nope, no rainbow. :(

... the aftermath flame war of "why didn't you say it earlier?" "because it's obvious!" "no it's not!" "yes it is!" "is not!" "but all mathematicians find it obvious!" "well then I am not a mathematician and will never be!"

And more to the point, an excellent learning environment requires empathy from all participants, juniors and seniors alike. Where diversity is celebrated as a source of new ideas and new ways to explain old ones. And where the coupling between the two is as gut-obvious as day and night. (2)

prove or disprove: for all e>0, there exists d>0, if 0

I grant you that clearly the implicit quantifiers for a and b are "for all". The unclear part is where to put them. There are essentially 4 choices

That's a stretch. It's all in the context, and here it's clearly a continuity verification exercise from freshman calculus. Unless being deliberately obtuse, one has no excuse not inferring where the quantifiers go if one knows about a theorem prover, what more wielding one to nuke this gnat of a proof. Moreover, if we grant the imaginary student the rudiments of logic, 3 of the 4 "choices" render the statement vacuously true. Almost. Set d to deny the antecedent, QED. I partly agree with Albert's main point, notwithstanding his choice of examples, that the absence of explicit quantifiers can lead to confusion. It all depends. On the other hand Alexander Solla is also on the money with his remark. A mathematician writes [1], "In particular, any given assertion in hard analysis usually comes with a number of unsightly quantifiers (For every there exists an N…) which can require some thought for a reader to parse." (3) Newspaper headline: Someone gets hit by a car every 6 seconds. A few months ago, a good chunk of Devlin's Intro to Math Thinking massive online course devoted itself to explicit and precise placement of quantifiers. So not only is the above headline judged improperly worded and hence badly ambiguous, but also commonplaces like "In case of fire do not use elevator." I'm a fervent believer against ambiguity: Let zealotry take its place. [1] http://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-a... -- Kim-Ee