Re: [Haskell-cafe] Re: Theory about uncurried functions

Ha. There's even a wiki page on the paradoxes of set theory http://en.wikipedia.org/wiki/Paradoxes_of_set_theory If I recall correctly, a math professor once told me that it is not yet proven if the cardinality of the power set of the natural numbers is larger or smaller or equal than the cardinality of the real numbers... But that is many many years ago so don't shoot me if I'm wrong :) 2009/3/4 Luke Palmer

On Wed, Mar 4, 2009 at 3:38 PM, Achim Schneider wrote:

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There's not much to understand about CT, anyway: It's actually nearly as trivial as set theory.

You mean that theory which predicts the existence of infinitely many infinities; in fact for any cardinal, there are at least that many cardinals? That theory in which aleph_1 and 2^aleph_0 are definitely comparable, but we provably cannot compare them? The theory which has omega_0 < omega_1 < omega_2 < ... omega_omega < ..., where obviously omega_a is much larger than a... except for when it catches its tail and omega_alpha = alpha for some crazy-ass alpha.

I don't think set theory is trivial in the least. I think it is complicated, convoluted, often anti-intuitive and nonconstructive.

Category theory is much more trivial, and that's what makes it powerful. (Although training yourself to think categorically is quite difficult, I'm finding)

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One part of the benefit starts when you begin to categorise different kind of categories, in the same way that understanding monads is easiest if you just consider their difference to applicative functors. It's a system inviting you to tackle a problem with scrutiny, neither tempting you to generalise way beyond computability, nor burdening you with formal proof requirements or shackling you to some other ball and chain.

Sadly, almost all texts about CT are absolutely useless: They tend to focus either on pure mathematical abstraction, lacking applicability, or tell you the story for a particular application of CT to a specific topic, loosing themselves in detail without providing the bigger picture. That's why I liked that Rosetta stone paper so much: I still don't understand anything more about physics, but I see how working inside a category with specific features and limitations is the exact right thing to do for those guys, and why you wouldn't want to do a PL that works in the same category.

Throwing lambda calculus at a problem that doesn't happen to be a DSL or some other language of some sort is a bad idea. I seem to understand that for some time now, being especially fond of automata[1] to model autonomous, interacting agents, but CT made me grok it. The future will show how far it will pull my thinking out of the Turing tarpit.

[1] Which aren't, at all, objects. Finite automata don't go bottom in any case, at least not if you don't happen to shoot them and their health drops below zero.

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