Comments for Mathematics and Computation http://math.andrej.com Mathematics for computers Sun, 25 Jul 2010 09:06:21 +0000 http://wordpress.org/?v=2.9.1 hourly 1 Comment on On programming language design by masklinn http://math.andrej.com/2009/04/11/on-programming-language-design/comment-page-2/#comment-13590 masklinn Sun, 25 Jul 2010 09:06:21 +0000 http://math.andrej.com/?p=194#comment-13590 <blockquote>Seth, but a Maybe Tree is not a tree, it’s a Maybe. How do you expect to call tree methods on it?</blockquote> Give them to the maybe, which will forward it or not and wrap the return itself in a <code>Maybe</code>? (essentially implementing method-level lifting natively in the language) Might want a different operator for lifted method calls in order to discriminate between calls on <code>Maybe</code> and lifted calls on whatever is within <code>Maybe</code>.

Seth, but a Maybe Tree is not a tree, it’s a Maybe. How do you expect to call tree methods on it?

Give them to the maybe, which will forward it or not and wrap the return itself in a Maybe? (essentially implementing method-level lifting natively in the language) Might want a different operator for lifted method calls in order to discriminate between calls on Maybe and lifted calls on whatever is within Maybe.

]]> Comment on A new style for the blog by Christian Perfect http://math.andrej.com/2010/01/08/a-new-style-for-the-blog/comment-page-1/#comment-13566 Christian Perfect Wed, 14 Jul 2010 19:06:04 +0000 http://math.andrej.com/?p=424#comment-13566 www.mathjax.org is very good and the successor to jsMath. I'm using it on my blog. http://www.mathjax.org is very good and the successor to jsMath. I’m using it on my blog.

]]> Comment on Constructive gem: irrational to the power of irrational that is rational by Cody Roux http://math.andrej.com/2009/12/28/constructive-gem-irrational-to-the-power-of-irrational-that-is-rational/comment-page-1/#comment-13550 Cody Roux Mon, 05 Jul 2010 11:56:01 +0000 http://math.andrej.com/?p=410#comment-13550 I see the proof of the above theorem in a somewhat different way than "a useless example of non-constructive proof". Let me make my point using a Car Analogy. You go to a car dealer, and he supplies a car to you, along with a guarantee: He must supply a -working- car to you, if needed. If you never use the car, the contract is implicitly satisfied, otherwise, if you do try to start the car, then two cases may occur: The car starts, and everything is fine, or the car does not start. In this case, you make use of your guarantee, and the dealer supplies a new, working car (or some other car with another guarantee). You do the same for your classical proof of existence of 2 irrational numbers a and b such that a^b is rational: act as if a=sqrt(2) and b=sqrt(2) satisfy your theorem. If anyone calls you out on it, by supplying a proof that sqrt(2)^sqrt(2) is irrational, you can "backtrack" by supplying new witnesses to your theorem: a=sqrt(2)^sqrt(2) and b=sqrt(2). In a sense, the non-constructive proof does contain "less information" than the constructive proof. However it does supply a "guarantee" as in the above sense, that is you may use your witnesses with the caveat of perhaps needing to substitute them for "correct" witnesses at a further point. I see the proof of the above theorem in a somewhat different way than “a useless example of non-constructive proof”. Let me make my point using a Car Analogy. You go to a car dealer, and he supplies a car to you, along with a guarantee: He must supply a -working- car to you, if needed. If you never use the car, the contract is implicitly satisfied, otherwise, if you do try to start the car, then two cases may occur: The car starts, and everything is fine, or the car does not start. In this case, you make use of your guarantee, and the dealer supplies a new, working car (or some other car with another guarantee). You do the same for your classical proof of existence of 2 irrational numbers a and b such that a^b is rational: act as if a=sqrt(2) and b=sqrt(2) satisfy your theorem. If anyone calls you out on it, by supplying a proof that sqrt(2)^sqrt(2) is irrational, you can “backtrack” by supplying new witnesses to your theorem: a=sqrt(2)^sqrt(2) and b=sqrt(2). In a sense, the non-constructive proof does contain “less information” than the constructive proof. However it does supply a “guarantee” as in the above sense, that is you may use your witnesses with the caveat of perhaps needing to substitute them for “correct” witnesses at a further point.

]]> Comment on Intuitionistic mathematics for physics by alex http://math.andrej.com/2008/08/13/intuitionistic-mathematics-for-physics/comment-page-1/#comment-13535 alex Sun, 27 Jun 2010 14:15:26 +0000 http://math.andrej.com/2008/08/13/intuitionistic-mathematics-for-physics/#comment-13535 This goes on my list of things which I would gain immeasurable benefit from being told 10 years ago. This goes on my list of things which I would gain immeasurable benefit from being told 10 years ago.

]]> Comment on The hydra game by Xamuel http://math.andrej.com/2008/02/02/the-hydra-game/comment-page-1/#comment-13518 Xamuel Fri, 18 Jun 2010 21:14:34 +0000 http://math.andrej.com/2008/02/02/the-hydra-game/#comment-13518 Hey Andrej, I just wrote up another similar application of ordinal arithmetic: a method of surfing the internet without getting infinitely sidetracked following links. Kind of like this hydra, except with webpages and links instead of hydra-heads. Here it is: http://www.xamuel.com/internet-surfing-ordinals/ Hey Andrej,

I just wrote up another similar application of ordinal arithmetic: a method of surfing the internet without getting infinitely sidetracked following links. Kind of like this hydra, except with webpages and links instead of hydra-heads. Here it is: http://www.xamuel.com/internet-surfing-ordinals/

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