"Bill Page" <[EMAIL PROTECTED]> writes: | On Tue, Apr 8, 2008 at 8:59 PM, Gabriel Dos Reis wrote: | > Bill Page writes: | > | > | I can not agree. Because something cannot always be computed | > | does not mean that it is therefore ill-defined. This is especially true | > | in mathematics. | > | > I believe you're putting confusing the issues. Function equality | > is well-defined in classic mathematics (set theoretical). But | > that definition is almost useless from algorithmic perspective. | > | | I do not think "almost unless" is synonymous with ill-defined even | from an algorithmic perspective.
So? | In reply to a similar comment by | Waldek in this thread I characterized the current fully computable | syntactic definition of equality for functions as an "approximation" | to the actual equality. The problem I have with that `characterization' is that it does not make any sense from principled computations point of view. If the game is to just hack up something, we know how to do that. But I cannot marry your `charaterization' with the nearly worship in category theory. | I think what it means to be approximate can in | principle be rigorously defined. I'm awaiting an *actual* rigorously defined one -- not one that `can be in principle'. Bcause, after all, I have feed the machine with an actual algorithm. | > | For example, the assumed equality of functions, i.e. the | > | concept of a commutative diagram, is essential for category theory. | > | > `category theory' is a medium, not the message. By itself it is void | > of content. | > | | You would have a hard time convincing me of this Then we just have to agree to disagree. I don't believe in worship in category theory. I do believe category is an extremely sophisticated and powerful tool... when one has actual *matter* to work with. [...] | Would you say that classical set theory is similarly "devoid of | content"? The problem with `set theory' is not that it is devoid of content. It is that it has a overweight non-constructive content. In essence, it concludes that `bubble sort' and `heap sort' are the same function. That is why it is so inadequate from (algorithmic) computational perspectives. -- Gaby ------------------------------------------------------------------------- This SF.net email is sponsored by the 2008 JavaOne(SM) Conference Don't miss this year's exciting event. There's still time to save $100. Use priority code J8TL2D2. http://ad.doubleclick.net/clk;198757673;13503038;p?http://java.sun.com/javaone _______________________________________________ open-axiom-devel mailing list open-axiom-devel@lists.sourceforge.net https://lists.sourceforge.net/lists/listinfo/open-axiom-devel
