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Re^4: What do you know, and how do you know that you know it?

by kabel (Chaplain)
on Aug 02, 2004 at 21:53 UTC ( #379442=note: print w/ replies, xml ) Need Help??


in reply to Re^3: What do you know, and how do you know that you know it?
in thread What do you know, and how do you know that you know it?

i think the proof problem is due to the application of classical logic. in intuitionistic logic, there is no such thing like an pure existence proof. instead, existence is showed by providing a method of computation -- an algorithm -- which constructs the thing whose existence is to be proofed.

provided that a) existence proofs are the problem and b) i understood your points.


Comment on Re^4: What do you know, and how do you know that you know it?
Re^5: What do you know, and how do you know that you know it?
by tilly (Archbishop) on Aug 02, 2004 at 22:56 UTC
    Conditions a) and b) are both not satisfied.

    The problem has nothing to do with formalism vs intuitionism. The problem is that mathematical proofs are written by humans and read by other humans, yet are supposed to adhere to an inhuman level of consistency. It is human to err, and as such errors compound, purported absolute proofs become..somewhat less so.

    I made the analogy to programming because this audience is one that has experienced first-hand the chasm between what you think that your program should do, and what it will do when put into a computer. I drew the analogy to a program that has only been specified because, at the research level, most proofs have significant gaps and steps that the reader is expected to be able to fill in. (Sometimes, of course, these gaps can't be filled in. At least not easily.) This is part of why a professional mathematician is not surprised to find that they can only read research papers at a rate of a page or two a day.

      so you are a step or two deeper to what i was thinking. thanks for clarifying. (sorry, i need to wrap this in my words to get a glimpse of it). second shot *peng*

      mathematical proofs are written by few and read by many. lets assume that a script is also read by many. sure, the writer must expect a reader to be at a certain level of skill. otherwise immediate understanding will not be possible. the writer comments all stuff he thinks that can be misinterpreted, and stuff he knows is his usus (and perhaps more); so that if you read the script, it makes perfect sense.

      but this seems to be the exception. with the effects of the personal belief system, everyone has an other view of the solution of a problem. so if the reader does not understand, he is either not qualified enough or he cannot see the intentions the writer had while writing it. in the latter case, there is an information loss; this information should actually be represented by a comment.

      a perfect script is one that its writer has all his belief knots undone and everything he cannot expect a reader to know is either transported by a good naming scheme or by comments. that means, there are no perfect scripts because it is dependent on the knowledgelevel/belief system of the reader.

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