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I'm finding it difficult to pinpoint where/why you think there's a difference. Absolutely there's a strong parallel between an oddly constructed theorem and an

by n4r9 9y ago

I'm finding it difficult to pinpoint where/why you think there's a difference. Absolutely there's a strong parallel between an oddly constructed theorem and an odd-shaped piece of reusable code. Geometry may be an eternal, perfect realm of ideas, but the actual codebase is the book, which is exceedingly good but is not perfect, especially given the restraints of the medium (written text). Someone who is practiced or naturally talented will be able to read that book with a similar ease with which you read elegantly written code (c.f. Einstein being amazed at the "lucidity" of Euclid's elements at the age of 12). For everyone else, I can imagine having a different, easier to grasp codebase for Euclid's elements: perhaps transcribe it into a wiki-like computer document which allows the reader to browse and flick between the referenced lemmas much more easily. Both mathematical exposition and coding, and in fact almost any sort of writing, involve a sort of data compilation from a very high-level language ("how I/we understand it in my/our head/s") to a lower-level language which is more rigorous but lacks an interactive element.

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hyperpallium 9y ago
Thinking about the beauty of proofs, as imdicated by simplicity, and how breifly they can be encoded (information-theoretic version of occam's razor): Another aspect of the symmetrical stackexchange diagram (two alternate interior angles), and Euclid's lop-sided one is that Euclid's is slightly simpler (one alternate interior angle, and one corresponding angle). But from another perspective, the symmetrical one is simpler, because its symmetry allows it to be remembered as symmetry, and only recording one side. Further, although stating it is symmetrical may require more information to record than just explicitly stating both sides if you only record one theorem, if you record a corpus of theorems and symmetry is common over that corpus (which it is), it can be stated with less information. (i.e, like using shorter words for frequently used concepts). But this is only for the proof details; I'm not sure how it applies to the theorem as a reusable component, where you don't have to worry about the details.
hyperpallium 9y ago
Well, I'm still struggling with the question! I wasn't thinking ideas vs their expression. I see you're saying: you can have different expressions of Euclid for people of different abilities. (Actually, even different proofs for different levels: the high school proof of the Inscribed Angle Theorem is different in part from Euclid's, III, 20 (though I found Euclid's clearer and simpler)). The difference I noted was just that geometry is like one codebase. But programmers deal with many codebases - there's no ugrad or high school course in a particular codebase! Though, in a large company, do you get experts in their key codebase, both practiced and talented. But there's still a reluctance to invest your all in a codebase - what if they rewrite? Or go out of business? Or you quit? Your investment is lost. Not so with geometry. There's also market forces for the adoption of third-party libraries (a library of theorems - like Elements' books!) - dominating the market will require being accessible to a range of abilities. A "pop culture", for better or worse. I guess you'd say: "fine, the easier expressions will dominate". But I don't think there'll be any higher expression at all... So... not a difference in essence, just in its role in the world. BTW I find Euclid very hard to read, but easy to understand (so far, anyway). BTW those links I gave are to a "wiki-like computer document".

by n4r9 9y ago

deleted 9y ago

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hyperpallium 9y ago

Thinking about the beauty of proofs, as imdicated by simplicity, and how breifly they can be encoded (information-theoretic version of occam's razor): Another aspect of the symmetrical stackexchange diagram (two alternate interior angles), and Euclid's lop-sided one is that Euclid's is slightly simpler (one alternate interior angle, and one corresponding angle). But from another perspective, the symmetrical one is simpler, because its symmetry allows it to be remembered as symmetry, and only recording one side. Further, although stating it is symmetrical may require more information to record than just explicitly stating both sides if you only record one theorem, if you record a corpus of theorems and symmetry is common over that corpus (which it is), it can be stated with less information. (i.e, like using shorter words for frequently used concepts). But this is only for the proof details; I'm not sure how it applies to the theorem as a reusable component, where you don't have to worry about the details.

hyperpallium 9y ago

Well, I'm still struggling with the question! I wasn't thinking ideas vs their expression. I see you're saying: you can have different expressions of Euclid for people of different abilities. (Actually, even different proofs for different levels: the high school proof of the Inscribed Angle Theorem is different in part from Euclid's, III, 20 (though I found Euclid's clearer and simpler)). The difference I noted was just that geometry is like one codebase. But programmers deal with many codebases - there's no ugrad or high school course in a particular codebase! Though, in a large company, do you get experts in their key codebase, both practiced and talented. But there's still a reluctance to invest your all in a codebase - what if they rewrite? Or go out of business? Or you quit? Your investment is lost. Not so with geometry. There's also market forces for the adoption of third-party libraries (a library of theorems - like Elements' books!) - dominating the market will require being accessible to a range of abilities. A "pop culture", for better or worse. I guess you'd say: "fine, the easier expressions will dominate". But I don't think there'll be any higher expression at all... So... not a difference in essence, just in its role in the world. BTW I find Euclid very hard to read, but easy to understand (so far, anyway). BTW those links I gave are to a "wiki-like computer document".