Telescopic Text, by Joe Davis.
The web page starts out with the words "I made tea. by Joe" and various words can be clicked on; when you click on them they expand, so "Joe" becomes "Joe Davis", for example, when you click on it. Clicking on "I" reveals the word "Yawning" preceding it; "tea" becomes "a cup of tea", and so on. As you expand the text, some biscuits that weren't there before, comments on how to make tea, and so on materialize.
This reminds me of something that's been thrown around the mathematical blogosphere as a possible way to write papers that might be well-adapted to our present computer technology; start with a very high-level sketch of a proof, and make each step clickable. Upon clicking on a word, the proof is expanded to remind you what that word means, how exactly one uses that particular technique here, etc.
This would require more work than writing a paper in the usual way, though; it's not clear whether it's worth the trouble. And there's always the issue that some people like to read papers away from the computer, they eventually end up in journals which are printed on paper, and so on; what level of detail should be published there?
20 August 2008
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That's kind of how I try to write lectures--start sketchy, but have prepared different levels of detail to sub in if I get a question about a particular step.
I recently took a course where the professor taught that way. Overall sketch of the proof with a lot of assumptions. Then he'd take each assumption one by one and prove them (possibly recursing the same procedure).
At each step, you know exactly where you are and more importantly, why you're trying to prove the current lemma. The usual style is to build things up, but that methodology lacks motivation all the way until you see the major result in the end.
Additionally, I also suspect that this also reflects a mathematician's thought processes while working on a problem.
I can categorically say that was the best math class I took, and this style of teaching was no doubt a big factor.
In a sense, it's also more work for the reader/audience. They have to do some bookkeeping to make sure nothing was left unproven. IMO, unless we're talking about something really complex, that's not at all difficult to manage.
I am going to get some of this wrong. Vladimir Nabokov wrote a translation of the Russian epic poem
Igor's tale . It was replete with scholarly foot notes so much so that it was quite unreadable --- sort of like Maharshi Yoga's translations and annotations of
the Bhagavad Gita . Our English class was told that Nabakov put in all the foot notes in order to satirize the trends in literary criticism at the time.
Jim Davis's piece is a masterpiece though!
Back to the math. I have often thought that the standard advanced calculus course has two main ideas: The real numbers are a complete totally ordered Archemedian field with the least upper bound property, and the various conclusions that one can draw by considering a continuous function on a compact set.
Unpacking those two concepts are the gist of the course.
Ooooh... that would be a GREAT way to write proofs! And maybe, like Scott Carter mentioned, it could even be scaled up to the course level to good effect.
As far as print publication, couldn't you just use indentation to show the tree structure, as is typically done in outlines?
Too much indentation, I think. The trickiest parts would be the most-squished!
It might be better to systematically differentiate between the terms "proposition", "lemma", "theorem", "fact", "claim", and derived terms like "sublemma".
Rather than inlining the proofs of the steps, why don't we make them more like subroutines? Like, we could prove some results in the abstract and then use them multiple times in different contexts as we need them.
And then we can extend the programming metaphor even further. We could take a bunch of standard results that people use a lot and throw them into "header" papers. Of course, we might have to state the results to keep them fresh in our minds, but we could keep the proofs "precompiled" so we don't have to bother with them. If we really need to know the internal mechanisms we can look up the original source.
In fact, we could amass collections of such previously-proven results and..
Barry Mazur has an interesting concept of theorems that prove temselves. They are prime candidates for being written as telescopic texts. Any volunteer to try?
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