gusl: (Default)
Mathematical books can be roughly divided into two parts: the obvious, which you breeze through; and the insufficiently explained parts, in which you can get stuck for hours. For any given book, this partition will be different for every reader.

When reading fiction, you're given a single perspective: you're seeing the world moment-to-moment, and your incomplete understanding is an important component of the fun. Just keep reading, and the mysteries of the past may be uncovered later.

When reading math, the same does not happen (partly because writers have no idea what background an individual reader has)... which is why readers of math may choose to be stuck for hours, so that they can completely understand what they're reading before moving on.
gusl: (Default)
Confession time: when proving stuff, I feel the need to justify things that I consider to be intuitively obvious. In the course of this, I sometimes look up theorems in a textbook that I couldn't possibly prove, and cite them. I feel slightly dishonest when I do this.

Why? Because the proof in my head didn't need the theorem in the textbook. So why should the paper expression of my thoughts use that theorem?

I am *sure* that the statement is true but I'm unable to justify this intuitive knowledge in a formal language, without the help of the theorem in the textbook. All this talk about eigenvalues feels artificial; but it's the only way I found to connect my intuitive idea with the "objective" mathematics that "everyone" must accept.

Maybe my mind is wrong in being so secure about intuitions whose foundations it has trouble fleshing out (but I doubt it). Instead, I think that my mathematical language skills are deficient, i.e. a kind of "aphasia". If you are a visual thinker, the "informal yet rigorous" language of sentential proofs is a foreign language.

I'd say that communication is one of the hardest and most important problems faced by humanity.
See Jukka Korpela on "How all human communication fails, except by accident".


Jun. 7th, 2007 07:25 pm
gusl: (Default)
Remember the joke about having LaTeX type-check your proof in order to catch any mistakes?

VeriMathDoc is dead serious. I'm really looking forward to the day when I have a "mathematical assistant" to let my brain focus on the more creative aspects of math.
gusl: (Default)
I have decided to read a mathematical textbook over the holidays, something I haven't done in a long time.

I keep wanting to click on terms that I don't recognize, to go to the definition.

Here's a useful programming/text-learning project: create a system that, given a mathematical document (LaTeX, PDF, scanned text), automatically turns it into a mathematical hypertext linking each technical term to its definition.
You can use a wiki to do the linking, as long as you have WikiTeX enabled to do the rendering. This should be a straightforward project if the original document is already in LaTeX. AFAIK, you can't put links on TeX-rendered images, but fortunately, terms that have definitions tend to be the kind that don't need any TeXing anyway.

For me, the next steps would be:
* variable substitution: the text lets the reader plug in values. This would be useful for understanding theorems.
* recognize whether a reader-given object satisfies a definition (along with a "WhyNot?" tool)

I think the main difficulty here is interpreting the written mathematical text into a formal language. Most people reading this are probably thinking about difficult questions, like which foundation they would choose. But the definitions that I have in mind could be decided with Prolog.

I should read OpenMath applications to see what people are up to. They have whole conferences on this stuff, so we should be seeing some progress in this area. I have to wonder why we don't yet see any interactive math books online (advanced, theorem-based math).

More ideas:
* interpreting diagrams as mathematical objects, which can (fail to) satisfy definitions.


gusl: (Default)

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