Can I read mathematics textbook like the music sheet?

Question

During my Masters degree, I tried doing a proof with a pen and paper in my hand. This took me days to figure out. Today, I saw a proof and began to follow each step with a full comprehension of what was done without having to solve.

This made me remember how I started learning how to sight-read/sing. Symbols such as crochets, quavers, mimins, and semibreves were not easy to read. Things changed after some time.

My Question: Will there ever be a time when reading mathematics textbook will be like reading from the music sheet?

Answer 1

This should probably be a comment, but it's too long. When I was in high school, I played the clarinet, so the sheet music I saw was usually the clarinet part of a piece written (or transcribed) for bands. I had no problem reading and playing that music.

But I've also seen the sheet music used by the conductor of the band, showing what every instrument is playing. That's obviously far more complicated to read.

On the one hand, I couldn't imagine sight-reading a conductor's score in the way I could sight-read a clarinet part (though I suppose experienced conductors develop that ability). On the other hand, studying the score would give the conductor a genuine understanding of the composer's ideas and how they fit together, an understanding that I could never get from the clarinet part alone, and that I only occasionally got after hearing the music several times in rehearsals.

What does this have to do with mathematics? When reading mathematics, one wants to get genuine understanding, analogous to what a conductor gets from the full score. The mathematical analog of the clarinet part would be a very superficial understanding, perhaps enough to do routine exercises but not much more. So it is reasonable to expect that learning mathematics would require work similar to studying a conductor's score. In particular, it's unrealistic to expect to learn mathematics by "sight-reading".

Answer 2

Some parallels can be drawn between music sight reading and regular word reading. For example, when you first learned to read the sort of words you're reading now, you started with the alphabet, and sounded out each letter of a word to try to determine what word it was. Eventually, you learned to recognize common words by sight and then common patterns of sequential words. To read this paragraph, you didn't have to sound out each letter; your eyes and brain and flowing through them.

Likewise, with sheet music, you no longer have to look at each and every note and count "Every Good Boy Does Fine" up the ledger lines, etc. You recognize common, repeating patterns. You can probably even look at a measure and play it while your eyes move over to the next measure.

So the question is, does this extend to the reading of mathematical symbols, like in a proof? I would say yes, at least sometimes. You may be able to run your eyes down a page of equations and say, "Oh yeah, this just looks like the derivation of the quadratic formula", or "Oh, they're just applying Integration By Parts in this next step". This can especially be true if you work within one specific field of mathematics, and keep seeing work from only that field. Not unlike how a musician who plays mostly in one particular genre will probably read similar music at a fast pace.

Even the best of mathematicians though will come across something that may take more time to dissect and understand. Some of that may be due to the writer being sloppy and leaving out parts, or using unconventional symbols (e.g., "x" for an Eigenvalue instead of lambda). But it may also be a symbol or function they're not too familiar with, particularly if it's out of their normal field.