how maps are like mathematics, and maps of mathematics

Yesterday I wrote about different ways of drawing the Interstate highway system, and introduced this map by Nat Case that clearly shows it as a grid. Case had written about the map at Map Head, and says there that people accused him of copying this map by Chris Yates.


Apparently some people aren't happy about Case's map, because they're claiming that Case just ripped off Yates' idea. I don't know anything about intellectual property law, but it seems to me that there are a lot of maps of the same area that look very similar and convey almost the same information -- maps that are much more similar than those of Case and Yates. For example, two ordinary road maps of the same area should look very similar, because they'll depict the same roads! The impression I have is that the information (where the roads are) is freely available but the particular way in which it's drawn (if this isn't just a simple transcription of the information) is not. To quote one of the commenters here:

Copyright law expressly does NOT apply to "ideas", only to the execution of ideas. If this dude had copied the colors, font, or any other specific element from Chris' map, then it would be a different story (and "straight lines" isn't quite a specific element). If he had made his own map look like a subway map, even then it might have been a fuzzy area.

But, what seems to have happened was that he looked at Chris' map, then started from scratch creating his own expression of a similar idea. While it may have been courteous for him to credit Chris for the inspiration more obviously than he did, he certainly isn't legally required to do so. In fact, this very process of re-interpreting others' work is basically how art progresses in society in general. To use your analogy, I think this guy looked at a Picasso, said, "Hmm, cubism is cool," and painted his own cubist painting of the same model that Picasso used.

To this I would add that mathematics works the same way. Most proofs of the same mathematical fact, most expositions of the same concept, etc. will look the same in outline, simply because they have to respect how the ideas stand in logical relation to each other. But the choice of notation, of words to go between the equations explaining what's going on, and so on is unique to each author, and probably has a lot to do with what else is going on in a particular book or article. One might choose to emphasize certain parts of a proof -- doing some simple algebra in full instead of just saying "the reader can verify this" -- because the same calculation will come in handy later. Notation seems analogous to graphic design elements like color; a good map uses colors which are clearly different for things which are different, and good notation does the same. (I've had professors who used a and α, u and μ, or v and ν in the same problem, and had bad handwriting. No! I'd even go so far as to say that using letters which look similar in typed work is bad practice, because the conscientious reader will be copying your notation by hand in order to check things.) I doubt there are two proofs out there of, say, the Fundamental Theorem of Calculus (to take a result that's been reproduced in a zillion textbooks) that are word-for-word and symbol-for-symbol the same, just like there probably aren't two maps of Manhattan that are pixel-for-pixel the same.

The point here is that there's good exposition and bad exposition, which coincide roughly with good maps and bad maps. A good map, or a good writer, will help you navigate a tricky area; a bad map will just confused you.

And what would a map of mathematics itself look like? Dave Rusin has given it a shot, but each area of mathematics is just a circle to him, and it's not clear to me how they're connected to each other, or why his circles are arranged the way they are. He says at his A Gentle Introduction to the Mathematics Subject Classification Scheme that "[t]he welcome page for this site shows an image of the areas of mathematics which shows the relative numbers of recent papers in each area (arranged so as to illustrate the affinities among related areas)." It's a decent job -- nothing seems too far from where it should be, and I imagine that projecting the whole thing down into two dimensions makes it quite difficult! -- and it's not just based on Rusin's prejudices. It seems to come from correlations between classifications in that scheme -- two classifications are "close together" if there are papers which are classified in both of them. This somehow gives rise to a 61-dimensional space (the 61 dimensions presumably corresponding to the 61 two-digit classes in that scheme) of which this map is a two-dimensional projection. The vertical direction seems to work out with "discrete" mathematics at the top and "continuous" mathematics at the bottom; I'm not sure what the horizontal direction represents. (I think one could make an argument for "pure" on the left versus "applied" on the right, but it's a weak one.)


The same can be done with individual papers, or with mathematicians; see, for example, the papers of eight Fields medalists.

I sense that more could be done. Which areas of mathematics do you need to learn before learning certain others? Which areas historically grew out of each other? Within an area, how are the important results related to each other? And how could this be illustrated pictorially? I'm not sure what good this would be, though.



(At least two people seem to have done something similar for motorways in England); not surprisingly they both did their maps in the style of the London Underground map. It probably won't surprise you to learn that Great Britian's road numbering scheme isn't a grid, but rather divides England and Wales into zones emanating from London, and Scotland into zones emanating from Edinburgh.)