John Langford has diagnosed a complexity illness that afflicts research in academia. One of its symptoms is what he calls “Math++”: the use of unnecessary and obfuscatory mathematics to improve the author’s chance of publication.
Having recently ploughed through a large number of math-heavy articles during the preparation of a COLT paper I have started to worry whether the illness is contagious. At present there is a rash of awkward notation breaking out in some sections of my draft. While I don’t think I can completely get rid of it I’m hoping that I can at least tidy it up and turn it into something presentable.
Wanting to tidy up awkward mathematical expression is definitely not the same as wanting removing it completely. To switch analogies, maths is akin to a communications channel. The aim of the encoder is to cram information down the line so it can be decoded a the other end. Good mathematical notation encodes frequently occurring concepts with short, memorable terms and takes advantage of uses relationships between concepts. Using a side-channel — e.g., the English text of the paper — to ease the burden of decoding is also a good strategy.
John also suggests treating Math++ (and other forms of complexity) with education. This doesn’t necessarily mean give a lecture on your research but any attempt at communication. I’ve found that attempting to describe what I’m working on over lunch - and without a whiteboard - can be a good way to focus on the story of your research rather than the technicalities. I find technical details of a paper much easier to understand when I understand their motivation.
Even if I don’t completely cure my paper of Math++, I take some solace from Fernano Pereira who points out that research is a form of dialogue and that dialogue is inherently messy which is sometimes the reason mathematical exposition is less than perfect. It’s only through repeated attempts to communicate ideas that one is able to figure out what is important.
Comments (3)
I doubt Math++ helps your career on the long run.
I have been guilty of Math++ myself, but given that I was trained as a mathematician, this is understandable.
Philosophers suffer from a similar syndrome: they tend to write very long sentences with big words.
My undergrad degree was in pure maths so I’m quite susceptible to bouts of Math++. I would have thought lots of exposure when I was young would have helped boost my immunity.
I can recommend Terry Tao’s tips on writing readable maths papers, even if I don’t always follow them:
http://terrytao.wordpress.com/advice-on-writing-papers/
Recently I spent (wasted more like) an afternoon reading a long and densely mathematical paper on Euclidean Motion Group Representations. After 32 pages I finally realised the point of the paper. To paraphrase (the equations) it proved that if one takes a pizza and cuts it into N separate sectors, then reassembling all the N sectors in the original configuration gives the original pizza. I wish the authors had given a similar summary in their work, interspersing the equations with some motivation and insight into their long, arduous and yet noble derivations. Then I’d have more time to contribute inciteful (sic) comments to blogs like this.
Postscript: I do not wish to undermine the importance of finding rigorous proofs for what appear to be deceptively simple and obvious geometrical “facts”, but I would like to be informed by clearer exposition. I guess that means I’ll never be pure, mathematically speaking!
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[...] In machine learning and artificial intelligence, I sometimes worry that there is an excess of math. Others have complained about “the use of unnecessary and obfuscatory mathematics to improve the [...]