Saturday, December 21, 2013

The over-mathification of Wikipedia

Wikipedia is an amazing resource. Truly amazing. Aside from personal interests, I think I use it at least several times a day for work-related science information. For biology, it's proven to be an invaluable resource for just getting a quick overview of a particular gene or disorder or other biological topic. It could also be a great resource for more theoretical stuff as well, and sometimes it is. For instance, I teach a fluid mechanics class, and one of the main confusions for students is the difference between streamlines, streaklines and pathlines. The Wikipedia page on this topic is just great, and has an animated GIF that gets the point across way better than I can on the chalkboard.

But for many of the theoretical topics, there's a big problem: somewhere along the line, it's clear that some mathematicians got involved. The issue is that they've inserted all sorts of mathematical technicalities into otherwise relatively simple (and useful) mathematical techniques that make the article essentially unreadable and useless. Take the Wikipedia entry on the Laplace transform. The Laplace transform is super useful in solving differential equations, basically because it turns derivatives into multiplication, thereby turning the differential equation into an algebraic equation. Very handy. But good luck getting that out of the Wikipedia article! Instead of starting with a simple application to show how people actually might use the Laplace transform in practice, the Wikipedia article begins with a lengthy and overly mathematical formal definition, with statements like:
One can define the Laplace transform of a finite Borel measure μ by the Lebesgue integral
While these conditions may be interesting to mathematicians, I don't think it is of any interest for the vast majority of people who use the Laplace transform. Then it gets into discussion of the region of convergence, which again begins with:
If f is a locally integrable function (or more generally a Borel measure locally of bounded variation), then the Laplace transform F(s) of f converges provided that the limit...
Perhaps you care about the Borel measure locally of bounded variation, but I'm guessing that most people who are interested in the Laplace transform haven't taken courses in point-set topology.

Overall, look at the organization of the page. It is just so... mathy! It starts with a formal definition (devoid of any real practical motivation), proceeds to some details about convergence, then a bunch of properties and theorems, then tables of transforms, and then, THEN, finally, some examples. The only part of this that has a bit of motivation for a person who doesn't already know about why the Laplace transform is useful is at the beginning of the "Properties and theorems" section:
The Laplace transform has a number of properties that make it useful for analyzing linear dynamical systems. The most significant advantage is that differentiation and integrationbecome multiplication and division, respectively, by s (similarly to logarithms changing multiplication of numbers to addition of their logarithms). Because of this property, the Laplace variable s is also known as operator variable in the L domain: either derivative operator or (for s−1) integration operator. The transform turns integral equations and differential equations to polynomial equations, which are much easier to solve. Once solved, use of the inverse Laplace transform reverts to the time domain.
Why is this not at the very beginning? The answer is that it was! Look at this version from 2005. Much better! Still not perfect, since it doesn't have an example, but at least it more clearly gets the main point across about why you would use the Laplace transform. To me, it's clear that at some point, mathematicians got involved and wanted to make the page "right", the same way that mathematicians make the delta function "right" with all kinds of stuff about distributions, in the process completely obscuring the basic point of the delta function for people who really want to use it practically (and many other such mathy topics have been similarly "rightified"). Just to be clear, I'm saying this as a person who has a Ph.D. in math and who loves math, and I think it's great that people far smarter than I have spent time to make sure that one can rigorously define these things. And it IS important, even in some practical contexts. But it's not good for exposition to a more general audience on a Wikipedia page.

Oh, and here's the original Wikipedia page for the Laplace transform. All else aside, we've come a long way, baby!

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