'I know what you're thinking about,' said Tweedledum: 'but it isn't so, nohow.'
'Contrariwise,' continued Tweedledee, 'if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic.'
'Can you do Addition?' the White Queen asked. 'What's one and one and one and one and one and one and one and one and one and one?'
'I don't know,' said Alice. 'I lost count.'
'She can't do Addition,' the Red Queen interrupted. 'Can you do Subtraction? Take nine from eight.'
'Nine from eight I can't, you know,' Alice replied very readily: 'but-'
'She can't do Subtraction,' said the White Queen. 'Can you do Division? Divide a loaf by a knife-what's the answer to that?'
Jordan Ellenberg defined mathiness as "a series of fervent gestures that gives the impression that mathematical ideas are being expressed, but doesn’t actually deliver the goods".
Let us examine some examples of mathiness, and some examples where honest attempts to deal with mathematical situations have foundered, and try to understand how we can be led astray by mathematical arguments.
I recently wrote about how in statistics, the measure that is often called skewness doesn't really mean what popular lore holds it to mean, and that it is often misused - for example, when people assert that zero skewness implies symmetry. I later pointed to several sites that made the kinds of errors I was talking about. In the brief time since then, new instances of the same issue have come up on some mathematics-related blogs. It's a case where the verbal "description" of the situation is not in agreement with the mathematical tools being used - mathematical ideas appear to be expressed, but the goods are not being delivered.
In another vein, bad statistical graphics, such as this
lie in graphical form
can mislead us, whether by accident, or as in this case, by design.
(via Andrew Gelman at Statistical Modeling, Causal Inference, and Social Science; there's other good examples to be found there.)
Examples abound in the media. Here's one from the NYT (via the Gallery of Data Visualization’s Missed Opportunities and Graphical Failures - click image for bigger version):
The top plot there is a graph of happiness against GNP-per-capita for a number of countries. The NYT has circled the countries in the top left hand corner, noting that many countries "had higher ... happiness than their economic situation would predict". This is the cardinal sin of treating inherently nonlinear relationships as linear - as they point out at the Gallery, an appropriate transformation - in this case looking at log-GNP, not raw GNP, makes these supposed "outliers" seem much more in keeping with the rest, and the apparent relationship more linear - and indeed, if anything, some entirely different points don't fit the general pattern. We seem to find nearly-linear relationships easier to understand, so transformation is often a useful strategy.
I have discussed the same issues - both the danger of treating nonlinear relationships as linear and the value of transformations in understanding relationships better in another context - relationships involving percentages. It's so easy to fall into the rut of linear thinking that we should consider taking advantage of the tendency to think that way and use transformation to reduce nonlinearity.
A common "nonlinear effect treated as linear" is when people try to average miles per gallon (or miles per hour, or a variety of other rates) - such as "I got 15 mpg going up and 45 mpg coming back, so I averaged 30 mpg overall" (when it's actually 22.5). In terms of transformations - the reciprocal (gallons per mile) - is linear and can be averaged.
Relying on a False Premise
Seemingly mathematical arguments may just be based on bad premises (such as one requiring selecting from the positive integers with equal probability - an impossibility that completely sinks the argument that relies on it).
That "infinity" thingy can be tricky - it seems to cause problems for journalists as well because they tend to underestimate how big it is.
Treating percentiles of distributions as if they were additive is unfortunately extremely common. In the case of official estimates of total oil reserves, it means that we probably have a fair bit more oil that we think.
What's the square root of that?
Or, sometimes, it seems, mathiness comes in because someone has no clue what the heck they're talking about, so we can be told that the Maya knew how to take the square root of a rectangle.
Mathematical arguments can feel unsually convincing, even unassailable, and we're awash in them for precisely that reason. It's too easy to forget that just because something seems to be laid out mathematically, it's not necessarily true - or even meaningful at all. Mathiness, like truthiness, is all around us. Even among skeptics, it's possible to put too much store in an argument couched in mathematical terms. We should be at least as skeptical of mathematical arguments - and in basically the same kinds of ways - as any other kinds of arguments, because we're all too often misled by them.
Unfortunately, it seems that we sometimes accept the (often implicit) conclusions of a mathematical argument without even realizing that an argument was being made.
If we fail to treat these arguments with the skepticism they deserve, we're open to being deceived by charlatans.