Fractal dimension retraction or converse error?

As Ivars Peterson and others have pointed out, Jackson Pollock’s paintings can be analyzed mathematically as fractals, and they turn out to have a distinctive fractal dimension. As various articles have pointed out, inauthentic (forged) Pollocks have incorrect fractal dimensions.

What does “incorrect” mean in this context? It means that a genuine Pollock painting will have a fractal dimension in a certain range, depending on the year in which it was produced. If a putative Pollock work has a significantly different fractal dimension, it must be a fake.

But then there has been a recent controversy on this subject. The claim that fractal dimension can be used for the detection of forged Pollocks has been called into question. But are the objections valid? The usual argument seems to go something like this:

  1. Genuine Pollocks have a fractal dimension in the range of a to b.
  2. This so-called Pollock has a fractal dimension of c, where c is not in that range.
  3. Therefore this painting is not a genuine Pollock.

This is certainly a valid syllogism — indeed a classical example of Aristotelian logic.

The only trouble is that recent objections — such as the one articulated on “Marketplace” on January 30 — do not follow this usual argument. Instead, they seem to reason something like this:

  1. Genuine Pollocks have a fractal dimension in the range of a to b.
  2. This so-called Pollock has a fractal dimension of c, where c is in that range.
  3. Therefore this painting is a genuine Pollock.

They then proceed to exhibit a counterexample — a fake Pollock with the appropriate fractal dimension — and thereby challenge step #1 of the revised syllogism.

But this is not a valid syllogism. It’s a classic fallacy: the fallacy of the converse. I don’t know whether the error was committed by the Marketplace guest, well-known physicist Lawrence Krauss, or by the article in Nature to which he refers, but Krauss claims that the article supports the following assertion:

…several childlike sketches drawn by one of the authors appeared to satisfy the criteria that had been claimed to distinguish Pollock from mere mortals.

This is akin to claiming that “all genuine Pollocks have a certain fractal dimension” is equivalent to “all works with that fractal dimension are genuine Pollocks.” They’re not equivalent! Finding a fake Pollock with the right fractal dimension in no way demolishes the original claim.



Categories: Math