According to that critique it isn't his math that's wrong, but his whole understanding of machine learning. He has apparently written a book on statistics and has a degree in math somewhere along the line, but I've seen people who are quite mathematically and statistically adept still draw unsupported conclusions of the kind Gottman is making.

What he has done apparently is shown that "there exists a set of characteristics that can be used to separate these entities into two classes", and then claimed "this set of characteristics is generalizable to all entities of the same kind (marriages)".

The problem with this is that if you look at any collection of entities (up vs down days in the stock market) and any reasonably large set of attributes of those data (daily rainfall, previous day's trading volume, etc) you'll find a way to separate the entities into two classes without much difficulty, particularly if you allow combinations of factors into your classifier (N attributes gives N*[N-1] pairs and so on, creating a combinatoric explosion of possibilities, one of which is nearly certain to be "accurate" by chance alone.

So he basically has nothing. No conclusion can be drawn from what he's done, if the Slate article is correct.