I got impatient, so I decided to see if he'd published any papers on it (which I would have known had I clicked on your wiki link). He did, in
Nature in 1993, and in the following year in response to critiques. I also recently wrote a blog post
Review of “Anthropic Bias Observation Selection Effects in Science and Philosophy” which reviews the (freely available) book
Anthropic Bias Observation: Selection Effects in Science and Philosophy. The author of that book not only goes into the Doomsday Argument (which is usually attributed to Leslie, who formulated the most popular version), but devotes quite a bit to Gott's argument:
Doomsayer Gott
The incorrectness of Gott’s argument
Personally, I find that the fallacy of the assumption of randomness (which for Gott is far more overt than in other versions of the Doomsday Argument) is the biggest problem (William Eckhardt has also points out the problem of retrocausality). We may not have a privileged position, or reference frame, or time at which we find ourselves alive, seeing (as in Gott's first example) the Berlin Wall, etc., but that doesn't make it random or justify locating
t-NOW randomly within his derived interval (which based upon an assumption about the probability distribution of a random variable given that we are experiencing/observing/etc. what we are when we are and that there is no reason to think this particular time "special"). One of the reasons that his equations seem to work at times is because they allow for an unreasonable duration. Using his math, if I happen to walk in on a stranger's birthday party, am invited in because of course I am famous and my presence is a present, I would predict with high probability using Gott's math that the woman should live to be ~125-150 years old. Other times, they will fail because we cannot derive a probability distribution from the assumption that our particular observation time isn't just not "special" but is actually equiprobable (in a way that can't be defined since we don't know the possibilities it is equiprobable with.
But I would suggest checking out the book given in the link. It's free, after all.