Even a cursory understanding of, say, the trials our ancestors went through in the 1300s makes it obvious that he's wrong to project back his little exponential curve that far.
See http://www.revisedhistory.org/view-garry-kasparov.htm for details of Kasparov's beliefs. See http://en.wikipedia.org/wiki/British_Agricultural_Revolution and http://en.wikipedia.org/wiki/Medieval_demography for basic historical facts that show some basic flaws in Kasparov's arguments. (About his points on Roman numerals, they are indeed entirely unsuited to calculation. The Romans did all calculations using abacuses, numerals were just their way of writing down the answers for which purpose they are just fine.)
Also, he sure is good at chess.
And yes, he's amazing at chess. However that is not necessarily an endorsement of his abilities at anything else. Paranoid delusions seem to be a significant occupational hazard at top levels of play. (See Bobby Fischer for a significant example.)
Any theories on why that is?
Maybe having super-charged pattern-making ability makes for great chess AND great paranoia.