Can Humans Really Discriminate 1 Trillion Odors?

A recent paper in a prominent science magazine claims to show that humans can discriminate at least 1 trillion odors. The authors reached that conclusion after performing just 260 comparisons of two smells, of which about half could be discriminated. Furthermore the paper claims that the human ability to discriminate smells vastly exceeds our abilities to discriminate colors or musical tones. Here I show that all these statements are wrong by astronomical factors. A reanalysis of the authors' experiments shows they are also consistent with humans discriminating just 10 odors. The paper's extravagant claims are based on errors of mathematical logic. Further analysis highlights the importance of establishing how many dimensions the perceptual odor space has. I review some arguments on the topic and propose experimental avenues towards an answer.