Alex, the African Grey Parrot, was likely one of the most demonstrably intelligent non-human animals in the world when he died prematurely in September of 2007. A final paper on his mathematical prowess has just been published, showing that the bird was likely capable adding sets of numbers in his feathery little head.
Alex came to the University of Arizona from a pet store in 1977 as part of an experiment in language comprehension. At that time, it was thought that only primates (like humans and apes) and other big-brained animals had the mental oomph required to understand the concepts underlying language: parrots could mimic sounds, but as far as the parrots themselves were concerned, the sounds didn't have any meaning.
This turned out to be totally, completely wrong. Alex proved himself capable of equating language with abstract concepts. He knew colors, numbers, shapes, objects, and materials. He understood what "similar" and "different" and "none" meant, effectively comprehending "zero." He could express thoughts and feelings in English, asking for things that he wanted, and he used words that he knew in original combinations to get his point across, at one point asking researchers (out of nowhere) what color he was, and then immediately learning the word "grey."
Math turned out to be one of Alex's strong points, and a paper accepted to Animal Cognition at the end of January details the last experiments that Alex was a part of. Apparently, Alex began to add sets of numbers out of the blue while overhearing a researcher performing an unrelated experiment on another parrot. He was able to add any two sets as long as the sum was eight or less, or any three sets as long as the sum was less than seven. He could add three sets when he was unable to see them all at the same time (meaning that he was adding them in his head and not just counting visible objects), and he even added Arabic numerals, implying that he could conceptually equate symbols with amounts.
The only other animal that has been able to do this was a chimpanzee named Sheba, who probably had a brain approximately the size of Alex's entire body. There's really no telling what other mental feats Alex might have been capable of had he not died so young, but the research team is currently working with a four-year old bird named Griffin to show that it's parrots in general, and not just Alex, who are smarter than we give them credit for.
The video below shows a similar experiment recorded with Alex in 2005, where he adds two sets of objects hidden under cups. For an overview of Alex's other skills, William Shatner narrates a ten minute video on him here.