The Monty Hall dilemma is a famously vexing mathematical puzzle. The premise of the problem, which takes its name from the former host of the game show “Let’s Make a Deal,” is this: Imagine that you are a game show contestant given a choice between three numbered doors. Behind one door is a spectacular prize, such as a car, while the other two doors hide booby prizes, such as live goats. After making your selection — door number 1, say — the host opens one of the remaining doors — number 2, for instance — to reveal a goat. Now, you have a choice: Do you stick with your original choice of door number 1 or switch to door number 3? Which gives you a better chance of winning the car?

If you sit down and work through the probabilities, you’ll discover that you should always switch doors; doing so doubles your odds of taking home the big prize. It’s a non-intuitive answer and many people get it wrong — the problem has made even accomplished mathematicians look a bit bird-brained.

Actual birds, on the other hand, grasp the probabilities quickly. In a 2010 study, psychologists at Whitman College found that pigeons outperformed college students on a variant of the Monty Hall problem. The birds recognized with some rapidity that they were most likely to earn a food reward when they switched their initial choice, while the humans, the researchers reported, “failed to adopt optimal strategies, even with extensive training.”

These prize-winning pigeons are just one data point in Jennifer Ackerman’s new book “The Genius of Birds,” which thoroughly and cheerfully dismantles the notion that birds should have dunce caps affixed to their feathered heads. “The expression ‘bird brain,’ for a stupid, foolish, or scatterbrained person, entered the English language in the early 1920s because people thought of birds as mere flying, pecking automatons, with brains so small they had no capacity for thought at all,” writes Ackerman, a long-time science and nature writer. “That view is a gone goose.” Decades of careful science have redeemed the avian mind; studies have revealed that birds can make and use tools, learn through observation, recognize their own reflections, delay gratification, deceive rivals, count, cooperate, and more.

And birds aren’t the only animals now being celebrated for their smarts. Scientists have documented sophisticated cognitive skills in an ever-expanding menagerie of creatures, including fish, lizards, spiders, and sheep. “The Remarkable Intelligence of Animal X” has become a full-blown sub-genre of popular science writing; we now have not only “The Genius of Birds,” but also “The Genius of Dogs,” “Dog Sense,” “Bird Sense,” “Mind of the Raven,” “Gifts of the Crow,” “The Soul of an Octopus,” “The Dolphin in the Mirror,” “The Cultural Lives of Whales and Dolphins,” “Honeybee Democracy,” and “The Inner World of Farm Animals.”

Together, this growing literature does more than simply showcase the keenness of animal minds. It also highlights the limitations of our own. For centuries, we Homo sapiens have struggled to understand animal intelligence — and have displayed a disappointing reluctance to explore it with real open-mindedness and integrity. “Even though for many people animal intelligence is self-evident, science never takes anything at face value,” writes Frans de Waal, a leading primatologist, whose provocative new book, “Are We Smart Enough to Know How Smart Animals Are?,” was published in late April. “We want proof, which with regard to animal cognition has now become overwhelming — so much so, in fact, that we risk forgetting the immense resistance that we had to overcome.”

Our own species has shown a historical tendency to believe that we sit at the center of the universe and at the top of the ladder of life. Many notable philosophers — including Aristotle, Kant, and Descartes — opined on the inferiority of animals. We were rational, conscious creatures; all other species were unthinking machines, mere bundles of reflexes and instincts. Descartes summed up his view starkly: “The greatest of all the prejudices we have retained from our infancy is that of believing that the beasts think.”

Even after Darwin began chipping away at the idea that humans stood apart from the rest of the animal kingdom, many continued to draw a bright, dividing line between “human” and “animal” cognition. De Waal dubs this belief “Neo-Creationism,” which, as he puts it, “accepts evolution but only half of it. Its central tenet is that we descend from the apes in body but not in mind. Without saying so explicitly, it assumes that evolution stopped at the human head.”

It wasn’t simply that we were ignorant of animal abilities; in many cases, we refused to even probe them. In 1866, as de Waal points out, the Linguistic Society of Paris forbade research into the origins of language. It would be easy to dismiss this denialism as a relic of a less enlightened time if modern linguists hadn’t, at a 1980 conference, proposed a ban on efforts to teach language to animals.

Even as scientists began to present evidence of animals’ cognitive prowess — reporting, for instance, that Japanese macaques had developed what appeared to be cultural traditions or that an African grey parrot named Alex seemed to understand abstract concepts, such as color, shape, and number — some scholars minimized the significance of the findings. De Waal often found himself at the receiving end of this sort of skepticism. “I can’t count the number of times I have been called naïve, romantic, soft, unscientific, anthropomorphic, anecdotal, or just a sloppy thinker for proposing that primates follow political strategies, reconcile after fights, empathize with others, or understand the social world around them,” he writes.

Why have we been so reluctant to give animals their due? De Waal’s book provides a long overdue reckoning. At the most basic level, the idea that animal minds might have something in common with our own is threatening, a challenge to “the fragile human ego,” as de Waal puts it. Canny animals give us existential fits; if monkeys have culture, then suddenly humans don’t seem quite so special.

We have also been fundamentally unimaginative about what “intelligence” means — about the forms it can take and the types of bodies and brains that can give rise to it. “The avian brain,” Ackerman points out, “had no cortex like ours, where all the ‘smart’ stuff happens.” The octopus is even more alien. How could a spineless, asocial, eight-armed sea creature with more neurons in its tentacles than in its brain possibly be capable of using tools or navigating mazes?

Our human-centered view of the world not only blinded us to the possibility of animal intelligence but also misled some of the scientists who actually did go looking for it. Often, the “IQ” tests we gave other species failed to reflect their unique biology, behavior, and lifestyles. For instance, Asian elephants flunked the classic mirror test — showing no sign of recognizing their own reflections — until researchers swapped the small, human-sized mirror they were using for a more, well, elephantine one. Likewise, octopuses initially seemed unable to learn how to open clear, plastic jars containing tasty crayfish. But the cephalopods often use chemical and tactile cues, rather than vision, to find prey; when the researchers smeared the outside of the jars with herring slime, the octopuses aced the task.

Even when studying our closest primate relatives, our anthropocentrism can set animals up to fail. Apes and monkeys, de Waal explains, appeared relatively unskilled at facial discrimination when researchers tested their ability to tell individual humans apart. But when de Waal and his colleagues presented chimps with photographs of other chimpanzees, the apes displayed a clear facility with faces. “In other words,” de Waal writes, “what is salient to us — such as our own facial features — may not be salient to other species. Animals often know only what they need to know.”

Indeed, it makes little sense to talk about “intelligence” as a single, uniform trait. Different species display wildly different cognitive talents, depending on the circumstances in which they live and the survival challenges they face. Though “anthropomorphism” has long been a dirty word in science, comparative psychologist Gordon Burghardt, of the University of Tennessee, has made a case for what he calls “critical anthropomorphism,” in which researchers try to project themselves into the lives of other creatures. Burghardt writes: “We need to use all our scientific and natural history knowledge about a species, including its physiology, ecology, and sensory abilities to develop testable hypotheses, which may indeed be based on ‘hmm, what would I do if I were in a similar situation to the other species?’”

Perhaps cold, hard scientific objectivity, to the extent that such a thing even exists — the assumption that animals are automatons is plenty value-laden — is not the best starting point when it comes to animal intelligence. Instead, maybe we should operate, as de Waal suggests, from a place of empathy and respect. That doesn’t mean we abandon the rigorous process of doing science. But — given how wrong we’ve been in the past and what we know now — perhaps we should begin with a real, good-faith attempt to imagine how other species experience the world and with the humble acknowledgement that there may be more going on in their minds than we will ever know.

Emily Anthes is a journalist and author who specializes in telling the stories of science. Her work has appeared in The New York Times, The New Yorker, Wired and Scientific American, among other publications.