We humans love to look at other species. The BBC series Planet Earth II is a huge hit. As well as the authority of David Attenborough’s voice, this is because it offers a series of incredible HD glimpses of the secret lives of animals. The latest sequence that had my family glued to the screen featured a pride of lions chasing a giraffe. Astounding.

Or is it? Is watching nature documentaries an enlightened attempt to comprehend our fellow creatures, or just another example of human beings imposing a gaze of power and knowledge on animals – classifying, controlling and ultimately skinning and stuffing them in the name of science?

Carl Linnaeus’s Systema Naturae, c 1735. Photograph: Wellcome Collection

An exhibition that has just opened at the Wellcome Collection in London mulls this over. Making Nature: How We See Animals is not about natural history in the sense of looking at a particular animal, or behaviour, or evolution. It is instead about the psychology of how we observe other species.

It begins with Carl Linnaeus (1707-78), the father of modern natural history, who first ordered knowledge of the natural world. Linnaeus classified flora and fauna into the great divisions and subdivisions of the natural world that are still used in science today: family, genus and species. Without Linnaeus we would not even have the scientific concept of a species, so Charles Darwin could never have theorised how species originated. In Making Nature there is a fish collected by Linnaeus himself, lent by the Linnaean Society of London. It has been stuck in a book, flattened and dried, looking like a brown fossil in a rock. It might easily be millions of years old. In fact, it swam in 18th-century seas.

Linnaeus in this exhibition represents a human urge to “order” nature, to make sense of it and perhaps by doing so master it. Museum displays – including that stupendous Victorian invention, the Natural History Museum – are examined not for what they teach but how they define and shape the way we see animals. The strange world of zoos and their architecture is also held up for scrutiny.

By contrast, the hero of the exhibition is a New Yorker called Antoine Yates. In 2003, Yates had to go to an ER after he was mauled by his pet tiger. It turned out that he had been keeping the tiger, called Ming, for years in a high-rise apartment in a public housing scheme where no pets were allowed. He also had an alligator. A video by Phillip Warnell reconstructs this unusual situation. A tiger prowls round an apartment, sniffing at the furniture, a surreal interloper in a human home.

Fox cubs by Peter Spicer, taxidermist of Leamington Spa, c 1875. Photograph: Wellcome Collection

If Linnaeus imposed order on the natural world, Yates created a hilarious chaos. He put humans and man-eaters in the same apartment, utterly subverting every rule and assumption going. Instead of being behind glass in a zoo, the tiger entered the human world. Imagine being his postman.

The idea that by ordering nature we control it goes back to the 1960s theories of Michel Foucault. It also, of course, speaks to our guilt about the harm we are doing to the planet. But looking at nature is surely the least destructive of all the things human beings do to the natural world.

Birdsong notated in the Athanasius Kircher, Musurgia Universalis, c 1650. Photograph: Wellcome Collection

Far from a pornography of power, the images of nature created by natural history museums and even zoos, at their best, bring us closer to the natural world. The more we know about nature, the less we want to hurt it.

Looking at animals is something we can’t help doing. Learning about them is a rational outgrowth of that curiosity. Linnaeus made it possible for a new understanding of nature to develop, and the latest fruit of that enlightened inquiry is Planet Earth II. Is that just a luxurious bit of infotainment as we ravage nature? I don’t think so. Seeing animals makes us care about what happens to them. Observation is not just a means of control. It can awaken the eye of love.

At the same time that Linnaeus was classifying nature, William Hogarth engraved a gruesome scene of human contempt for our fellow animals. Hogarth’s print The First Stage of Cruelty (1751) shows boys in a London street inflicting monstrous tortures on cats and dogs. Was animal cruelty of this kind as common in premodern times as Hogarth makes it look? The point for him, as for us, is that it is abhorrent. From being cruel to cats, his antihero Tom Nero progresses in this print series to murdering a woman. One violence begets another.

Such ideas were new in the 18th century. The painter George Stubbs goes one further than Hogarth, portraying a racehorse with the sensitivity of a Rembrandt self-portrait. So does Antoine Watteau when he depicts a donkey’s melancholy eye in his masterpiece Gilles.

These artists who looked at animals with such compassion and respect were exact contemporaries of Linnaeus and belonged to the same intellectual movement: the Enlightenment.

It is to the Enlightenment and its unruly child, Romanticism, that we owe our affection for animals. As the historian Keith Thomas has shown in a classic book, before the 18th century people were taught to look down on “brute creation”. Pets, vegetarianism and green politics are modern creations that all originate in the Enlightenment.

Linnaeus, like Hogarth, Stubbs and Watteau, should be a hero to anyone who cares about our planet. You can’t save what you can’t name. Science and art look at nature not out of an urge to dominate but a longing to understand. Stubbs could place himself in the mind of a horse only after he had dissected one, for knowledge awakens sympathy.

• Making Nature: How We See Animals is at the Wellcome Collection, London, until 21 May.