# A pile of dough

Reading Is Maths Real? by Eugenia Cheng, 17 May 2023

Let’s start with an obvious trick question: why does 1 plus 1 equal 2? Well, it often doesn’t. Add one pile of dough to one pile of dough and you get, well, one pile of dough.

This looks like a twisty and trivial point, but it isn’t. Mathematics describes the logical operations of logical worlds, but you can dream up any number of those, and you’re going to need many more than one of them to even come close to modelling the real world.

“Deep down,” writes mathematician Eugenia Cheng, “maths isn’t about clear answers, but about increasingly nuanced worlds in which we can explore different things being true.”

Cheng wants the reader to ask again all those “stupid” questions they asked about mathematics as kids, and so discover what it feels like to be a real mathematician. Sure enough, mathematicians turn out to be human beings, haunted by doubts, saddled with faulty memories, blessed with unsuspected resources of intuition, guided by imagination. Mathematics is a human pursuit, depicted here from the inside.

We begin in the one-dimensional world of real numbers, and learn in what kinds of worlds numbers can be added together in any order (“commutativity”) and operations grouped together any-old-how (“associativity”). Imaginary numbers (which can’t be expressed as digits; think pi) add a second dimension to our mathematical world, and sure enough there are now patterns we can see that we couldn’t see before, “when we were all squashed into one dimension”.

Keep adding dimensions. (The more we add to our mathematical universe, however, the less we can rely on our visual imagination, and the more we come to rely on algebra.) Complex numbers (which have a real part and an imaginary part) give us the field of complex analysis, on which modern physics depends.

And we don’t stop there. Cheng’s object is not to teach us maths, but to show us what we don’t know; we eventually arrive at a terrific description of mathematical braids in higher dimensions that at the very least we might find interesting, even if we don’t understand it. This is the generous impulse driving this book, and it’s splendidly realised.

Alas, Is Maths Real?, not content with being a book about what it is like to be a mathematician, also wants to be a book about what it is like to be Eugenia Cheng, and success, in this respect, leads to embarrassment.

There’s Cheng’s inner policeman, reminding her, as she discusses the role of pictures in mathematics “to acknowledge that this is thus arguably ableist and excludes those who can’t see.”

There are narcissistic exclamations that defy parody, as when Cheng explains that “the only thing I want everyone to care about is reducing human suffering, violence, hunger, prejudice, exclusion and heartbreak.” (Good to know.)

There are the Soviet-style political analogies for everything. Imaginary and complex numbers took a while to be accepted as numbers because, well, you know people: “some people lag behind, perhaps accepting women and black people but not gay people, or maybe accepting gay, lesbian and bisexual people but not transgender people.”

A generous reader may simply write these irritations off, but then Cheng’s desire to smash patriarchal power structures with the righteous hammer of ethnomathematics (which looks for “other types of mathematics” overlooked, undervalued or suppressed by the colonialist mainstream) tips her into some depressingly hackneyed nonsense. “Contemporary culture,” she tells us, “is still baffled by how ancient cultures were able to do things like build Stonehenge or construct the pyramids.”

Really? The last time I looked, the answers were (a) barges and (b) organised labour.

Cheng tells us she is often asked how she comes up with explanations and diagrams that bring clarity “to various sensitive, delicate, nuanced and convoluted social arguments.” Her training in the discipline of abstract mathematics, she explains, “makes those things come to me very smoothly.”

How smoothly? Well, quite early in the book, “intolerance of intolerance” becomes “tolerance” through a simple mathematical operation — a pratfall in ethics that makes you wonder what kind of world Cheng lives in. Cheng’s abstract mathematics may well be able solve her real-world problems — but I suspect most other people’s worlds feel a deal less tractable.

# On not being a horrible person

Reading The Human Mind by Paul Bloom for New Scientist, 11 May 2023

Inspired, he tells us, by The Origin of the Universe, John Barrow’s 1994 survey of what was then known about cosmology, the Canadian American psychologist Paul Bloom set about writing an introductory tome of his own: a brief yet comprehensive guide to the human mind.

Emulating Barrow’s superb survey has been hard because, as Bloom cheekily points out, “the mysteries of space and time turn out to be easier for our minds to grasp than those of consciousness and choice.”

The first thing to say — though hardly the most interesting — is that Bloom nevertheless succeeds, covering everything from perception and behaviour to language and development; there’s even a small but very worthwhile foray into abnormal psychology. It’s an account that is positive, but never self-serving. Problems in reproducing some key studies, the field’s sometimes scandalous manipulation of statistics, and the once prevailing assumption that undergrad volunteers could accurately represent the diversity of the entire human species, are serious problems, dealt with seriously.

Of course Bloom does more than simply set out the contents of the stall (with the odd rotten apple here and there); he also explores psychology’s evolving values. He recalls his early behaviourist training, in a climate hostile to (then rather woolly) questions about consciousness. “If we were asked to defend our dismissal of consciousness,” he recalls, “we would point out that intelligence does not require sentience.”

Intelligence is no longer the field’s only grail, and consciousness is now front and centre in the science of the mind. This is not only a technical advance; it’s an ethical one. In 1789 Jeremy Bentham asked whether the law could ever refuse its protection to “any sensitive being”, and pointed out that “The question is not, Can [certain beings] reason?, nor Can they talk? but, Can they suffer?”

Suffering requires consciousness, says Bloom; understanding one enables us to tackle the other; so the shift in interest to consciousness itself is a welcome and humanising move.

This strong belief in the humanitarian potential of psychology allows Bloom to defend aspects of his discipline that often discomfort outside observers. He handles issues of environmental and genetic influences on the mind very well, and offers a welcome and robust defence of Alfred Binet’s 1905 invention, the measure of general intelligence or “intelligence quotient”. Bloom shows that the IQ test is as robust a metric as anything in social science. We know that a full half of us score less than 100 on that test; should this knowledge not fill us with humility and compassion? (Actually our responses tend to be more ambiguous. Bloom points out that Nazi commentators hated the idea of IQ because they thought Jews would score better than they would.)

Bloom is concerned to demonstrate that minds do more than think. The privileging of thinking over feeling and intuiting and suffering is a mistake. “A lot depends on what is meant by ‘rational.’ Bloom writes. If you’re stepping outside and it’s raining and you don’t want to get wet, it’s rational to bring an umbrella. But rationality defined in this manner is separate from goodness. “Kidnapping a rich person’s child might be a rational way to achieve the goal of getting a lot of money quickly,” Bloom observes, “so long as you don’t have other goals, such as obeying the law and not being a horrible person.”

Bloom’s ultimate purpose is to explain how a robustly materialistic view of the mind is fully compatible with the existence of choice and morality and responsibility. This middle-of-the-road approach may disappoint intellectual storm-chasers, but the rest of us can can be assured of an up-to-the-minute snapshot of the field, full of unknowns and uncertainties, yes, and speculations, and controversies — but guided by an ever-more rounded idea of what it is to be human.

# Dreams of a fresh crab supper

Reading David Peña-Guzmán’s When Animals Dream for New Scientist, 17 August 2022

Heidi the octopus is dreaming. As she sleeps, her skin changes from smooth and white to flashing yellow and orange, to deepest purple, to a series of light greys and yellows, criss-crossed by ridges and spiky horns. Heidi’s human carer David Scheel has seen this pattern before in waking octopuses: Heidi, he says, is dreaming of catching and eating a crab.

The story of Heidi’s dream, screened in 2019 in the documentary “Octopuses: Making Contact”, provides the starting point for When Animals Dream, an exploration of non-human imaginations by David Pena-Guzman, a philosopher at San Francisco State University.

The Roman philosopher-poet Lucretius thought animals dreamt. So did Charles Darwin. The idea only lost its respectability for about a century, roughly between 1880 to 1980, when the reflex was king and behaviourism ruled the psychology laboratory.

In the classical conditioning developed by Ivan Pavlov, it is possible to argue that your trained salivation to the sound of a bell is “just a reflex”. But later studies in this mould never really banished the interior, imaginative lives of animals. These later studies relied on a different kind of conditioning, called “operant conditioning”, in which you behave in a certain way before you receive a reward or avoid a punishment. The experimenter can claim all they want that the trained rat is “conditioned”; still, that rat running through its maze is acting for all the world as though it expects something.

In fact, there’s no “as though” about it. Pena-Guzman, in a book rich in laboratory and experimental detail, describes how rats, during their exploration of a maze, will dream up imaginary mazes, and imaginary rewards — all as revealed by distinctive activity in their hippocampuses.

Clinical proofs that animals have imaginations are intriguing enough, but what really dragged the study of animal dreaming back into the light was our better understanding of how humans dream.

From the 1950s to the 1970s we were constantly being assured that our dreams were mere random activity in the pons (the part of the brainstem that connects the medulla to the midbrain). But we’ve since learned that dreaming involves many more brain areas, including the parietal lobes (involved in the representation of physical spaces) and frontal lobes (responsible among other things for emotional regulation).

At this point, the sight of a dog dreaming of chasing a ball became altogether too provocative to discount. The dog’s movements while dreaming mirror its waking behaviours too closely for us to say that they lack any significance.

Which animals dream? Pena-Guzman’s list is too long to quote in its entirety. There are mice, dogs and platypuses, beluga whales and ostriches, penguins, chameleons and iguanas, cuttlefish and octopuses — “the jury is still out on crocodiles and turtles.”

The brain structures of these animals may be nothing like our own; nonetheless, studies of sleeping brains throw up startling commonalities, suggesting, perhaps, that dreaming is a talent to which many different branches of the evolutionary tree have converged.

Pena-Guzman poses big questions. When did dreaming first emerge and why? By what paths did it find its way into so many branches of the evolutionary tree? And — surely the biggest question of all — what are we do with this information?

Pena-Guzman says dreams are morally significant “because they reveal animals to be both carriers and sources of moral value, which is to say, beings who matter and for whom things matter.”

In short, dreams imply the existence of a self. And whether or not that self can think rationally, act voluntarily, or produce linguistic reports, just like a human, is neither here nor there. The fact is, animals that dream “have a phenomenally charged experience of the world… they sense, feel and perceive.”

Starting from the unlikely-sounding assertion that Heidi the octopus dreams of fresh crab suppers, Pena-Guzman assembles a short, powerful, closely argued and hugely well evidenced case for animal personhood. This book will change minds.

# Pig-philosophy

Reading Science and the Good: The Tragic Quest for the Foundations of Morality
by James Davison Hunter and Paul Nedelisky (Yale University Press) for the Telegraph, 28 October 2019

Objective truth is elusive and often surprisingly useless. For ages, civilisation managed well without it. Then came the sixteenth century, and the Wars of Religion, and the Thirty Years War: atrocious conflicts that robbed Europe of up to a third of its population.

Something had to change. So began a half-a-millennium-long search for a common moral compass: something to keep us from ringing each other’s necks. The 18th century French philosopher Condorcet, writing in 1794, expressed the evergreen hope that empiricists, applying themselves to the study of morality, would be able “to make almost as sure progress in these sciences as they had in the natural sciences.”

Today, are we any nearer to understanding objectively how to tell right from wrong?

No. So say James Davison Hunter, a sociologist who in 1991 slipped the term “culture wars” into American political debate, and Paul Nedelisky, a recent philosophy PhD, both from the University of Virginia. For sure, “a modest descriptive science” has grown up to explore our foibles, strengths and flaws, as individuals and in groups. There is, however, no way science can tell us what ought to be done.

Science and the Good is a closely argued, always accessible riposte to those who think scientific study can explain, improve, or even supersede morality. It tells a rollicking good story, too, as it explains what led us to our current state of embarrassed moral nihilism.

“What,” the essayist Michel de Montaigne asked, writing in the late 16th century, “am I to make of a virtue that I saw in credit yesterday, that will be discredited tomorrow, and becomes a crime on the other side of the river?”

Montaigne’s times desperately needed a moral framework that could withstand the almost daily schisms and revisions of European religious life following the Protestant Reformation. Nor was Europe any longer a land to itself. Trade with other continents was bringing Europeans into contact with people who, while eminently businesslike, held to quite unfamiliar beliefs. The question was (and is), how do we live together at peace with our deepest moral differences?

The authors have no simple answer. The reason scientists keep trying to formulate one is same reason the farmer tried teaching his sheep to fly in the Monty Python sketch: “Because of the enormous commercial possibilities should he succeed.” Imagine conjuring up a moral system that was common, singular and testable: world peace would follow at an instant!

But for every Jeremy Bentham, measuring moral utility against an index of human happiness to inform a “felicific calculus”, there’s a Thomas Carlyle, pointing out the crashing stupidity of the enterprise. (Carlyle called Bentham’s 18th-century utilitarianism “pig-philosophy”, since happiness is the sort of vague, unspecific measure you could just as well apply to animals as to people.)

Hunter and Nedelisky play Carlyle to the current generation of scientific moralists. They range widely in their criticism, and are sympathetic to a fault, but to show what they’re up to, let’s have some fun and pick a scapegoat.

In Moral Tribes (2014), Harvard psychologist Joshua Greene sings Bentham’s praises:”utilitarianism becomes uniquely attractive,” he asserts, “once our moral thinking has been objectively improved by a scientific understanding of morality…”

At worst, this is a statement that eats its own tail. At best, it’s Greene reducing the definition of morality to fit his own specialism, replacing moral goodness with the merely useful. This isn’t nothing, and is at least something which science can discover. But it is not moral.

And if Greene decided tomorrow that we’d all be better off without, say, legs, practical reason, far from faulting him, could only show us how to achieve his goal in the most efficient manner possible. The entire history of the 20th century should serve as a reminder that this kind of thinking — applying rational machinery to a predetermined good — is a joke that palls extremely quickly. Nor are vague liberal gestures towards “social consensus” comforting, or even welcome. As the authors point out, “social consensus gave us apartheid in South Africa, ethnic cleansing in the Balkans, and genocide in Armenia, Darfur, Burma, Rwanda, Cambodia, Somalia, and the Congo.”

Scientists are on safer ground when they attempt to explain how our moral sense may have evolved, arguing that morals aren’t imposed from above or derived from well-reasoned principles, but are values derived from reactions and judgements that improve the odds of group survival. There’s evidence to back this up and much of it is charming. Rats play together endlessly; if the bigger rat wrestles the smaller rat into submission more than three times out of five, the smaller rat trots off in a huff. Hunter and Nedelisky remind us that Capuchin monkeys will “down tools” if experimenters offer them a reward smaller than that they’ve already offered to other Capuchin monkeys.

What does this really tell us, though, beyond the fact that somewhere, out there, is a lawful corner of necessary reality which we may as well call universal justice, and which complex creatures evolve to navigate?

Perhaps the best scientific contribution to moral understanding comes from studies of the brain itself. Mapping the mechanisms by which we reach moral conclusions is useful for clinicians. But it doesn’t bring us any closer to learning what it is we ought to do.

Sociologists since Edward Westermarck in 1906 have shown how a common (evolved?) human morality might be expressed in diverse practices. But over this is the shadow cast by moral skepticism: the uneasy suspicion that morality may be no more than an emotive vocabulary without content, a series of justificatory fabrications. “Four legs good,” as Snowball had it, “two legs bad.”

But even if it were shown that no-one in the history of the world ever committed a truly selfless act, the fact remains that our mythic life is built, again and again, precisely around an act of self- sacrifice. Pharaonic Egypt had Osiris. Europe and its holdings, Christ. Even Hollywood has Harry Potter. Moral goodness is something we recognise in stories, and something we strive for in life (and if we don’t, we feel bad about ourselves). Philosophers and anthropologists and social scientist have lots of interesting things to say about why this should be so. The life sciences crew would like to say something, also.

But as this generous and thoughtful critique demonstrates, and to quite devastating effect, they just don’t have the words.