It is the time of year when newly arrived students gather around the university in uneasy groups, shuffling like swallows waiting to migrate. All have passed, quite recently, through the trial of school exams. Meeting them, I remember all too well the shock of exam papers whose questions bore little relation to anything I had learned. The key thing my teachers told me was not to panic: read the rubric on the paper, check you know how many questions to answer, and finish each question off as best you can.

I had a particular dislike of maths tests at school: complex questions constructed around unlikely scenarios such as a baths with running taps and the plug out, or a weightless block sliding down an inclined plane with strange frictional qualities. Who could arrange such a thing, and why would they do it? That was not, as successive maths teachers explained, a relevant way to think. It was the units, the numbers and the equations that mattered.

The New Scientist Feedback column has had a lot of fun over the years with the odd units people use – lengths expressed in double decker buses, or weights in whales (clearly a highly variable unit), or areas as multiples of Wales. Their correspondents rejoice in the bizarre, but the underlying message is always the clarity that sensible SI units and a bit of careful thought would bring.

Environmentalists love to cite statistics of all kinds, and they too like striking metrics. They too have fallen in love with a new mega-unit: Earths. Ever since the moon shots of the 1960s, the idea of Only One Earth has been at the heart of environmentalist argument about the shape of human economy and society (a ‘one world environmental ontology’ as Chris Sandbrook calls it).   But the current fashion for the earth as unit is a little more specific. Two framings have become particularly dominant.

The first is the idea of ‘Half Earth’. Nature, we are told, needs 0.5 Earths. The 20% of land in protected areas that comprises the CBD 2020 target is too little: conservationists are urged to place 50% of the earth in protected areas. As Chris Sandbrook has pointed out, and as the wider literature has discussed, this wonderfully disguises a lot of tricky politics (since people already own and live on most of the land conservationists want, so ‘saving’ those areas is likely to be expensive and unpopular as well as being unfair and unjust).

The second is the idea of measuring human consumption in Earths. Humans, we are told, are using 1.7 Earths a year: the Global Footprint Network calculates that ‘Earth Overshoot Day’ was 1 August in 2018. The idea of Global Footprint provides a metric of the ecological resources and services consumed by the economy through pollution, overfishing, unsustainable agriculture, overharvesting of forests, and emission of carbon dioxide.

The metrics behind Half Earth or Earth Overshoot Day are complicated, and you probably have to be a bit of a science geek to dig in to the algebra.   But in a sense the detail does not matter greatly. Neither is really intended to be scientific. They are both metaphors, framing devices in environmentalist arguments about future actions.

The problem is the mathematics pushes in two very different directions – the two ‘Earth’ metaphors reflect very different ideas about the future direction of human society and economy.

Half Earth proposes a separation of nature and human society, half an Earth of dense human settlement and efficient production, and half of biodiverse ecosystems and little human presence (an essentially ecomodernist vision). The Global Footprint proposes a reduction of the energy and material basis of production and consumption with a redistribution towards poor countries (a kind of degrowth vision).

The tricky thing is that both these equations need to be solved at the same time. It is no good trying to pretend that one is more important than the other. Conservationists running the numbers and supporting the idea of Half Earth are doing the arithmetic right. But so are other environmentalists calculating the Global Footprint.

There is only one Earth to play with. Nature needs space, but it also needs a significant reduction in human consumption. Space where non-human lives can flourish needs to be doubled. But net consumption (in all its forms) also needs to be halved.

The calculations we need to navigate forwards are much more complicated than either crude ‘Earth Unit’ headline might suggest. Earth mathematics is going to be complicated: tessellating economic production and countryside, trading off reductions in energy and material use and the restoration of ecosystems, the integration of human society with non-human nature at every scale from the biotechnology vat to the productive ocean, the garden to the biosphere.

The Twenty First century offers a tricky exam paper for humanity, and we need to get the answers right if we are to make it through with any space for human and non-human flourishing.

The alert student would be well advised to tackle more than one question.