The question is of course provocative, but the answer (negative!?) is not so obvious after all, as we shall try to show in this article. Jokes aside, we will address a serious subject of growing and dramatic relevance: the debate on the possible scenarios for the sustainability of planet Earth in the 21st century.
We take our cue from an article on the front page of the Corriere della Sera of 16 June 2008. Its author, the economist Giovanni Sartori, comments with these bitter words on the outcome of the FAO (Food and Agriculture Organization of the United Nations) meeting held in Rome (3-5 June 2008): "The great FAO carnival has ended with the laughable and irresponsible promise of defeating hunger by 2050. Let us hope the FAO is shut down first." Referring to the conference held during those same days by the Aurelio Peccei Foundation to celebrate the 40th anniversary of the Club of Rome, Sartori remarks: "the serious discussions are held elsewhere" (16-17 June 2008), and continues: "Peccei was the first prophet of the impossibility of unlimited growth on planet Earth, just as two centuries ago the worthy Reverend Malthus was the first to glimpse the population bomb. He calculated that while population could grow in geometric progression (1,2,4,8), agricultural production can only grow in arithmetic progression (1,2,3,4). His Essay on the Principle of Population appeared in 1798, before the industrial revolution. And it is mechanized agriculture, which Malthus could not foresee, that has postponed the reckoning by two centuries. But now we are there.
"Peccei's concern, and that of the Club of Rome, was different: it warned of the imminent depletion of natural resources, and of oil in particular. Understandably, we consume too much because we are too many. But in 1972, when the first report, The Limits to Growth, appeared, the world population was 3.85 billion. Do you realise? In less than forty years it has almost doubled. […] we now live under a blanket that is too short, and pulling it to one side leaves another side uncovered."
Already from this excerpt one can glimpse how mathematics plays a far from marginal role in the heated debate over the sustainability of our planet. It is worth briefly introducing the figures Sartori mentions.

Thomas Malthus

Thomas R. Malthus (London 1766 – Haileybury 1834) was professor of Political Economy at the college of Haileybury, where the officials of the East India Company were trained. In 1798 he published the volume Essay on the Principle of Population, which Darwin himself cites in his autobiography as an important theoretical reference for his research.
In this treatise Malthus introduces an early mathematical model for the study of the evolution of a population. The question could be summarised in these terms: is there a "formula" that allows the number of individuals in a population to be estimated as time varies?
His studies led him to foresee the halt of economic development because of the incompatibility between demographic growth and the availability of food (growth of the population in geometric progression, increase of agricultural production in arithmetic progression).

Aurelio Peccei and the Club of Rome

Aurelio Peccei (Turin 1908 – Rome 1984), an industrial executive (FIAT and Olivetti) in Italy and Latin America; from his privileged vantage point on international affairs he understood, as early as the 1960s, that planetary changes were under way that would have to be faced with care and courage. In 1968 he founded the Club of Rome, with the aim of bringing together a group of scholars for a scientific analysis of the situation. The research was commissioned from the System Dynamics Group of the Sloan School of Management at MIT (Massachusetts Institute of Technology). The results, published in the treatise The Limits to Growth (1972), caused a storm, particularly among economists. At a time when the economy was pushing towards a continuous process of growth, it seemed absurd to speak of limits and control.
The book was a publishing success, with more than thirty million copies sold worldwide, but the attacks and the mockery of the academic world embittered Peccei for the rest of his life. Today, more than thirty years on, the figures seem to prove him right.

The Dennis Meadows group

The research commissioned by the Club of Rome was carried out by an international group of young researchers led by Dennis Meadows, a lecturer in System Management at the University of New Hampshire.
The problem addressed can be summarised in the following questions: do current policies lead to a sustainable future or to collapse? What can be done to create a human economy that provides for everyone's needs?
The study was conducted using suitable mathematical models processed by a powerful computer built for the purpose, World3.
In the treatise The Limits to Growth twelve scenarios are analysed, illustrating different possible models of global development over the span of two centuries, from 1900 to 2100. The analysis focuses above all on the physical limits of the planet, in particular on exhaustible natural resources and on the Earth's finite capacity to absorb industrial and agricultural emissions. Every realistic scenario shows that these limits would force growth to stop during the 21st century. In the most pessimistic scenario the book places the end of growth around 2015, a span of time that seemed sufficient to reflect, choose and adopt corrective measures even on a global scale. Two updates followed the first treatise and both substantially confirm its predictions.
In the study Beyond the Limits (1992) global developments between 1970 and 1990 are analysed, highlighting an important new fact: humanity had by then exceeded the limits of the Earth's carrying capacity and was moving into the territory of unsustainability.
The latest report, Limits to Growth: The 30-Year Update (2006), offers an even more pessimistic view than the previous ones.

Carrying capacity of the Earth – Ecological footprint

The ecological footprint is the portion of the Earth's surface that would be needed to produce the resources (cereals, fodder, timber, fish and urban land) and to absorb the emissions (carbon dioxide) of the global population.
The diagram alongside shows that human demand exceeded natural resources as early as the 1980s, while at the start of the new millennium it reaches values 20% above them.
This research enabled Dennis Meadows to win, last April, the prestigious Japan Prize (US$500,000) of the Science and Technology Foundation of Japan, with the citation: "for transformation towards a sustainable society in harmony with nature". Unfortunately, collective awareness of the problem is severely limited, as Meadows himself states in the video interview that can be downloaded from the site www.japanprize.jp

Malthus's model of evolution
The model sets out to study the dynamics of the evolution of a population. More precisely, it sets out to identify a formula that allows the number of individuals to be estimated as time varies.
Let us assume, for example, that time is measured in years and let us denote by 0P the initial size (number of individuals at stage 0). At the end of the first year the population will have passed from P0 to P1, with an increment ΔP=P 1−P0, called the growth rate.
After the second year the population will have reached P2 individuals with growth rate ΔP=P2 −P1, and so on. The growth rate will be positive when the population increases, negative when it decreases, or zero when the number of individuals remains constant (stationary situation).
The theoretical evolution of the population, described by the sequence P0, P1, P2,….., Pn, … depends decisively on which type of growth rate is assumed.
The simplest case is the one represented by a constant growth rate.

ΔP= Pn+1 – Pn = k
n=0,1,2….

The number of individuals, as time varies, is therefore described by the arithmetic progression:

P0, P1 = P0 + k, P2 = P0 + 2k P3 = P0 + 3k

The formula that concisely describes the evolution process is therefore:

Pn = P0 + n*k n= 0,1,2,…

From the formula it is clear that growth (relative to the initial value) is proportional to time:

Pn – P0 = n*k n= 0,1,2,…

Thus, for example, one can estimate that in ten years the population will have grown ten times as much as it increased in one year.
Indeed, as the graph shows, the sequence (Pn)n lies along a straight line, precisely the line with equation:

y = P0 + k*x

The parameter k (the slope of the line) describes the speed of growth, as is clear from the graph alongside, in which evolution models with different growth rates are compared . The model also makes it possible to evaluate how much time is needed for the population to reach a given threshold S.
To answer this question it is enough to solve the equation:

P0 + n*k = S n = P0 / S*k

The time required is therefore inversely proportional to the growth factor. In other words, a higher growth rate makes it possible to reach the threshold in a shorter time.
Malthus puts forward the following hypothesis: the growth rate is directly proportional to the number of individuals.

The hypothesis is reasonable, particularly when applied to an isolated population, that is, a population that has no exchanges with the outside either in terms of resources (limitation of food or of natural resources, pollution, economic constraints, limits induced by the forms of social organisation) or in terms of individuals (immigration, emigration).
Malthus's hypothesis translates in mathematical terms into the relation:

ΔP = Pn+1 – Pn = a* Pn

or into the equivalent formula:

Pn+1 = Pn + a* Pn = (1+a) Pn

where a is a constant of proportionality.

The evolution of an isolated population is therefore described by the geometric progression P0, P1 = (1+a)P0, P2 = (1+a)2P0, P3 = (1+a)3P0, … that is, by the following formula Pn = (1+a)nP0 n = 0,1,2,…

The constant a is called the growth factor.
In this case the sequence (Pn)n has as its "support curve" an exponential curve, precisely the graph of the function:

f(x) = P0*(1+a)x

The speed of growth depends strongly on the parameter a, as is clear from the graph alongside, in which the different evolutions of populations with the same initial size but different growth factors are compared.
Taking into account the properties of the exponential function, the possible evolutions are summarised in Limits to Growth: The 30-Year Update and in the following graph.

Evolution of a population:

Growth factor-Evolution of the process

-1 < a < 0 extinction

a = 0 stationary (zero growth)

a > 0 explosion

Linear growth and exponential growth
The second chapter of Limits to Growth: The 30-Year Update is titled Exponential Growth as a Driving Force, which shows how important the concept is to the subject at hand.
"The prime cause of overshooting the limits is growth, acceleration, the rapidity of change. For more than a century many physical characteristics of the global system have been growing rapidly. For example, population, food production, industrial production, resource consumption and pollution are all increasing, in many cases ever faster. Their increase follows a pattern that mathematicians call exponential growth. […]Extreme weather events, economic swings, technological transformations, epidemics and civil unrest are all phenomena that can break, with slight ups and downs, the regularity of the curves; but, on the whole, exponential growth has dominated the behaviour of the socioeconomic system ever since the industrial revolution [that is, exactly since Malthus introduced his model].
"On a finite planet, physical growth cannot go on for ever […]. This kind of growth has surprising features that make it very difficult to control. […]Quantities that grow exponentially are deceptive because most of us conceive of growth as a linear process".
The profound difference between linear growth and exponential growth can be illustrated by the graph alongside, which shows the two sequences expressly cited by Peccei: in red the arithmetic progression 1,2,3,4,… that is, Xn = 1 + n n = 0, 1, 2, … in blue the geometric progression 1,2,4,8,… that is, Xn = 2n n=0, 1, 2, …
From the second stage onwards exponential growth prevails over linear growth, and the gap widens as time passes.

Doubling time
Sartori provides a telling figure on the growth of the world population, recalling that in forty years it has almost doubled. This is not a mere observation, since the doubling time (or the halving time in the case of decline) is a fundamental parameter for assessing how rapidly a process evolves.
More precisely, the doubling time T of an expanding population (a>0) is the time the process takes to reach twice the initial value.
In the case of Malthus's model, thanks to the closed formula, the doubling time is the solution of an exponential equation:

xo (1+a)T = 2×0 → T = log2/log(1+a) a > 0

This last relation shows that the doubling time is independent of the initial value, but depends only on the growth factor "a". It is a significant index of the process and is adopted as a genuine unit of measurement.
Let us tabulate some values of the correspondence growth rate – doubling time:

In Limits to Growth: The 30-Year Update the variation in the growth rate and the corresponding doubling time of the world population over the centuries are reported:

a (%): 0,2 0,4 0,6 0,8 1 2

T (years): 347174116 87 70 35

year 1650-1900 1965-2000

Population (billions) 0,5 1,6 3,3 6

Growth rate (%) 0,3 0,7-0,82 1,2

Doubling time (years) 240 100 36 58

Malthus's model is an elementary example of a dynamic model. To use a metaphor, we can say that it does not merely describe a situation by taking a photograph, but follows the evolution of the phenomenon in itinere, as if filming it. The group of researchers led by Meadows adopted precisely these dynamic models (somewhat more complex) for the study of the planet's sustainability.

Some phenomena of exponential growth taken from Limits to Growth: The 30-Year Update

World population 1650-2050

Since the beginning of the industrial revolution, the Earth's population has grown exponentially, as the shape of the curve and the increase in the variation over time clearly show. An estimate of the doubling time of the world population, made in 1994, put it at just 38 years. Today, however, the growth rate is declining. In 2001 it was 1.3% per year, corresponding to a doubling time of 55 years.

Industrial production 1930-2010

Concentration of carbon dioxide in the atmosphere 1700-2050
The concentration of carbon dioxide (CO2):

Food production index 1950-2010
The total food production index has doubled or tripled over the last fifty years in the areas of the world most afflicted by hunger, but the per capita production index, in those same areas, has changed very little because the population has grown almost as rapidly. In the case of Africa, per capita food production actually fell by 9% between 1996 and 2001.

Costs of pollution reduction
It is possible to reduce nitrogen oxide (NOx) in the atmosphere significantly and at low cost, but beyond a certain level of abatement the costs of further reductions rise steeply.
In short, the key theme of the latest report, Limits to Growth: The 30-Year Update, is: time is the true limit of the real world.

"Given enough time, humanity is able to solve almost any problem. Growth, and especially exponential growth, is extremely insidious because it reduces the time available to act effectively. It drives a system at an ever faster pace, until the structures meant to cope with it, suited to slower rates of change, begin to falter."

Conclusions
From what has been shown here, it is clear that mathematics is not only a powerful tool for specialists, but also serves as an indispensable support for the transmission of knowledge. In recent years mathematics has proved to be a powerful language of mass communication, ever more frequently used and at times misused by the media.
The sustainability of planet Earth is a subject of great topical importance today; we hope that this short note may enable non-experts too to take in the information with a critical spirit and to face the question with greater awareness.
We like to conclude with a very telling remark by Dennis Meadows: "all any of us can do, day after day, through our work, our science, our teaching, is to try to make things a little better than they would have been otherwise. And I know I can do that".

Works consulted
Primo Brandi, Anna Salvadori, Matematica&Realtà, Didactic innovation workshops 2006-2007, University of Perugia

Thomas R. Malthus, An Essay on the Principle of the Population as it Affects the Future Improvement of Society, 1798

Donella H. Meadows, Dennis L. Meadows, Jørgen Randers, William W. Behrens III, I limiti dello sviluppo, Mondadori, Milano 1972

Donella H. Meadows, Dennis L. Meadows, Jørgen Randers, Oltre i limiti dello sviluppo, Il Saggiatore, Milano 1993

Donella H. Meadows, Dennis L. Meadows, Jørgen Randers, I nuovi limiti dello sviluppo, Mondadori, Milano 2006