But the areas where mathematics intersect with human affairs is another matter entirely. The obscurities of human activity lead to mathematical formulas, models, and projections that are tenuous at best, unreliable at worst, and deliberately obfuscatory at even worse than worst (which is sometimes why economics is called "the dismal science"); while the clarity of mathematics expresses the insolubility of certain human problems with such consternating clarity as to make a pessimist of anyone hopeful enough of a solution to conduct an investigation (which is why denial remains the best-selling coping mechanism for 10,000 years and counting).
Here are some numbers I find troubling.
Where I = (environmental) impact, P = population, A = affluence (read: rate of consumption), and T = the technology factor.
This is the Paul Ehrlich Equation, a fairly early effort to slap together a mathematical expression of humanity's bootprint on the environment. In short: environmental degradation is a product of the number of people on the planet, the amount of resources gobbled up per head, and the efficiency (or wastefulness) of the technological processes used to extract, refine, and transport those resources.
For the moment we will abstain from discussing my feelings about the size of the anthropic shit pile steaming on the Earth's surface being mathematically expressed by the construction "I, Pat."
In 2002, the United Nations Environment Programme produced a document titled Sustainable Consumption: A Global Status Report, where the Paul Ehrlich equation is refined to:
TEI is total environmental impact, P is population. UC/hp is (average) units of consumption of products and services per head of population and EE is the environmental efficiency of the production, use and disposal of those units.
This equation makes it easy to visualise the importance of considering levels of consumption of goods and services (per head) and the resources used (and waste generated) to produce those goods and services. Patterns of consumption is a term that intends to capture both these variables. Consumption pressure per head describes the (aggregated) product of the two terms UC/hd and the inverse of EE.
It is from such an equation that the concept of factor 4 (etc) emerges——being the level of change in EE that can be achieved through technical and organisational improvements (cleaner production; product re-design etc) If the intent is to reach some specific level of TEI (say for CO2 production) in a given period, then estimates of the likely population growth over that period, as well as the likely rise in the average level of consumption per head (from development, GDP growth etc), will define the factor of improvement in EE necessary to compensate for this rise.
Arguments that arise over the role of population growth in environmental degradation can also be clarified with reference to this equation, since it is clear that the issue is the product of population numbers times the average consumption pressure per head. Rebound effects arise from a relationship between UC/hd and EE, where improvements in EE generate increased consumption per head.
The paper makes a cogent case for sustainable consumption habits—but come on, when was the last time anyone listened to good advice from the United Nations?
If someone is sufficiently concerned about the potentially disruptive or even cataclysmic consequences that might arise from changes to the environment wrought by human activity (here are some reasons for concern), what is the most effective way to proceed?
Becoming a better consumer is a nice start—although it might not be enough to grow your own produce, buy only locally-made goods, give up disposable products, stop eating meat, trade your car for a bicycle, and turn off your air conditioner in the summer unless you can persuade other people to change their lifestyles too. (Good luck with that.)
Developing more efficient technology would be an extremely helpful contribution—but most of us aren't engineers or venture capitalists, and aren't afforded that option. What's more, a purely technological solution might only amount to a palliative in the long run. If increases in population growth and global consumption habits outstrip technological improvements (which may well be the case), all we are doing is reducing EE's rate of growth, not EE itself.
It might seem the only effective (and, on the face of it, the least disruptive) method of reducing EE is by a reduction of P. Fewer humans times more efficient technology = smaller impact. Thus, the ardent environmentalist might choose to abstain from breeding (or to only have one child, or choose to adopt) as his contribution to the anthropocene problem.
Perhaps the individual can't change the consumption habits of the aggregate. Perhaps he's incapable of inventing an engine powered by a greywater purification process. But he can choose not bring another carbon footprint onto the Earth.
This brings us to another mathematical formula, cited by Thomas Piketty as the second fundamental law of capitalism in Capital in the Twenty-First Century:
Where β = the capital/income ration, s = savings rate, and g = growth rate.
What β basically expresses is the value of an economy's capital stock (land, buildings, machinery, business holdings, etc.) over the value of its income (basically GDP minus expenses) during a given year. If β = 7, that means an economy's capital is equal to 700% of its national income that year. It would be useless trying to paraphrase arguments that Piketty wrote an entire book to make, but an essential point is that a higher β value tends to correspond to greater economic inequality. (In the first fundamental law of capitalism, β is multiplied by the rate of return on capital to produce α, capital's share of the national income. Think of it as a mathematical translation of the phrase "the rich get richer.")
The second fundamental law is less useful for gauging the capital/income ratio of a given year than for projecting how it will develop if savings and growth trends remain relatively stable over the long run. Piketty explains:
This formula...reflects an obvious but important point: a country that saves a lot and grows slowly over will over the long run accumulate an enormous stock of capital (relative to its income), which can in turn have a significant effect on the social structure and distribution of wealth.
Let me put it another way: in a quasi-stagnant society, wealth accumulated in the past will inevitably acquire disproportionate importance.
The return to a structurally high capital/income ratio in the twenty-first century, close to the levels observed in the eighteenth and nineteenth centuries, can therefore be explained by the return to a slow-growth regime. Decreased growth——especially demographic growth——is thus responsible for capital's comeback.
Read: decreased population growth.
To understand what is at issue here and its relation to the convergence process and the dynamics of inequality, it is important to decompose the growth of output into two terms: population growth and per capita output growth. In other words, growth always includes a purely demographic component and a purely economic component, and only the latter allows for an improvement in the standard of living....In 2013–2014, for example, global economic growth will probably exceed 3 percent, thanks to very rapid progress in the emerging countries. But global population is still growing at an annual rate close to 1 percent, so that global output per capita is actually growing at a rate barely above 2 percent (as is global income per capita).
Other things being equal, strong demographic growth tends to play an equalizing role because it decreases the importance of inherited wealth: every generation must in some sense construct itself.
To take an extreme example, in a world in which each couple has ten children, it is clearly better as a general rule not to count too much on inherited wealth, because the family wealth will be divided by ten with each new generation. In such a society, the overall influence of inherited wealth would be strongly diminished, and most people would be more realistic to rely on their own labor and savings....
Conversely, a stagnant or, worse, decreasing population increases the influence of capital accumulated in previous generations.
How funny. By choosing to help mitigate environmental impact by limiting population growth, the ardent environmentalist would bring about a deprecation of the g value in Piketty's formula, thus facilitating an increase in β—contributing to an increase in economic inequality, a decrease in economic mobility, a deprecation of the democratic process, and the bolstering of capital—of people who have a considerable interest in whetting the population's appetite for excessive consumption.
Choose your poison.