Green mathematics

Summary: Modeling, analysis, and computation are used to promote environmentally conscious practices.

Green mathematics is the use of mathematical modeling, analysis, and computation to promote ecologically sound practices, such as sustainable production or reduction of pollution. Green mathematics is an increasingly popular and controversial topic with ties to other contentious social, scientific, and political issues, such as recycling laws and global warming. It is a rich area of research and development for mathematicians and scientists. For example, computer scientists Young Choon Lee and Albert Zomaya have developed and patented an Energy Conscious Scheduling (ECS) algorithm. The ECS software maps the assignment of computational tasks in high-performance computer systems as a function of the dynamic voltage scaling capability of the processors. It optimizes scheduling to decrease task completion time and energy use. Green mathematics also appeals to many mathematics educators at all levels for its apparent applicability, real-world connections, and the ability to connect to academic curriculum in other areas like history and science. In 2010, Roger Williams University’s student mathematics fair was organized around the theme “Designer Math Goes Green! Mathematics and the Environment!” College programs for ecology and sustainable development rely heavily on mathematics and statistics for research and applications. On the other hand, “green math” can have negative connotations for some people, especially when it affects taxpayer dollars and restrictive changes in public policy. In some cases, this reflects an incomplete understanding regarding the basis for such calculations and the methods by which final figures are derived, often because such information is not presented to the public. In others, this may result from inappropriate extrapolation or the political “spin” attached to such calculations.

Green Measurements and Metrics

Measurements of sustainability and environmental impact apply to persons, groups, products, and events. Carbon footprint, for example, is the measure of total emission of greenhouse gases, mostly carbon dioxide, involved in an event or in the lives of people over a given time, usually a year. Energy consumption measures how much energy a process or product takes over its lifetime and accounts for sources of energy, such as atomic or fossil fuels. Ecological metrics may also include emissions of chemical pollutants, such as heavy metals, strength of potentially harmful electromagnetic fields, and intensity of light pollution. The units of measure vary by type of pollution; for example, weight per volume is used for water quality measurements, but light pollution is measured in changes in sky brightness.

Quality standards in ecological measurements include some of the same principles that apply to measurement in general, such as precision and accuracy, together constituting validity. In addition, the measurement of ecological impact requires holistic, systemic approaches, taking into account interactions among multiple variables and their relative weight for particular ecosystems. For example, different ecosystems have different resource balances. Polluting a scarce resource, such as the only water source in a desert oasis, has higher environmental impacts than polluting an abundant resource. This can be reflected in mathematics equations by applying different coefficients to different types of impacts, according to the particular situation within each ecosystem.

It is difficult to weigh different types of environmental impacts against one another. For example, producing paper from trees grown for that specific purpose takes less energy than recycling paper, but involves more water and air pollutants.

Computational Modeling in Ecology

A mathematical model is an idealized system of variables, parameters, and equations governing relationships, assumed to be close enough to a real system for the purposes of prediction or explanation. Mathematical models in ecology are typically based on observations of sets of data from real environments and involve hypothesizing about sets of data that would result if variables changed.

Models can predict developments of ecological systems if the outputs of models, taken over time, fit the corresponding changes in variables of the real ecosystem closely enough. Evaluation of a model includes its accuracy, based on a statistical metric of closeness between observed and predicted data. Nonparametric statistics is the field that deals with evaluating the accuracy of models when the data is limited and not all mathematical assumptions can be tested.

The explanatory power of a model is based on the claim that the model preserves cause-effect relationships within the ecosystem. In mathematical models, such relationships are expressed as algebraic or differential equations among variables of the model.

The possibility of general patterns (models) in ecology has to do with two global problems, or hypotheses: contingency and complexity. The contingency hypothesis says that causal relationships in any given ecosystem are so numerous that projection from one system to another is not possible. The complexity problem is that the number of variables and their weak interactions in any given ecosystem are beyond the computational power theoretically available, making systems immeasurable, their equations insoluble, and the models unable to be interpreted. That is, contingency and complexity are theoretical and philosophical challenges to the possibility and validity of ecological modeling.

Environmental Considerations by Type of Mathematics

Different areas of mathematics allow different approaches to environmental problems. Algebraic reasoning, for example, assumes functional dependencies among variables and known operations. It is most appropriate in cases where algebraic relationships among variables are stable over time and can be established with empirical measurements. For example, producing one megajoule of energy by burning coal emits 92 grams of carbon dioxide. One can compute the carbon footprint of heating a house by coal algebraically by measuring the energy consumption and multiplying it by 92 grams of coal.

Calculus is the study of rates of change in variables and limits of change. In green mathematics, calculus methods are most appropriate when algebraic relationships between variables and their changes over time are measurable. For example, rocket propulsion consumes fuel stored within the vehicle, making the vehicle lighter with time. The efficiency of rocket engines can be computed by applying integrals over time to equations connecting changes in mass and momentum resulting from the engine.

Differential equations involve the study of unknown functions by known values and their rates of change, that is, derivatives—a situation frequently encountered in ecology. Differential equations are extensively used in green mathematics to model interactions within systems, such as predator-prey dynamics, fluid dynamics in natural and human-made water and gas systems, radioactive decay, or economic growth.

Statistical methods deal with the organization and interpretation of data that include random elements. Descriptive statistics summarizes patterns in data collected from some group of objects or events, called “population.” It may include data calculations such as mean or frequency. Descriptive statistics is useful for comparing systems that include randomness, such as per capita consumption of energy in different countries or recycling behaviors in neighborhoods of a city.

Inferential statistics predicts patterns in the whole population based on data observed in a sample of the population. It is extensively used in biology, ecology, and economics because collecting data about every element in the population is rarely possible. One of the most powerful methods of inferential statistics is the analysis of correlations within data. For example, the levels of air pollution in cities correlate with the incidence of asthma among the population. Notably, even strong correlations between two variables do not necessarily mean particular cause-effect relationships. The first variable may depend on the second, or the second on the third, or both may depend on another factor. For example, in children younger than 6, problem-solving abilities strongly correlate with foot size. The reason is that both foot size and problem-solving abilities increase with age.

Data visualization is an interdisciplinary area spanning descriptive statistics; grid and graph use from algebra and calculus; specific representation methods from more narrow areas of mathematics, such as tree diagrams from combinatorics; psychology of perception and learning; and design. Visual literacy combines the ability to understand and critically analyze visualizations produced by others and to create quality visualizations for the purposes of analyzing and sharing messages. Because green mathematics frequently deals with controversial issues, individuals and groups promoting different agendas use and often abuse data visualization to make their point. Visual literacy is one of the “twenty-first-century skills” whose importance is growing with heavier use of mathematics in ecology and growing emphasis on ecological approaches in all areas of life.

Green Economics and Sustainability

Mathematics is used to describe, plan, model, and predict green economy, which is economy based on ecological and social sustainability. Sustainability is a system’s capacity to endure over time, measured by a variety of indices and metrics. For example, the biodiversity index measures the number of plant and animal species in an ecosystem. Using an old-growth forest for lumber and replanting trees may produce the same amount of biomass, but such “farmed” forest typically has a much-lower biodiversity index. Air quality indices assign point values to combinations of air pollutants, such as dust, ground-level ozone, and sulfur dioxide. Higher values of an air quality index correlate with higher incidents of asthma and other adverse health effects. Factories and other entities and events can be evaluated by their effects on an air quality index.

Carrying capacity of an environment, with respect to a species, is the number of individuals the environment can sustain. In differential equations, carrying capacity is the stable state of the system: populations over carrying capacities decrease over time, and populations under carrying capacities grow. Carrying capacity for humans changes depending on their practices. For example, hunter-gatherer tribes need larger areas for sustenance than groups that practice agriculture. The classic mathematical models of carrying capacity were developed for animal populations in relatively small and closed ecosystems. Because people actively change their environments, travel, and exchange resources globally, such models need significant modifications for applications to humans. Current mathematical models are based on evaluating population growth and resource use over time. For example, mining for groundwater can dramatically increase agricultural outputs and thus support population growth until the water runs out, at which time famine can lead to a population collapse.

Bibliography

Fusaro, B. A., and P. C. Kenschaft. Environmental Mathematics in the Classroom. Washington, DC: Mathematical Association of America, 2003.

Hanebuth, Eddie. A Geospatial Industry Series in Science, Technology, Engineering, & Mathematics: Green & Sustainability Focus. Ridgeland, MS: Digital Quest, 2010.

Pfaff, Tom. “Mathematics and Sustainability.” http://www.ithaca.edu/tpfaff/sustainability.htm.