Agriculture and mathematics

Summary: As a fundamentally important human activity, agriculture has long been a motivator for mathematical and statistics research.

Farming, also called “agriculture,” is the production and distribution of plant and animal products. Farming methods range from organic farming to industrial agriculture. Farming operations are also categorized by their products, including foods, pets, decorative plants, pharmaceuticals, building materials, fibers, resins, and bioplastics. Agriculture has long been a motivator for mathematical and statistic research. Mathematical concepts and models have helped advance many agricultural methods beyond simple arithmetic calculations of quantities of seed and fertilizer. Many consider Ronald Fisher to be the father of modern statistics. Much of his research in statistical methods originated from his work with more than 60 years of agricultural data at Rothamsted Experimental Station, one of the oldest agricultural research institutions in the world. Methods pioneered by Fisher are still widely used in the twenty-first century, including hypothesis testing, analysis of variance, maximum likelihood estimation, and factorial experimental design. Mathematician Michael Weiss has worked in several mathematical areas with applications in agriculture, including nonlinear and chaotic dynamics, fuzzy set theory, and topological and algebraic entropy. Some applications of his work include a model of crop yields as a two-dimensional stochastic process, called “random surfaces,” and assessing revenue risk as a probabilistic function of foodborne disease outbreaks. Precision farming models spatial variability in farmland and the resulting changes in yields as geometric surfaces.

Numerical characteristics of the farmland, such as fertilizer needs, are assigned to surfaces by functions and mapped to other surfaces by operators using modeling software. The so-called cobweb theorem relates price and production for situations in which there is a time lag between the marketing of a product and initially obtaining price information to determine production. This is common in agricultural markets, since prices in one year tend to influence planting in subsequent years.

The Role of Agriculture in the History of Mathematics and Science

Agricultural development shaped the history of humankind, including the growth of science in mathematics. This impact is acknowledged in the historical tradition of naming major farming breakthroughs “revolutions,” since the changes they produced in society were large and relatively fast. The neolithic revolution started circa 8000 b.c.e. and included the development of permanent settlements. The resulting architecture and centralized management systems required abstract thought and systems of knowledge, including writing, mathematics, and science. The Arab agricultural revolution took place in the eighth through the thirteenth centuries c.e. and included the development and distribution of international knowledge exchange, sophisticated algebra and geometry, and astronomy for farming and navigation, as well as the scientific method and the modern number and computational system in mathematics. The British agricultural revolution started in the seventeenth century. It codeveloped with the Industrial Revolution and included the heavy use of mechanical tools and developments in the natural sciences, including chemistry and biology. This industrialization of agriculture continued into the twentieth century, driving the research in organic chemistry and genetics known as the green revolution.

Domestication of local crops, such as rice in China around 8000 b.c.e., allowed for both population growth and population concentration in villages and, later, towns. Planting, harvesting, and other timed activities required relatively exact time and weather observation, which in turn led to the development of astronomy and the development of sophisticated time measurement tools and calendars. Circa 5000 b.c.e., the people of Mesopotamia employed intensive farming methods, including monocrop fields, aggregation of crops for trade, and complex irrigation. Such methods called for and enabled major technological developments, such as better plows. It is hypothesized that the complex division of labor, distribution, and observation of water levels and calendars required for this type of agriculture led to the development and relatively widespread use of writing.

Mesopotamian clay tables show that quadratic and cubic equations, the Pythagorean theorem, and other topics currently found in algebra, geometry, and calculus were already widely used circa 2000 b.c.e. in problems related to agriculture, such as astronomy-based calendars to time flooding and harvesting or the distribution of products. Some of this knowledge later was lost and then rediscovered by other cultures, and some continued to be used in the original form. For example, the practice of measuring time based on 60 minutes in an hour and 60 seconds in a minute comes from the Babylonian sexadecimal (base 60) number system. The number “60” was a convenient one for the Babylonians being highly composite (with more divisors than any number less than 60).

Agriculture promoted the development and spread of increasingly complex mechanisms, such as waterwheels in China. Excess crops supported the development of trade and transportation, from the domestication of draft and pack animals in ancient times to sophisticated spice trade fleets circa the sixteenth century. Starting in the eighth century c.e., Muslim traders established an extensive network of trade routes among Asia, Europe, and Africa, enabling the diffusion of agricultural techniques and crops beyond their places of origin. This Arab agricultural revolution led to the development and distribution of science and mathematics, including the Arabic numerals used around the world in the twenty-first century. For example, one of the first documented uses of the scientific method comes from thirteenth-century work on medicinal plants and agronomy (the farming of plants).

The Industrial Revolution, starting in the eighteenth century, included the increasing mechanization of agriculture. Agricultural machines, such as the tractor, both decreased the number of people required for farming and increased productivity. The scientific advances associated with these developments primarily took place in engineering and chemistry. The green agricultural revolution of the second half of the twentieth century promoted advances in chemistry, genetics, and bioengineering, which led to high-yield, disease- and pest-resistant cultivation of major crops. The sustainability of these practices is not yet clear at the start of the twenty-first century.

Measurements in Agriculture

Metrics used in farming focus on average production of different cultivars of plants, breeds of animals, or farming methods; resource intensity of practices; efficiency of distribution; nutritional value of food products and industry-specific values of fibers, fuels, and lumber; environmental impact and sustainability; and the role of agriculture in local and global economy.

The global production levels, by crop type, are measured in tons per year. For example, cereals was the number one category of agricultural product, with worldwide production at around 2 billion tons per year in the early twenty-first century, while meat production at this same time was around 250 million tons per year. The total and per capita rates of production are frequently compared between years. For example, the total agricultural production grew by a factor of 16 between the early 1800s and 1970, while the world population grew by a factor of seven. This means that per capita consumption of agricultural products more than doubled during that period but not necessarily because of food items. Fiber or farmed trees for paper and construction are also included.

Farm yields are measured in crop weight per area for plants; in the ratio of seed input to seed output for grains; or in meat, fiber, or egg production per animal for animals. The yields are estimated using statistical methods of random sampling, or total outputs of a farm. In the United States, for example, corn yields averaged about 30 bushels per acre in the early 1900s and around 130 bushels per acre in the early 2000s. Food anthropologists estimate the minimal ratio of grain input to output necessary for sustaining farming as the main source of food as 1:3. For each grain planted, farmers get three grains, one of which is planted and two of which are either eaten by people or fed to farm animals. Yield metrics can be used to compare different methods of farming. For example, irrigation can raise corn yields by a factor of four or five. Industrial farming in developed countries produces yields that are about 10% greater than organic farming in nondrought years and about 70% less in drought years, netting about the same average yields over decades.

Resource intensity is measured by the outside input required per area of crops, per individual animal in meat or egg farms, or per unit of farm product output. For example, it takes about 1000 liters of water to produce 1 liter of corn-based ethanol. Resource intensity is one of many sustainability metrics used in farming. Other mathematical metrics of sustainability include nutrient leaching into water systems, which may cause proliferation of algae; biodiversity of farms; and pollution of soil, water, and air with herbicide and pesticide residues; as well as the carbon footprint of farming practices. For example, livestock production is currently responsible for about one-fifth of the total carbon footprint of humanity.

Farming and the Economy

Agricultural systems include production, processing, packaging, distribution, marketing, and consumption. The proportion of resources and energy required for these activities varies with farming practices. For example, eating local foods reduces the resources expended in transportation; operating monocrop farms reduces labor per unit of production; eating processed foods increases packaging costs.

Agricultural economics is the study of resource allocation and distribution related to agriculture. It uses mathematical statistics for data analysis and trend prediction and mathematical modeling for research and development. Many general economic mathematical models were first developed in agricultural economics, for example, the cobweb model, which explains the cycles of price fluctuations through analyzing lags within the production chains, such as planting and harvesting.

Factory farming uses economies of scale by raising livestock in confinement and with high population densities. The calculations involved in factory farming include cost-output analysis and bioengineering of animals to optimize product output as well as the logistics of supplying food in to each animal in place and disease prevention through administering antibiotics. There are several measurements of factory farming impacts. For example, there are metrics involved with animal welfare, such as the degree of confinement, measured in area of pen per animal. Human health impact measures and research include studies of pesticide, antibiotic, and growth hormone levels in farm products and statistical studies of the impact of food on human health. Environmental impact measures are standard for all operations and include levels of specific air, water, and soil pollutants produced by the farm and its carbon footprint. Capital redistribution is the measure of movement of money among communities, which is relatively high for factory farming because of its centralized nature.

Industrial marketing and distribution models do not work well for organic farming because most organic products are not scalable. In the early 2000s, organic farmers developed a variety of peer-to-peer credence and distribution models, network marketing models, and sharing economy (mesh) models. Such modern models support decentralized production and disintermediated distribution. Some organic farmers join together in cooperatives and use economies of scale. Community-supported agriculture (CSA) is an economic model that provides a way to share the benefits and risks of farming. In a typical CSA, consumers buy farm shares and receive a weekly delivery of farm outputs.

Bibliography

Alspaugh, Shawn. “Farmer Ted Goes 3D.” Mathematics Magazine 78, no. 3 (2005).

Glen, John. “Mathematical Models in Farm Planning: A Survey.” Operations Research 35, no. 5 (1987).

Street, Deborah. “Fisher’s Contributions to Agricultural Statistics.” Biometrics 46 (1990).

Weiss, Michael. “Precision Farming and Spatial Economic Analysis: Research Challenges and Opportunities.” American Journal of Agricultural Economics 78, no. 5 (1996).