Mathematics in Environmental Science

Mathematics is a discipline based on deductive reasoning, allowing for logically proven conclusions through numbers, equations, and structured logic. In environmental science, mathematics enables climate modeling, species population tracking, pollution analysis, and geographic data visualization-making it essential for understanding and solving environmental challenges. From basic arithmetic to advanced calculus, mathematical principles underpin every environmental discipline from ecology to atmospheric science.

Welcome to a deeper understanding of how mathematics shapes our ability to protect and study the natural world. Whether you're considering an environmental science degree or curious about the quantitative skills behind environmental research, this guide explores mathematics as both a universal language and an essential tool for addressing environmental challenges. We'll trace its evolution from ancient civilizations to modern computing, examine its applications across environmental disciplines, and show you why mathematical literacy matters for anyone passionate about environmental careers.

Jump to Section

• What is Mathematics?
• Why is Mathematics Important to Science?
• Mathematics and Environmental Science
• The History of Mathematics
• What are the Areas of Mathematics?
• The Future of Mathematics
• Frequently Asked Questions
• Key Takeaways

What is Mathematics?

Mathematics is often described as a discipline based on deductive reasoning and logical proofs, often considered the purest of the sciences. The term comes from the Greek translation of máthema, meaning "learning" or "study."

Modern atmospheric sciences and architecture rely heavily on mathematics for modeling and design. The same is true for physics, chemistry, biology, and many other fields. That's because mathematics is the examination and application of numbers, including quantities, volumes, structures, patterns, and order.

Mathematics uses concepts such as abstraction and logic, numbering and calculation, measurement of volume and distance, and quantification of shape and motion (including speed). This universal language transcends national boundaries and was even included on the Voyager space probes in hopes of communicating with potential alien civilizations.

We know that humans have applied mathematics since the dawn of time, even if past generations did not recognize it. They understood volume and distance, and why certain structures would perform better when built with specific exterior or interior angles. As an applied science, it can take many years of trial and error to work out a mathematical solution-that is as true for building a mine as it is for calculating the distance between our planet and other planetary bodies, and for technology as it is for finance investment.

Mathematics is also a natural science. Many aspects of the universe adhere to mathematical laws. Its status as a pure science has led to its elevation as a universal language.

Why is Mathematics Important to Science?

Mathematics as a Scientific Framework

Mathematics is as much of a tool as it is a science in itself. It creates simple concepts around which other sciences are built and provides a quantitative framework for building hypotheses and theories, not always dealing in absolutes, but always reaffirming evidence.

Many of our sciences would not exist were it not for the mathematical framework, and especially without the areas of mathematics listed in a later section of this guide. Mathematical concepts were independently developed in several early civilizations, including Mesopotamia, Egypt, India, and China. Even those who practiced it did not necessarily see it as a science as we would see it-but as a method of solving problems, facts about the world around them, using logic and trial error. Eventually, this would lead to the study of mathematics as a formal subject, as it did in Ancient Egypt, but it predates even their society.

Mathematics in Modern Science

Today, while some modern scientists don't need more than a basic grasp of mathematics-high school level or slightly beyond-many cannot conduct their work without it. Most do need to understand data science, and any researcher in any field needs more than a passing understanding of statistics.

Arguably, most people should know more about statistics than they actually do. But most need the thought processes of mathematics. That includes statistics, but it also includes arithmetic that we all need-proportions, percentages, and calculating how much change to get from a $10 bill for an item that costs $6.49.

Mathematics in the Three Major Sciences

As far as the three major sciences go, math is fundamental:

• Chemistry: Uses mathematical principles, such as stoichiometry and quantum mechanics, to classify and predict molecular behavior. The number of protons, electrons, and neutrons can change a molecule's composition and behavior; mathematics is used to count and classify these relationships.
• Physics: Uses mathematics to calculate distances, plot orbital trajectories, and determine optimal launch times for space missions. You may have heard about "launch windows" for any space-bound mission. These windows represent times when planetary alignments minimize travel distance and maximize mission success.
• Biology: One area of biology that uses mathematics is population biology. This is the examination of species numbers, prevalence, and distribution. Biology can deal with predictability in many ways-as we will see below, this applies particularly strongly with the environmental sciences to,o and especially where they overlap with biology.

Mathematics and Environmental Science

The environmental sciences concern everything about the world around us, from the land on which we live to the subsurface rock, our waterways and soils, our built environment, and the nature and conservation of plant and animal species. Many broad fields classified as environmental sciences use mathematics in one way or another.

Architecture: Architecture uses applied mathematics, particularly geometry and proportion, in structural design. It has done so since the dawn of civilization. Working out ratios and proportions of buildings has allowed architects throughout history to build impressive structures, not least the Giza Pyramids, whose angles and bottom-heavy design have stood the test of time. Architecture relies heavily on angles and mathematical precision.

Atmospheric Sciences: Climatology and other atmospheric sciences, such as meteorology, focus on analyzing data and making predictions using simulation models. Both concepts require mathematics, and climatology cannot make predictions without modeling. At best, they would be educated guesswork and highly inaccurate most of the time. That is as true for today's environment as it is for the paleoclimate. Atmospheric greenhouse gas concentrations and their effects, local weather patterns, and precipitation levels all require mathematical data.

Ecology: Ecology is the science of balance and relationships between living things. That can include the human digestive system and the flora and microbes that live within it, entire ecosystems, predator-prey relationships, and the interdependence between plant life and animals in that landscape. More recently, ecology has developed a subdiscipline called mathematical ecology that uses more quantitative data in its analyses.

Geography: Essentially, geography is a science concerned with space and area, distributions, and relationships. That means it uses mathematics in many ways-to measure the size or volume of land and bodies of water, the expansion and contraction of planetary features over time (such as glacial retreat or water levels during excess or low precipitation-i.e., flooding or drought). It's most prevalent in the subarea of geophysics, which deals with measurements.

Zoology: The primary application of zoology in environmental science is species monitoring and conservation. We know roughly the balance of predator to prey to maintain equilibrium. We also use mathematics to monitor populations and distribution. This data is used to classify the conservation status of species: not evaluated, data deficient, least concern, near threatened, vulnerable, endangered, critically endangered, extinct in the wild, extinct. Each of these grades is based on numbers, distribution, and availability of habitat.

Environmental Engineering: Engineering and math go hand in hand thanks to the necessities of physics in building projects. Engineers need a solid foundation of advanced mathematics, just as architects do, regardless of the field of engineering in which they work. Which aspects of math they need to understand the most varies, though. Mathematics and technological innovation require physics and engineering.

Geographic Data Sciences: Increasingly, many environmental sciences are turning to hard data and analytics. This can include statistical analysis (which is in itself a mathematical science), but as such information is required by multiple stakeholders, even non-scientists and members of the public, it is being combined with visual output. The most common example here is geographic information systems, which analyze datasets and generate maps for visual reference.

This is a small sample of the environmental sciences, but it demonstrates, with broad and varied examples, how important mathematics is to environmental science, as it is to other sciences. That's not to say that all of the environmental sciences require a large degree of math, and even within disciplines that may vary. For example, archaeology and anthropology may require that those working on spatial analysis and modeling understand mathematics and data science to support time and distribution analysis, whereas those examining artifacts may need less.

The History of Mathematics

Pre-Civilization Mathematics

Mathematics is such a base concept; it is hard to do anything without its application. Even though prehistoric peoples didn't necessarily know they were doing mathematics, that doesn't mean they weren't applying it. The history of math begins with several discoveries and inventions, namely, numbering.

We have the first physical evidence of numerical understanding in Africa, dating to around 37,000 years ago, in Swaziland. It was a baboon bone with 29 carved lines. The purpose of this is unclear, but it resembles many traditional African bushmen calendar sticks. Perhaps understandably, it is generally believed that the idea of numbering originated from the needs of hunter-gathering societies, particularly for counting the number of seasons and ensuring there was enough food for the kinship group. Therefore, it's possible that it far predates this artifact.

More controversial is the Ishango bone, which contains sequences that some scholars interpret as evidence of early mathematical thinking, including possible prime number recognition, though this remains debated.

Similarly, the stone circles of Europe have been cited by various experts as evidence for the invention of mathematics. No single, universally accepted explanation has ever been fully accepted by archaeologists, although many believe there is a reference to the changing of the seasons for agricultural purposes and possibly to alignment with certain stars.

The mathematics angle has never fully been explained or agreed. But we do know that by the time of the first civilizations in Babylonia and Egypt, mathematics was firmly established. That is not a subject of disagreement, unlike these prehistoric artifacts and monuments.

Math in the First Civilizations

While much of the last section is up for debate on whether mathematical abstractions or concepts were present merely beyond a vague understanding, the true beginning of theoretical and applied mathematics came in Mesopotamia. We know that the Cradle of Civilization was responsible for the world's first cities.

That meant surplus agriculture, complex social organizations and hierarchy, and the first true economic systems. Mathematical principles such as algebra, arithmetic and geometry were required for many areas of these new societies but particularly for the new complex buildings, taxation and finance, trade and barter (the concept of valuation), land division and management, and even the crafts such as metalwork and weaving-and of course, divisions of time into hours, days, and astronomical seasons.

It's perhaps unsurprising that mathematics, regardless of when it was invented, became necessary as bureaucracy increased.

While it was highly likely that Mesopotamia had a concept of number, Egypt appears to have been the first to develop written numeration, using symbols to denote specific numerical values. Hieratic Script predates the ornate hieroglyphs of Egypt's later writing systems. Undoubtedly, the Mesopotamian civilization influenced Egypt greatly, as both used a sexagesimal numbering system-that is, base 60 rather than base 10.

We still use this system today, including in geometry, time, and even map coordinates. But how this developed is intriguing, and we still see it today in the Middle East and the Indian subcontinent. The right hand is used to demonstrate multiples of 5 in a base 10 system; the left hand is used-not to count fingers, but knuckles. The fingers of the human hand possess 12.

The suggestion for this development is that 60 is a diverse number; in fact, it is divisible by 1, 2, 3, 4, 5, and 6, making it useful in early numerical systems. Egypt is also renowned for developing arithmetic, algebra, and geometry to advanced levels-this we know from the Rhind Mathematical Papyrus, a document that directly references the mathematics of the Giza Pyramids.

Greece and Rome

We look to Greek civilization to see mathematics as an area of study in its own right. That began around 600BC, and Greek culture revolutionized mathematics, making it more complex than ever before, perhaps setting it on the path to becoming a true science in the modern sense of the word. It was practiced at the Greek Academy for approximately 1,100 years.

While pre-Greek math is known largely for "inductive reasoning"-essentially ongoing observations used to determine the facts-the Greeks introduced "deductive reasoning"-essentially using logic to derive conclusions from a premise and using math to prove it. Ancient Greece was a golden age of early mathematics.

Thales, for example, used the new mathematical science of geometry to calculate the dimensions of the Giza Pyramids. By this time, Egypt was under Greek rule-both peoples learned much from each other's advances.

Greek specialists from the period include Pythagoras (responsible for the doctrine that "math rules the universe"), who was also, incidentally, a great astronomer, Euclid (responsible for mathematical rigor and axiomatic method from his book Elements), and Archimedes (responsible for the method of exhaustion and discovering the number "pi").

All these figures are familiar, and perhaps uniquely for ancient science, their discoveries and teachings still form the crux of the modern discipline. Some argue that their works were inspired by the Egyptians and Babylonians, which is likely true, but their contributions in the field are not disputed. Also of note here are Diophantus, who made significant contributions to algebra, and Pappus of Alexandria, who developed the hexagon and centroid theorems.

Most of Rome's mathematics simply adopted Greek discoveries, but its major shift came with the adoption of base 10, which did not introduce the (later controversial) concept of zero. The Romans did not invent it, but they popularized it, and we have maintained that usage ever since for everyday use.

They also introduced the 12-month calendar used in the early Roman Kingdom, which was modified during the Republic and the Empire, when the Julian Calendar was introduced-still used in Eastern Orthodox countries today, but eventually supplanted in the West by today's Gregorian Calendar.

The Far East

Many of the sciences we take for granted today have ancient traditions. At about the same time, the Roman Empire was conquering Western Europe and parts of the Middle East, while Chinese thinkers were working on other areas of basic mathematical principles.

During the Han Dynasty, many mathematical concepts developed independently of the discoveries in the West, and it is believed that there was no trade of knowledge. The best-known invention in China from this period is the odometer-the first known device to measure distance. This invention is also credited with the later clockwork. Shortly afterwards, the Romans invented it as well, and it's believed the two civilizations did so independently.

Although we credit the Romans with the decimal system, the Chinese were using it from very early on. Most interesting and important is the concept of the decimal place. They used sticks numbered 1-9 to represent base-10 numbers. It also seems they developed geometry independent of the Greeks.

While Chinese civilization was early or contemporary with the Greeks and Romans in many mathematical areas, in other areas it was quite late. For example, algebra did not develop in the Far East until the late 13th century.

There is a similar story in ancient India-geometry and shape are common across the Indus Valley civilization, but only, it seems, as a religious or semi-mystical approach to the exterior architecture of temples and other complexes. This is a tradition that modern Hindus continue today, honoring the natural shapes of fractals.

Indian civilization would continue to develop important mathematical concepts and theorems, although the extent of their influence from contact with the Greeks and Romans remains debated. Inventions include trigonometry, primarily for astronomical observation, as in most places around the world where mathematical concepts were developed; mathematical notation; and the binary number system, used in computational mathematics today.

By the time of the Islamic Conquest, many of the Indian subcontinent's discoveries were tied up in the works of the Islamic scholars, who were influenced a great deal by the Romans and the Greeks.

The New World

There is some debate over the extent of contact between China and India, but not to the same extent as in the New World. We know that the Maya had a complex mathematical system, which other contemporary, earlier, and later civilizations in the Americas also used (e.g., the Olmec and the Aztecs).

They used a base-20 numbering system and a complex calendar to create significant "ages" and cycles, which were later misinterpreted as predicting the end of the world in 2012. It is understandable that they developed mathematics to such a high level. After all, the Maya were expert astronomers.

They even-contrary to what the Old World civilizations were doing at the time-created the number "0" or at the very least, factored it into their mathematical system. This proved highly controversial in Christian Europe, as is described in a later section.

The Inca, much like the Mesopotamians in the Middle East, developed complex mathematics for administrative and bureaucratic purposes. They were the largest civilization in pre-Columbian South America and used something called a "quipu". Considered a counting device, it was a piece of string or rope with knots at various intervals and colors to denote numerical values.

Two Medieval Worlds, Two Different Approaches

The Islamic World: Most developments in mathematics occurred during the Islamic Golden Age, but even then, little was added. Islamic scholars continued the ancient Greek and Roman traditions in the sciences of the time, as well as Indian developments through the Arab and Persian worlds.

They introduced the decimal numeral system through the Persian Muḥammad ibn Mūsā al-Khwārizmī, who wrote several important books on numerals and mathematics, directly inspired by Indian works. It is also from his name that we get the word "algorithm". Most of the work in algebra came during this period.

In Egypt, Abu Kamil developed irrational numbers and square roots, which remain highly influential to this day. The Islamic World was obsessed with Euclid, solving some of his equations, identifying flaws, and developing solutions to his most complex problems. Little work had been done until this time, and it's thanks to these North African and Middle Eastern scholars that the ancient Greek and Roman mathematicians are still celebrated today.

The Christian World: Developments in mathematics focused on looking for evidence of the natural order of things and, by definition, God's fingerprints. The reason is that The Book of Wisdom defines the world as being ordered in "measure, number, and weight".

One of the earliest Christian mathematicians was Boethius, who lived in the 6th century. He defined the "quadrivium" of mathematical subjects as arithmetic, astronomy, geometry, and music. Two of these are not subjects of modern mathematics (astronomy and music), although both rely heavily on mathematical principles.

Unlike many other areas, Christian Europe was eager to adopt Islamic ideas and concepts in mathematics, and much work in Spain sought to harness the knowledge of the Moors both before and after their expulsion from Iberia. Not that the medieval period lacked mathematical progress.

Math in the Renaissance

Renaissance mathematics is largely concerned with the expansion of bureaucracy and taxation and the advent of accounting, driven by the new areas and methods of business that emerged in this period. This was the age of the first true modern banks and the calculation of interest and dividends. For that, we required complex algebra and arithmetic.

But mathematics, as a scientific discipline, also developed during this period, thanks to an invention that emerged around the beginning of the period-the printing press. Suddenly, publishing books was faster and easier than ever before, enabling the dissemination of printed material, and some great works on mathematics were published in the few hundred years that followed.

• Summa de Arithmetica, Geometria, Proportioni et Proportionalita by Luca Pacioli is a detailed work on bookkeeping. His most popular work was Particularis de Computis et Scripturis, a reference book for merchants, but also contained possibly the world's first mathematical puzzles
• A prolific and diverse writer was Piero della Francesca, who wrote about geometry, mathematics, and perspective in painting (because mathematical proportion has always been important in the arts)
• Pacioli also wrote Summa Arithmetica, in which were the first uses of representative symbols for plus and minus. We still use the same symbols today, but they originated during the Italian Renaissance. This was also the first book to contain algebra, published in Italy, although it was later discovered that Pacioli plagiarized Piero Della Francesca

Also, in the early Renaissance, Scipione del Ferro and Niccolò Fontana Tartaglia independently discovered solutions to the cubic equation. This was also the great age of the "artist-designer," a precursor to scientific engineering. The likes of Leonardo Da Vinci designed some fantastical machines, including the parachute and a flying machine. Although science would later refine these early concept designs, they demonstrate how mathematics and engineering were becoming fundamental sciences.

This is also the age of trigonometry with practical applications in navigation and the age of sail. Trigonometry is the science of measuring relationships, angles, distances, elevation, and bearing. The word and the concept arrived around this time.

During the Enlightenment and Scientific Revolution

The importance of mathematics only increased during the scientific revolution, and new applications were continually discovered. In astronomy, although the link had always been there, figures such as Tycho Brahe, Johannes Kepler, and Galileo Galilei made major discoveries with the telescope and used mathematics to calculate distances and relationships among heavenly bodies.

Much of their conclusions about astronomy during the Enlightenment were based on complex mathematical measurements, particularly the work of Kepler, who used a solid foundation of mathematics to define laws of planetary motion, and René Descartes developed analytical geometry to plot those motions in graphs. Engineering aside, there are a few areas of science that have received as much development as astronomy and astrophysics. Isaac Newton would later build on the work of both men and formulate the first laws of physics.

But there were other areas for applied mathematics, too. Pierre de Fermat and Blaise Pascal set the wheels in motion for the development of probability theory and combinatorics. Pascal contributed to the foundation of probability theory, partly through philosophical discussions such as Pascal's Wager, which posited that belief in God was a rational wager given infinite potential rewards.

This thinking would eventually lead to utility theory, which emerged later in the Enlightenment and was largely the work of philosophers Jeremy Bentham and John Stuart Mill, although they did not develop it until the modern age.

The 18th century was dominated by the works of Leonhard Euler. He was responsible for developing mathematics in many ways, not least of all the development of graph theory and solving what were (until then) seemingly unsolvable mathematical conundrums such as the square root of minus 1, and the first use of the Greek letter that now represents the number pi (3.141592654…), topology, calculus and complex analysis-all of which have real-world applications too.

He is most famous for the Seven Bridges of Königsberg problem. The last great names of the Enlightenment before the modern age include Joseph-Louis Lagrange. His work in algebra, mathematical theory, calculus (differential calculus and variations), and Laplace, who was famous for statistics and celestial mechanics.

Mathematics in The Modern Age

The modern age, roughly beginning with the Industrial Revolution in the early 19th century, is marked by the practical application of mathematics to the emerging field of engineering. Great new machines, building materials, and processes simply built on what went before, although the advent of the combustion engine is one of the greatest inventions of its age.

But the 19th century is also one in which mathematics becomes increasingly abstract. One example of this trend is Carl Friedrich Gauss. His work on functions of complex variables for geometry, the fundamental theorem of algebra, and the quadratic reciprocity law was far less practical for everyday life, but put mathematics on the road to becoming an area of pursuit in itself rather than a means to an end.

Most importantly, he claimed to have developed non-Euclidean geometry, although he never published the work, and the development was credited to rivals Nikolai Ivanovich Lobachevsky and János Bolyai. This principle challenges the Euclidean assumption that straight and parallel lines will never meet. Yet this is not true, especially when dealing with spherical objects.

The next important development is abstract algebra.

• In Germany, mathematician Hermann Grassmann defined vector spaces and linear algebra
• Irish mathematician William Rowan Hamilton developed something called noncommutative algebra, which studies properties of non-commutative bodies
• British mathematician George Boole devised Boolean algebra, which uses the binary counting system of 0s and 1s, highly important today for computing
• The reformulation of calculus by Augustin-Louis Cauchy, Bernhard Riemann, and Karl Weierstrass

By the turn of the 20th century, mathematics had become a viable academic and industrial career choice. Its applications in engineering, physics, chemistry, and biology made it a useful subject to study and teach, while the new technologies of the Industrial Revolution and urbanization meant that math graduates around the world had options to work in a variety of areas.

The 20th century would see two major disruptions. The first was the career of Albert Einstein, who applied advanced mathematics, such as tensor calculus and differential geometry, to develop theories like General Relativity. His work demonstrated how mathematical frameworks could unlock new understandings of physical reality.

The second is computing,g which developed during the Second World War and now permeates our lives so heavily that we cannot imagine living without information technology. We now live in the Computing Age, and much of the technology would not have been possible without programming and coding necessary to develop the theory behind computing-everything from the first calculators to Cloud Computing and Big Data.

Controversy and Confusion over the Number "0"

The key debate over "0" today is whether it is a number or merely a concept. But go back a few hundred years, and it is far more controversial than some internal disagreement between mathematical scientists. The concept of zero did not exist in the Western world until around 1,200.

Although many societies in both the Old World (Babylonia and later the Arabs) and New World (Maya) had it, many considered it merely a "placeholder" rather than a workable number-a way to represent the absence of something. There is some debate among European historians about the use of zero, largely because the major ancient powers of Greece and Rome did not use it-this is most evident in the Roman numeral system, which never had a representation of zero.

We know the concept entered the West from the Arab world, which was-eventually-hugely influential on Renaissance and later Enlightenment mathematics.

We know there was some resistance among conservative elements in Catholic Europe at the time to the inclusion of zero in mathematics. Recent claims that they feared the idea of nothingness as a precursor to the non-existence of God are not supported by evidence, while others have claimed their opposition was simply the fear of concepts from Islamic thinking and culture.

Evidence for the former is lacking, though it is often repeated, while we do know that the city of Florence banned the use of Arabic numerals due to concerns about fraud, particularly in trade and accounting, rather than religiously motivated opposition. It is likely that opposition was simply an emotional one, either against something new or over concerns about enabling fraud. The simplicity with which a number can be changed to something far less or far greater with the addition of one or more zeroes fueled this problem.

Ultimately, zero would enter common use. When it did, it allowed for the highly important developments in calculus. With the adoption of zero, we could calculate distance and speed, and without that, we would not have had many aspects of physics or even engineering. "0" may have been a source of controversy or confusion, but without it, many of our scientific advances and discoveries would not have been possible.

What are the Areas of Mathematics?

Algebra

Algebra is one of the most important areas of mathematics and is usually taught in its most basic form to elementary school children. Despite this, it can get complex. It deals with properties and their operations, and also their structures, to solve often quite simple problems.

Many people use it as a way of solving simple or slightly more complex arithmetic, although that only becomes algebra with the introduction of "variables"-usually represented by letters or other symbols that do not have set, defined values. They vary because they may depend on the variable nature of other items in the equation.

It's divided into many subareas, with elementary algebra being the most commonly used in everyday life. As discussed above, it is named after the Persian thinker who developed several concepts related to the idea.

Arithmetic

To many people, arithmetic is the ability to-as most people would call it-do sums in one's head without needing paper and pen, a calculator, or any other device to work it out. But it's actually much simpler than that. Arithmetic concerns numbers, their values and relationships, and operations-addition, subtraction, multiplication, and division.

It is a fundamental of number theory, deals with decimals, and underpins any economy, budgeting, bookkeeping, and even household finance. Arguably, it is the one area of math that we all need every minute, every day of our lives to work out times, measurements, distances, and so on.

Calculus

It is another area of math that many people believe is more complicated than it actually is. It's the study of change, including distance and time. It differs from the arithmetic specified above in that it examines changes rather than seeking an answer based on integers by decomposing quantities into infinitesimals.

There are two branches of calculus: differential calculus, which is the study of instantaneous rates such as curves and slopes, and integral calculus, which studies quantities and accumulations and the areas around and underneath curves. Although proto-calculus may have existed earlier, it is generally believed that both Isaac Newton and Gottfried Wilhelm Leibniz independently invented it. It has many uses, including and especially in science, design, engineering, and economics.

Combinatorics

Lesser known to many outside of mathematics and science, combinatorics is the science of things with finite combinations or structures. For example, it aims to solve simple questions with difficult answers, such as the number of possible combinations of cash change from all available coins when given a $1 bill, or the total number of Sudoku puzzles we can devise.

There is no set standard definition, but it is largely described as being concerned with the enumeration theory, and with combinations and permutations, existing in a finite set of possible answers to a puzzle. This typically includes graph theory, amongst other things.

Computational Mathematics

For reasons that are understandable, this is a relatively new part of mathematics, although it did exist before the age of information technology. How we use the term varies depending on which aspect of the relationship between computing and mathematics is under discussion.

It could refer to applied mathematics and the use of math principles in driving computing or it may refer to the use of computing power to solve or drive complex mathematical equations. Applied computational mathematics can be used in any area of science and is particularly useful for data modeling and analysis, such as atmospheric carbon measurements over time or across geographic areas.

Foundational Mathematics

Many areas of science have a "foundational" aspect, that is, the study of the core principles. Mathematics is no different. The field is dedicated to examining the algorithmic, logical, and philosophical basis of math.

It is simply returning to the core of a subject, and in this instance it means studying the most basic concepts of mathematics: numbering, sets, functions, geometric figures, and the framework of mathematical language. It has undergone several fundamental shifts with each new problem and addition of theories to solve them, most notably in Euclid's era, Newton's, and then Einstein's.

Geometry

The study of the shapes of objects-lines, points, circles, angles, circumference, and volumes in both two and three dimensions. This means it is integral to a number of areas, including (and especially) cartography, route planning and navigation, civil engineering, particularly in the built environment, computer-aided design, MRI and other medical imaging, and many other areas.

Topology

This is the subarea of geometry in mathematics concerned with the properties of space under pressure and deformation, including crumpling, bending, stretching, compression, and tearing. It is particularly useful for geologic landscape analysis.

Mechanics

Mechanics, through mathematics, examines the behavior of physical bodies under specified pressures, forces, and displacements. It also examines the permanent effects of the changes on those bodies and on the environment.

It began with Aristotle and Archimedes and expanded significantly during the Renaissance and the pre-Enlightenment era. Kepler, Newton, and Galileo are largely credited with its application in physics and why the two disciplines are so intertwined today, but that is not the limit of its influence. More recently, it has moved to incorporate motion and force on objects.

Operation Research

In analysis, this refers to the application of advanced methods to decision-making, using a wide range of techniques. It is a method in applied mathematics that uses many principles and ideas to solve real-world problems. It is sometimes called "management science" or "decision science" and uses computational science, modeling, and statistical analysis, often leveraging powerful technologies.

Operational research originated in military strategy during World War II and has since expanded into business, engineering, environmental science, industry, and operations management. It can use both quantitative and qualitative data, depending on the application area.

Probability

Devised by Blaise Pascal, probability is the mathematics of the likelihood of an outcome. Although it is often associated with gambling, it also has applications in investment and business risk management, insurance premium calculation, and a range of other financial services, including the stock market.

But probability also has broader applications. Often expressed as a percentage (with 0% being impossible and 100% being certain), this is not always the case. In probability, many calculations are expressed as a range of 0-1.

Statistics

Statistics is about the collection, processing, interpretation, and presentation of hard data. It is fundamental to scientific research, including measuring demographics, population change, health issues, and resource allocation, to calculate percentage changes.

Today it includes the methods of data collection as well as the process of collection in order to get a proper and fully accurate sample size. For example, standing outside a church on a Sunday and asking people who come out about their church attendance and whether they are Christians might lead us to conclude that 100% of the population attends church and that 100% attend every Sunday.

Therefore, statistics is as much a philosophy of data collection as a tool for collecting data.

The Future of Mathematics

Remaining Visible

There has been some criticism that government science and technology budgets have focused on areas with a high likelihood of return on investment. Because mathematics has little to no quantifiable ROI, budgets for research and applications here are at the mercy of engineering and science projects that might better warrant it.

Part of the problem, according to Congressman George E. Brown Jr., speaking in 1997, is that the ways in which mathematics underpins science are poorly understood in government, which tends to enjoy the benefits of engineering and math investment while continually failing to realize why math is important. The overwhelming majority of our discoveries in science have been due to breakthroughs in math.

Keeping Math Exciting

There has been a considerable drop in the number of students taking hard sciences, particularly mathematics. Many teachers and teaching decision-makers in the Western world are noticing the trend and believe that the way math is taught needs to be updated and made more relevant to today's students.

Maybe the problem is too much theory; maybe the issue is that the way it is taught lacks sufficient practical applications and 21st-century problem-solving exercises. Also, there is insufficient focus on the reasons and critical thinking in math that people need in their everyday lives, rooted in archaic thinking and practices, while the rest of the world has moved on.

Falling Standards and Myths About Math

Despite having a much smaller education budget, China has some of the best math standards in the world. There is a similar attitude in Japan. Both countries believe that even the least academically gifted student can be taught advanced mathematics, probability, analysis, and critical thinking, and that this is taught from an early age.

This is a stark contrast to the US, Canada, and many European countries, where math is often viewed as difficult and reserved for exceptional students. If Western nations are to succeed in improving math standards this century, it may require a revolution at the education board level, as well as greater efforts to dispel myths students hold about mathematics.

Solving Equations

The history of modern mathematics is full of complex equations and conundrums that have eventually been solved. Today, there are as many as several hundred unsolved mathematical problems, and just six of the Millennium Math Problems now remain to be solved.

Anyone who solves them will win a $1m prize. The fact that there are so many and we keep making headway in solving them shows there is a desire to solve them, despite declining student interest in the field. Some of these problems could be the key to unlocking some complex new advances in technology.

Frequently Asked Questions

What level of mathematics do environmental science majors need?

Most environmental science programs require mathematics through calculus, along with strong foundations in statistics and data analysis. Bachelor's degree programs typically mandate Calculus I and II, while advanced programs may require differential equations and linear algebra. However, the specific math requirements vary by specialization-ecology-focused students need more statistics and modeling, while environmental engineering students require advanced calculus and physics-based mathematics.

Which areas of mathematics are most important for environmental careers?

Statistics is the most critical mathematical skill for environmental professionals, used in data collection, analysis, and interpretation across all specializations. Calculus is essential for understanding rates of change in climate models and pollutant dispersion. Geometry and spatial analysis are fundamental for GIS work and landscape assessment. Environmental scientists also benefit from computational mathematics for modeling ecosystems and predicting environmental trends.

Can I succeed in environmental science if I'm not strong in math?

Yes, though you'll need to develop basic quantitative skills. Many environmental science subfields-such as environmental policy, education, and communication-require less intensive mathematics than research or engineering positions. Even in math-light specializations, you'll still need competency in data interpretation and basic statistics. Most universities offer math support services, and many students find that environmental applications make mathematical concepts more engaging and understandable than abstract classroom exercises.

What's the difference between applied and theoretical mathematics in environmental science?

Applied mathematics in environmental science focuses on using mathematical tools to solve real-world problems-calculating pollution concentrations, modeling species populations, or analyzing climate data. Theoretical mathematics involves developing new mathematical frameworks and proofs. Most environmental careers rely heavily on applied mathematics, using established principles to address practical challenges rather than advancing mathematical theory.

How is mathematics used in climate change research?

Climate science depends on complex mathematical models that simulate Earth's atmospheric, oceanic, and land systems. Researchers use differential equations to represent energy transfer, statistical methods to analyze temperature trends, and computational algorithms to run simulations on supercomputers. Mathematical modeling helps predict future climate scenarios, assess the impact of greenhouse gas emissions, and evaluate potential mitigation strategies. Without advanced mathematics, climate scientists couldn't make the quantitative predictions that inform environmental policy.

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Matthew Mason

MG Mason has a BA in Archaeology and MA in Landscape Archaeology, both from the University of Exeter. A personal interest in environmental science grew alongside his formal studies and eventually formed part of his post-graduate degree where he studied both natural and human changes to the environment of southwest England; his particular interests are in aerial photography. He has experience in GIS (digital mapping) but currently works as a freelance writer as the economic downturn means he has struggled to get relevant work. He presently lives in southwest England.

Latest posts by Matthew Mason (see all)

• What Is Parasitology? The Science of Parasites Explained - November 19, 2018
• Desert Ecosystems: Types, Ecology, and Global Importance - November 19, 2018
• Conservation: History and Future - September 14, 2018

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