What is Mathematics (And Why Does it Matter)
It's time to stop treating you like a child and explain exactly what math is about (even if you're in college)!
Mathematics has long been essential for intellectual growth, but its teaching today often focuses on rote problem-solving rather than true understanding. To genuinely grasp math, it should be seen as a key part of scientific thinking and problem-solving. Effective teaching should emphasize clear, motivated explanations over technical drills, making learning transformative.
Contrary to the perception of math as static, it's dynamic, with major advancements like the Four Color Theorem and Fermat’s Last Theorem. Engaging students requires direct interaction with concepts, not just indirect methods like biographies. Mathematics isn't just a system of conclusions; it's about the relationships and operations of objects that exist uniquely between the real and abstract.
Think of this article as a banquet: savor each part until satisfied. Its purpose is to provoke thought and disquiet. Mathematics is an infinite field, with countless individuals dedicating their lives to its development.
I hope you come away from this article with the feeling that something is missing. The feeling that you need something more. That something more exists. For there is always more to discover.
Mathe-What?
Mathematics is the science of quantity, shape, and arrangement, covering pure mathematics (abstract concepts) and applied mathematics (methods used in science and engineering).
Pure Mathematics: Focuses on abstract concepts and theoretical frameworks without necessarily having a practical application in mind.
Applied Mathematics: Uses mathematical methods to solve practical problems in other fields such as physics, engineering, economics, and biology.
It combines logic, intuition, analysis, construction and generality, forming the essence of mathematical science. Different traditions may focus on various elements, but it's the blend of these forces that gives mathematics its unique vitality and value.
Historically, practical needs sparked mathematical development. However, it soon transcended its practical origins, evolving into a theoretical science.
Oh-OK-Mathematics
What we learn in school is just a small part of a vast and fascinating field. The perception that mathematics is a rigid and boring subject, limited to the formulas and equations we memorize in class barely scratches the surface of the incredible depth and beauty that mathematics possesses.
Mathematics began with basic counting, a skill seen even in animals, with prehistoric evidence marked on bones.
Throughout history, various civilizations made notable contributions: Egyptians created the first equation. Greeks advanced geometry and numerology, Chinese invented negative numbers, and India introduced zero.
During the Golden Age of Islam, Persian mathematicians made significant strides in algebra. The Renaissance saw a surge in mathematical progress alongside scientific advancements.
Today's mathematics is a vast and dynamic field, constantly evolving and expanding into new areas of inquiry. It encompasses a wide range of topics, from classical branches like algebra, geometry, and calculus to more contemporary fields such as computational mathematics, mathematical biology, and data science.
Do you remember the main branches of pure mathematics and applied mathematics?
Pure Mathematics
It’s often perceived as an exploration of theoretical landscapes without immediate real-world application, delves deep into the essence of mathematical thought.
Algebra uses symbols and rules to solve equations and understand variable relationships. Linear algebra, a subfield, focuses on vectors and matrices, solving systems of linear equations and transforming geometric objects.
Number theory explores properties and relationships of integers, such as prime numbers, divisibility, greatest common divisors, and modular arithmetic.
Combinatorics studies counting, arranging, and structuring sets of objects. It covers permutations, combinations, partitions, and structures like trees and graphs.
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Geometry studies shapes, sizes, and space properties. It includes Euclidean geometry, dealing with flat surfaces and shapes like triangles and circles, and non-Euclidean geometries exploring curved spaces.
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Fractal geometry examines complex patterns showing self-similarity at different scales, found in nature like tree branches, snowflakes, and coastlines.
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Topology studies properties of space preserved under continuous deformations, like stretching and bending but not tearing or gluing. It involves concepts like continuity, compactness, and connectedness.
Complex analysis studies functions involving complex numbers, extending calculus concepts to the complex plane.
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Calculus studies change, involving integrals and differentials. It analyzes motion, growth, and dynamic processes. Integral calculus focuses on accumulation and area under curves, while differential calculus examines rates of change and slopes.
Order theory looks at arranging objects according to rules, involving concepts like partial orders, lattices and ordered sets.
Group theory examines algebraic structures called groups, sets with a single operation satisfying specific axioms.
Chaos theory studies systems highly sensitive to initial conditions, where small differences lead to vastly different outcomes.
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Applied Mathematics
Applied Mathematics emphasizes both developing new mathematical methods and addressing practical issues. Problems arise from various fields like physical and biological sciences, engineering, and social sciences.
Numerical analysis involves using approximations to solve complex mathematical problems, often with the aid of computers. This field focuses on developing algorithms that provide accurate numerical solutions to equations that cannot be solved analytically.
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Game theory studies optimal decision-making in competitive situations with defined rules. It analyzes strategies where individuals or groups seek to maximize their outcomes, finding applications in economics, politics, and evolutionary biology.
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Probability and statistics are fundamental for understanding and analyzing random events and data. Probability theory models uncertainty and predicts the likelihood of events, while statistics involves collecting, analyzing, and interpreting data. Both fields are essential in finance, helping assess risks and make informed decisions.
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Optimization is the mathematical discipline focused on finding the best possible solution among a set of available options. It is used in various fields to maximize efficiency, minimize costs, or achieve the best outcomes in constrained situations.
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Cryptography applies mathematical principles from combinatorics and number theory to secure communication. It ensures data confidentiality, integrity, and authenticity, playing a critical role in cybersecurity.
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As you can see, mathematics is an endless field. There is always more to be discovered and the feeling that we don't know enough will last for the rest of our studies.
This is the first article in a series I intend to write with the aim of answering the question: what is mathematics anyway? Welcome aboard.