Mathematics as a Language: Understanding the Rules and Assumptions
How Mathematical Structures Mirror Natural Language and the Power of Assumptions in Shaping Truths
This is something I have been thinking about this for some time now, and these are my conclusions: Many people view mathematics as a rigid, mechanical system filled with numbers, equations, and abstract symbols. However, a powerful way to approach math is to see it as a language, one with its own grammar, vocabulary, and logical structure. Just like learning a new spoken language, mastering mathematics requires understanding its rules, recognizing its patterns, and practicing its application.
Mathematics and Natural Language: A Structural Comparison
Mathematics shares several fundamental characteristics with natural languages:
Grammar and Syntax: Just as languages have grammatical rules that dictate how words and sentences are formed, mathematics has syntax that determines how numbers, symbols, and equations are structured. Misplacing a symbol in math is as problematic as misplacing a word in a sentence. For example, in English, "She eats quickly" follows a subject-verb-adverb structure. In math, an equation like 3 + 5 × 2 follows order of operations (PEMDAS), meaning multiplication happens before addition, yielding 3 + 10 = 13 rather than (3 + 5) × 2 = 16.
Vocabulary: In natural language, we have words with specific meanings. In math, we have numbers, operations (+, -, ×, ÷), and functions (sin, log, ∑) that each have precise definitions. Just as words can have synonyms (e.g., "big" and "large"), math often has equivalent expressions (e.g., 1/2 and 0.5 mean the same thing).
Logical Flow: Sentences in a language must follow a logical sequence to be understood. Similarly, mathematical expressions and proofs must follow step-by-step reasoning, ensuring consistency and coherence. A misused conjunction in English (e.g., "Let's eat, Grandma!" vs. "Let's eat Grandma!") may change meaning, just as skipping a logical step in a proof can render it invalid.
Applying This Perspective to Improve Mathematical Skills
By treating math as a language, we can adopt effective learning techniques:
Learn the Rules Like Grammar: Master the foundational principles, order of operations (PEMDAS), algebraic properties, and mathematical notation, just as one learns sentence structure in a new language.
Think in Mathematical Sentences: When solving problems, construct your reasoning as a logical sequence, much like forming a coherent paragraph.
Practice "Speaking" Math: Regularly engage with math by solving problems, writing out full explanations, and even teaching concepts to others, reinforcing comprehension.
Translate Between Math and Natural Language: Many struggle with word problems because they haven't practiced converting statements into mathematical expressions. For example, "The sum of two numbers is ten" translates to x + y = 10. Developing this skill makes math more accessible.
The Role of Assumptions: Math as a System Built on Agreements
Much of modern mathematics is built on fundamental assumptions, just as natural language is based on agreed-upon definitions of words. These assumptions, or axioms, are the starting points for logical deductions.
Axioms and Theorems: Euclidean geometry assumes that parallel lines never intersect. But in non-Euclidean geometry, different assumptions lead to entirely different mathematical truths.
Logical Proofs = Argument Structures: In both math and language, changing initial assumptions changes the conclusions. This is why mathematical systems like Boolean logic can be applied to fields like computing, cryptography, and even philosophy.
Recognizing Hidden Assumptions: In everyday discussions, people often make hidden assumptions that shape their conclusions. The same happens in math, understanding the foundational assumptions helps in solving complex problems and critically evaluating proofs.
Flawed Assumptions in Modern Math: Many of the most advanced mathematical concepts rely on assumptions that may not always hold in all contexts. For instance, classical set theory assumes that infinite sets behave similarly to finite ones, leading to paradoxes like Russell’s Paradox. Similarly, the foundations of calculus assume the existence of infinitely small quantities, a concept that was later rigorously redefined with limits.
Proving Almost Anything by Choosing the "Right" Assumptions: Because mathematical systems depend on initial axioms, it's possible to prove seemingly contradictory or counterintuitive results if one starts with a different set of assumptions. For example, assuming that 1 = 0 leads to every number being equal, an obviously flawed conclusion. This highlights the importance of scrutinizing the foundations before accepting results.
Other Key Similarities Between Math and Language
Ambiguity vs. Precision
Natural languages allow for ambiguity. Words like bank can mean a financial institution or the side of a river. Mathematics, however, aims for precision, ensuring that every symbol and equation has a single, clear meaning. But despite this goal, ambiguity still exists in math:
Symbols with Multiple Meanings:
The symbol "i" represents the imaginary unit (−1−1) in complex numbers, but in summation notation, it’s often just an index variable. In electrical engineering, i represents electric current.
The Greek letter "Σ" represents summation in calculus but can also denote a set or system in other branches of math.
Equations That Depend on Context:
The expression "1 ÷ 2 × 3" can be misinterpreted if parentheses are omitted. Some might read it as (1÷2)×3=1.5, while others may mistakenly interpret a different order of operations.
Einstein’s famous equation E=mc2E=mc2 is actually a simplified form of a more general equation used in relativity:E2=(mc2)2+(pc)2 which includes an additional term for momentum (pp), showing how equations can be context-dependent.
Undefined or Context-Dependent Meaning:
1/01/0 is generally considered undefined, but in certain mathematical frameworks, it can be interpreted as approaching infinity.
Infinity (∞) has different meanings across mathematics, whether in calculus (as a limit), in set theory (as different sizes of infinity), or in projective geometry (where parallel lines "meet at infinity").
Thus, while mathematics strives for precision, the meaning of symbols and expressions can shift depending on context, just as words do in natural language.
Evolution Over Time
Just as languages evolve, so does math and our scientific understanding. Words change meaning over centuries, and new grammar rules emerge. Similarly, mathematical and physical concepts develop over time:
Calculus, now a cornerstone of modern math, did not exist before Newton and Leibniz formalized it in the 17th century.
Set Theory, fundamental to modern mathematics, was controversial when introduced by Georg Cantor in the 19th century because it challenged traditional notions of infinity.
Negative Numbers were once seen as meaningless or unnatural. Ancient mathematicians rejected the idea of "less than zero," but today they are essential to algebra and finance.
The Speed of Light (c), now fixed at exactly 299,792,458 meters per second, was once hotly debated. Aristotle believed light traveled instantaneously, while 17th-century astronomers like Rømer measured it at 220,000,000 m/s (26% too low). By 1983, it was so precisely understood that science redefined the meter using c, turning a measured quantity into a universal constant.
Mathematical and scientific ideas are not fixed; they grow and adapt, just as languages do. Even "constants" can shift in definition as our knowledge deepens, yesterday’s heresy becomes today’s textbook truth.
Creativity in Expression
Just as different cultures develop unique dialects, mathematicians create new branches of math to describe the world in novel ways:
Topology studies properties of shapes that remain unchanged under stretching or bending, much like a mathematical "language" for understanding space.
Game Theory introduced a new way to analyze decision-making and competition, influencing fields from economics to biology.
Category Theory provides a high-level abstract framework that connects different areas of math, almost like a universal grammar for mathematical structures.
Mathematics, like language, is more than just a system of rigid rules, it’s a living, evolving tool for understanding the world.
Conclusion
If we view math as a language, then mastering it is similar to achieving fluency in any spoken language, it requires practice, immersion, and an appreciation of structure. By embracing this perspective, we can make math more accessible, engaging, and useful for solving real-world problems.
Understanding that math is not just a set of arbitrary rules but a language with its own structure and assumptions can help demystify it for learners. Whether you’re a student struggling with equations or someone looking to sharpen your problem-solving skills, thinking of math as a language can make all the difference.