Mathematics: History and Applications

• Mathematics explores quantity (numbers), structure, space, and change.
• Mathematicians and philosophers hold diverse views regarding the precise scope and definition of mathematics.
• It serves as a crucial tool across various disciplines globally, including natural science, engineering, medicine, finance, and social sciences.
• Applied mathematics, applying mathematical principles to fields, both draws from and stimulates fresh mathematical discoveries.
• Pure mathematics involves the pursuit of mathematics for its inherent value.

History

• The history of mathematics features a continuously expanding sequence of abstractions
• The notions of number, magnitude, and form represent the most ancient instances.
• Mathematical writings have been discovered from as far back as 1800 BC (Babylonia) and 1600 BC (Egypt).
• Mathematical progress occurred irregularly until the 12th-century Renaissance, when newly available texts fueled further development.
• Certain mathematical domains, such as statistics and game theory, are propelled by practical, real-world uses.

Areas of mathematics

Quantity

• Quantity begins with numbers, initially with natural numbers and integers and subsequently with rational and real numbers.
• Further number generalizations led to complex numbers, including the square root of -1.
• Hypercomplex numbers exist, sometimes preserving number properties but not consistently.
• Arithmetic is the study of numbers
• Operations like addition, subtraction, multiplication, division, exponentiation, and root extraction fall under arithmetic.
• Number theory delves into integer properties.
• Fermat's Last Theorem falls under number theory.
• The twin prime conjecture (whether infinitely many twin primes exist) and Goldbach's conjecture (whether every even integer > 2 can be expressed as the sum of two primes) remain unsolved in number theory.

Structure

• Numerous mathematical entities, like sets of numbers and functions, possess internal structure.
• Axioms can define structure.
• Structure is analyzed within group theory, ring theory, field theory, and other systems of abstraction.
• The classification of finite simple groups exemplifies a structural theorem.

Space

• Space involves the study of shapes.
• Geometry, trigonometry, and topology are encompassed by the concept of space.
• Geometry originated from ancient measurements of distances and areas.
• Trigonometry explores relationships between triangle sides and angles and is employed for distance and height calculations.
• Topology examines properties preserved through continuous transformations such as stretching and bending.
• Space includes higher-dimensional extensions of conventional geometry.

Change

• Understanding and describing change is a common theme in the natural sciences.
• Calculus was created to analyze change.
• For a precise understanding of change, calculus uses real numbers and functions.
• The derivative quantifies instantaneous change rates, while the integral measures a quantity's accumulation.
• Differential equations link a function to its derivatives and are utilized to simulate time-varying processes.
• Chaos theory elucidates systems exhibiting unpredictable behavior despite deterministic governing laws.

Foundations and philosophy

• Mathematical logic and set theory arose to clarify mathematics' foundations.
• Mathematical logic studies formal logic's application to mathematics.
• Set theory studies sets, or collections of objects.
• Set theory provides a basis for contemporary mathematics.
• Category theory represents a more abstract study area compared to set theory.
• Category theory aims to capture structural connections between mathematical theories.

Discrete mathematics

• Discrete mathematics explores discrete, rather than continuous, mathematical structures.
• Combinatorics, graph theory, and coding theory are included in discrete mathematics.
• Combinatorics involves counting the ways specific arrangements can be made.
• Graph theory examines networks depicted as vertex collections interconnected by edges.

Applied mathematics

• Applied mathematics focuses on employing mathematical tools for problem-solving in science, engineering, business, and other domains.
• Mathematical physics, numerical analysis, optimization, and statistics fall under applied mathematics.
• Mathematical physics utilizes mathematics to address physics problems.
• Numerical analysis studies algorithms for addressing mathematical problems via computers.
• Optimization seeks the best problem solution, like minimizing expenses or maximizing profits.
• Statistics uses data to infer and predict population characteristics.

Mathematical notation, terminology, and style

• Mathematics relies on specialized notation and terminology.
• Symbols, rather than words, are used in mathematical statements.
• Often, mathematical statements are expressed in a formal language.
• Mathematical proofs serve to confirm the validity of mathematical statements.
• Proofs progress from axioms and previously validated theorems to establish novel findings.
• Mathematical journals and books systematically communicate new mathematical research.

Influence and applications

• Mathematics holds significance in the natural sciences, engineering, medicine, finance, and social sciences.
• Theoretical mathematics has unexpectedly led to applications in other fields.
• Mathematics forms a foundation for numerous technological advancements including algorithm design, encryption, and data analysis.
• Mathematical models simulate intricate systems and anticipate future behavior.

Awards

• Major mathematical honors include the Fields Medal and the Abel Prize.
• These awards acknowledge exceptional contributions to the field.