Introduction to Mathematics

• Mathematics studies quantity, structure, space, and change.
• No universally accepted definition exists for mathematics.
• Mathematicians identify patterns and create conjectures.
• Mathematical proofs are used to determine the truth or falsity of conjectures.
• Solving mathematical problems can require years or centuries of sustained research.
• Since the late 19th century, mathematical research is viewed as establishing truth through rigorous deduction from axioms and definitions, which has its roots in the work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943).
• Mathematics is a crucial tool in natural science, engineering, medicine, finance, and social sciences worldwide.
• Applied mathematics applies mathematics to various fields, inspiring new discoveries and leading to new disciplines.
• Mathematics plays an essential role in natural science, engineering, medicine, finance, computer science, and social sciences.
• Applied mathematics has spurred the creation of new mathematical areas like statistics and game theory.
• Mathematicians engage in pure mathematics for its intrinsic value, without specific applications in mind.
• The distinction between pure and applied mathematics is blurred, as practical uses for pure mathematics are often found.
• Mathematics is broadly divided into pure and applied branches.

Quantity

• Quantity begins with numbers, starting with natural numbers, integers, and arithmetic codified in algebra.
• Number theory explores deeper integer properties.

Structure

• Mathematical objects such as sets of numbers and functions, have internal structure due to operations or relations defined on the set.
• Studies involve properties of sets expressible in terms of this structure.
• Abstract algebra studies algebraic structures like groups, rings, and fields that follow arithmetic axioms.
• Mathematical structure can be axiomatized, independent of concrete realization
• Combinatorics focuses on counting objects that meet specific criteria.

Space

• Geometry studies space.
• Trigonometry combines space and numbers using sine, cosine, and tangent functions.
• Modern geometry integrates calculus and abstract algebra to study local and global space properties.
• Space includes: geometry, trigonometry, topology, differential geometry, and fractal geometry.

Change

• Understanding and describing change is a key theme in the natural sciences, and calculus was created to study it.
• The concept of a function is central to calculus.
• Differential equations describe rates of change.
• Functions are studied using differential equations due to the continuous nature of many changing phenomena.
• Numerical analysis develops methods to approximate differential equation solutions.
• Calculus encompasses calculus, vector calculus, differential equations, dynamical systems and chaos theory, and numerical analysis.

Foundations and Philosophy

• Mathematical logic and set theory were developed to clarify the foundations of mathematics.
• Mathematical logic establishes a rigorous axiomatic framework for mathematics and analyzes its outcomes.
• Set theory studies sets as mathematical objects.
• Category theory, focuses on mathematical structures and their relations in an abstract way, and is still being developed.
• There is no universal agreement on the exact scope and definitions of mathematical logic and set theory.
• Many mathematicians do not focus on the foundations of mathematics, leaving it to philosophers or logicians.
• Foundations and philosophy include: mathematical logic, set theory, category theory, and philosophy of mathematics.

Discrete Mathematics

• Discrete mathematics includes the mathematical fields most applicable to computer science.
• Computability theory, computational complexity theory, and information theory are included.
• Computability theory studies the limitations of theoretical computer models, including the Turing machine.
• Complexity theory analyzes the time algorithms take to solve problems.
• Information theory deals with quantifying data storage on a medium, covering compression and entropy.
• Discrete mathematics includes: computability theory, computational complexity theory, information theory, cryptography, graph theory, and order theory.