Introduction to Mathematics

• Mathematics explores quantity, structure, space, and change.
• Mathematicians and philosophers hold varying perspectives on the precise scope and definition of mathematics.
• Mathematics aims to identify patterns and create new conjectures.
• Mathematicians use mathematical proofs to validate or invalidate conjectures.
• Mathematical reasoning offers insights or predictions about nature when mathematical structures accurately represent real-world phenomena.
• Globally, mathematics serves as a vital tool across diverse fields like natural science, engineering, medicine, finance, and social sciences.
• Applied mathematics focuses on employing mathematical tools to address challenges in natural science, engineering, medicine, finance, business, computer science, and social sciences.
• Pure mathematics explores mathematical questions without considering practical applications.
• Pure and applied mathematics have a history of significant interaction.
• Mathematical research often seeks mathematical principles applicable to real-world problem-solving.
• Real-world challenges can inspire new mathematical fields.
• Galileo (1564–1642) stated, "The book of nature is written in the language of mathematics."
• Mathematics is essential in many fields, including natural science, engineering, medicine, finance, and social sciences.
• Applied mathematics and statistics have overlapping aspects.
• Mathematicians seek out patterns and formulate conjectures.
• Mathematics is used to solve real-world problems and is an essential tool.

Areas of mathematics

• Quantity encompasses numbers, number systems, arithmetic, and algebra.
• Structure involves order, relations, sets, and algebraic structures.
• Space includes geometry, trigonometry, and topology.
• Change covers calculus, differential equations, and dynamical systems.

History

• The history of mathematics is an ever-evolving progression of abstractions.
• The concept of abstract numbers started to emerge.
• Representing time was the first abstract application of numbers.
• It is plausible that some form of mathematics was integral to all human cultures.
• Ancient Egypt and Mesopotamia provide the oldest known mathematical texts.
• The Rhind Mathematical Papyrus (c. 1650 BC) exemplifies Egyptian mathematics.
• Mesopotamian mathematics featured systems for measuring angles, areas, and volumes.
• Babylonian mathematics employed a sexagesimal (base-60) numeral system.
• This system underlies the modern use of 60 minutes in an hour and 360 degrees in a circle.
• Mathematics from the Hellenistic era is known as Greek mathematics.
• Greek mathematics began with Thales of Miletus (c. 624–c. 546 BC) and Pythagoras (c. 582–c. 507 BC).
• Euclid (c. 300 BC) introduced the axiomatic method, still used in mathematics today, which consists of definition, axiom, theorem, and proof.
• Euclid's Elements is considered one of history's most successful and influential textbooks.
• Archimedes (c. 287–c. 212 BC) developed methods for calculating surface area and volume of solids of revolution.
• The term "mathematics" originates from the Ancient Greek μάθημα (máthēma), meaning "that which is learnt."
• The Hindu–Arabic numeral system was the most significant development of Indian mathematics.
• The Hindu–Arabic numeral system first appeared in Indian mathematics.
• Indian mathematicians contributed to trigonometry, algebra, arithmetic, and negative numbers.
• Early advances in Chinese mathematics included a place value system.
• The mathematicians Brahmagupta, Aryabhata, and Bhaskara made fundamental contributions.
• Arabic mathematics greatly advanced algebra.
• Persian mathematician Muhammad ibn Musa al-Khwarizmi wrote about comprehensively solving quadratic equations.
• Al-Khwarizmi is regarded as the founder of algebra.
• Islamic scholars preserved and further developed Greek mathematics.
• Mathematics saw increased interaction with other fields during the Renaissance.
• Isaac Newton (1643–1727) and Gottfried Wilhelm Leibniz (1646–1716) developed infinitesimal calculus.
• Leonhard Euler (1707–1783) made discoveries in mathematical analysis, number theory, and graph theory.
• Mathematics became increasingly abstract in the 19th century.
• Nikolai Lobachevsky (1792–1856), János Bolyai (1802–1860), and Carl Friedrich Gauss (1777–1855) did revolutionary work on non-Euclidean geometries.
• David Hilbert (1862–1943) presented a list of 23 unsolved problems at the International Congress of Mathematicians in 1900.
• Many of these problems have been solved since and continue to inspire mathematical research.
• Kurt Gödel (1906–1978) demonstrated that within any sufficiently complex mathematical system, there exist statements that cannot be proven true or false based on the system's axioms.

Notation, terminology, and style

• Mathematics uses notation to express mathematical concepts concisely.
• Mathematical notation has evolved over centuries.
• Most mathematics today uses the notation introduced by Leonhard Euler (1707–1783).
• A mathematical proof is sufficient to ensure truth.
• Rigor is expected in mathematical proofs.
• Mathematics follows its own terminology.
• Mathematical proofs rely on logic and notation.

Pure mathematics

• Pure mathematics develops mathematics for its own sake.
• Pure mathematics is done without application in mind.
• However, practically useful applications for what began as pure mathematics are often discovered later.
• Pure mathematics includes number theory, geometry, and mathematical analysis.
• Number theory began with the study of whole numbers.
• Geometry is one of the oldest branches of mathematics.
• Geometry is concerned with shape, size, relative position of figures, and the properties of space.
• Mathematical analysis includes calculus, differential equations, and real analysis.

Applied mathematics

• Applied mathematics is concerned with the use of mathematical tools to solve problems in science, engineering, and other fields.
• Applied mathematics overlaps with the discipline of statistics.
• Applied mathematics includes mathematical physics, numerical analysis, optimization, and control theory.
• Statistics uses math to analyze data and make inferences.
• Numerical analysis is concerned with developing algorithms for solving mathematical problems.
• Optimization is the mathematical study of finding the "best" solution.
• Control theory is concerned with controlling the behavior of dynamical systems.

Foundations and philosophy

• It is difficult to give a simple definition of mathematics.
• The philosophy of mathematics is concerned with the epistemological, metaphysical, and aesthetic dimensions of mathematics.
• The foundations of mathematics concern the axiomatic method and attempts to put mathematics on a firm foundation.
• Key schools of thought in the philosophy of mathematics include Platonism, formalism, and constructivism.
• Platonism holds that mathematical entities are abstract, existing independently of human minds.
• Formalism holds that mathematics is concerned with the manipulation of symbols according to certain rules.
• Constructivism holds that mathematical objects must be constructed in some sense before they can be said to exist.

Mathematics explores quantity, structure, space, and change, seeking patterns and new ideas. Proofs validate these conjectures. It provides key tools and insights for science, engineering, and more.