Overview of Mathematics – Key Concepts & Figures

These flashcards summarize the definitions, historical periods, major branches, and influential figures discussed in the lecture "Overview of Mathematics." They are formatted in a question-and-answer style to aid exam preparation.

What two Greek roots give us the word "mathematics"?

mánthanein, meaning knowledge/study/learning, and techne, meaning art or technique.

How did Aristotle define mathematics?

As "the science of quantity."

What 19th-century development forced new definitions of mathematics?

The rise of rigorous, abstract areas (e.g., group theory, projective geometry) not directly tied to quantity and measurement.

Which three philosophical schools offer leading definitions of mathematics?

Logicist, intuitionist, and formalist.

What was Benjamin Peirce’s 1870 logicist definition of mathematics?

"The science that draws necessary conclusions."

Finish Bertrand Russell’s famous claim: "All mathematics is _."

Symbolic logic.

How do intuitionists characterize mathematics?

As certain mental constructions or phenomena carried out in succession.

Give an example of an intuitionist definition of mathematics.

"Mathematics is the mental activity which consists in carrying out constructs one after the other."

How do formalists define mathematics?

As the study of symbols and the rules for operating on them.

State Haskell Curry’s short formalist definition of mathematics.

"The science of formal systems."

According to Immanuel Kant, why is mathematics indispensable?

He called it "the indispensable instrument of all physical resources."

What did Francis Bacon call mathematics?

"The gateway and key to all science."

How does the Oxford Dictionary (traditional) define mathematics?

The abstract science that investigates, deductively, the conclusions implicit in elementary spatial and numerical relations (geometry, arithmetic, algebra).

Provide the American Heritage Dictionary (2000) definition of mathematics.

The study of the measurement, properties, and relationships of quantities and sets using numbers and symbols.

State the broad modern definition of mathematics.

The study of quantity, structure, space, and change.

What years mark the Ancient Period of mathematics?

Roughly 3600 BC – 260 AD.

Name two major needs that drove ancient mathematics.

Numeration/arithmetic for trade and measurement/geometry for construction and astronomy.

Which clay tablet lists Pythagorean triples in Babylonian numerals?

The Plimpton 322 tablet.

What base did the Babylonian numeral system use?

Base 60 (sexagesimal).

Which historical period (600 BC – 450 AD) is noted for logic, philosophy, and Euclidean geometry?

The Greek Period.

What is Euclid’s famous treatise that systematized geometry?

Elements.

Which five regular polyhedra did Plato link to the elements?

Tetrahedron, cube, octahedron, icosahedron, and dodecahedron.

What exclamation is Archimedes said to have shouted upon a key discovery?

"Eureka!"

Give the date range of the Hindu-Arabian Period of mathematics.

About 200 BC – 1250 AD.

List three key features of the Hindu-Arabic numeral system.

Base-10, positional notation, and a symbol for zero.

Which Persian scholar’s name gave rise to the word "algorithm"?

Abu Abdullah Muhammad ibn Musa al-Khwarizmi.

What is the title of al-Khwarizmi’s book that led to the term "algebra"?

Hisab al-jabr wa al-muqabala.

Which numeral system slowly replaced the abacus across Europe during 1000-1500 AD?

The Hindu-Arabic numeration system.

Who is often depicted rediscovering Euclid’s geometry in the 15th century?

Luca Pacioli.

Who introduced logarithms in 1614?

John Napier.

Which French mathematician pioneered symbolic algebra around 1591?

François Viète.

Name the two thinkers who developed analytic geometry (1630-1640).

René Descartes and Pierre Fermat.

Which two scientists independently co-invented calculus?

Isaac Newton and Gottfried Wilhelm Leibniz.

What period of mathematics spans 1450 – 1800 AD?

The Early Modern Period.

What is the title of Newton’s 1687 work that mathematically described gravity?

Philosophiæ Naturalis Principia Mathematica (Principia).

Whose notation (dx, dy, ∫) underpins modern calculus?

Leibniz’s.

Which 19th-century giants advanced non-Euclidean geometry?

Gauss, Lobachevsky, and Riemann.

Which branch, founded by Cantor, studies infinite sets and transfinite numbers?

Set theory.

What technological invention opened new mathematical research avenues in the 20th century?

The electronic computer.

Into what four broad areas is mathematics traditionally subdivided?

Quantity, structure, space, and change.

What is the main distinction between pure and applied mathematics?

Pure maths studies concepts for their own sake; applied maths uses them to solve problems in other fields.

Which foundational discipline studies formal reasoning rules in mathematics?

Mathematical logic.

Which branch of pure mathematics deals with numbers and the basic operations on them?

Arithmetic.

What branch studies mathematical symbols and the rules for manipulating them?

Algebra.

Which discipline focuses on shapes, sizes, and spatial properties?

Geometry.

Which branch of mathematics is fundamentally about change and motion?

Calculus.

Name two applied-math fields concerned with uncertainty and decision making.

Probability theory and statistics (including decision science).

Which applied field models financial markets mathematically?

Mathematical finance.

What area of applied math underpins encryption and coding theory?

Computer science / cryptography.

Who introduced the Cartesian coordinate system?

René Descartes.

What does Descartes’ rule of signs tell us about a polynomial?

The possible number of positive and negative real roots based on sign changes.

Which ancient scholar is called the "Father of Geometry"?

Euclid of Alexandria.

State Euclid’s famous theorem about prime numbers.

There are infinitely many prime numbers.

Who is nicknamed the "Prince of Mathematics"?

Carl Friedrich Gauss.

Which theorem, rigorously treated by Gauss, guarantees that every non-constant polynomial has a root in the complex numbers?

The Fundamental Theorem of Algebra.

Which Greek thinker gave the first proof of the Pythagorean Theorem?

Pythagoras (or his school).

Give one key mathematical contribution of Archimedes.

Accurate estimation of π (pi), formulas for areas/volumes, or the method of exhaustion.

Which Italian author of Liber Abaci popularized Hindu-Arabic numerals in Europe?

Leonardo of Pisa, known as Fibonacci.

What famous integer sequence bears Fibonacci’s name?

The Fibonacci numbers (1, 1, 2, 3, 5, 8, …).

Which prolific Swiss mathematician introduced symbols such as e, i, f(x), and Σ?

Leonhard Euler.

What abbreviations for trigonometric functions were popularized by Euler?

sin, cos, tan, cot, sec, and csc.

Which mathematician co-invented calculus and formalized the binary number system?

Gottfried Wilhelm Leibniz.

What symbol did Leibniz introduce to denote integration?

The elongated S ( ∫ ).