Nature, Characteristics and Definition of Mathematics

Flashcards covering the nature, essence, characteristics, and systems of mathematics, including patterns in nature and mathematical language terminology.

Mathematics

A tool to quantify, organize and control our world, predict phenomena and make life easier for us.

Science of Discovery

The discovery of relationships and the expression of those relationships in symbolic form – in words, in numbers, in letters, by diagrams or by graphs.

Intellectual Game

A characteristic of mathematics suitable for child development because its conclusions are certain, it permits gradual progression from easy to difficult, and its conclusions are applicable to other subjects.

Mathematics as a Tool Subject

Also known as the 'handmaiden to the sciences,' it is a powerful and incisive tool of wide applicability used as a means to other ends.

System of Logical Processes

A system where conclusions are deduced from certain fundamental assumptions and definitions that have been hypothesized.

Intuition

The act of grasping the meaning or structure of a problem without explicit reliance on the analytic apparatus of one’s craft, which often yields hypotheses quickly.

Logical Sequence

A characteristic where mathematical learning proceeds from simple to complex and from concrete to abstract, often influenced by previous knowledge.

Structure

The formation, arrangement, and articulation of parts in anything built up by nature or art.

Precision

The 'exact' nature of mathematics where results are either right or wrong, accepted or rejected, with no midway possible.

Accuracy

An evaluation of a measure or computation in terms of the relative error or percent of error made.

Abstractness

A characteristic indicating that mathematics does not deal with actual objects and its questions cannot be settled by direct appeal to experiment.

Mathematical Language and Symbolism

A system of communication that cuts short lengthy statements and helps express ideas in an exact form free from verbosity.

Generalization and Classification

The practice of widening conceptions, combining various results under one head, and making schematic arrangements.

Undefined Terms

Arbitrary terms such as point, line, set, variable, or plane that serve as the starting point for a mathematical system.

Defined Terms

Terms in a mathematical system defined using undefined terms, such as an equilateral triangle being defined using 'triangle,' 'equal,' and 'sides.'

Axioms

Statements in a mathematical system taken for granted as self-evident truths that describe relationships among undefined terms.

Rule of Implication

A form of argument stating that if statement $p$ implies statement $q$ and statement $p$ is true, then statement $q$ will be true.

Theorem

A statement arrived at by successive application of the rule of implication to the axioms and previously established statements.

Rigor

The formal formulation of concepts and reasoning, which is secondary in pedagogical importance to intuitive understanding.

Golden Ratio

A special number found by dividing a line into two parts so the longer part divided by the smaller part equals the whole length divided by the longer part, symbolized by phi ($\phi$).

Phi ($\phi$)

In the context of the lecture, the ratio of the circumference of a circle to its diameter, usually rounded off to $1.618$.

Fibonacci Sequence

A numerical sequence beginning with $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \text{etc.}$ found in various natural patterns like pineapples and tree branches.

Bilateral Symmetry

A condition in which an object has two sides that are mirror images of each other, exemplified by the human body.

Radial Symmetry

A condition where there is a center point and numerous lines of symmetry can be drawn, exemplified by a circle or a cut orange.

Sphere

A perfectly round geometrical object in three-dimensional space; the Earth is close to an oblate spheroid version of this shape.

Hexagons

Six-sided polygons that fit together without gaps, such as the wax cells created by bees in a honeycomb.

Cone

A three-dimensional geometric shape that tapers smoothly from a flat, circular base to an apex or vertex.

Parallel Lines

Lines that stretch to infinity without ever converging or diverging.

Fibonacci Spiral

A curve formed by tracing a line through the diagonals of a series of squares with lengths equal to the Fibonacci numbers.

Expression

The mathematical analogue of an English noun; a correct arrangement of symbols representing a mathematical object that does not state a complete thought.

Mathematical Sentence

The mathematical analogue of an English sentence; a correct arrangement of symbols that states a complete thought and can be determined as true or false.