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Mathematics

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Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline.

This is the Ulam spiral, which illustrates the distribution of prime numbers. The dark diagonal lines in the spiral hint at the hypothesized approximate independence between being prime and being a value of a quadratic polynomial, a conjecture now known as Hardy and Littlewood's Conjecture F.

On the surface of a sphere, Euclidean geometry only applies as a local approximation. For larger scales the sum of the angles of a triangle is not equal to 180°.

The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC

A page from al-Khwārizmī's Algebra

Knowledge

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Knowledge is an awareness of facts, a familiarity with individuals and situations, or a practical skill. Knowledge of facts, also called propositional knowledge, is often characterized as true belief that is distinct from opinion or guesswork by virtue of justification. While there is wide agreement among philosophers that propositional knowledge is a form of true belief, many controversies focus on justification. This includes questions like how to understand justification, whether it is needed at all, and whether something else besides it is needed. These controversies intensified in the latter half of the 20th century due to a series of thought experiments called Gettier cases that provoked alternative definitions.

The owl of Athena, a symbol of knowledge in the Western world

Declarative knowledge can be stored in books.

Knowing how to ride a bicycle is one form of non-propositional knowledge.

Perception relies on the senses to acquire knowledge.