Number in the context of "Pi"

A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, relationship, or mathematical formula. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise different concepts and experiences. All communication is achieved through the use of symbols: for example, a red octagon is a common symbol for "STOP"; on maps, blue lines often represent rivers; and a red rose often symbolizes love and compassion. Numerals are symbols for numbers; letters of an alphabet may be symbols for certain phonemes; and personal names are symbols representing individuals. The academic study of symbols is called semiotics.

In the arts, symbolism is the use of a concrete element to represent a more abstract idea. In cartography, an organized collection of symbols forms a legend for a map.

Platonism is the philosophy of Plato and philosophical systems closely derived from it, though later and contemporary Platonists do not necessarily accept all of Plato's own doctrines. Platonism has had a profound effect on Western thought. At the most fundamental level, Platonism affirms the existence of abstract objects, which are asserted to exist in a third realm distinct from both the sensible external world and from the internal world of consciousness, and is the opposite of nominalism. This can apply to properties, types, propositions, meanings, numbers, sets, truth values, and so on (see abstract object theory). Philosophers who affirm the existence of abstract objects are sometimes called Platonists; those who deny their existence are sometimes called nominalists. The terms "Platonism" and "nominalism" also have established senses in the history of philosophy. They denote positions that have little to do with the modern notion of an abstract object.

In a narrower sense, the term might indicate the doctrine of Platonic realism, a form of mysticism. The central concept of Platonism, a distinction essential to the Theory of Forms, is the distinction between the reality which is perceptible but unintelligible, associated with the flux of Heraclitus and studied by the likes of physical science, and the reality which is imperceptible but intelligible, associated with the unchanging being of Parmenides and studied by the likes of mathematics. Geometry was the main motivation of Plato, and this also shows the influence of Pythagoras. The Forms are typically described in dialogues such as the Phaedo, Symposium and Republic as perfect archetypes of which objects in the everyday world are imperfect copies. Aristotle's Third Man Argument is its most famous criticism in antiquity.

In philosophy and the arts, a fundamental distinction exists between abstract and concrete entities. While there is no universally accepted definition, common examples illustrate the difference: numbers, sets, and ideas are typically classified as abstract objects, whereas plants, dogs, and planets are considered concrete objects.

Philosophers have proposed several criteria to define this distinction:

A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a symbol, and therefore can be involved in formulas. Commonly encountered mathematical objects include numbers, expressions, shapes, functions, and sets. Mathematical objects can be very complex; for example, theorems, proofs, and even formal theories are considered as mathematical objects in proof theory.

In philosophy of mathematics, the concept of "mathematical objects" touches on topics of existence, identity, and the nature of reality. In metaphysics, objects are often considered entities that possess properties and can stand in various relations to one another. Philosophers debate whether mathematical objects have an independent existence outside of human thought (realism), or if their existence is dependent on mental constructs or language (idealism and nominalism). Objects can range from the concrete: such as physical objects usually studied in applied mathematics, to the abstract, studied in pure mathematics. What constitutes an "object" is foundational to many areas of philosophy, from ontology (the study of being) to epistemology (the study of knowledge). In mathematics, objects are often seen as entities that exist independently of the physical world, raising questions about their ontological status. There are varying schools of thought which offer different perspectives on the matter, and many famous mathematicians and philosophers each have differing opinions on which is more correct.

In metaphysics, nominalism is the view that universals and abstract objects do not actually exist other than being merely names or labels. There are two main versions of nominalism. One denies the existence of universals—that which can be instantiated or exemplified by many particular things (e.g., strength, humanity). The other version specifically denies the existence of abstract objects as such—objects that do not exist in space and time.

Most nominalists have held that only physical particulars in space and time are real, and that universals exist only post res, that is, subsequent to particular things. However, some versions of nominalism hold that some particulars are abstract entities (e.g., numbers), whilst others are concrete entities – entities that do exist in space and time (e.g., pillars, snakes, and bananas). Nominalism is primarily a position on the problem of universals. It is opposed to realist philosophies, such as Platonic realism, which assert that universals do exist over and above particulars, and to the hylomorphic substance theory of Aristotle, which asserts that universals are immanently real within them; however, the name "nominalism" emerged from debates in medieval philosophy with Roscellinus.

A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.

A numerical digit (often shortened to just digit) or numeral is a single symbol used alone (such as "1"), or in combinations (such as "15"), to represent numbers in positional notation, such as the common base 10. The name "digit" originates from the Latin digiti meaning fingers.

For any numeral system with an integer base, the number of different digits required is the absolute value of the base. For example, decimal (base 10) requires ten digits (0 to 9), and binary (base 2) requires only two digits (0 and 1). Bases greater than 10 require more than 10 digits, for instance hexadecimal (base 16) requires 16 digits (usually 0 to 9 and A to F).

Graphics (from Ancient Greek γραφικός (التصميم) 'pertaining to drawing, painting, writing, etc.') are visual images or designs on some surface, such as a wall, canvas, screen, paper, or stone, to inform, illustrate, or entertain. In contemporary usage, it includes a pictorial representation of data, as in design and manufacture, in typesetting and the graphic arts, and in educational and recreational software. Images that are generated by a computer are called computer graphics.

Examples are photographs, drawings, line art, mathematical graphs, line graphs, charts, diagrams, typography, numbers, symbols, geometric designs, maps, engineering drawings, or other images. Graphics often combine text, illustration, and color. Graphic design may consist of the deliberate selection, creation, or arrangement of typography alone, as in a brochure, flyer, poster, web site, or book without any other element. The objective can be clarity or effective communication, association with other cultural elements, or merely the creation of a distinctive style.