Mathematical ideas {mathematics} include intuition, empiricism, and abstraction; relations, functions, and transformations; logic, proof, and rigor; truth, certainty, and uncertainty; existence and quantity; and formalism, symbolic languages, and formal languages.
mathematical objects
In realism, mathematical objects can be abstract acausal things. In structuralism, they are structures and systems. In idealism, constructivism, and intuitionism, they are mental conventions. Perhaps, mathematics has no objects but is only analytic or linguistic conventions used by the mathematics community.
mathematical operations
Arithmetic main operations {arithmetic operation} {fundamental operation} are add, subtract, multiply, and divide. Division by 0 has no definition. Algebra operations {algebraic operation} include all arithmetic operations and their combinations. Exponential operations manipulate constant and variable powers to find exponents {evolution, arithmetic} and logarithms {involution, arithmetic}. Operations {trigonometric operation} can find sines, cosines, and tangents. Calculus operations {analytic operation} differentiate and integrate. Geometric operations {symmetry operation} are for transformation, translation, rotation, reflection, and inversion.
mathematical foundations
Mathematical objects, such as circles and tori, have reality or are concepts.
realism
Realism says that object essences are real. Mathematical Realism, Idealism, and Platonism [Penrose, 2004] say mathematical objects are abstract essences: always existing, unchanging, timeless, without spatial location, sometimes with spatial extension, and metaphysical. They are not physical or mental. Mathematical objects are Ideals or Forms, fundamental reality that underlies physical and mental things. The physical follows mathematical laws and objects. The mental can comprehend mathematical laws and objects.
objectivism
Objectivism says that people can know the real world. People can know things' essences and underlying reality.
language
Because they share definitions and concepts, which are accessible reality or built by language, people can communicate about knowledge [Peirce, 1878].
positivism
Positivism [Ayer, 1940] says that people can experience the real world, and can know what statements about it are right or wrong based on their and other people's measurements to determine corroborations or denials.
nominalism
Nominalism says that object essences are only mental concepts, derived by human perception and reasoning from physical examples. Outside of language are only particulars. Conceptualism [Abelard, 1120] says that categories and rules are mental concepts shared by people that respond to similar world with similar minds, so universals are real insofar as they express similarities or essential object characteristics to which people respond to make concepts or dispositions. The physical world has no universals. Mathematical objects are human constructs, definitions, or concepts.
intuitionism
Mathematical intuitionism [Brouwer, 1927] says that people, using cognitive skills and experience, develop understandings of mathematical ideas. Analysis of the real world reveals mathematical objects and concepts. People then describe mathematical objects and concepts using language and logic. People cannot know things' essences or if there is underlying reality.
constructivism
Constructivism [Piaget, 1954] says that people use their, and other people's, innate and learned cognitive skills on perceptions to build concepts about their mental world, as opposed to discovering concepts of the real world. As opposed to just "telling", people use "showing" or proving [Wittgenstein, 1922]. Statements are not true but are provable or constructable. Statements are not false but have a counterexample.
formalism
Formalism [Russell, 1919] says that mathematical laws and objects are mental logical/axiomatic systems. Logic and reasoning are unchanging, timeless, meta-mathematical, and abstract. Mathematical laws and objects can describe the physical.
Mathematical Sciences>Mathematics
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0224