Mathematics, Form and Function is a survey of the whole of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, including its origins and deep structure, by the American

United States

The United States of America is a federal constitutional republic comprising fifty states and a federal district...

 mathematician Saunders Mac Lane

Saunders Mac Lane

Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...

.

Mac Lane's relevance to the philosophy of mathematics

Mac Lane founded (with Samuel Eilenberg

Samuel Eilenberg

Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor...

) category theory

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, which enables a unified treatment

Unifying theories in mathematics

There have been several attempts in history to reach a unified theory of mathematics. Some of the greatest mathematicians have expressed views that the whole subject should be fitted into one theory.-Historical perspective:...

 of mathematical structures and of the relations among them, at the cost of breaking away from their cognitive grounding

Abstract nonsense

In mathematics, abstract nonsense, general abstract nonsense, and general nonsense are terms used facetiously by some mathematicians to describe certain kinds of arguments and methods related to category theory. roughly speaking, category theory is the study of the general form of mathematical...

. Nevertheless, his views—however informal—are a valuable contribution to the philosophy

Philosophy of mathematics

The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...

 and anthropology

Anthropology

Anthropology is the study of humanity. It has origins in the humanities, the natural sciences, and the social sciences. The term "anthropology" is from the Greek anthrōpos , "man", understood to mean mankind or humanity, and -logia , "discourse" or "study", and was first used in 1501 by German...

 of mathematics. His views anticipate, in some respects, the much richer and more detailed account of the cognitive basis of mathematics given by George Lakoff

George Lakoff

George P. Lakoff is an American cognitive linguist and professor of linguistics at the University of California, Berkeley, where he has taught since 1972...

 and Rafael E. Núñez

Rafael E. Núñez

Rafael E. Núñez is a professor of cognitive science at the University of California, San Diego and a proponent of embodied cognition. He co-authored Where Mathematics Comes From with George Lakoff.-External links:*...

 in their Where Mathematics Comes From

Where Mathematics Comes From

Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being is a book by George Lakoff, a cognitive linguist, and Rafael E. Núñez, a psychologist...

. Lakoff and Núñez argue that mathematics emerges via conceptual metaphor

Conceptual metaphor

In cognitive linguistics, conceptual metaphor, or cognitive metaphor, refers to the understanding of one idea, or conceptual domain, in terms of another, for example, understanding quantity in terms of directionality . A conceptual domain can be any coherent organization of human experience...

s grounded in the human body, its motion through space

Space

Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

 and time

Time

Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

, and in human sense perceptions.

Mathematics and human activities

Throughout his book, and especially in chapter I.11, Mac Lane informally discusses how mathematics is grounded in more ordinary concrete and abstract human activities. The following table is adapted from one given on p. 35 of Mac Lane (1986). The rows are very roughly ordered from most to least fundamental. For a bullet list that can be compared and contrasted with this table, see section 3 of Where Mathematics Comes From

Where Mathematics Comes From

Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being is a book by George Lakoff, a cognitive linguist, and Rafael E. Núñez, a psychologist...

.

Human Activity	Related Mathematical Idea	Mathematical Technique
Collecting	Collection	Set; class Class (set theory) In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context... ; multiset Multiset In mathematics, the notion of multiset is a generalization of the notion of set in which members are allowed to appear more than once... ; list; family
Connecting	Cause and effect Causality Causality is the relationship between an event and a second event , where the second event is understood as a consequence of the first....	ordered pair Ordered pair In mathematics, an ordered pair is a pair of mathematical objects. In the ordered pair , the object a is called the first entry, and the object b the second entry of the pair... ; relation Relation (mathematics) In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple... ; function; operation Operation (mathematics) The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....
"	Proximity; connection Connectedness In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected...	Topological space Topological space Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion... ; mereotopology Mereotopology In formal ontology, a branch of metaphysics, and in ontological computer science, mereotopology is a first-order theory, embodying mereological and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries between parts....
Following	Successive actions	Function composition Function composition In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x... ; transformation group
Comparing	Enumeration	Bijection Bijection A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements... ; cardinal number Cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite... ; order Order theory Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
Timing	Before & After	Linear order
Counting Counting Counting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking those elements to avoid visiting the same element more than once,...	Successor Successor A successor can refer to* Someone who, or something which succeeds or comes after * Successor , an American Thoroughbred racehorseIn history:* The Diadochi, or Successors to Alexander the GreatIn mathematics:...	Successor function; ordinal number Ordinal number In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
Computing	Operations Operation (mathematics) The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....  on number Number A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers.... s	Addition, multiplication recursively defined Peano axioms In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano... ; abelian group Abelian group In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers... ; rings Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
Looking at objects	Symmetry Symmetry Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...	Symmetry group Symmetry group The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation... ; invariance Invariant (mathematics) In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used... ; isometries
Building; shaping	Shape Shape The shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material... ; point Point (geometry) In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...	Sets of points Point (geometry) In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics... ; geometry Geometry Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers .... ; pi Pi ' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
Rearranging	Permutation Permutation In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...	Bijection Bijection A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements... ; permutation group Permutation group In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...
Selecting; distinguishing	Parthood Mereology In philosophy and mathematical logic, mereology treats parts and the wholes they form...	Subset Subset In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment... ; order Order theory Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions... ; lattice theory Lattice (order) In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities... ; mereology Mereology In philosophy and mathematical logic, mereology treats parts and the wholes they form...
Arguing	Proof Proof Proof may refer to:* Proof , sufficient evidence or argument for the truth of a proposition* Formal proof* Mathematical proof, a convincing demonstration that some mathematical statement is necessarily true...	First-order logic First-order logic First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
Measuring Measurement Measurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the metre, second or degree Celsius...	Distance Distance Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance... ; extent	Rational number Rational number In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number... ; metric space Metric space In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
Endless repetition	Infinity Infinity Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity... ; Recursion Recursion Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...	Recursive set Recursive set In computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithm which terminates after a finite amount of time and correctly decides whether or not a given number belongs to the set.... ; Infinite set
Estimating	Approximation Approximation An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...	Real number Real number In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π... ; real field Real field Real field may refer to:* Real numbers, the numbers that can be represented by infinite decimals* Formally real field, an algebraic field that has the so-called "real" property* Realfield, a spoof of Garfield...
Moving through space Space Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...  & time Time Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects.... :	curvature Curvature In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...	calculus Calculus Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem... ; differential geometry
--Without cycling	Change	Real analysis Real analysis Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real... ; transformation group
--With cycling	Repetition	pi Pi ' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants... ; trigonometry Trigonometry Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves... ; complex number Complex number A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part... ; complex analysis Complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
--Both		Differential equations; mathematical physics Mathematical physics Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
Motion through time alone	Growth & decay	e E (mathematical constant) The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base... ; exponential function Exponential function In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,... ; natural logarithms;
Altering shapes	Deformation	Differential geometry; topology Topology Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
Observing patterns	Abstraction Abstraction Abstraction is a process by which higher concepts are derived from the usage and classification of literal concepts, first principles, or other methods....	Axiomatic set theory; universal algebra Universal algebra Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures.... ; category theory Category theory Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions... ; morphism Morphism In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
Seeking to do better	Optimization Optimization (mathematics) In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....	Operations research Operations research Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations... ; optimal control theory; dynamic programming Dynamic programming In mathematics and computer science, dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. It is applicable to problems exhibiting the properties of overlapping subproblems which are only slightly smaller and optimal substructure...
Choosing; gambling Gambling Gambling is the wagering of money or something of material value on an event with an uncertain outcome with the primary intent of winning additional money and/or material goods...	Chance Probability Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...	Probability theory Probability theory Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single... ; mathematical statistics Mathematical statistics Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis... ; measure Measure (mathematics) In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

Also see the related diagrams appearing on the following pages of Mac Lane (1986): 149, 184, 306, 408, 416, 422-28.

Mac Lane (1986) cites a related monograph by Lars Gårding

Lars Gårding

Lars Gårding is a Swedish mathematician. He has made notable contributions to the study of partial differential operators. He is a professor emeritus of mathematics at Lund University in Sweden...

(1977).