Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most to objects that are relevant to mathematics; the language of set theory can be used to define nearly all mathematical objects. The modern study of set theory was initiated by Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, such as Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with or without the axiom of choice, are the best-known. Set theory is employed as a foundational system for mathematics in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
Mathematical topics emerge and evolve through interactions among many researchers. Set theory, was founded by a single paper in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers". Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1870–1874 and was motivated by Cantor's work in real analysis. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper. Cantor's work polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. Cantorian set theory became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, the "infinity of infinities" resulting from the power set operation.
This utility of set theory led to the article "Mengenlehre" contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia. The next wave of excitement in set theory came around 1900, when it was discovered that some interpretations of Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes. Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself and not a member of itself. In 1899 Cantor had himself posed the question "What is the cardinal number of the set of all sets?", obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics. In 1906 English readers gained the book Theory of Sets of Points by husband and wife William Henry Young and Grace Chisholm Young, published by Cambridge University Press.
The momentum of set theory was such. The work of Zermelo in 1908 and the work of Abraham Fraenkel and Thoralf Skolem in 1922 resulted in the set of axioms ZFC, which became the most used set of axioms for set theory; the work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is used as a foundational system, although in some areas—such as algebraic geometry and algebraic topology—category theory is thought to be a preferred foundation. Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member of A, the notation o. Since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the subset relation called set inclusion. If all the members of set A are members of set B A is a subset of B, denoted A ⊆ B. For example, is a subset of, so is but is not; as insinuated from this definition, a set is a subset of itself.
For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B. Note that 1, 2, 3 are members of the set but are not subsets of it. Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The: Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both; the union of and is the set. Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B; the intersection of and is the set. Set difference of U and A, denoted U \ A, is the set of all members of U that are not members of A; the set difference \ is, conversely, the set difference \ is. When A is a subset of U, the set difference U \ A is called the complement of A in U. In this case, if the choice of U is clear from the context, the notation Ac is sometimes used instead of U \ A if U is a universal set as in the study of Venn diagrams.
Symmetric difference of sets A and B, denoted A △ B or A ⊖ B, is
In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; the concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s.
After contributions from other fields such as number theory and geometry, the group notion was generalized and established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory; the modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.
The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots. The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884; the third field contributing to group theory was number theory.
Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae, more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers; the convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques. Walther von Dyck introduced the idea of specifying a group by means of generators and relations, was the first to give an axiomatic definition of an "abstract group", in the terminology of the time; as of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, more locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.
Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley and by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004; this project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification; these days, group theory is still a active mathematical branch, impacting many other fields. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition. The following properties of integer addition serve as a model for the group axioms given in the definition below.
For any two integers a and b, the sum a + b is an integer. That is, addition of integers always yields an integer; this property is known as closure under addition. For all integers a, b and c, + c = a +. Expressed in words
Algebraic varieties are the central objects of study in algebraic geometry. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.:58Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility; the concept of an algebraic variety is similar to that of an analytic manifold. An important difference is that an algebraic variety may have singular points, while a manifold cannot; the fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial in one variable with complex number coefficients is determined by the set of its roots in the complex plane.
Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory; this correspondence is a defining feature of algebraic geometry. An affine variety over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define quasi-projective varieties in a similar way; the most general definition of a variety is obtained by patching together smaller quasi-projective varieties. It is not obvious that one can construct genuinely new examples of varieties in this way, but Nagata gave an example of such a new variety in the 1950s. For an algebraically closed field K and a natural number n, let An be affine n-space over K; the polynomials f in the ring K can be viewed as K-valued functions on An by evaluating f at the points in An, i.e. by choosing values in K for each xi.
For each set S of polynomials in K, define the zero-locus Z to be the set of points in An on which the functions in S vanish, to say Z =. A subset V of An is called an affine algebraic set if V = Z for some S.:2 A nonempty affine algebraic set V is called irreducible if it cannot be written as the union of two proper algebraic subsets.:3 An irreducible affine algebraic set is called an affine variety.:3 Affine varieties can be given a natural topology by declaring the closed sets to be the affine algebraic sets. This topology is called the Zariski topology.:2Given a subset V of An, we define I to be the ideal of all polynomial functions vanishing on V: I =. For any affine algebraic set V, the coordinate ring or structure ring of V is the quotient of the polynomial ring by this ideal.:4 Let k be an algebraically closed field and let Pn be the projective n-space over k. Let f in k be a homogeneous polynomial of degree d, it is not well-defined to evaluate f on points in Pn in homogeneous coordinates.
However, because f is homogeneous, meaning that f = λd f , it does make sense to ask whether f vanishes at a point. For each set S of homogeneous polynomials, define the zero-locus of S to be the set of points in Pn on which the functions in S vanish: Z =. A subset V of Pn is called a projective algebraic set if V = Z for some S.:9 An irreducible projective algebraic set is called a projective variety.:10Projective varieties are equipped with the Zariski topology by declaring all algebraic sets to be closed. Given a subset V of Pn, let I be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.:10A quasi-projective variety is a Zariski open subset of a projective variety. Notice that every affine variety is quasi-projective. Notice that the complement of an algebraic set in an affine variety is a quasi-projective variety. In classical algebraic geometry, a
In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting and bending, but not tearing or gluing. An n-dimensional topological space is a space with certain properties of connectedness and compactness; the space discrete. It can be closed. Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space and transformation; such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems; the term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics. Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries.
Among these are certain questions in geometry investigated by Leonhard Euler. His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750, Euler wrote to a friend that he had realised the importance of the edges of a polyhedron; this led to his polyhedron formula, V − E + F = 2. Some authorities regard this analysis as the first theorem. Further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term "Topologie" in Vorstudien zur Topologie, written in his native German, in 1847, having used the word for ten years in correspondence before its first appearance in print; the English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated". The term "topologist" in the sense of a specialist in topology was used in 1905 in the magazine Spectator.
Their work was corrected and extended by Henri Poincaré. In 1895, he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a special case of a general topological space, with any given topological space giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined the term "topological space" and gave the definition for what is now called a Hausdorff space. A topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski. Modern topology depends on the ideas of set theory, developed by Georg Cantor in the part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series.
For further developments, see point-set topology and algebraic topology. Topology can be formally defined as "the study of qualitative properties of certain objects that are invariant under a certain kind of transformation those properties that are invariant under a certain kind of invertible transformation." Topology is used to refer to a structure imposed upon a set X, a structure that characterizes the set X as a topological space by taking proper care of properties such as convergence and continuity, upon transformation. Topological spaces show up in every branch of mathematics; this has made topology one of the great unifying ideas of mathematics. The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects and both separate the plane into two parts, the part inside and the part outside.
In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg that would cross each of its seven bridges once. This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks; this Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory. The hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick." This fact is convincing to most people though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg, the result does not depend on the shape of t
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques from commutative algebra, for solving geometrical problems about these sets of zeros; the fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, parabolas, hyperbolas, cubic curves like elliptic curves, quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.
Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis and number theory. A study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, it becomes more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution. In the 20th century, algebraic geometry split into several subareas; the mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more to the points with coordinates in an algebraically closed field. Real algebraic geometry is the study of the real points of an algebraic variety. Diophantine geometry and, more arithmetic geometry is the study of the points of an algebraic variety with coordinates in fields that are not algebraically closed and occur in algebraic number theory, such as the field of rational numbers, number fields, finite fields, function fields, p-adic fields.
A large part of singularity theory is devoted to the singularities of algebraic varieties. Computational algebraic geometry is an area that has emerged at the intersection of algebraic geometry and computer algebra, with the rise of computers, it consists of algorithm design and software development for the study of properties of explicitly given algebraic varieties. Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way, similar to its use in the study of differential and analytic manifolds; this is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring.
This means that a point of such a scheme may be either a subvariety. This approach enables a unification of the language and the tools of classical algebraic geometry concerned with complex points, of algebraic number theory. Wiles' proof of the longstanding conjecture called Fermat's last theorem is an example of the power of this approach. In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that satisfy one or more polynomial equations. For instance, the two-dimensional sphere of radius 1 in three-dimensional Euclidean space R3 could be defined as the set of all points with x 2 + y 2 + z 2 − 1 = 0. A "slanted" circle in R3 can be defined as the set of all points which satisfy the two polynomial equations x 2 + y 2 + z 2 − 1 = 0, x + y + z = 0. First we start with a field k. In classical algebraic geometry, this field was always the complex numbers C, but many of the same results are true if we assume only that k is algebraically closed.
We consider the affine space of dimension n over denoted An. When one fixes a coordinate system, one may identify An with kn; the purpose of not working with kn is to emphasize that one "forgets" the vector space structure that kn carries. A function f: An → A1 is said to be polynomial if it can be written as a polynomial, that is, if there is a polynomial p in k such that f = p for every point M with coordinates in An; the property of a function to be polynomial does not depend on the choice of a coordinate system in An. When a coordinate system is chosen, the regular functions on the affine n-space may be identified with the ring of polynomial functions in n variables over k. Therefore, the set of the
In mathematics abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients and subobjects. Versions of the theorems exist for groups, vector spaces, Lie algebras, various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences; the isomorphism theorems were formulated in some generality for homomorphisms of modules by Emmy Noether in her paper Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern, published in 1927 in Mathematische Annalen. Less general versions of these theorems can be found in work of Richard Dedekind and previous papers by Noether. Three years B. L. van der Waerden published his influential Algebra, the first abstract algebra textbook that took the groups-rings-fields approach to the subject. Van der Waerden credited lectures by Noether on group theory and Emil Artin on algebra, as well as a seminar conducted by Artin, Wilhelm Blaschke, Otto Schreier, van der Waerden himself on ideals as the main references.
The three isomorphism theorems, called homomorphism theorem, two laws of isomorphism when applied to groups, appear explicitly. We first state the three isomorphism theorems in the context of groups. Note that some sources switch the numbering of the second and third theorems. Another variation encountered in the literature in Van der Waerden's Algebra, is to call first isomorphism theorem the Fundamental Homomorphism Theorem and to decrement the numbering of the remaining isomorphism theorems by one. In the most extensive numbering scheme, the lattice theorem is sometimes referred to as the fourth isomorphism theorem. Let G and H be groups, let φ: G → H be a homomorphism. Then: The kernel of φ is a normal subgroup of G, The image of φ is a subgroup of H, The image of φ is isomorphic to the quotient group G / ker. In particular, if φ is surjective H is isomorphic to G / ker. Let G be a group. Let S be a subgroup of G, let N be a normal subgroup of G; the following hold: The product S N is a subgroup of G, The intersection S ∩ N is a normal subgroup of S, The quotient groups / N and S / are isomorphic.
Technically, it is not necessary for N to be a normal subgroup, as long as S is a subgroup of the normalizer of N in G. In this case, the intersection S ∩ N is not a normal subgroup of G, but it is still a normal subgroup of S; this isomorphism theorem has been called the "diamond theorem" due to the shape of the resulting subgroup lattice with S N at the top, S ∩ N at the bottom and with N and S to the sides. It has been called the "parallelogram theorem" because in the resulting subgroup lattice the two sides assumed to represent the quotient groups / N and S / are "equal" in the sense of isomorphism. An example of the second isomorphism theorem gives an identity of projective linear groups. Setting G = G L 2, the group of invertible 2x2 complex matrices, S = S L 2, the subgroup of determinant 1 matrices, N the normal subgroup of scalar matrices C × I =, we have S ∩ N =, where I is the identity matrix, S N = G L 2; the second isomorphism theorem states that: P G L 2:= G L 2 / ≅ S L 2 / =: P S L
Serge Lang was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the influential Algebra, he was a member of the Bourbaki group. As an activist, he campaigned against the nomination of the political scientist Samuel P. Huntington to the National Academies of Science, descended into AIDS denialism, claiming that HIV had not been proven to cause AIDS and protesting Yale's research into HIV/AIDS. Lang was born in Saint-Germain-en-Laye, close to Paris, in 1927, he had a twin brother who became a sister who became an actress. Lang moved with his family to California as a teenager, where he graduated in 1943 from Beverly Hills High School, he subsequently graduated from the California Institute of Technology in 1946, received a doctorate from Princeton University in 1951. He held faculty positions at the University of Chicago, Columbia University, Yale University. Lang studied under Emil Artin at Princeton University, writing his thesis on quasi-algebraic closure, worked on the geometric analogues of class field theory and diophantine geometry.
He moved into diophantine approximation and transcendental number theory, proving the Schneider–Lang theorem. A break in research while he was involved in trying to meet 1960s student activism halfway caused him difficulties in picking up the threads afterwards, he wrote on modular forms and modular units, the idea of a'distribution' on a profinite group, value distribution theory. He made a number of conjectures in diophantine geometry: Mordell–Lang conjecture, Bombieri–Lang conjecture, Lang–Trotter conjecture, the Lang conjecture on analytically hyperbolic varieties, he introduced the Lang map, the Katz–Lang finiteness theorem, the Lang–Steinberg theorem in algebraic groups. Lang was a prolific writer of mathematical texts completing one on his summer vacation. Most are at the graduate level, he wrote calculus texts and prepared a book on group cohomology for Bourbaki. Lang's Algebra, a graduate-level introduction to abstract algebra, was a influential text that ran through numerous updated editions.
His Steele prize citation stated, "Lang's Algebra changed the way graduate algebra is taught... It has affected all subsequent graduate-level algebra books." It contained ideas of Artin. Lang was noted for his eagerness for contact with students, he was described as a passionate teacher who would throw chalk at students who he believed were not paying attention. One of his colleagues recalled: "He would rave in front of his students, he would say,'Our two aims are truth and clarity, to achieve these I will shout in class.'" He won a Leroy P. Steele Prize for Mathematical Exposition from the American Mathematical Society. In 1960, he won the sixth Frank Nelson Cole Prize in Algebra for his paper Unramified class field theory over function fields in several variables. Lang spent much of his professional time engaged in political activism, he was a staunch socialist and active in opposition to the Vietnam War, volunteering for the 1966 anti-war campaign of Robert Scheer. Lang quit his position at Columbia in 1971 in protest over the university's treatment of anti-war protesters.
Lang engaged in several efforts to challenge anyone he believed was spreading misinformation or misusing science or mathematics to further their own goals. He attacked the 1977 Survey of the American Professoriate, an opinion questionnaire that Seymour Martin Lipset and E. C. Ladd had sent to thousands of college professors in the United States, accusing it of containing numerous biased and loaded questions; this led to a public and acrimonious conflict. In 1986, Lang mounted what the New York Times described as a "one-man challenge" against the nomination of political scientist Samuel P. Huntington to the National Academy of Sciences. Lang described Huntington's research, in particular his use of mathematical equations to demonstrate that South Africa was a "satisfied society", as "pseudoscience", arguing that it gave "the illusion of science without any of its substance." Despite support for Huntington from the Academy's social and behavioral scientists, Lang's challenge was successful, Huntington was twice rejected for Academy membership.
Huntington's supporters argued that Lang's opposition was political rather than scientific in nature. Lang kept his political correspondence and related documentation in extensive "files", he would send letters or publish articles, wait for responses, engage the writers in further correspondence, collect all these writings together and point out what he considered contradictions. He mailed these files to people he considered important, his extensive file criticizing Nobel laureate David Baltimore was published in the journal Ethics and Behaviour in January 1993. Lang fought the decision by Yale University to hire Daniel Kevles, a historian of science, because Lang disagreed with Kevles' analysis in The Baltimore Case. Lang's most controversial political stance was as an AIDS denialist.