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Thursday, January 25, 2024

Metalogic

From Wikipedia, the free encyclopedia

Metalogic is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths.

The basic objects of metalogical study are formal languages, formal systems, and their interpretations. The study of interpretation of formal systems is the branch of mathematical logic that is known as model theory, and the study of deductive systems is the branch that is known as proof theory.

Overview

Formal language

A formal language is an organized set of symbols, the symbols of which precisely define it by shape and place. Such a language therefore can be defined without reference to the meanings of its expressions; it can exist before any interpretation is assigned to it—that is, before it has any meaning. First-order logic is expressed in some formal language. A formal grammar determines which symbols and sets of symbols are formulas in a formal language.

A formal language can be formally defined as a set A of strings (finite sequences) on a fixed alphabet α. Some authors, including Rudolf Carnap, define the language as the ordered pair <α, A>. Carnap also requires that each element of α must occur in at least one string in A.

Formation rules

Formation rules (also called formal grammar) are a precise description of the well-formed formulas of a formal language. They are synonymous with the set of strings over the alphabet of the formal language that constitute well formed formulas. However, it does not describe their semantics (i.e. what they mean).

Formal systems

A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation rules (also called inference rules) or a set of axioms, or have both. A formal system is used to derive one expression from one or more other expressions.

A formal system can be formally defined as an ordered triple <α,,d>, where d is the relation of direct derivability. This relation is understood in a comprehensive sense such that the primitive sentences of the formal system are taken as directly derivable from the empty set of sentences. Direct derivability is a relation between a sentence and a finite, possibly empty set of sentences. Axioms are so chosen that every first place member of d is a member of and every second place member is a finite subset of .

A formal system can also be defined with only the relation d. Thereby can be omitted and α in the definitions of interpreted formal language, and interpreted formal system. However, this method can be more difficult to understand and use.

Formal proofs

A formal proof is a sequence of well-formed formulas of a formal language, the last of which is a theorem of a formal system. The theorem is a syntactic consequence of all the well formed formulae that precede it in the proof system. For a well formed formula to qualify as part of a proof, it must result from applying a rule of the deductive apparatus of some formal system to the previous well formed formulae in the proof sequence.

Interpretations

An interpretation of a formal system is the assignment of meanings to the symbols and truth-values to the sentences of the formal system. The study of interpretations is called Formal semantics. Giving an interpretation is synonymous with constructing a model.

Important distinctions

Metalanguage–object language

In metalogic, formal languages are sometimes called object languages. The language used to make statements about an object language is called a metalanguage. This distinction is a key difference between logic and metalogic. While logic deals with proofs in a formal system, expressed in some formal language, metalogic deals with proofs about a formal system which are expressed in a metalanguage about some object language.

Syntax–semantics

In metalogic, 'syntax' has to do with formal languages or formal systems without regard to any interpretation of them, whereas, 'semantics' has to do with interpretations of formal languages. The term 'syntactic' has a slightly wider scope than 'proof-theoretic', since it may be applied to properties of formal languages without any deductive systems, as well as to formal systems. 'Semantic' is synonymous with 'model-theoretic'.

Use–mention

In metalogic, the words 'use' and 'mention', in both their noun and verb forms, take on a technical sense in order to identify an important distinction. The use–mention distinction (sometimes referred to as the words-as-words distinction) is the distinction between using a word (or phrase) and mentioning it. Usually it is indicated that an expression is being mentioned rather than used by enclosing it in quotation marks, printing it in italics, or setting the expression by itself on a line. The enclosing in quotes of an expression gives us the name of an expression, for example:

'Metalogic' is the name of this article.
This article is about metalogic.

Type–token

The type-token distinction is a distinction in metalogic, that separates an abstract concept from the objects which are particular instances of the concept. For example, the particular bicycle in your garage is a token of the type of thing known as "The bicycle." Whereas, the bicycle in your garage is in a particular place at a particular time, that is not true of "the bicycle" as used in the sentence: "The bicycle has become more popular recently." This distinction is used to clarify the meaning of symbols of formal languages.

History

Metalogical questions have been asked since the time of Aristotle. However, it was only with the rise of formal languages in the late 19th and early 20th century that investigations into the foundations of logic began to flourish. In 1904, David Hilbert observed that in investigating the foundations of mathematics that logical notions are presupposed, and therefore a simultaneous account of metalogical and metamathematical principles was required. Today, metalogic and metamathematics are largely synonymous with each other, and both have been substantially subsumed by mathematical logic in academia. A possible alternate, less mathematical model may be found in the writings of Charles Sanders Peirce and other semioticians.

Results

Results in metalogic consist of such things as formal proofs demonstrating the consistency, completeness, and decidability of particular formal systems.

Major results in metalogic include:

Scheme (mathematics)

From Wikipedia, the free encyclopedia

In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).

Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise Éléments de géométrie algébrique; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem.

Formally, a scheme is a topological space, together with commutative rings for all of its open sets, that arises from gluing together spectra (spaces of prime ideals) of commutative rings along their open subsets. In other words, it is a ringed space that is locally a spectrum of a commutative ring.

The relative point of view is that much of algebraic geometry should be developed for a morphism XY of schemes (called a scheme X over Y), rather than for an individual scheme. For example, in studying algebraic surfaces, it can be useful to consider families of algebraic surfaces over any scheme Y. In many cases, the family of all varieties of a given type can itself be viewed as a variety or scheme, known as a moduli space.

For some of the detailed definitions in the theory of schemes, see the glossary of scheme theory.

Development

The origins of algebraic geometry mostly lie in the study of polynomial equations over the real numbers. By the 19th century, it became clear (notably in the work of Jean-Victor Poncelet and Bernhard Riemann) that algebraic geometry was simplified by working over the field of complex numbers, which has the advantage of being algebraically closed. Two issues gradually drew attention in the early 20th century, motivated by problems in number theory: how can algebraic geometry be developed over any algebraically closed field, especially in positive characteristic? (The tools of topology and complex analysis used to study complex varieties do not seem to apply here.) And what about algebraic geometry over an arbitrary field?

Hilbert's Nullstellensatz suggests an approach to algebraic geometry over any algebraically closed field k: the maximal ideals in the polynomial ring k[x1,...,xn] are in one-to-one correspondence with the set kn of n-tuples of elements of k, and the prime ideals correspond to the irreducible algebraic sets in kn, known as affine varieties. Motivated by these ideas, Emmy Noether and Wolfgang Krull developed the subject of commutative algebra in the 1920s and 1930s. Their work generalizes algebraic geometry in a purely algebraic direction: instead of studying the prime ideals in a polynomial ring, one can study the prime ideals in any commutative ring. For example, Krull defined the dimension of any commutative ring in terms of prime ideals. At least when the ring is Noetherian, he proved many of the properties one would want from the geometric notion of dimension.

Noether and Krull's commutative algebra can be viewed as an algebraic approach to affine algebraic varieties. However, many arguments in algebraic geometry work better for projective varieties, essentially because projective varieties are compact. From the 1920s to the 1940s, B. L. van der Waerden, André Weil and Oscar Zariski applied commutative algebra as a new foundation for algebraic geometry in the richer setting of projective (or quasi-projective) varieties. In particular, the Zariski topology is a useful topology on a variety over any algebraically closed field, replacing to some extent the classical topology on a complex variety (based on the topology of the complex numbers).

For applications to number theory, van der Waerden and Weil formulated algebraic geometry over any field, not necessarily algebraically closed. Weil was the first to define an abstract variety (not embedded in projective space), by gluing affine varieties along open subsets, on the model of manifolds in topology. He needed this generality for his construction of the Jacobian variety of a curve over any field. (Later, Jacobians were shown to be projective varieties by Weil, Chow and Matsusaka.)

The algebraic geometers of the Italian school had often used the somewhat foggy concept of the generic point of an algebraic variety. What is true for the generic point is true for "most" points of the variety. In Weil's Foundations of Algebraic Geometry (1946), generic points are constructed by taking points in a very large algebraically closed field, called a universal domain. Although this worked as a foundation, it was awkward: there were many different generic points for the same variety. (In the later theory of schemes, each algebraic variety has a single generic point.)

In the 1950s, Claude Chevalley, Masayoshi Nagata and Jean-Pierre Serre, motivated in part by the Weil conjectures relating number theory and algebraic geometry, further extended the objects of algebraic geometry, for example by generalizing the base rings allowed. The word scheme was first used in the 1956 Chevalley Seminar, in which Chevalley was pursuing Zariski's ideas. According to Pierre Cartier, it was André Martineau who suggested to Serre the possibility of using the spectrum of an arbitrary commutative ring as a foundation for algebraic geometry.

Origin of schemes

Grothendieck then gave the decisive definition of a scheme, bringing to a conclusion a generation of experimental suggestions and partial developments. He defined the spectrum X of a commutative ring R as the space of prime ideals of R with a natural topology (known as the Zariski topology), but augmented it with a sheaf of rings: to every open subset U he assigned a commutative ring OX(U). These objects Spec(R) are the affine schemes; a general scheme is then obtained by "gluing together" affine schemes.

Much of algebraic geometry focuses on projective or quasi-projective varieties over a field k; in fact, k is often taken to be the complex numbers. Schemes of that sort are very special compared to arbitrary schemes; compare the examples below. Nonetheless, it is convenient that Grothendieck developed a large body of theory for arbitrary schemes. For example, it is common to construct a moduli space first as a scheme, and only later study whether it is a more concrete object such as a projective variety. Also, applications to number theory rapidly lead to schemes over the integers that are not defined over any field.

Definition

An affine scheme is a locally ringed space isomorphic to the spectrum Spec(R) of a commutative ring R. A scheme is a locally ringed space X admitting a covering by open sets Ui, such that each Ui (as a locally ringed space) is an affine scheme. In particular, X comes with a sheaf OX, which assigns to every open subset U a commutative ring OX(U) called the ring of regular functions on U. One can think of a scheme as being covered by "coordinate charts" that are affine schemes. The definition means exactly that schemes are obtained by gluing together affine schemes using the Zariski topology.

In the early days, this was called a prescheme, and a scheme was defined to be a separated prescheme. The term prescheme has fallen out of use, but can still be found in older books, such as Grothendieck's "Éléments de géométrie algébrique" and Mumford's "Red Book".

A basic example of an affine scheme is affine n-space over a field k, for a natural number n. By definition, An
k
is the spectrum of the polynomial ring k[x1,...,xn]. In the spirit of scheme theory, affine n-space can in fact be defined over any commutative ring R, meaning Spec(R[x1,...,xn]).

The category of schemes

Schemes form a category, with morphisms defined as morphisms of locally ringed spaces. (See also: morphism of schemes.) For a scheme Y, a scheme X over Y (or a Y-scheme) means a morphism XY of schemes. A scheme X over a commutative ring R means a morphism X → Spec(R).

An algebraic variety over a field k can be defined as a scheme over k with certain properties. There are different conventions about exactly which schemes should be called varieties. One standard choice is that a variety over k means an integral separated scheme of finite type over k.

A morphism f: XY of schemes determines a pullback homomorphism on the rings of regular functions, f*: O(Y) → O(X). In the case of affine schemes, this construction gives a one-to-one correspondence between morphisms Spec(A) → Spec(B) of schemes and ring homomorphisms BA. In this sense, scheme theory completely subsumes the theory of commutative rings.

Since Z is an initial object in the category of commutative rings, the category of schemes has Spec(Z) as a terminal object.

For a scheme X over a commutative ring R, an R-point of X means a section of the morphism X → Spec(R). One writes X(R) for the set of R-points of X. In examples, this definition reconstructs the old notion of the set of solutions of the defining equations of X with values in R. When R is a field k, X(k) is also called the set of k-rational points of X.

More generally, for a scheme X over a commutative ring R and any commutative R-algebra S, an S-point of X means a morphism Spec(S) → X over R. One writes X(S) for the set of S-points of X. (This generalizes the old observation that given some equations over a field k, one can consider the set of solutions of the equations in any field extension E of k.) For a scheme X over R, the assignment SX(S) is a functor from commutative R-algebras to sets. It is an important observation that a scheme X over R is determined by this functor of points.

The fiber product of schemes always exists. That is, for any schemes X and Z with morphisms to a scheme Y, the fiber product X×YZ (in the sense of category theory) exists in the category of schemes. If X and Z are schemes over a field k, their fiber product over Spec(k) may be called the product X × Z in the category of k-schemes. For example, the product of affine spaces Am and An over k is affine space Am+n over k.

Since the category of schemes has fiber products and also a terminal object Spec(Z), it has all finite limits.

Examples

Here and below, all the rings considered are commutative:

  • Every affine scheme Spec(R) is a scheme.
  • A polynomial f over a field k, fk[x1, ..., xn], determines a closed subscheme f = 0 in affine space An over k, called an affine hypersurface. Formally, it can be defined as
    For example, taking k to be the complex numbers, the equation x2 = y2(y+1) defines a singular curve in the affine plane A2
    C
    , called a nodal cubic curve.
  • For any commutative ring R and natural number n, projective space Pn
    R
    can be constructed as a scheme by gluing n + 1 copies of affine n-space over R along open subsets. This is the fundamental example that motivates going beyond affine schemes. The key advantage of projective space over affine space is that Pn
    R
    is proper over R; this is an algebro-geometric version of compactness. A related observation is that complex projective space CPn is a compact space in the classical topology (based on the topology of C), whereas Cn is not (for n > 0).
  • A homogeneous polynomial f of positive degree in the polynomial ring R[x0, ..., xn] determines a closed subscheme f = 0 in projective space Pn over R, called a projective hypersurface. In terms of the Proj construction, this subscheme can be written as
    For example, the closed subscheme x3 + y3 = z3 of P2
    Q
    is an elliptic curve over the rational numbers.
  • The line with two origins (over a field k) is the scheme defined by starting with two copies of the affine line over k, and gluing together the two open subsets A1 − 0 by the identity map. This is a simple example of a non-separated scheme. In particular, it is not affine.
  • A simple reason to go beyond affine schemes is that an open subset of an affine scheme need not be affine. For example, let X = An − 0, say over the complex numbers C; then X is not affine for n ≥ 2. (The restriction on n is necessary: the affine line minus the origin is isomorphic to the affine scheme Spec(C[x, x−1]). To show that X is not affine, one computes that every regular function on X extends to a regular function on An, when n ≥ 2. (This is analogous to Hartogs's lemma in complex analysis, though easier to prove.) That is, the inclusion f: X → An induces an isomorphism from O(An) = C[x1, ...., xn] to O(X). If X were affine, it would follow that f was an isomorphism. But f is not surjective and hence not an isomorphism. Therefore, the scheme X is not affine.
  • Let k be a field. Then the scheme is an affine scheme whose underlying topological space is the Stone–Čech compactification of the positive integers (with the discrete topology). In fact, the prime ideals of this ring are in one-to-one correspondence with the ultrafilters on the positive integers, with the ideal corresponding to the principal ultrafilter associated to the positive integer n. This topological space is zero-dimensional, and in particular, each point is an irreducible component. Since affine schemes are quasi-compact, this is an example of a quasi-compact scheme with infinitely many irreducible components. (By contrast, a Noetherian scheme has only finitely many irreducible components.)

Examples of morphisms

It is also fruitful to consider examples of morphisms as examples of schemes since they demonstrate their technical effectiveness for encapsulating many objects of study in algebraic and arithmetic geometry.

Arithmetic surfaces

If we consider a polynomial then the affine scheme has a canonical morphism to and is called an arithmetic surface. The fibers are then algebraic curves over the finite fields . If is an elliptic curve then the fibers over its discriminant locus generated by where

[16] are all singular schemes. For example, if is a prime number and
then its discriminant is . In particular, this curve is singular over the prime numbers .

Motivation for schemes

Here are some of the ways in which schemes go beyond older notions of algebraic varieties, and their significance.

  • Field extensions. Given some polynomial equations in n variables over a field k, one can study the set X(k) of solutions of the equations in the product set kn. If the field k is algebraically closed (for example the complex numbers), then one can base algebraic geometry on sets such as X(k): define the Zariski topology on X(k), consider polynomial mappings between different sets of this type, and so on. But if k is not algebraically closed, then the set X(k) is not rich enough. Indeed, one can study the solutions X(E) of the given equations in any field extension E of k, but these sets are not determined by X(k) in any reasonable sense. For example, the plane curve X over the real numbers defined by x2 + y2 = −1 has X(R) empty, but X(C) not empty. (In fact, X(C) can be identified with C − 0.) By contrast, a scheme X over a field k has enough information to determine the set X(E) of E-rational points for every extension field E of k. (In particular, the closed subscheme of A2
    R
    defined by x2 + y2 = −1 is a nonempty topological space.)
  • Generic point. The points of the affine line A1
    C
    , as a scheme, are its complex points (one for each complex number) together with one generic point (whose closure is the whole scheme). The generic point is the image of a natural morphism Spec(C(x)) → A1
    C
    , where C(x) is the field of rational functions in one variable. To see why it is useful to have an actual "generic point" in the scheme, consider the following example.
  • Let X be the plane curve y2 = x(x−1)(x−5) over the complex numbers. This is a closed subscheme of A2
    C
    . It can be viewed as a ramified double cover of the affine line A1
    C
    by projecting to the x-coordinate. The fiber of the morphism X → A1 over the generic point of A1 is exactly the generic point of X, yielding the morphism
    This in turn is equivalent to the degree-2 extension of fields
    Thus, having an actual generic point of a variety yields a geometric relation between a degree-2 morphism of algebraic varieties and the corresponding degree-2 extension of function fields. This generalizes to a relation between the fundamental group (which classifies covering spaces in topology) and the Galois group (which classifies certain field extensions). Indeed, Grothendieck's theory of the étale fundamental group treats the fundamental group and the Galois group on the same footing.
  • Nilpotent elements. Let X be the closed subscheme of the affine line A1
    C
    defined by x2 = 0, sometimes called a fat point. The ring of regular functions on X is C[x]/(x2); in particular, the regular function x on X is nilpotent but not zero. To indicate the meaning of this scheme: two regular functions on the affine line have the same restriction to X if and only if they have the same value and first derivative at the origin. Allowing such non-reduced schemes brings the ideas of calculus and infinitesimals into algebraic geometry.
  • For a more elaborate example, one can describe all the zero-dimensional closed subschemes of degree 2 in a smooth complex variety Y. Such a subscheme consists of either two distinct complex points of Y, or else a subscheme isomorphic to X = Spec C[x]/(x2) as in the previous paragraph. Subschemes of the latter type are determined by a complex point y of Y together with a line in the tangent space TyY. This again indicates that non-reduced subschemes have geometric meaning, related to derivatives and tangent vectors.

Coherent sheaves

A central part of scheme theory is the notion of coherent sheaves, generalizing the notion of (algebraic) vector bundles. For a scheme X, one starts by considering the abelian category of OX-modules, which are sheaves of abelian groups on X that form a module over the sheaf of regular functions OX. In particular, a module M over a commutative ring R determines an associated OX-module ~M on X = Spec(R). A quasi-coherent sheaf on a scheme X means an OX-module that is the sheaf associated to a module on each affine open subset of X. Finally, a coherent sheaf (on a Noetherian scheme X, say) is an OX-module that is the sheaf associated to a finitely generated module on each affine open subset of X.

Coherent sheaves include the important class of vector bundles, which are the sheaves that locally come from finitely generated free modules. An example is the tangent bundle of a smooth variety over a field. However, coherent sheaves are richer; for example, a vector bundle on a closed subscheme Y of X can be viewed as a coherent sheaf on X that is zero outside Y (by the direct image construction). In this way, coherent sheaves on a scheme X include information about all closed subschemes of X. Moreover, sheaf cohomology has good properties for coherent (and quasi-coherent) sheaves. The resulting theory of coherent sheaf cohomology is perhaps the main technical tool in algebraic geometry.

Generalizations

Considered as its functor of points, a scheme is a functor that is a sheaf of sets for the Zariski topology on the category of commutative rings, and that, locally in the Zariski topology, is an affine scheme. This can be generalized in several ways. One is to use the étale topology. Michael Artin defined an algebraic space as a functor that is a sheaf in the étale topology and that, locally in the étale topology, is an affine scheme. Equivalently, an algebraic space is the quotient of a scheme by an étale equivalence relation. A powerful result, the Artin representability theorem, gives simple conditions for a functor to be represented by an algebraic space.

A further generalization is the idea of a stack. Crudely speaking, algebraic stacks generalize algebraic spaces by having an algebraic group attached to each point, which is viewed as the automorphism group of that point. For example, any action of an algebraic group G on an algebraic variety X determines a quotient stack [X/G], which remembers the stabilizer subgroups for the action of G. More generally, moduli spaces in algebraic geometry are often best viewed as stacks, thereby keeping track of the automorphism groups of the objects being classified.

Grothendieck originally introduced stacks as a tool for the theory of descent. In that formulation, stacks are (informally speaking) sheaves of categories. From this general notion, Artin defined the narrower class of algebraic stacks (or "Artin stacks"), which can be considered geometric objects. These include Deligne–Mumford stacks (similar to orbifolds in topology), for which the stabilizer groups are finite, and algebraic spaces, for which the stabilizer groups are trivial. The Keel–Mori theorem says that an algebraic stack with finite stabilizer groups has a coarse moduli space that is an algebraic space.

Another type of generalization is to enrich the structure sheaf, bringing algebraic geometry closer to homotopy theory. In this setting, known as derived algebraic geometry or "spectral algebraic geometry", the structure sheaf is replaced by a homotopical analog of a sheaf of commutative rings (for example, a sheaf of E-infinity ring spectra). These sheaves admit algebraic operations that are associative and commutative only up to an equivalence relation. Taking the quotient by this equivalence relation yields the structure sheaf of an ordinary scheme. Not taking the quotient, however, leads to a theory that can remember higher information, in the same way that derived functors in homological algebra yield higher information about operations such as tensor product and the Hom functor on modules.

Pure mathematics

From Wikipedia, the free encyclopedia
Pure mathematics studies the properties and structure of abstract objects, such as the E8 group, in group theory. This may be done without focusing on concrete applications of the concepts in the physical world.

Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles.

While pure mathematics has existed as an activity since at least ancient Greece, the concept was elaborated upon around the year 1900, after the introduction of theories with counter-intuitive properties (such as non-Euclidean geometries and Cantor's theory of infinite sets), and the discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable, and Russell's paradox). This introduced the need to renew the concept of mathematical rigor and rewrite all mathematics accordingly, with a systematic use of axiomatic methods. This led many mathematicians to focus on mathematics for its own sake, that is, pure mathematics.

Nevertheless, almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories. Also, many mathematical theories, which had seemed to be totally pure mathematics, were eventually used in applied areas, mainly physics and computer science. A famous early example is Isaac Newton's demonstration that his law of universal gravitation implied that planets move in orbits that are conic sections, geometrical curves that had been studied in antiquity by Apollonius. Another example is the problem of factoring large integers, which is the basis of the RSA cryptosystem, widely used to secure internet communications.

It follows that, presently, the distinction between pure and applied mathematics is more a philosophical point of view or a mathematician's preference rather than a rigid subdivision of mathematics.

History

Ancient Greece

Ancient Greek mathematicians were among the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between "arithmetic", now called number theory, and "logistic", now called arithmetic. Plato regarded logistic (arithmetic) as appropriate for businessmen and men of war who "must learn the art of numbers or [they] will not know how to array [their] troops" and arithmetic (number theory) as appropriate for philosophers "because [they have] to arise out of the sea of change and lay hold of true being." Euclid of Alexandria, when asked by one of his students of what use was the study of geometry, asked his slave to give the student threepence, "since he must make gain of what he learns." The Greek mathematician Apollonius of Perga was asked about the usefulness of some of his theorems in Book IV of Conics to which he proudly asserted,

They are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason.

And since many of his results were not applicable to the science or engineering of his day, Apollonius further argued in the preface of the fifth book of Conics that the subject is one of those that "...seem worthy of study for their own sake."

19th century

The term itself is enshrined in the full title of the Sadleirian Chair, "Sadleirian Professor of Pure Mathematics", founded (as a professorship) in the mid-nineteenth century. The idea of a separate discipline of pure mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of the kind, between pure and applied. In the following years, specialisation and professionalisation (particularly in the Weierstrass approach to mathematical analysis) started to make a rift more apparent.

20th century

At the start of the twentieth century mathematicians took up the axiomatic method, strongly influenced by David Hilbert's example. The logical formulation of pure mathematics suggested by Bertrand Russell in terms of a quantifier structure of propositions seemed more and more plausible, as large parts of mathematics became axiomatised and thus subject to the simple criteria of rigorous proof.

Pure mathematics, according to a view that can be ascribed to the Bourbaki group, is what is proved. "Pure mathematician" became a recognized vocation, achievable through training.

The case was made that pure mathematics is useful in engineering education:

There is a training in habits of thought, points of view, and intellectual comprehension of ordinary engineering problems, which only the study of higher mathematics can give.

Generality and abstraction

An illustration of the Banach–Tarski paradox, a famous result in pure mathematics. Although it is proven that it is possible to convert one sphere into two using nothing but cuts and rotations, the transformation involves objects that cannot exist in the physical world.

One central concept in pure mathematics is the idea of generality; pure mathematics often exhibits a trend towards increased generality. Uses and advantages of generality include the following:

  • Generalizing theorems or mathematical structures can lead to deeper understanding of the original theorems or structures
  • Generality can simplify the presentation of material, resulting in shorter proofs or arguments that are easier to follow.
  • One can use generality to avoid duplication of effort, proving a general result instead of having to prove separate cases independently, or using results from other areas of mathematics.
  • Generality can facilitate connections between different branches of mathematics. Category theory is one area of mathematics dedicated to exploring this commonality of structure as it plays out in some areas of math.

Generality's impact on intuition is both dependent on the subject and a matter of personal preference or learning style. Often generality is seen as a hindrance to intuition, although it can certainly function as an aid to it, especially when it provides analogies to material for which one already has good intuition.

As a prime example of generality, the Erlangen program involved an expansion of geometry to accommodate non-Euclidean geometries as well as the field of topology, and other forms of geometry, by viewing geometry as the study of a space together with a group of transformations. The study of numbers, called algebra at the beginning undergraduate level, extends to abstract algebra at a more advanced level; and the study of functions, called calculus at the college freshman level becomes mathematical analysis and functional analysis at a more advanced level. Each of these branches of more abstract mathematics have many sub-specialties, and there are in fact many connections between pure mathematics and applied mathematics disciplines. A steep rise in abstraction was seen mid 20th century.

In practice, however, these developments led to a sharp divergence from physics, particularly from 1950 to 1983. Later this was criticised, for example by Vladimir Arnold, as too much Hilbert, not enough Poincaré. The point does not yet seem to be settled, in that string theory pulls one way, while discrete mathematics pulls back towards proof as central.

Pure vs. applied mathematics

Mathematicians have always had differing opinions regarding the distinction between pure and applied mathematics. One of the most famous (but perhaps misunderstood) modern examples of this debate can be found in G.H. Hardy's 1940 essay A Mathematician's Apology.

It is widely believed that Hardy considered applied mathematics to be ugly and dull. Although it is true that Hardy preferred pure mathematics, which he often compared to painting and poetry, Hardy saw the distinction between pure and applied mathematics to be simply that applied mathematics sought to express physical truth in a mathematical framework, whereas pure mathematics expressed truths that were independent of the physical world. Hardy made a separate distinction in mathematics between what he called "real" mathematics, "which has permanent aesthetic value", and "the dull and elementary parts of mathematics" that have practical use.

Hardy considered some physicists, such as Einstein and Dirac, to be among the "real" mathematicians, but at the time that he was writing his Apology, he considered general relativity and quantum mechanics to be "useless", which allowed him to hold the opinion that only "dull" mathematics was useful. Moreover, Hardy briefly admitted that—just as the application of matrix theory and group theory to physics had come unexpectedly—the time may come where some kinds of beautiful, "real" mathematics may be useful as well.

Another insightful view is offered by American mathematician Andy Magid:

I've always thought that a good model here could be drawn from ring theory. In that subject, one has the subareas of commutative ring theory and non-commutative ring theory. An uninformed observer might think that these represent a dichotomy, but in fact the latter subsumes the former: a non-commutative ring is a not-necessarily-commutative ring. If we use similar conventions, then we could refer to applied mathematics and nonapplied mathematics, where by the latter we mean not-necessarily-applied mathematics... [emphasis added]

Friedrich Engels argued in his 1878 book Anti-Dühring that "it is not at all true that in pure mathematics the mind deals only with its own creations and imaginations. The concepts of number and figure have not been invented from any source other than the world of reality". He further argued that "Before one came upon the idea of deducing the form of a cylinder from the rotation of a rectangle about one of its sides, a number of real rectangles and cylinders, however imperfect in form, must have been examined. Like all other sciences, mathematics arose out of the needs of men...But, as in every department of thought, at a certain stage of development the laws, which were abstracted from the real world, become divorced from the real world, and are set up against it as something independent, as laws coming from outside, to which the world has to conform."

Introduction to entropy

From Wikipedia, the free encyclopedia https://en.wikipedia.org/wiki/Introduct...