- Unifying theories in mathematics
There have been several attempts in history to reach a

**unified theory of mathematics**. Some of the greatestmathematician s have expressed views that the whole subject should be fitted into one theory.**Historical perspective**The process of unification might be seen as helping to define what constitutes mathematics as a discipline.

For example,

mechanics andmathematical analysis were commonly combined into one subject during the 18th century, united by thedifferential equation concept; whilealgebra andgeometry were considered largely distinct. Now we consider analysis, algebra, and geometry, but not mechanics, as parts of mathematics because they are primarily deductiveformal science s, while mechanics likephysics must proceed from observation. There is no major loss of content, withanalytical mechanics in the old sense now expressed in terms ofsymplectic topology , based on the newer theory ofmanifold s.**Mathematical theories**The term "

theory " is used informally within mathematics to mean a self-consistent body ofdefinition s,axiom s,theorem s, examples, and so on. (Examples includegroup theory ,Galois theory ,control theory , andK-theory .) In particular there is no connotation of "hypothetical". Thus the term "unifying theory" is more like asociological term used to study the actions of mathematicians. It may assume nothing conjectural that would be analogous to an undiscovered scientific link. There is really no cognate within mathematics to such concepts as "Proto-World" inlinguistics or theGaia hypothesis .Nonetheless there have been several episodes within the history of mathematics in which sets of individual theorems were found to be special cases of a single unifying result, or in which a single perspective about how to proceed when developing an area of mathematics could be applied fruitfully to multiple branches of the subject.

**Geometrical theories**A well-known example was the development of

analytic geometry , which in the hands of mathematicians such asDescartes andFermat showed that many theorems aboutcurve s andsurface s of special types could be stated in algebraic language (then new), each of which could then be proved using the same techniques. That is, the theorems were very similar algebraically, even if the geometrical interpretations were distinct.At the end of the 19th century,

Felix Klein noted that the many branches of geometry which had been developed during that century (affine geometry ,projective geometry ,hyperbolic geometry , etc.) could all be treated in a uniform way. He did this by considering the groups under which the objects were invariant. This unification of geometry goes by the name of theErlangen programme .**Through-axiomatisation**Early in the 20th century, many parts of mathematics began to be treated by delineating useful sets of axioms and then studying their consequences. Thus, for example, the studies of "

hypercomplex number s", such as considered by theQuaternion Society (1899 - 1913) , were put onto an axiomatic footing as branches ofring theory (in this case, with the specific meaning ofassociative algebra s over the field of complex numbers.) In this context, thequotient ring concept is one of the most powerful unifiers.This was a general change of methodology, since the needs of applications had up until then meant that much of mathematics was taught by means of

algorithm s (or processes close to being algorithmic).Arithmetic is still taught that way. It was a parallel to the development ofmathematical logic as a stand-alone branch of mathematics. By the 1930ssymbolic logic itself was adequately included within mathematics.In most cases, mathematical objects under study can be defined (albeit non-canonically) as sets or, more informally, as sets with additional structure such as an addition operation.

Set theory now serves as a "lingua franca " for the development of mathematical themes.**Bourbaki**The cause of axiomatic development was taken up in earnest by the

Bourbaki group of mathematicians. Taken to its extreme, this attitude was thought to demand mathematics developed in its greatest generality. One started from the most general axioms, and then specialized, for example, by introducing modules overcommutative ring s, and limiting tovector space s over thereal number s only when absolutely necessary. The story proceeded in this fashion, even when the specializations were the theorems of primary interest.In particular, this perspective placed little value on fields of mathematics (such as

combinatorics ) whose objects of study are very often special, or found in situations which can only superficially be related to more axiomatic branches of the subject.**Category theory as a rival**An alternative (mostly complementary) to set theory but also serving to give a consistent approach to most of axiomatic mathematics is

category theory , developed in the second half of the 20th century. A key theme from this point of view is that mathematics studies not only certain kinds of objects (Lie group s,Banach space s, etc.) but also the mappings between them.In particular, this clarifies exactly what it means for mathematical objects to be considered to be "the same". (For example, are all equilateral triangles "the same", or does size matter?)

Saunders Mac Lane proposed that any concept with enough 'ubiquity' (occurring in various branches of mathematics) deserved isolating and studying in its own right. Category theory is arguably better adapted to that end than any other current approach. The disadvantages of relying on so-called "abstract nonsense " are a certain blandness and abstraction in the sense of breaking away from the roots in concrete problems. Nevertheless, the methods of category theory have steadily advanced in acceptance, in numerous areas (fromD-module s tocategorical logic ).**Uniting theories**On a less grandiose scale, there are frequent instances in which it appears that sets of results in two different branches of mathematics are similar, and one might ask whether there is a unifying framework which clarifies the connections. We have already noted the example of analytic geometry, and more generally the field of

algebraic geometry thoroughly develops the connections between geometric objects (algebraic varieties , or more generally schemes) and algebraic ones (ideals); the touchstone result here isHilbert's Nullstellensatz which roughly speaking shows that there is a natural one-to-one correspondence between the two types of objects.One may view other theorems in the same light. For example the

fundamental theorem of Galois theory asserts that there is a one-to-one correspondence between extensions of a field and subgroups of the field'sGalois group . TheTaniyama-Shimura conjecture for elliptic curves (now proven) establishes a one-to-one correspondence between curves defined one way (themodular forms ) andelliptic curve s defined a different way (at least, over the field ofrational number s). A research area sometimes nicknamedMonstrous Moonshine developed connections between modular forms and the finite simple group known as the Monster, starting solely with the surprise observation that in each of them the rather unusual number 196884 would arise very naturally. Another field, known as theLanglands program , likewise starts with apparently haphazard similarities (in this case, between number-theoretical results and representations of certain groups) and looks for constructions from which both sets of results would be corollaries.**Reference list of major unifying concepts**A short list of these theories might include:

*

Cartesian geometry

*Calculus

*Complex analysis

*Galois theory

*Erlangen programme

*Lie group

*Set theory

*Hilbert space

*Computable function

*Characteristic class es

*Homological algebra

*Homotopy theory

*Grothendieck's schemes

*Langlands program

*Non-commutative geometry .We illustrate the concept by discussing some of these themes in detail.

**Recent developments in relation with modular theory**A well-known example is the

Taniyama-Shimura conjecture , now themodularity theorem , which proposed that eachelliptic curve over the rational numbers can be translated into amodular form (in such a way as to preserve the associatedL-function ). There are difficulties in identifying this with an isomorphism, in any strict sense of the word. Certain curves had been known to be both elliptic curves (of genus 1) andmodular curve s, before the conjecture was formulated (about 1955). The surprising part of the conjecture was the extension to factors of Jacobians of modular curves of genus > 1. It had probably not seemed plausible that there would be 'enough' such rational factors, before the conjecture was enunciated; and in fact the numerical evidence was slight until around 1970, when tables began to confirm it. The case of elliptic curves withcomplex multiplication was proved by Shimura in 1964. This conjecture stood for decades before being proved in generality.In fact the

Langlands program (or philosophy) is much more like a web of unifying conjectures; it really does postulate that the general theory ofautomorphic form s is regulated by theL-group s introduced byRobert Langlands . His "principle of functoriality" with respect to the L-group has a very large explanatory value with respect to known types of "lifting" of automorphic forms (now more broadly studied asautomorphic representation s). While this theory is in one sense closely linked with the Taniyama-Shimura conjecture, it should be understood that the conjecture actually operates in the opposite direction. It requires the existence of an automorphic form, starting with an object that (very abstractly) lies in a category of motives.Another significant related point is that the Langlands approach stands apart from the whole development triggered by

monstrous moonshine (connections betweenelliptic modular function s asFourier series , and thegroup representation s of theMonster group and othersporadic group s). The Langlands philosophy neither foreshadowed nor was able to include this line of research.**Isomorphism conjectures in K-theory**Another case, which so far is less well-developed but covers a wide range of mathematics, is the conjectural basis of some parts of

K-theory . TheBaum-Connes conjecture , now a long-standing problem, has been joined by others in a group known as theisomorphism conjectures in K-theory . These include theFarrell-Jones conjecture andBost conjecture .**ee also***

philosophy of mathematics

*foundations of mathematics

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