The final part of the book contains general reflections on the nature of mathematics and the mathematical life. A first part of the editorial process naturally consisted of planning the book and finding authors.

## Contributors

Della Fenster, Professor, Department of Mathematics and Computer Science, University of Richmond, Virginia emil artin[VI.86]. Gillispie, Dayton-Stockton Professor of History of Science, Emeritus, Princeton University pierre-simon laplace[VI.23].

The Princeton Companion to Mathematics

## Introduction

### What Is Mathematics About?

*Algebra versus Geometry**Algebra versus Analysis**Algebra**Number Theory**Geometry**Algebraic Geometry**Analysis**Logic**Combinatorics**Theoretical Computer Science**Probability**Mathematical Physics*

In fact, one of the most important branches of mathematics is even called algebraic geometry [IV.4]. This concrete example of a group (or rather, a class of groups, one for each polynomial) played a very important role in the development of abstract group theory.

### The Language and Grammar of Mathematics

*Sets**Functions**Relations**Binary Operations**Quantiﬁers**Negation**Free and Bound Variables*

Let us now turn our attention from the word "is" to some other parts of sentences (1)–(3), focusing first on the term "square root of" in sentence (1). So one might think it inappropriate to ask what kind of object a "square root" is.

### Some Fundamental Mathematical Deﬁnitions

*The Natural Numbers**The Integers**The Rational Numbers**The Real Numbers**The Complex Numbers**Groups**Fields**Vector Spaces**Rings**Substructures**Products**Quotients**Homomorphisms, Isomorphisms, and Automorphisms**Eigenvalues and Eigenvectors**Limits**Continuity**Diﬀerentiation**Partial Diﬀerential Equations**Integration**Holomorphic Functions**Geometry and Symmetry Groups**Euclidean Geometry**Aﬃne Geometry**Topology**Spherical Geometry**Hyperbolic Geometry**Projective Geometry**Lorentz Geometry**Manifolds and Diﬀerential Geometry**Riemannian Metrics*

A second view of the projective plane is that it is the set of all lines in R3 that pass through the origin. This can be thought of as the square of the distance between the point (x, y) and the infinitely close point (x+dx, y+dy).

### The General Goals of Mathematical Research

*Linear Equations**Polynomial Equations**Polynomial Equations in Several Variables Suppose that we are faced with an equation such as**Diophantine Equations**Diﬀerential Equations**Identifying Building Blocks and Families There are two situations that typically lead to inter-**Equivalence, Nonequivalence, and Invariants There are many situations in mathematics where two**Weakening Hypotheses and Strengthening Conclusions**Proving a More Abstract Result**Identifying Characteristic Properties**Generalization after Reformulation**Higher Dimensions and Several Variables We have already seen that the study of polynomial**Exact Counting**Estimates**Averages**Extremal Problems**Conditional Results**Numerical Evidence*

If an ideal is of the form (γ) for some number γ, then it is called a principal ideal. But any line that meets all Odds is one of the M(s) lines and must pass through P or Q (since the M(s) lines lie in Sand and L4 meets S only at these two points). In this case, each of the two arguments tells us something that the other does not.

One might imagine that the reverse of the reverse side of a number is the number itself, but this is not the case.

## The Origins of

### From Numbers to Number Systems

In the second part of the book, which deals with geometry and measurement, one even sees an approach to a square root: “The product is one thousand eight hundred and seventy-five; take root, this is the territory; it's forty-three and a bit.”. The most influential of these wasstevin[VI.10], a Flemish mathematician and engineer who popularized the system in a booklet called De Thiende ("The Tenth"), first published in 1585. The integers took pride of place, followed by a nested hierarchy consisting of the rational numbers (i.e. the fractions), the real numbers (Stevin's decimal, now carefully formalized) and the complex numbers.

Today it is no longer so easy to decide what counts as a "number". The objects from the original order of "integer, rational, real, and complex" are certainly numbers, but so are the p-adici.

### Geometry Jeremy Gray

It turns out that the size of the angle depends on the length of the perpendicular from P to m. Bolyai and Lobachevskii also knew that spheres could be drawn in their three-dimensional space, and they showed (though they were not original in this) that the formulas of spherical geometry hold independently of the parallel postulate. This is a measure of the attitude that Euclidean geometry still held in the minds of most people at the time.

Finally, suppose the conclusion is that the sum of the angles of a triangle is actually less than two right angles by an amount proportional to the area of the triangle.

### The Development of Abstract Algebra

Moreover, the Islamic conquest of most of the Iberian Peninsula in the eighth century and the subsequent establishment of a court there, a library and. But could we extend this technique to the case of quintic polynomials and higher degree polynomials. As Lagrange's analysis seemed to point out, the answer to this question in the cubic and quartic cases critically involved the cube, or the fourth root of unity.

By the time Sylvester coined the term, the invariant phenomenon had also appeared in the work of the English mathematician Boole [VI.43], and had attracted the attention of Cayley.

### Algorithms Jean-Luc Chabert

*Abacists and Algorists**Finiteness**Euclid’s Algorithm: Iteration**The Newton–Raphson Method**Hilbert’s Tenth Problem**Recursive Functions: Church’s Thesis**Turing Machines**Complexity**The Inﬂuence of Algorithms on Contemporary Mathematics*

1984. The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory, translated by A. The formulation above is of course not a mathematical definition in the classical sense of the term. That is, it is important that the values that come out of the iteration come arbitrarily close to π.

We can define the function "prod" in terms of the function "sum" by means of the following two rules: prod(1, y)equals and prod(x+1, y)equals sum(prod(x, y), y).

### The Development of Rigor in Mathematical Analysis

*Euler**Responses from the Late Eighteenth Century One signiﬁcant response to Berkeley in Britain was that**Riemann, the Integral, and Counterexamples Riemann is indelibly associated with the foundations of**The Aftermath of Weierstrass and Riemann Analysis became the model subdiscipline for rigor for*

These alleged mysteries have made the infinitesimal calculus quite suspect to many people. Many writers contributed to debates about the rigor of analysis in the first decades of the nineteenth century. Cauchy revised the definition and stated that the sum of an infinite series is the limit of a sequence of partial sums.

If the functionφ(z) increases or decreases continuously between = b and = b , the value of the integral becomes [b.

### The Development of the Idea of Proof

The same six parts of the proof and the diagram invariably appear in these types of statements. And at the same time, the essence and role of the axioms from which the derivations in question arise also underwent a dramatic change. On the one hand, there was a trend that flourished in the United States at the turn of the twentieth century, led by Eliakim H.

Epilogue: Evidence in the Twentieth Century The new concept of evidence that stabilized at the beginning of the twentieth century provided an idealized model—.

### The Crisis in the Foundations of Mathematics

*Paradoxes and Consistency**Predicativity**Choices**Intuitionism**Hilbert’s Program**Personal Disputes*

Cantor's proofs in set theory have also become the most important examples of the modern methodology of existential proofs. One propagation law defines the nodes in the tree, while the other maps them to objects. They were difficult to digest, but eventually led to the re-establishment of fundamental terms for fundamental studies.

In the process, our understanding of the characteristics, possibilities, and limitations of formal systems has been greatly clarified.

## Mathematical Concepts

### The Axiom of Choice

Again, we can't specify exactly how we do this, so we use the axiom of choice (although we don't say so explicitly). In general, the axiom of choice states that if we have a family of nonempty sets Xi, then we can choose an elementxi from each. An example of a statement whose proof involves the axiom of choice is the Banach–Tar paradox.

There are two forms of the axiom of choice that are used more often in everyday mathematical life than the basic form we have discussed.

The Axiom of Determinacy

## Banach Spaces

### Bayesian Analysis

In particular, as in the example just given, there is not always an obvious prior distribution, and it is a subtle and interesting mathematical problem to devise methods for choosing prior distributions that are different in different ways. be optimal'. As the diagrams suggest, we insist that the strings go from left to right without “doubling back”; a knotted string, for example, is not permitted. In our example, Y =X−1, since “pulling all the strings tight” shows that

As a group, B is generated by elements(σi)1in−1, where σi is formed from the trivial braid by crossing the ith string over the (i+1)st, as in Figure 4.

### Buildings Mark Ronan

The groups described here are, strictly speaking, plane meshing groups, the plane being the drilled object. For example, groups of type Anary the groups of invertible linear transformations inn+1 dimensions. The buildings for the Lie-type groups are all spherical and, just as A3 is related to the tetrahedron, their apartments are related to the regular and semi-regular indimensions of the polyhedron, where is the signature in the Lie notation given earlier.

The theory of spherical buildings not only provides a nice geometric basis for Lie-type groups: it can also be used to construct those of types E6, E7, E8 and F4, for an arbitrary field, without the need for sophisticated machinery such as EG like Lie algebras.

### Calabi–Yau Manifolds Eric Zaslow

Given a complex orientation—that is, the nonmetric definition of the Calabi–Yau manifold given above—the compatible Kähler structure leads to a holonomy lying in SU(n)⊂U(n), the natural analogue of the real orientation case. Thus, the metric notion of the Calabi–Yau manifold represents a powerful nonlinear partial differential equation for. Thus, in fact, the metric definition of a Calabi-Yau manifold is uniquely determined by its Kähler structure and its complex orientation.

The two conditions together mean that M is a complex manifold with holonomy group SU(n): that is, a Calabi–Yau manifold.

## The Calculus of Variations

Cardinals

### Categories Eugenia Cheng

The general framework and language of "objects and morphisms" enables us to search for and study structural features that depend only on the "form" of the category, that is, on its morphisms and on the equations they satisfy. It expresses the fact that giving a function from the disjoint union to another set is exactly the same as giving a function from each of the individual sets; this fully characterizes a split union (which we consider to be defined up to isomorphism). Furthermore, there is a notion of morphism between functions called unnatural transformation, which is analogous to the notion of homotopy between maps of topological spaces.

There is also a commuting condition analogous to the fact that, in the case of homotopy, a path inX must have its image underF continuously transformed to its image underG without passing over any "holes" in the spaceY.

## Class Field Theory

This avoidance of holes is expressed in the case of a category by the commutativity of certain squares in the target category Y, which is known as the “natural state”. One example of a natural transformation encodes the fact that every vector space is canonically isomorphic to its counterpart; there is a functor from the category of vector spaces to itself that takes each vector space to its dual dual, and there is an invertible natural transformation from this functor to the identity functor via canonical isomorphisms. Conversely, every finite-dimensional vector space is isomorphic to its dual, but not canonically, because isomorphism involves an arbitrary choice of basis; if we try to construct a natural transformation in this case, we find that the naturalness condition fails.

In the presence of natural transformations, categories actually form a 2-category, which is a two-dimensional generalization of a category, with objects, morphisms, and morphisms between morphisms.

## Cohomology

### Compactness and Compactiﬁcation

Thus, using this compression of the real line, we can generalize the notion of a limit to one that no longer needs to be a real number. For example, using the device of stereographic projection, one can topologically identify the real line with a circle from which a single point has been removed. For example, one can see a straight line in the plane as the limit of ever-larger circles by describing an appropriate compression of the space of the circles that includes lines.

Compactifications can also be used to view continuity as the limit of the discrete: for example, it is possible to compactify the sequence Z/2Z,Z/3Z,Z/4Z,.

### Computational Complexity Classes

The resulting object, known as the extended real line[−∞,+∞], can be given a topology in a natural way that basically defines what it means to converge to+∞ or to . Thus, one can view the Dirac delta function as a limit of classical functions, and this can be very useful in manipulating it. We briefly mention two other important complexity classes. PSPACE consists of all problems that can be solved using an amount of memory that grows polynomially at most with the size of the input.

Another remarkable fact is that almost all complexity classes have "hardest problems" within them: that is, problems for which a solution can be converted into a solution for any other problem in the class.

## Continued Fractions

*Countable and Uncountable Sets**C ∗ -Algebras**Curvature**Designs**Determinants**Diﬀerential Forms and Integration*

As the above example shows, this is exactly the same as saying that we can enumerate the elements of a set. So the set of all subsets of real numbers is "strictly greater" than the set of real numbers, and so on. It is also an integral part of general relativity (discussed in general relativity and Einstein's equations [IV.13]).

It goes without saying that integration is one of the fundamental concepts of calculus with one variable.