The Mathematical Sciences: A Report (1968) / Chapter Skim
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4 Core Mathematics
4 Core Mathematics
Pages 57-83
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From page 57... ...
As a matter of fact, the confluence of mathematical ideas and the interpenetration of the various fields are the most characteristic and most rewarding features of contemporary core mathematics. A sketchy presentation couched in nontechnical language can convey only a pale reflection of the excitement felt by those who participate in these developments.
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From page 58... ...
A major reason for the flowering of mathematical logic lies in the ~ ~;~ ~£ ^~ ~ ^~ ~ r ~ If =' ~11~ L`~ly ~' ~1111~11~e ants, During tne same period, by Georg Cantor. This theory influenced all mathematics.
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From page 59... ...
By reducing mathematical reasoning, and in fact all deductive reasoning, to purely mechanical manipulation of symbols according to definite rules, mathematical logic prepared for the development of digital computers. Indeed, a mathematical theory has been completely formalized if any argument in it could in principle be checked by computer.
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From page 60... ...
The interplay between mathematical logic and computing continues. Training in mathematical logic is often a way to creative work in computer sciences, and the problems on which computer scientists and logicians work sometimes have certain features in common.
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From page 61... ...
A characteristic phenomenon of recent decades is the application of probability theory to number theory and, very recently, the applications of mathematical logic. Modern number theory is concerned not only with integers but also with so-called algebraic Integers, that is, with numbers that can be roots of algebraic equations with integer coefficients and highest coefficient 1.48 In studying algebraic integers one often uses a geometric language hardly distinguishable from that of algebraic geometry (see below)
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From page 62... ...
Modern algebra is largely a devel
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From page 63... ...
Its techniques and mode of thought play an important role not only in other mathematical disciplines but also in various areas of the biological, physical, and social sciences. Modern algebra is concerned with the study of so-called "algebraic structures." Most of these may be considered as generalizations of the basic structures of elementary mathematics: the system of integers, the system of rational numbers, the system of real numbers, the system of complex numbers, and the system of vectors (arrows)
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From page 64... ...
It is convenient to clothe linear algebra in a geometric language. The basic concept is that of a vector space over a field.
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From page 65... ...
Finite dimensional vector spaces occur in many parts of engineering, physics, statistics, numerical analysis, and the social sciences. Linear algebra is the study of finite dimensional vector spaces.
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From page 66... ...
One of the most elementary examples of a group is the set of all positive rational numbers ire which one considers only the operation of multiplication. A group is a set of elements in which one operation, called group operation or group multiplication, is defined; this operation must obey the same formal rules as ordinary multiplication of rational numbers, except that we do not require the commutative law (that is, ab arid ba need not be the same)
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From page 67... ...
Homological A [ge bra and Category Theory Modern mathematics is characterized by an ever-increasing range of applications of algebra to other mathematical subjects (see Eilenberg's essay in reference 7~. A particularly striking example is topology, a branch of geometry concerned with qualitative rather than quantitative aspects of shapes of geometric figures.
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From page 68... ...
The second task consists of deriving from the simple laws governing what happens in the small the incomparably more complicated laws describing what happens in the large. This second step usually involves solving differential equations a purely mathematical task.
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From page 69... ...
For instance, this extension unifies the theories of elementary functions like trigonometric functions, logarithms, and exponentials. Complex analysis is the theory of analytic functions of complex variables.
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From page 70... ...
Just as the sound produced by a musical instrument can be analyzed into a superposition of pure tones, so physicists can describe many complicated processes and states as a superposition of eigenfunctions solutions of dill erential equations corresponding in some sense to pure tones. The mathematics of such a representation provides one of the basic tools for solving the differential equations of mathematical physics and has been a source of interesting and difficult problems for generations of workers in pure mathematics.
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From page 71... ...
Problems in functional analysis often arise from integral equations, differential equations, and other parts of so-called classical analysis. An important application of functional analysis, or more specifically of Hilbert space, occurs in modern physics.
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From page 72... ...
In geometric language, to solve an ordinary differential equation means to find a curve satisfying certain conditions; to solve a partial differential equation means to find a surface or a higher dimensional manifold satisfying certain conditions. Differential equations are either linear or nonlinear.
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From page 73... ...
From the mathematical point of view, the linear theory is always simpler. The modern theory of ordinary differential equations includes control theory, which is closely related to technical and engineering applications.
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From page 74... ...
Mathematicians originally did not recognize this as a legitimate function, but in the late 1940's Schwartz developed a theory of generalized functions, or distributions, in which not only the Dirac function but also much weirder entities occupy a legitimate place. A new generation of mathematicians learned to look at partial differential equations from the viewpoint of distributions, and they have created a new branch of this old discipline that abounds in beautiful and powerful results.
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From page 75... ...
Important breakthroughs have been achieved, but an immense amount of work remains to be done. Much of the modern work in partial differential equations looks highly esoteric, and only a few years ago such work would have horn ~ A: ~ ~ ~_ ~, ~ J ~^ ~.
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From page 76... ...
Examples of these disciplines are measure theory, function spaces, classical potential theory, Hilbert spaces, information theory, and partial differential equations. This list could be extended, but it should suffice to add that probabilistic methods have been used for proofs in logic.
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From page 77... ...
The second event was the discovery of analytic geometry by Fermat and Descartes, who showed how points can be represented by pairs or triples of numbers and geometric figures by algebraic equations. This discovery bridged the chasm between geometric and arithmetic concepts that dominated Greek mathematics and prepared for the invention of calculus.
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From page 78... ...
Algebraic geometry is concerned with solutions of systems of algebraic equations using a geometric language. A basic aim is to achieve a full understanding of the totality of solutions.
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From page 79... ...
. ~ ~ ~ Point Set Topology This branch of geometry is first of all the study of very general geometric figures in ordinary space.
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From page 80... ...
In studying qualitative aspects of geometry, one considered first figures composed of the simplest conceivable building blocks points, straight segments, triangles, tetrahedra, for example. Brouwer, one of the founders of topology, named these building blocks "simplexes." Certain basic figures studied in algebraic topology are called "complexes." They are composed of simplexes in such a way that two simplexes that touch each other have a whole face in common.
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From page 81... ...
The recent extension of this theory to infinite dimensional spaces (by Smale and others? promises to he of .si~nifi cance for solving nonlinear partial differential equations.
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From page 82... ...
Differential topology has now become one of the most active mathematical disciplines. It is, more than any other branch of topology, connected with differential geometry, the theory of differential equations, and algebraic geometry.
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From page 83... ...
Lie groups play a dominating and unifying role in modern mathematics; they have stimulated significant research in algebra and topology, and such widely different fields as finite-group theory and differential geometry are today strongly influenced by Lie group theory. The "representation theory" of Lie groups, which on occasions has been the source of inspiration of important discoveries in physics, has also suggested entirely new directions in analysis, thereby incorporating and revitalizing large parts of classical analysis.
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