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Reference Work Entry In depth
Reverse Mathematics
Reverse mathematics is a new take on an old idea: asking which axioms are necessary to prove a given theorem. This question was first asked about the parallel axiom in Euclid’s geometry and later about the axi...
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Chapter
Learning from the Masters (and Some of Their Pupils)
Historians are trained to read original sources, and mathematicians in general are also advised to “study the masters.” In practice, this is difficult to do, because some masters (and some languages) are easie...
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Article
Book Review: How Geography Changed the World and My Small Part in it. M. C. Clarke. Published by Sweet Design (UK) Limited, Henleaze, Bristol, 2020. 305 pages. ISBN: 978–0–9,567,541-1-0
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Living Reference Work Entry In depth
Reverse Mathematics
Reverse mathematics is a new take on an old idea: asking which axioms are necessary to prove a given theorem. This question was first asked about the parallel axiom in Euclid’s geometry and later about the axi...
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Article
Open AccessMeasuring and Assessing Regional Education Inequalities in China under Changing Policy Regimes
China’s uneven regional economic development and decentralisation of its education system have led to increasing regional education disparities. Here, we introduce a new multidimensional index, the Index of Re...
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Chapter
The Theorem of Pythagoras
The is the most appropriate starting point for a book on mathematics and its history. It is not only the oldest mathematical theorem, but also the source of three great streams of mathematical thought: numbe...
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Greek Number Theory
Number theory is the second large field of mathematics that comes to us from the Pythagoreans via Euclid. The Pythagorean theorem led mathematicians to the study of squares and sums of squares; Euclid drew att...
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Chapter
Topology
In Chapter 11 we saw how Riemann found the topological concept of genus to be important in the study of algebraic curves. In the present chapter we will see how topology became a major field of mathematics, with ...
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Chapter
Infinity in Greek Mathematics
Perhaps the most interesting—and most modern—feature of Greek mathematics is its treatment of infinity. The Greeks feared infinity and tried to avoid it, but in doing so they laid the foundations for a ri...
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Chapter
Algebraic Geometry
The first field of mathematics to benefit from the new language of equations was geometry. Around 1630, both and realized that geometric problems could be translated into algebra by means of coordinates, tha...
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Chapter
Complex Numbers and Curves
revisits polynomial equations and algebraic curves, observing how these topics are simplified by introducing complex numbers. That’s right: the so-called “complex” numbers actually make things simpler.
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Chapter
Non-Euclidean Geometries
of the new frontiers in geometry opened up by calculus was the study of curvature. The concept of curvature is particularly interesting for surfaces, because it can be defined intrinsically. The intrinsic cur...
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Chapter
Polynomial Equations
The first phase the history of algebra was the search for solutions of polynomial equations. The “degree of difficulty” of an equation corresponds rather well to the degree of the corresponding polynomial.
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Chapter
Sets, Logic, and Computation
In the 19th century, perennial concerns about the role of infinity in mathematics were finally addressed by the development of set theory and formal logic. Set theory was proposed as a mathematical theory of infi...
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Chapter
Projective Geometry
At about the same time as the algebraic revolution in classical geometry, a new kind of geometry also came to light: projective geometry. Based on the idea of projecting objects from space to a plane, or from one...
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Chapter
Infinite Series
As we saw in the previous chapter, many calculus problems have a solution that can be expressed as an infinite series. It is therefore useful to be able to recognize important individual series and to understa...
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Chapter
Elliptic Curves and Functions
Number theory revived in Europe with the rediscovery of Diophantus by Bombelli, and the publication of a new edition by Bachet de Méziriac (1621). It was this book that inspired Fermat and launched number theo...
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Chapter
Complex Numbers and Functions
The insight into algebraic curves afforded by complex coordinates—that a complex curve is topologically a surface—has important implications for functions defined as integrals of algebraic functions, such as t...
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Chapter
Greek Geometry
Geometry was the first branch of mathematics to become highly developed. The concepts of “theorem” and “proof” originated in geometry, and most mathematicians until recent times were introduced to their subjec...
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Chapter
Group Theory
Group theory was the first branch of modern, or abstract, algebra to emerge from the old algebra of equations. Group theory today is often described as the theory of symmetry, and indeed groups have been inherent...
