"In fact, the answer to the question 'What is mathematics?' has changed several times during the course of history [...] It was only in the last twenty years or so that a definition of mathematics emerged on which most mathematicians agree: mathematics is the science of patterns." (Keith Devlin, "Sets, Functions, and Logic: An Introduction to Abstract Mathematics", 1979)
"In set theory, perhaps more than in any other branch of mathematics, it is vital to set up a collection of symbolic abbreviations for various logical concepts. Because the basic assumptions of set theory are absolutely minimal, all but the most trivial assertions about sets tend to be logically complex, and a good system of abbreviations helps to make otherwise complex statements." (Keith Devlin, "Sets, Functions, and Logic: An Introduction to Abstract Mathematics", 1979)
"Just as music comes alive in the performance of it, the same is true of mathematics. The symbols on the page have no more to do with mathematics than the notes on a page of music. They simply represent the experience." (Keith Devlin, "Mathematics: The Science of Patterns", 1994)
"The increased abstraction in mathematics that took place during the early part of this century was paralleled by a similar trend in the arts. In both cases, the increased level of abstraction demands greater effort on the part of anyone who wants to understand the work." (Keith Devlin, "Mathematics: The Science of Patterns", 1994)
"Mathematics is a product - a discovery - of the human mind. It enables us to see the incredible, simple, elegant, beautiful, ordered structure that lies beneath the universe we live in. It is one of the greatest creations of mankind - if it is not indeed the greatest." (Keith Devlin, "Life By the Numbers", 1998)
"At this stage you might be thinking that there is no justification for calling something of the form a+bi a number, even if you are prepared to countenance i = √-1 in the first place. But remember, it is not what numbers are that matters, but how they behave. Provided the complex numbers have a workable and useful (either in mathematics itself or possibly in a wider context) arithmetic, possibly forming a field, then they have as much right to be called 'numbers' as do any others." (Keith Devlin, "Mathematics: The New Golden Age", 1998)
"In fact the complex numbers form a field. [...] So however strange you may feel the very notion of a complex number to be, it does turn out to provide a 'normal' type of arithmetic. In fact it gives you a tremendous bonus not available with any of the other number systems. [...] The fundamental theorem of algebra is just one of several reasons why the complex-number system is such a 'nice' one. Another important reason is that the field of complex numbers supports the development of a powerful differential calculus, leading to the rich theory of functions of a complex variable." (Keith Devlin, "Mathematics: The New Golden Age", 1998)
"Sometimes the solution of a long-standing problem marks the end (or the beginning of the end) of a mathematical era or field, the culmination of years of effort. On other occasions it may open up an entire new area of research, possibly previously undreamt of." (Keith Devlin, "Mathematics: The New Golden Age", 1998)
"The reason why a 'crude', experimental approach is not adequate for determining mathematical truth lies in the nature of what mathematics is and is intended to be. Though its roots lie in the physical world, mathematics is a precise and idealized discipline. The 'points', 'lines', 'planes', and other ideal constructs of mathematics have no exact counterpart in reality. What the mathematician does is to take a totally abstract, idealized view of the world, and reason with his abstractions in an entirely precise and rigorous fashion." (Keith Devlin, "Mathematics: The New Golden Age", 1998)
"The whole apparatus of the calculus takes on an entirely different form when developed for the complex numbers." (Keith Devlin, "Mathematics: The New Golden Age", 1998)
"Arithmetic and number theory study patterns of number and counting. Geometry studies patterns of shape. Calculus allows us to handle patterns of motion. Logic studies patterns of reasoning. Probability theory deals with patterns of chance. Topology studies patterns of closeness and position." (Keith Devlin, "The Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip", 2000)
"A neural network is a particular kind of computer program, originally developed to try to mimic the way the human brain works. It is essentially a computer simulation of a complex circuit through which electric current flows." (Keith J Devlin & Gary Lorden, "The Numbers behind NUMB3RS: Solving crime with mathematics", 2007)
"Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence." (Keith J Devlin)
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