"No branch of number theory is more saturated with mystery than the study of prime numbers: those exasperating, unruly integers that refuse to be divided evenly by any integers except themselves and 1. Some problems concerning primes are so simple that a child can understand them and yet so deep and far from solved that many mathematicians now suspect they have no solution. Perhaps they are 'undecideable'. Perhaps number theory, like quantum mechanics, has its own uncertainty principle that makes it necessary, in certain areas, to abandon exactness for probabilistic formulations." (Martin Gardner, "The remarkable lore of the prime numbers", Scientific American, 1964)
"In many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance." (Martin Gardner, "Mathematical Games", Scientific American, 1973)
"Surreal numbers are an astonishing feat of legerdemain. An empty hat rests on a table made of a few axioms of standard set theory. Conway waves two simple rules in the air, then reaches into almost nothing and pulls out an infinitely rich tapestry of numbers that form a real and closed field. Every real number is surrounded by a host of new numbers that lie closer to it than any other 'real' value does. The system is truly 'surreal.'" (Martin Gardner, "Mathematical Magic Show", 1977)
“All mathematical problems are solved by reasoning within a deductive system in which basic laws of logic are embedded.” (Martin Gardner, “Aha! Insight”, 1978)
"At the heart of mathematics is a constant search for simpler and simpler ways to prove theorems and solve problems. [...] The sudden hunch, the creative leap of the mind that ‘sees’ in a flash how to solve a problem in a simple way, is something quite different from general intelligence." (Martin Gardner, "Aha! Insight", 1978)
“Combinatorial analysis, or combinatorics, is the study of how things can be arranged. In slightly less general terms, combinatorial analysis embodies the study of the ways in which elements can be grouped into sets subject to various specified rules, and the properties of those groupings. […] Combinatorial analysis often asks for the total number of different ways that certain things can be combined according to certain rules.” (Martin Gardner, "Aha! Insight", 1978)
"Every branch of mathematics has its combinatorial aspects […] There is combinatorial arithmetic, combinatorial topology, combinatorial logic, combinatorial set theory-even combinatorial linguistics, as we shall see in the section on word play. Combinatorics is particularly important in probability theory where it is essential to enumerate all possible combinations of things before a probability formula can be found." (Martin Gardner, "Aha! Insight", 1978)
"Every branch of geometry can be defined as the study of properties that are unaltered when a specified figure is given specified symmetry transformations. Euclidian plane geometry, for instance, concerns the study of properties that are 'invariant' when a figure is moved about on the plane, rotated, mirror reflected, or uniformly expanded and contracted. Affine geometry studies properties that are invariant when a figure is 'stretched' in a certain way. Projective geometry studies properties invariant under projection. Topology deals with properties that remain unchanged even when a figure is radically distorted in a manner similar to the deformation of a figure made of rubber." (Martin Gardner, "Aha! Insight", 1978)
“Geometry is the study of shapes. Although true, this definition is so broad that it is almost meaningless. The judge of a beauty contest is, in a sense, a geometrician because he is judging […] shapes, but this is not quite what we want the word to mean. It has been said that a curved line is the most beautiful distance between two points. Even though this statement is about curves, a proper element of geometry, the assertion seems more to be in the domain of aesthetics rather than mathematics.” (Martin Gardner, "Aha! Insight", 1978)
“Graph theory is the study of sets of points that are joined by lines.” (Martin Gardner, “Aha! Insight”, 1978)
“The great revolutions in science are almost always the result of unexpected intuitive leaps. After all, what is science if not the posing of difficult puzzles by the universe? Mother Nature does something interesting, and challenges the scientist to figure out how she does it. In many cases the solution is not found by exhaustive trial and error […] or even by a deduction based on the relevant knowledge.” (Martin Gardner, "Aha! Insight", 1978)
“The word ‘induction’ has two essentially different meanings. Scientific induction is a process by which scientists make observations of particular cases, such as noticing that some crows are black, then leap to the universal conclusion that all crows are black. The conclusion is never certain. There is always the possibility that at least one unobserved crow is not black." (Martin Gardner, “Aha! Insight”, 1978)
"Mathematical induction […] is an entirely different procedure. Although it, too, leaps from the knowledge of particular cases to knowledge about an infinite sequence of cases, the leap is purely deductive. It is as certain as any proof in mathematics, and an indispensable tool in almost every branch of mathematics.” (Martin Gardner, “Aha! Insight”, 1978)
“Geometry is the study of shapes. Although true, this definition is so broad that it is almost meaningless. The judge of a beauty contest is, in a sense, a geometrician because he is judging […] shapes, but this is not quite what we want the word to mean. It has been said that a curved line is the most beautiful distance between two points. Even though this statement is about curves, a proper element of geometry, the assertion seems more to be in the domain of aesthetics rather than mathematics.” (Martin Gardner, "Aha! Insight", 1978)
“Graph theory is the study of sets of points that are joined by lines.” (Martin Gardner, “Aha! Insight”, 1978)
“The word ‘induction’ has two essentially different meanings. Scientific induction is a process by which scientists make observations of particular cases, such as noticing that some crows are black, then leap to the universal conclusion that all crows are black. The conclusion is never certain. There is always the possibility that at least one unobserved crow is not black." (Martin Gardner, “Aha! Insight”, 1978)
"Mathematical induction […] is an entirely different procedure. Although it, too, leaps from the knowledge of particular cases to knowledge about an infinite sequence of cases, the leap is purely deductive. It is as certain as any proof in mathematics, and an indispensable tool in almost every branch of mathematics.” (Martin Gardner, “Aha! Insight”, 1978)
"The external world exists; the structure of the world is ordered; we know little about the nature of the order, nothing at all about why it should exist." (Martin Gardner, "Order and Surprise", 1983)
"People who have a casual interest in mathematics may get the idea that a topologist is a mathematical playboy who spends his time making Möbius bands and other diverting topological models. If they were to open any recent textbook in topology, they would be surprised. They would find page after page of symbols, seldom relieved by a picture or diagram." (Martin Gardner, "Hexaflexagons and Other Mathematical Diversions", 1988)
"People who have a casual interest in mathematics may get the idea that a topologist is a mathematical playboy who spends his time making Möbius bands and other diverting topological models. If they were to open any recent textbook in topology, they would be surprised. They would find page after page of symbols, seldom relieved by a picture or diagram." (Martin Gardner, "Hexaflexagons and Other Mathematical Diversions", 1988)
"Besides being essential in modern physics, the complex-number field provides pure mathematics with a multitude of brain-boggling theorems. It is worth keeping in mind that complex numbers, although they include the reals.as a subset, differ from real numbers in startling ways. One cannot, for example, speak of a complex number as being either positive or negative: those properties apply only to the reals and the pure imaginaries. It is equally meaningless to say that one complex number is larger or smaller than another." (Martin Gardner, "Fractal Music, Hypercards and More... Mathematical Recreations from Scientific American Magazine", 1992)
"The seemingly preposterous assumption that there is a square root of -1 was justified on pragmatic grounds: it simplified certain calculations and so could be used as long as 'real' values were obtained at the end. The parallel with the rules for using negative numbers is striking. If you are trying to determine how many cows there are in a field (that is, if you are working in the domain of positive integers), you may find negative numbers useful in the calculation, but of course the final answer must be in terms of positive numbers because there is no such thing as a negative cow." (Martin Gardner, "Fractal Music, Hypercards and More... Mathematical Recreations from Scientific American Magazine", 1992)
"I enjoy mathematics so much because it has a strange kind of unearthly beauty. There is a strong feeling of pleasure, hard to describe, in thinking through an elegant proof, and even greater pleasure in discovering a proof not previously known." (Martin Gardner, 2008)
"[…] if all sentient beings in the universe disappeared, there would remain a sense in which mathematical objects and theorems would continue to exist even though there would be no one around to write or talk about them. Huge prime numbers would continue to be prime, even if no one had proved them prime." (Martin Gardner, "When You Were a Tadpole and I Was a Fish", 2009)
"A surprising proportion of mathematicians are accomplished musicians. Is it because music and mathematics share patterns that are beautiful?" (Martin Gardner, The Dover Math and Science Newsletter, 2011)
"All mathematicians share […] a sense of amazement over the infinite depth and the mysterious beauty and usefulness of mathematics." (Martin Gardner)
"Mathematics is not only real, but it is the only reality. [The] entire universe is made of matter, obviously. And matter is made of particles. It's made of electrons and neutrons and protons. So the entire universe is made out of particles. Now what are the particles made out of? They're not made out of anything. The only thing you can say about the reality of an electron is to cite its mathematical properties. So there's a sense in which matter has completely dissolved and what is left is just a mathematical structure." (Martin Gardner)
"One would be hard put to find a set of whole numbers with a more fascinating history and more elegant properties surrounded by greater depths of mystery - and more totally useless - than the perfect numbers." (Martin Gardner)
"There are some traits all mathematicians share. An obvious one is a sense of amazement over the infinite depth and the mysterious beauty and usefulness of mathematics." (Martin Gardner)
"I enjoy mathematics so much because it has a strange kind of unearthly beauty. There is a strong feeling of pleasure, hard to describe, in thinking through an elegant proof, and even greater pleasure in discovering a proof not previously known." (Martin Gardner, 2008)
"[…] if all sentient beings in the universe disappeared, there would remain a sense in which mathematical objects and theorems would continue to exist even though there would be no one around to write or talk about them. Huge prime numbers would continue to be prime, even if no one had proved them prime." (Martin Gardner, "When You Were a Tadpole and I Was a Fish", 2009)
"A surprising proportion of mathematicians are accomplished musicians. Is it because music and mathematics share patterns that are beautiful?" (Martin Gardner, The Dover Math and Science Newsletter, 2011)
"All mathematicians share […] a sense of amazement over the infinite depth and the mysterious beauty and usefulness of mathematics." (Martin Gardner)
"Mathematics is not only real, but it is the only reality. [The] entire universe is made of matter, obviously. And matter is made of particles. It's made of electrons and neutrons and protons. So the entire universe is made out of particles. Now what are the particles made out of? They're not made out of anything. The only thing you can say about the reality of an electron is to cite its mathematical properties. So there's a sense in which matter has completely dissolved and what is left is just a mathematical structure." (Martin Gardner)
"One would be hard put to find a set of whole numbers with a more fascinating history and more elegant properties surrounded by greater depths of mystery - and more totally useless - than the perfect numbers." (Martin Gardner)
"There are some traits all mathematicians share. An obvious one is a sense of amazement over the infinite depth and the mysterious beauty and usefulness of mathematics." (Martin Gardner)
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