"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)
"Ordinary numbers have immediate connection to the world around us; they are used to count and measure every sort of thing. Adding, subtracting, multiplying and dividing all have simple interpretations in terms of the objects being counted and measured. When we pass to complex numbers, though, the arithmetic takes on a life of its own. Since -1 has no square root, we decided to create a new number game which supplies the missing piece. By adding in just this one new element √-1. we created a whole new world in which everything arithmetical, miraculously, works out just fine." (David Mumford, Caroline Series & David Wright, "Indra’s Pearls: The Vision of Felix Klein", 2002)
"Probably the first clear insight into the deep nature of control […] was that it is not about pulling levers to produce intended and inexorable results. This notion of control applies only to trivial machines. It never applies to a total system that includes any kind of probabilistic element - from the weather, to people; from markets, to the political economy. No: the characteristic of a non-trivial system that is under control, is that despite dealing with variables too many to count, too uncertain to express, and too difficult even to understand, something can be done to generate a predictable goal. Wiener found just the word he wanted in the operation of the long ships of ancient Greece. At sea, the long ships battled with rain, wind and tides - matters in no way predictable. However, if the man operating the rudder kept his eye on a distant lighthouse, he could manipulate the tiller, adjusting continuously in real-time towards the light. This is the function of steersmanship. As far back as Homer, the Greek word for steersman was kubernetes, which transliterates into English as cybernetes." (Stafford Beer, "What is cybernetics?", Kybernetes, 2002)
"Be careful about infinity: we want an infinite number of geometrically distinct geodesics. As a kinematic motion, running twice along a periodic geodesic is different from running only once, but it is not geometrically distinct. For example when working on counting functions one should be careful to distinguish between the counting function for geometric periodic geodesics and that for parameterized ones. The question is difficult because the standard ways to prove existence of periodic geodesics consider them as motions." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
"Any statistic based on more than a guess requires some sort of counting. Definitions specify what will be counted. Measuring involves deciding how to go about counting. We cannot begin counting until we decide how we will identify and count instances of a social problem. [...] Measurement involves choices. [...] Often, measurement decisions are hidden." (Joel Best, "Damned Lies and Statistics: Untangling Numbers from the Media, Politicians, and Activists", 2001)
"Every number has its limitations; every number is a product of choices that inevitably involve compromise. Statistics are intended to help us summarize, to get an overview of part of the world’s complexity. But some information is always sacrificed in the process of choosing what will be counted and how. Something is, in short, always missing. In evaluating statistics, we should not forget what has been lost, if only because this helps us understand what we still have." (Joel Best, "More Damned Lies and Statistics: How numbers confuse public issues", 2004)
"Good statistics are not only products of people counting; the quality of statistics also depends on people’s willingness and ability to count thoughtfully and on their decisions about what, exactly, ought to be counted so that the resulting numbers will be both accurate and meaningful." (Joel Best, "More Damned Lies and Statistics: How numbers confuse public issues", 2004)
"In much the same way, people create statistics: they choose what to count, how to go about counting, which of the resulting numbers they share with others, and which words they use to describe and interpret those figures. Numbers do not exist independent of people; understanding numbers requires knowing who counted what, why they bothered counting, and how they went about it." (Joel Best, "More Damned Lies and Statistics: How numbers confuse public issues", 2004)
"Statistics depend on collecting information. If questions go unasked, or if they are asked in ways that limit responses, or if measures count some cases but exclude others, information goes ungathered, and missing numbers result. Nevertheless, choices regarding which data to collect and how to go about collecting the information are inevitable." (Joel Best, "More Damned Lies and Statistics: How numbers confuse public issues", 2004)
"When people use statistics, they assume - or, at least, they want their listeners to assume - that the numbers are meaningful. This means, at a minimum, that someone has actually counted something and that they have done the counting in a way that makes sense. Statistical information is one of the best ways we have of making sense of the world’s complexities, of identifying patterns amid the confusion. But bad statistics give us bad information." (Joel Best, "More Damned Lies and Statistics: How numbers confuse public issues", 2004)
"With four outcomes and two players, a 2 × 2 game is completely described by eight numbers. An array with eight numbers is just an address in an 8-dimensional Cartesian payoff space, and there are uncountably many 2 × 2 games, each fully described by an 8-number address." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"Although it is not difficult to count the holes in a real pretzel in your hand, prior to eating it, when a surface pops out of an abstract mathematical construction it can be very difficult to figure out its properties, such as how many holes it has. The cohomology groups can help us to do so." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)
"Histograms use area to represent counts of a distribution. This makes them somewhat related to barcharts and mosaic plots, although the number or the width of the bins of a histogram is not determined a priori and the bins are drawn without gaps between them reflecting the continuous scale of the data. Whereas barcharts and mosaic plots show the exact distribution of the sample, a histogram is always just one approximation to the distribution of the data. Sometimes histograms are also used as crude density estimators for some 'true', but usually unknown, underlying distribution for the data. There are much better density estimation methods that produce smooth distribution displays." (Antony Unwin et al [in "Graphics of Large Datasets: Visualizing a Million"], 2006)
"The definition of homeomorphism was motivated by the idea of preserving the general shape or configuration of a geometric figure. Since path components are significant characteristics of a space, it is certainly reasonable that a homeomorphism will preserve the decomposition of a space into path components. […] Suppose we are given two geometric figures that we suspect are not topologically equivalent. If both of the figures are path-connected, counting components will not distinguish the spaces. However, we might be able to remove a special subset of one of the figures and count the number of components of the remainder. If no comparable set can be removed from the other space to leave the same number of components, we will then know that the two spaces are not homeomorphic." (Robert Messer & Philip Straffin, "Topology Now!", 2006)
"[myth:] Counting can be done without error. Usually, the counted number is an integer and therefore without (rounding) error. However, the best estimate of a scientifically relevant value obtained by counting will always have an error. These errors can be very small in cases of consecutive counting, in particular of regular events, e.g., when measuring frequencies." (Manfred Drosg, "Dealing with Uncertainties: A Guide to Error Analysis", 2007)
"Euclid sharply distinguished between number and magnitude, associating the former with the operation of counting and the latter with a line segment. So, for Euclid, only integers were numbers; even the notion of fractions as numbers had not yet been conceived of. He represented a whole number n by a line segment that was n times the chosen unit line segment. However, the opposite procedure of distinguishing all line segments by labeling them with numerals representing counting numbers was not possible. Obviously, this one-way correspondence of counting number with magnitude implies that the latter concept was more general than the former. The sharp distinction between counting number and magnitude, made by Euclid, was an impediment to the development of the concept of number." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)
"In the binary digital world of computers, all information is reduced to sequences of zeros and ones. But there’s a space between zero and one, between the way the machine counts and thinks and the way we count and think." (Scott Rosenberg, "Dreaming in Code", 2007
"A little ingenuity is involved, but once a couple of tricks are learnt, it is not hard to show many sets of numbers are countable, which is the term we use to mean that the set can be listed in the same fashion as the counting numbers. Otherwise a set is called uncountable." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)
"Mathematicians call them twin primes: pairs of prime numbers that are close to each other, almost neighbors, but between them there is always an even number that prevents them from truly touching. […] If you go on counting, you discover that these pairs gradually become rarer, lost in that silent, measured space made only of ciphers. You develop a distressing presentiment that the pairs encountered up until that point were accidental, that solitude is the true destiny. Then, just when you’re about to surrender, you come across another pair of twins, clutching each other tightly." (Paolo Giordano,"The Solitude of prime numbers", 2008)
"The rationals therefore also form a countable set, as do the euclidean numbers, and indeed if we consider the set of all numbers that arise from the rationals through taking roots of any order, the collection produced is still countable. We can even go beyond this: the collection of all algebraic numbers, which are those that are solutions of ordinary polynomial equations∗ form a collection that can, in principle, be arrayed in an infinite list: that is to say it is possible, with a little more crafty argument, to describe a systematic listing that sweeps them all out." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)
"There are various ways in which one can try to define combinatorics. None is satisfactory on its own, but together they give some idea of what the subject is like. A first definition is that combinatorics is about counting things. [...] Combinatorics is sometimes called ‘discrete mathematics’ because it is concerned with ‘discrete’ as opposed to ‘continuous’ structures. Roughly speaking, an object is discrete if it consists of points that are isolated from each other and continuous if you can move from one point to another without making sudden jumps. [...] There is a close affinity between combinatorics and theoretical computer science (which deals with the quintessentially discrete structure of sequences of 0s and 1s), and combinatorics is sometimes contrasted with analysis, though in fact there are several connections between the two. A third definition is that combinatorics is concerned with mathematical structures that have ‘few constraints’. This idea helps to explain why number theory, despite the fact that it studies (among other things) the distinctly discrete set of all positive integers, is not considered a branch of combinatorics." (Timothy Gowers, June Barrow-Green & Imre Leader, "The Princeton Companion to Mathematics", 2008)
"The problem of identifying the subset of good moves is much more complicated than simply counting the total number of possibilities and falls completely into the domain of strategy and tactics of chess as a game." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)
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