On Counting (1975-1989)

"Numbers are not just counters; they are elements in a system." (Scott Buchanan,"Poetry and Mathematics", 1975)

"Concepts form the basis for any science. These are ideas, usually somewhat vague" (especially when first encountered), which often defy really adequate definition. The meaning of a new concept can seldom be grasped from reading a one-paragraph discussion. There must be time to become accustomed to the concept, to investigate it with prior knowledge, and to associate it with personal experience. Inability to work with details of a new subject can often be traced to inadequate understanding of its basic concepts." (William C Reynolds & Harry C Perkins, "Engineering Thermodynamics", 1977)

"Numbers are the product of counting. Quantities are the product of measurement. This means that numbers can conceivably be accurate because there is a discontinuity between each integer and the next. Between two and three there is a jump. In the case of quantity there is no such jump, and because jump is missing in the world of quantity it is impossible for any quantity to be exact. You can have exactly three tomatoes. You can never have exactly three gallons of water. Always quantity is approximate." (Gregory Bateson, "Number is Different from Quantity", CoEvolution Quarterly, 1978)

"Automation is certainly one way to improve the leverage of all types of work. Having machines to help them, human beings can create more output. But in both widget manufacturing and administrative work, something else can also increase the productivity of the black box. This is called work simplification. To get leverage this way, you first need to create a flow chart of the production process as it exists. Every single step must be shown on it; no step should be omitted in order to pretty things up on paper. Second, count the number of steps in the flow chart so that you know how many you started with. Third, set a rough target for reduction of the number of steps." (Andrew S Grove, "High Output Management", 1983)

"Combinatorics can be classified into three types: enumerative, existential, and constructive. Enumerative combinatorics deals with the counting of combinatorial objects. Existential combinatorics studies the existence or nonexistence of combinatorial configurations. Constructive combinatorics deals with methods for actually finding specific configurations" (as opposed to merely demonstrating their existence theoretically). [...] In constructive combinatorics, the problem is usually one of finding a solution efficiently, [...] using a reasonable length of time." (George Pólya, Robert E Tarjan & Donald R Woods, "Notes on Introductory Combinatorics", 1983)

"The logarithm is an extremely powerful and useful tool for graphical data presentation. One reason is that logarithms turn ratios into differences, and for many sets of data, it is natural to think in terms of ratios. […] Another reason for the power of logarithms is resolution. Data that are amounts or counts are often very skewed to the right; on graphs of such data, there are a few large values that take up most of the scale and the majority of the points are squashed into a small region of the scale with no resolution." (William S. Cleveland, "Graphical Methods for Data Presentation: Full Scale Breaks, Dot Charts, and Multibased Logging", The American Statistician Vol. 38 (4) 1984)

"Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing: you don’t get a feeling of having done an honest day’s work. Don’t get the wrong idea - combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow undertaking, with sophisticated techniques." (Gian-Carlo Rota, "Mathematics, Philosophy and Artificial Intelligence", Los Alamos Science No. 12, 1985)

"Recognition of the idea that a programming language should have a precise mathematical meaning or semantics dates from the early 1960s. The mathematics provides a secure, unambiguous, precise and stable specification of the language to serve as an agreed interface between its users and its implementors. Furthermore, it gives the only reliable grounds for a claim that different implementations are implementations of the same language. So mathematical semantics are as essential to the objective of language standardisation as measurement and counting are to the standardisation of nuts and bolts." (C Anthony R Hoare, "Communicating Sequential Processes", 1985)

"[zero is] A mysterious number, which started life as a space on a counting board, turned into a written notice that a space was present, that is to say that something was absent, then confused medieval mathematicians who could not decide whether it was really a number or not, and achieved its highest status in modern abstract mathematics in which numbers are defined anyway only by their properties, and the properties of zero are at least as clear, and rather more substantial, than those of many other numbers." (David Wells, "The Penguin Dictionary of Curious and Interesting Numbers", 1986)

"Indeed, it is hard to see how mathematics could exist without the notion of infinity, for the very first thing a child learns about mathematics - how to count - is based on the tacit assumption that every integer has a successor. The notion of a straight line, so fundamental in geometry, is based on a similar assumption - that we can, at least in principle, extend a line indefinitely in both directions. Even in such seemingly 'finite' branches of mathematics as probability, the notion of infinity plays a subtle role: when we toss a coin ten times, we may get five 'heads' and five 'tails', or we may get six 'heads' and four 'tails', or in fact any other outcome; but when we say that the probability of getting 'heads' or 'tails' is even, we tacitly assume that an infinite number of tosses would produce an equal outcome." (Eli Maor, "To Infinity and Beyond: A Cultural History of the Infinite", 1987)