On Counting (2010-2019)

"Zero is such a weird idea, having 'something' represent 'nothing', and it eluded the Romans. Complex numbers are similar - it’s a new way of thinking. But both zero and complex numbers make math much easier. If we never adopted strange, new number systems, we’d still be counting on our fingers." (Kalid Azal, "Math,Better Explained", 2011)

"[…] humans make mistakes when they try to count large numbers in complicated systems. They make even greater errors when they attempt - as they always do - to reduce complicated systems to simple numbers." (Zachary Karabell, "The Leading Indicators: A short history of the numbers that rule our world", 2014)

"The taming of chance created mathematical probability. [...] Probability is not one of a kind; it was born with three faces: frequency, physical design, and degrees of belief. [...] in the first of its identities, probability is about counting. [...] Second, probability is about constructing. For example, if a die is constructed to be perfectly symmetrical, then the probability of rolling a six is one in six. You don’t have to count. [...] Probabilities by design are called propensities. Historically, games of chance were the prototype for propensity. These risks are known because people crafted, not counted, them. [...] Third, probability is about degrees of belief. A degree of belief can be based on anything from experience to personal impression." (Gerd Gigerenzer, "Risk Savvy: How to make good decisions", 2014)

"Moreover, there is still another important observation that seems to be essential for the idea to group objects into a set: This is the human ability to recognize similarities in different objects. Usually, a collection, or group, consists of objects that somehow belong together, objects that share a common property. While a mathematical set could also be a completely arbitrary collection of unrelated objects, this is usually not what we want to count. We count coins or hours or people, but we usually do not mix these categories." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"The act of counting is governed by five principles. They describe the conditions and prerequisites that make counting possible. We call them the 'BOCIA' principles - from the words Bijection, Ordinality, Cardinality, Invariance, and Abstraction." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"The invariance principle states that the result of counting a set does not depend on the order imposed on its elements during the counting process. Indeed, a mathematical set is just a collection without any implied ordering. A set is the collection of its elements - nothing more." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"When we extend the system of natural numbers and counting to embrace infinite cardinals, the larger system need not have all of the properties of the smaller one. However, familiarity with the smaller system leads us to expect certain properties, and we can become confused when the pieces don’t seem to fit. Insecurity arose when the square of a complex number violated the real number principle that all squares are positive. This was resolved when we realised that the complex numbers cannot be ordered in the same way as their subset of reals." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"Keep in mind that a weighted average may be different than a simple (non- weighted) average because a weighted average - by definition - counts certain data points more heavily. When you’re thinking about an average, try to determine if it’s a simple average or a weighted average. If it’s weighted, ask yourself how it’s being weighted, and see which data points count more than others." (John H Johnson & Mike Gluck, "Everydata: The misinformation hidden in the little data you consume every day", 2016)

"Often a yes/no problem in mathematics becomes more interesting when it turns into a counting problem. One reason for this is that counting may introduce more structure, or a finer structure, into the data or the conceptual framework of the problem. Similarly, sometimes the counting problem becomes more interesting if it can be turned into a group theory problem. Sometimes, the yes/no problem has an obvious answer but the counting problem still yields a fascinating and beautiful theory. Another reason to solve counting problems is that we are more naturally interested in a more precise and more quantitative question than the simple binary puzzle that we began with. Of course, only with success in achieving an interesting answer will we be satisfied with the more complex approach." (Avner Ash & Robert Gross, "Summing It Up : From one plus one to modern number theory", 2016)

"Statistics, because they are numbers, appear to us to be cold, hard facts. It seems that they represent facts given to us by nature and it’s just a matter of finding them. But it’s important to remember that people gather statistics. People choose what to count, how to go about counting, which of the resulting numbers they will share with us, and which words they will use to describe and interpret those numbers. Statistics are not facts. They are interpretations. And your interpretation may be just as good as, or better than, that of the person reporting them to you." (Daniel J Levitin, "Weaponized Lies", 2017)

"Boltzmann has shown that entropy exists because we describe the world in a blurred fashion. He has demonstrated that entropy is precisely the quantity that counts how many are the different configurations that our blurred vision does not distinguish between. Heat, entropy, and the lower entropy of the past are notions that belong to an approximate, statistical description of nature. The difference between past and future is deeply linked to this blurring." (Carlo Rovelli, "The Order of Time", 2018)