Peter M Higgins - Collected Quotes

"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)

"Algebraic symbols carry a universality of interpretation that allows them to be manipulated in a way that words cannot. Indeed, this was the key breakthrough that allowed mathematics to flourish in a way that was not possible until the advent of algebra. All higher mathematics relies on constant use of algebraic manipulation and would be impossible without it." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"For mathematics to be applicable in any sense at all we need to be able to do something with it. In practice this nearly always means developing forms of calculation, and this imperative channels its practitioners into algebraic manipulations of one form or another and ultimately into producing numbers. To the modern mind, this might seem natural and inevitable." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"If we refused to use complex numbers out of stubbornness disguised as some kind of bogus philosophical objection, a solution to a whole range of important problems would remain forever out of reach.[...] The plane of the complex numbers is the natural arena of discourse for much if not most of mathematics." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"Nonetheless, some hesitation persisted. After all, the very word imaginary betrays ambivalence, and suggests that in our heart of hearts we do not believe these numbers exist. On the other hand, by calling every number representable by a decimal expansion real, we are making the psychological distinction more stark. Indeed the adjective imaginary is a somewhat unfortunate one - although an intriguing name, some students’ perceptions are so colored by the word that they consequently fail to come to grips with the idea." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"Perhaps the greatest legacy of the solution of the cubic was the arrival, without invitation, of the imaginary number i into the world of mathematics." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"Since they represent so natural a sequence, it is almost irresistible to search for patterns among the primes. There are however no genuinely useful formulas for prime numbers. That is to say there is no rule that allows you to generate all prime numbers or even to calculate a sequence that consists entirely of different primes." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"The beauty of the complex plane is that we may finally carry out all our mathematical work in a single number arena. However, although there may be no pressing mathematical difficulty that is driving us further, we can ask the question whether or not it is possible to go beyond the complex plane into some larger realm of number." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"The importance of continued fractions in approximation of irrationals by rationals is that the so called convergents of the fraction, which are the rational approximations of the original number that result from truncating the representation at some point and working out the corresponding rational number, are the best approximation possible in the sense that any better approximation will have a larger denominator than that of the convergents." (Peter M Higgins, "Number Story: From Counting to Cryptography", 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)

"The simple algebraic numbers, like √2, seem closest in nature to the rationals, while we might expect that non-algebraic numbers, the transcedentals, to live apart and not to have close rational neighbors. Surprisingly, the opposite is true. On the one hand, it can be proved that any irrational number that can be well-approximated by rationals (in a sense that can be made precise) must be transcendental. Indeed this affords one of the standard techniques for showing that a number is transcendental." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"[...] the use of complex numbers reveals a connection between the exponential, or power function and the seemingly unrelated trigonometric functions. Without passing through the portal offered by the square root of minus one, the connection may be glimpsed, but not understood. The so-called hyperbolic functions arise from taking what are known as the even and odd parts of the exponential function."  (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"[...] transcendental numbers, those numbers that lie beyond those that arise through euclidean geometry and ordinary algebraic equations. [...] The transcendentals are the numbers that fill the huge void between the more familiar algebraic numbers and the collection of all decimal expansions: to use an astronomical comparison, the transcendentals are the dark matter of the number world." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"Transcendental numbers then are numerous but exceedingly slippery. As a rule of thumb, a number that arises in mathematics is almost always transcendental unless it is obvious that it is not. However, showing that a particular number is transcendental can be exceedingly difficult. Number theory throws up endless problems of this kind where everyone feels sure what the answer must be but at the same time no-one has any real idea how it could ever by proved." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

The Infinite and Its Difficulties II

“The existence of an actual infinite multitude is impossible. For any set of things one considers must be a specific set. And sets of things are specified by the number of things in them. Now no number is infinite, for number results from counting through a set of units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited.” (Thomas Aquinas, “Summa theologia”, 13th century) 

"[Paradoxes of the infinite arise] only when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but this […] is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another.” (Galileo Galilei, "Two New Sciences", 1638)

“Infinities and indivisibles transcend our finite understanding, the former on account of their magnitude, the latter because of their smallness; Imagine what they are when combined. In spite of this men cannot refrain from discussing them.” (Galileo Galilei, "Two New Sciences", 1638)

“Whatever we imagine is finite. Therefore, there is no idea or conception of anything we call finite. No man can have in his mind an image of infinite magnitude; nor conceive infinite swiftness, infinite time, or infinite force, or inmate power.” (Thomas Hobbes, "Of Man", 1658)

“Man is equally incapable of seeing the nothingness from which he emerges and the infinity in which he is engulfed.” (Blaise Pascal, "Pensées", 1670)

“Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use." (Leonhard Euler, "Introduction to Analysis of the Infinite", 1748)

 “A great deal of misunderstanding is avoided if it be remembered that the terms infinity, infinite, zero, infinitesimal must be interpreted in connexion with their context, and admit a variety of meanings according to the way in which they are defined.” (George B Mathews, “Theory of Numbers”, 1892)

“Like children who are not permitted to do certain things, we are not permitted by nature to think in terms of infinity.” (Robert Tuttle Morris, “Microbes and Men”, 1916)

 "The infinite in mathematics is always unruly unless it is properly treated."  (Edward Kasner & James Newman, “Mathematics and the Imagination”, 1940)

“I am incapable of conceiving infinity, and yet I do not accept finity.” (Simone de Beauvoir, “La Vieillesse”, 1970)

On Theorems (1980-1989)

“Some people believe that a theorem is proved when a logically correct proof is given; but some people believe a theorem is proved only when the student sees why it is inevitably true.” (Wesley R Hamming, “Coding and Information Theory”, 1980)

“We become quite convinced that a theorem is correct if we prove it on the basis of reasonably sound statements about numbers or geometrical figures which are intuitively more acceptable than the one we prove.” (Morris Kline, “Mathematics: The loss of certainty”, 1980)

“For what is important when we give children a theorem to use is not that they should memorize it. What matters most is that by growing up with a few very powerful theorems one comes to appreciate how certain ideas can be used as tools to think with over a lifetime. One learns to enjoy and to respect the power of powerful ideas. One learns that the most powerful idea of all is the idea of powerful ideas.” (Seymour Papert, “Mindstorms: Children, Computers and Powerful Ideas”, 1980)

"The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them." (George Pólya, "Mathematical Discovery", 1981)

"To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples [...]" (John B Conway, “Subnormal Operators”, 1981)

“Proof serves many purposes simultaneously […] Proof is respectability. Proof is the seal of authority. Proof, in its best instance, increases understanding by revealing the heart of the matter. Proof suggests new mathematics […] Proof is mathematical power, the electric voltage of the subject which vitalizes the static assertions of the theorems.” (Reuben Hersh, “The Mathematical Experience”, 1981)

“There are no deep theorems - only theorems that we have not understood very well.” (Nicholas P Goodman, “Reflections on Bishops Philosophy of Mathematics”, 1983)

“Mathematics is not a deductive science - that's a cliche. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork." (Paul Halmos, “I Want to Be a Mathematician”, 1985)

„The pursuit of pretty formulas and neat theorems can no doubt quickly degenerate into a silly vice, but so can the quest for austere generalities which are so very general indeed that they are incapable of application to any particular.“ (Eric T Bell, „Men of Mathematics“, 1986)

“Mathematics is more than doing calculations, more than solving equations, more than proving theorems, more than doing algebra, geometry or calculus, more than a way of thinking. Mathematics is the design of a snowflake, the curve of a palm frond, the shape of a building, the joy of a game, the frustration of a puzzle, the crest of a wave, the spiral of a spider's web. It is ancient and yet new. Mathematics is linked to so many ideas and aspects of the universe.” (Theoni Pappas, “More Joy of Mathematics: Exploring Mathematics All Around You”, 1986)

“Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems - general and specific statements - can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences.” (Sir Erik C Zeeman, “Private Games”, 1988)

5 Books 10 Quotes IV: On Complex Numbers IV

Ian Stewart, "Why Beauty Is Truth: The History of Symmetry", 2007
	“A complex number is just a pair of real numbers, manipulated according to a short list of simple rules. Since a pair of real numbers is surely just as ‘real’ as a single real number, real and complex numbers are equally closely related to reality, and ‘imaginary’ is misleading.”
“The complex numbers extend the real numbers by throwing in a new kind of number, the square root of minus one. But the price we pay for being able to take square roots of negative numbers is the loss of order. The complex numbers are a complete system but are spread out across a plane rather than aligned in a single orderly sequence.”
David Mumford, Caroline Series & David Wright, "Indra’s Pearls: The Vision of Felix Klein", 2002
	“Complex numbers are really not as complex as you might expect from their name, particularly if we think of them in terms of the underlying two dimensional geometry which they describe. Perhaps it would have been better to call them 'nature's numbers'. Behind complex numbers is a wonderful synthesis between two dimensional geometry and an elegant arithmetic in which every polynomial equation has a solution.”
“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.”
Paul J Nahin, "An Imaginary Tale: The History of √-1", 1998
	“The discovery of complex numbers was the last in a sequence of discoveries that gradually filled in the set of all numbers, starting with the positive integers (finger counting) and then expanding to include the positive rationals and irrational reals, negatives, and then finally the complex.”
“When we try to take the square root of -1 (a real number), for example, we suddenly leave the real numbers, and so the reals are not complete with respect to the square root operation. We don’t have to be concerned that something like that will happen with the complex numbers, however, and we won’t have to invent even more exotic numbers (the ‘really complex’!) Complex numbers are everything there is in the two-dimensional plane.”
Jerry R Muir Jr., “Complex Analysis: A Modern First Course in Function Theory”, 2015
	“Complex analysis should never be underestimated as simply being calculus with complex numbers in place of real numbers and is distinguished from being so at every possible opportunity.”
“The upgrade from the real numbers to the complex numbers has both algebraic and analytic motivation. The real numbers are not algebraically complete, meaning there are polynomial equations such as x^2 = −1 with no solutions. The incorporation of  √-1 […] is a direct response to this.”
Tobias Dantzig, “Number: The Language of Science”, 1930
	“[…] extensions beyond the complex number domain are possible only at the expense of the principle of permanence. The complex number domain is the last frontier of this principle. Beyond this either the commutativity of the operations or the rôle which zero plays in arithmetic must be sacrificed.”
“And so it was that the complex number, which had its origin in a symbol for a fiction, ended by becoming an indispensable tool for the formulation of mathematical ideas, a powerful instrument for the solution of intricate problems, a means for tracing kinships between remote mathematical disciplines.” Previous Post <<||>> Next Post See also: More Quotes on Complex Numbers III More Quotes on Complex Numbers II More Quotes on Complex Numbers I Complex Numbers

On Numbers: The Infinity in Numbers

“When the consequences of either assumption are the same, we should always assume that things are finite rather than infinite in number, since in things constituted by nature that which is infinite and that which is better ought, if possible, to be present rather than the reverse […]” (Aristotle)

“But of all other ideas, it is number, which I think furnishes us with the clearest and most distinct idea of infinity we are capable of.” (John Locke)

"For any number there exists a corresponding even number which is its double. Hence the number of all numbers is not greater than the number of even numbers, that is, the whole is not greater than the part." (Gottfried W Leibniz)

“I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetical act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding […]” (Richard Dedekind, “On Continuity and Irrational Numbers”, 1872)

 “If you can take away some of the terms of a collection, without diminishing the number of terms, then there is an infinite number of terms in the collection.” (Bertrand Russell)

"The prototype of all infinite processes is repetition. […] Our very concept of the infinite derives from the notion that what has been said or done once can always be repeated.” (Tobias Dantzig, “Number: The Language of Science”, 1930)

“The sequence of numbers which grows beyond any stage already reached by passing to the next number is a manifold of possibilities open towards infinity, it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties […]” (Hermann Weyl, “Mathematics and Logic”, 1946)

"[…] infinity is not a large number or any kind of number at all; at least of the sort we think of when we say 'number'. It certainly isn't the largest number that could exist, for there isn't any such thing." (Isaac Asimov)

"Each act of creation could be symbolized as a particular product of infinity and zero. From each such product could emerge a particular entity of which the appropriate symbol was a particular number." (Srinivasa Ramanujan)

“Mathematics is the only infinite human activity. It is conceivable that humanity could eventually learn everything in physics or biology. But humanity certainly won't ever be able to find out everything in mathematics, because the subject is infinite. Numbers themselves are infinite.” (Paul Erdős)

On Numbers: Defining Numbers

“Number is the bond of the eternal continuance of things.” (Plato)

“Measure, time and number are nothing but modes of thought or rather of imagination.” (Baruch Spinoza, [Letter to Ludvicus Meyer] 1663)

"[…] if number is merely the product of our mind, space has a reality outside our mind whose laws we cannot a priori completely prescribe" (Carl F Gauss, 1830)

"Numbers are intellectual witnesses that belong only to mankind, and by whose means we can achieve an understanding of words." (Honore de Balzac)

"Numbers constitute the only universal language." (Nathanael West, “Miss Lonelyhearts”, 1933)

“[…] there are terms which cannot be defined, such as number and quantity. Any attempt at a definition would only throw difficulty in the student’s way, which is already done in geometry by the attempts at an explanation of the terms point, straight line, and others, which are to be found in treatise on that subject.” (Augustus de Morgan, “On the Study and Difficulties of Mathematics”, 1943)

“Numbers have neither substance, nor meaning, nor qualities. They are nothing but marks, and all that is in them we have put into them by the simple rule of straight succession.” (Hermann Weyl, “Mathematics and the Laws of Nature”, 1959)

“[…] numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things.” (Richard Dedekind, “Essays on the Theory of Numbers”, 1963)

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

“Number is therefore the most primitive instrument of bringing an unconscious awareness of order into consciousness.” (Marie-Louise von Frany, “Creation Myths”, 1995)

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On Statistics: Some Historical Definitions (1901-1950)

"[…] statistics is the science of the measurement of the social organism, regarded as a whole, in all its manifestations." (Sir Arthur L Bowley, "Elements of Statistics", 1901)

"Statistics may rightly be called the science of averages. […] Great numbers and the averages resulting from them, such as we always obtain in measuring social phenomena, have great inertia. […] It is this constancy of great numbers that makes statistical measurement possible. It is to great numbers that statistical measurement chiefly applies." (Sir Arthur L Bowley, "Elements of Statistics", 1901)

"Statistics may, for instance, be called the science of counting. Counting appears at first sight to be a very simple operation, which any one can perform or which can be done automatically; but, as a matter of fact, when we come to large numbers, e.g., the population of the United Kingdom, counting is by no means easy, or within the power of an individual; limits of time and place alone prevent it being so carried out, and in no way can absolute accuracy be obtained when the numbers surpass certain limits." (Sir Arthur L Bowley, "Elements of Statistics", 1901)

"[...] statistics is the science that, through calculation, leads to an understanding of the characteristics of human societies, and its purpose is the study of masses through the enumeration of the units that compose them." (Armand Julin, "Summary for a Course of Statistics, General and Applied, 1910)

"The science of Statistics is the method of judging collective, natural or social phenomenon from the results obtained from the analysis or enumeration or collection of estimates." (Willford I King, "The Elements of Statistical Method", 1912)

"By Statistics we mean aggregate of facts affected to a marked extent by multiplicity of factors [...] and placed in relation to each other." (Horace Secrist, "An Introduction to Statistical Methods", 1917)

"Statistics may be defined as numerical statements of facts by means of which large aggregates are analyzed, the relations of individual units to their groups are ascertained, comparisons are made between groups, and continuous records are maintained for comparative purposes." (Melvin T Copeland. "Statistical Methods" [in: Harvard Business Studies, Vol. III, Ed. by Melvin T Copeland, 1917])

"Statistics may be regarded as (i) the study of populations, (ii) as the study of variation, and (iii) as the study of methods of the reduction of data." (Sir Ronald A Fisher, "Statistical Methods for Research Worker", 1925)

"Statistics is a scientific discipline concerned with collection, analysis, and interpretation of data obtained from observation or experiment. The subject has a coherent structure based on the theory of Probability and includes many different procedures which contribute to research and development throughout the whole of Science and Technology." (Egon Pearson, 1936)

"[Statistics] is both a science and an art. It is a science in that its methods are basically systematic and have general application; and an art in that their successful application depends to a considerable degree on the skill and special experience of the statistician, and on his knowledge of the field of application, e.g. economics." (Leonard H C Tippett, "Statistics", 1943)

"Statistics is the branch of scientific method which deals with the data obtained by counting or measuring the properties of populations of natural phenomena. In this definition 'natural phenomena' includes all the happenings of the external world, whether human or not " (Sir Maurice G Kendall, "Advanced Theory of Statistics", Vol. 1, 1943)

"To some people, statistics is ‘quartered pies, cute little battleships and tapering rows of sturdy soldiers in diversified uniforms’. To others, it is columns and columns of numerical facts. Many regard it as a branch of economics. The beginning student of the subject considers it to be largely mathematics." (The Editors, "Statistics, The Physical Sciences and Engineering", The American Statistician, Vol. 2, No. 4, 1948) [Link]

Further definitions:
1800-1900
1951-2000
2001- …

On Statistics: What is Statistics?

“Statistics is a science which ought to be honourable, the basis of many most important sciences; but it is not to be carried on by steam, this science, any more than others are; a wise hand is requisite for carrying it on. Conclusive facts are inseparable from unconclusive except by a head that already understands and knows.” (Thomas Carlyle, “Critical and Miscellaneous Essays”, 1838)

“To some people, statistics is ‘quartered pies, cute little battleships and tapering rows of sturdy soldiers in diversified uniforms’. To others, it is columns and columns of numerical facts. Many regard it as a branch of economics. The beginning student of the subject considers it to be largely mathematics.” (The Editors, “Statistics, The Physical Sciences and Engineering”, The American Statistician, Vol. 2, No. 4, 1948)

"Statistics is that branch of mathematics which deals with the accumulation and analysis of quantitative data." (David B MacNeil, "Modern Mathematics for the Practical Man", 1963)

“Statistics is the branch of scientific method which deals with the data obtained by counting or measuring the properties of populations of natural phenomena.” (Sir Maurice G Kendall & Alan Stuart, “The Advanced Theory of Statistics”, 1963)

“Statistics may be defined as the discipline concerned with the treatment of numerical data derived from groups of individuals.” (Peter Armitage, “Statistical Methods in Medical Research”, 1971)

“[…] statistics is the science that deals with distributions and proportions in actual (large but finite) classes (also called ‘populations’, ‘aggregates’, ‘ensembles’) of actual things.” (Bas C van Frassen, “The Scientic Image”, 1980)

“We provisionally define statistics as the study of how information should be employed to reflect on, and give guidance for action in, a practical situation involving uncertainty.” (Vic Barnett, “Comparative Statistical Inference” 2nd Ed., 1982)

“[Statistics] is the technology of extracting meaning from data.” (David J Hand, “Statistics: A Very Short Introduction”, 2008)

“[Statistics] is the technology of handling uncertainty.” (David J Hand, “Statistics: A Very Short Introduction”, 2008)

“[…] statistics is the key discipline for predicting the future or for making inferences about the unknown, or for producing convenient summaries of data.” (David J Hand, “Statistics: A Very Short Introduction”, 2008)

“[Statistics: used with a plural verb] facts or data, either numerical or nonnumerical, organized and summarized so as to provide useful and accessible information about a particular subject.” (Neil A Weiss, "Introductory Statistics" 10th Ed., 2017)

“[Statistics: used with a singular verb] the science of organizing and summarizing numerical or nonnumerical information.” (Neil A Weiss, "Introductory Statistics" 10th Ed., 2017)

“Statistics is the science of finding relationships and actionable insights from data.” (Nate Silver)

“Statistics is the science, the art, the philosophy, and the technique of making inferences from the particular to the general.” (John W Tukey)