On Calculus: On Integrals (1975-1999)

"Apart from its own intrinsic interest, a knowledge of differentiable manifolds has become useful-ven mandatory-in an ever-increasing number of areas of mathematics and of its applications. This is not too surprising, since differentiable manifolds are the underlying, if unacknowledged, objects of study in much of advanced calculus and analysis. Indeed, such topics as line and surface integrals, divergence and curl of vector fields, and Stokes's and Green's theorems find their most natural setting in manifold theory. But however natural the leap from calculus on domains of Euclidean space to calculus on manifolds may be to those who have made it, it is not at all easy for most students." (William M Boothby, "Riemannian geometry: An introduction to differentiable Manifolds and Riemannian Geometry", 1975)

"The chief difficulty of modern theoretical physics resides not in the fact that it expresses itself almost exclusively in mathematical symbols, but in the psychological difficulty of supposing that complete nonsense can be seriously promulgated and transmitted by persons who have sufficient intelligence of some kind to perform operations in differential and integral calculus […]" (Celia Green, "The Decline and Fall of Science", 1976)

"In the path-integral formulation, the essence of quantum physics may be summarized with two fundamental rules: (1). The classical action determines the probability amplitude for a specific chain of events to occur, and (2) the probability that either one or the other chain of events occurs is determined by the probability amplitudes corresponding to the two chains of events. Finding these rules represents a stunning achievement by the founders of quantum physics." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"The acceptance of complex numbers into the realm of algebra had an impact on analysis as well. The great success of the differential and integral calculus raised the possibility of extending it to functions of complex variables. Formally, we can extend Euler's definition of a function to complex variables without changing a single word; we merely allow the constants and variables to assume complex values. But from a geometric point of view, such a function cannot be plotted as a graph in a two-dimensional coordinate system because each of the variables now requires for its representation a two-dimensional coordinate system, that is, a plane. To interpret such a function geometrically, we must think of it as a mapping, or transformation, from one plane to another." (Eli Maor, "e: The Story of a Number", 1994)

"By studying analytic functions using power series, the algebra of the Middle Ages was connected to infinite operations" (various algebraic operations with infinite series). The relation of algebra with infinite operations was later merged with the newly developed differential and integral calculus. These developments gave impetus to early stages of the development of analysis. In a way, we can say that analyticity is the notion that first crossed the boundary from finite to infinite by passing from polynomials to infinite series. However, algebraic properties of polynomial functions still are strongly present in analytic functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"One of the basic tasks of topology is to learn to distinguish nonhomeomorphic figures. To this end one introduces the class of invariant quantities that do not change under homeomorphic transformations of a given figure. The study of the invariance of topological spaces is connected with the solution of a whole series of complex questions: Can one describe a class of invariants of a given manifold? Is there a set of integral invariants that fully characterizes the topological type of a manifold? and so forth." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

♾️On Complex Numbers XIX (Euler's Formula II)

"The equation e^πi+1 = 0 is true only by virtue of a large number of profound connections across many fields. It is true because of what it means! And it means what it means because of all those metaphors and blends in the conceptual system of a mathematician who understands what it means. To show why such an equation is true for conceptual reasons is to give what we have called an idea analysis of the equation." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"The equation e^πi =-1 says that the function w= e^z, when applied to the complex number πi as input, yields the real number -1 as the output, the value of w. In the complex plane, πi is the point [0,π) - π on the i-axis. The function w=e^z maps that point, which is in the z-plane, onto the point (-1, 0) - that is, -1 on the x-axis-in the w-plane. […] But its meaning is not given by the values computed for the function w=e^z. Its meaning is conceptual, not numerical. The importance of  e^πi =-1 lies in what it tells us about how various branches of mathematics are related to one another - how algebra is related to geometry, geometry to trigonometry, calculus to trigonometry, and how the arithmetic of complex numbers relates to all of them." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"The significance of e^πi+1 = 0 is thus a conceptual significance. What is important is not just the numerical values of e, π, i, 1, and 0 but their conceptual meaning. After all, e, π, i, 1, and 0 are not just numbers like any other numbers. Unlike, say, 192,563,947.9853294867, these numbers have conceptual meanings in a system of common, important nonmathematical concepts, like change, acceleration, recurrence, and self-regulation. [...] They are not mere numbers; they are the arithmetizations of concepts. When they are placed in a formula, the formula incorporates the ideas the function expresses as well as the set of pairs of complex numbers it mathematically determines by virtue of those ideas." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"We will now turn to e^πi+1 = 0. Our approach will be there as it was here. e^πi+1 = 0 uses the conceptual structure of all the cases we have discussed so far - trigonometry, the exponentials, and the complex numbers. Moreover, it puts together all that conceptual structure. In other words, all those metaphors and blends are simultaneously activated and jointly give rise to inferences that they would not give rise to separately. Our job is to see how e^πi+1 = 0 is a precise consequence that arises when the conceptual structure of these three domains is combined to form a single conceptual blend." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"[…] the equation’s five seemingly unrelated numbers (e, i, π, 1, and 0) fit neatly together in the formula like contiguous puzzle pieces. One might think that a cosmic carpenter had jig-sawed them one day and mischievously left them conjoined on Euler’s desk as a tantalizing hint of the unfathomable connectedness of things.[…] when the three enigmatic numbers are combined in this form, e^iπ, they react together to carve out a wormhole that spirals through the infinite depths of number space to emerge smack dab in the heartland of integers." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Thus, while feelings may be the essence of subjectivity, they are by no means part of our weaker nature - the valences they automatically generate are integral to our thought processes and without them we’d simply be lost. In particular, we’d have no sense of beauty at all, much less be able to feel (there’s that word again) that we’re in the presence of beauty when contemplating a work such as Euler’s formula. After all, e^iπ + 1 = 0 can give people limbic-triggered goosebumps when they first peer with understanding into its depths." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Today, Euler’s formula is a tool as basic to electrical engineers and physicists as the spatula is to short-order cooks. It’s arguable that the formula’s ability to simplify the design and analysis of circuits contributed to the accelerating pace of electrical innovation during the twentieth century." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"[…] when the three enigmatic numbers are combined in this form, e^iπ, they react together to carve out a wormhole that spirals through the infinite depths of number space to emerge smack dab in the heartland of integers." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Euler’s formula - although deceptively simple - is actually staggeringly conceptually difficult to apprehend in its full glory, which is why so many mathematicians and scientists have failed to see its extraordinary scope, range, and ontology, so powerful and extensive as to render it the master equation of existence, from which the whole of mathematics and science can be derived, including general relativity, quantum mechanics, thermodynamics, electromagnetism and the strong and weak nuclear forces! It’s not called the God Equation for nothing. It is much more mysterious than any theistic God ever proposed." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

Random Numbers

“A random sequence is a vague notion embodying the idea of a sequence in which each term is unpredictable to the uninitiated and whose digits pass a certain number of tests traditional with statisticians and depending somewhat on the uses to which the sequence is to be put.” (Derrick H Lehmer, 1951)

“Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number - there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method.” (John von Neumann, "Various techniques used in connection with random digits", 1951)

"[A] sequence is random if it has every property that is shared by all infinite sequences of independent samples of random variables from the uniform distribution." (J. N. Franklin (1962)

“[…] random numbers should not be generated with a method chosen at random. Some theory should be used.” (Donald E. Knuth, “The Art of Computer Programming” Vol. II, 1968)

"The generation of random numbers is too important to be left to chance." (Robert R. Coveyou, 1969)

“What will prove altogether remarkable is that some very simple schemes to produce erratic numbers behave identically to some of the erratic aspects of natural phenomena.” (Mitchell Figenbaum, “Universal Behavior in Nonlinear Systems”, 1980)

“[In statistics] you have the fact that the concepts are not very clean. The idea of probability, of randomness, is not a clean mathematical idea. You cannot produce random numbers mathematically. They can only be produced by things like tossing dice or spinning a roulette wheel. With a formula, any formula, the number you get would be predictable and therefore not random. So as a statistician you have to rely on some conception of a world where things happen in some way at random, a conception which mathematicians don’t have.” (Lucien LeCam, [interview] 1988)

"It is evident that the primes are randomly distributed but, unfortunately, we don't know what 'random' means.'' (Rob C Vaughan, 1990)

"According to the narrower definition of randomness, a random sequence of events is one in which anything that can ever happen can happen next. Usually it is also understood that the probability that a given event will happen next is the same as the probability that a like event will happen at any later time. [...] According to the broader definition of randomness, a random sequence is simply one in which any one of several things can happen next, even though not necessarily anything that can ever happen can happen next." (Edward N Lorenz, "The Essence of Chaos", 1993)

“Suppose that we think of the integers lined up like dominoes. The inductive step tells us that they are close enough for each domino to knock over the next one, the base case tells us that the first domino falls over; the conclusion is that they all fall over. The fault in this analogy is that it takes time for each domino to fall and so a domino which is a long way along the line won't fall over fora long time. Mathematical implication is outside time.” (Peter J Eccles, “An Introduction to Mathematical Reasoning”, 1997)

“Sequences of random numbers also inevitably display certain regularities. […] The trouble is, just as no real die, coin, or roulette wheel is ever likely to be perfectly fair, no numerical recipe produces truly random numbers. The mere existence of a formula suggests some sort of predictability or pattern.” (Ivars Peterson, “The Jungles of Randomness: A Mathematical Safari”, 1998)

“The practical definitions of randomness - a sequence is random by virtue of how many and which statistical tests it satisfies and a sequence is random by virtue of the length of the algorithm necessary to describe it [...].” (Deborah J. Bennett, “Randomness”, 1998)

On Numbers: Large Numbers I

"The calculation of probabilities is of the utmost value, […] but in statistical inquiries there is need not so much of mathematical subtlety as of a precise statement of all the circumstances. The possible contingencies are too numerous to be covered by a finite number of experiments, and exact calculation is, therefore, out of the question. Although nature has her habits, due to the recurrence of causes, they are general, not invariable. Yet empirical calculation, although it is inexact, may be adequate in affairs of practice." (Gottfried W Leibniz [letter to Bernoulli], 1703)

"Further, it cannot escape anyone that for judging in this way about any event at all, it is not enough to use one or two trials, but rather a great number of trials is required. And sometimes the stupidest man - by some instinct of nature per se and by no previous instruction (this is truly amazing) - knows for sure that the more observations of this sort that are taken, the less the danger will be of straying from the mark." (Jacob Bernoulli, "The Art of Conjecturing", 1713)

"If thus all events through all eternity could be repeated, by which we would go from probability to certainty, one would find that everything in the world happens from definite causes and according to definite rules, and that we would be forced to assume amongst the most apparently fortuitous things a certain necessity, or, so to say, FATE." (Jacob Bernoulli, "The Art of Conjecturing", 1713)

"And thus in all cases it will be found, that although Chance produces Irregularities, still the odds will be infinitely great that in the process of time, those Irregularities will bear no proportion to the recurrency of that Order which naturally results from ORIGINAL DESIGN." (Abraham de Moivre, "The Doctrine of Chances", 1718)

"Things of all kinds are subject to a universal law which may be called the law of large numbers. It consists in the fact that, if one observes very considerable numbers of events of the same nature, dependent on constant causes and causes which vary irregularly, sometimes in one direction, sometimes in the other, it is to say without their variation being progressive in any definite direction, one shall find, between these numbers, relations which are almost constant." (Siméon-Denis Poisson, "Poisson’s Law of Large Numbers", 1837)

"Huge numbers are commonplace in our culture, but oddly enough the larger the number the less meaningful it seems to be." (Albert Sukoff, "Lotsa Hamburgers", Saturday Review of the Society, 1973)

"We know the laws of trial and error, of large numbers and probabilities. We know that these laws are part of the mathematical and mechanical fabric of the universe, and that they are also at play in biological processes. But, in the name of the experimental method and out of our poor knowledge, are we really entitled to claim that everything happens by chance, to the exclusion of all other possibilities?" (Albert Claude, [Nobel Prize Lecture], 1974)

"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)

"The trouble with integers is that we have examined only the small ones. Maybe all the exciting stuff happens at really big numbers, ones we can’t get our hand on or even begin to think about in any very definite way. So maybe all the action is really inaccessible and we’re just fiddling around. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions." (Paul Hauffman, "The Man Who Loves Only Numbers", The Atlantic Magazine, Vol 260, No 5, 1987)

"The law of truly large numbers states: With a large enough sample, any outrageous thing is likely to happen." (Frederick Mosteller, "Methods for Studying Coincidences Journal of the American Statistical Association, Volume 84, 1989)

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