Discrete Mathematics Quotes

Let me then remind you that justice is an immutable, natural principle; and not anything that can be made, unmade, or altered by any human power. It is also a subject of science, and is to be learned, like mathematics, or any other science. It does not derive its authority from the commands, will, pleasure, or discretion of any possible combination of men, whether calling themselves a government, or by any other name. It is also, at all times, and in all places, the supreme law. And being everywhere and always the supreme law, it is necessarily everywhere and always the only law.

Perhaps the rivers of ink that have been expended discussing the nature of the “continuous” over the centuries, from Aristotle to Heidegger, have been wasted. Continuity is only a mathematical technique for approximating very finely grained things. The world is subtly discrete, not continuous. The good Lord has not drawn the world with continuous lines: with a light hand, he has sketched it in dots, like the painter Georges Seurat.

I am pretty strongly convinced that there is an ongoing reversal in the collective consciousness of mathematicians: the right hemispherical and homotopical picture of the world becomes the basic intuition, and if you want to get a discrete set, then you pass to the set of connected components of a space defined only up to homotopy.

I had written this paper on pancake numbers with help from my adviser, Manuel Blum, who's a well-known computer scientist, and we submitted it to a journal called Discrete Applied Mathematics. I subsequently left graduate school to come and write for The Simpsons. After the paper was accepted, there was an extremely long lag between it being submitted, revised, and published. So, by the time the paper was published, I had been working at The Simpsons for a while, and Ken Keeler had also been hired at that point. So, finally the research article appeared, and I came in with the reprints of this article and I said, 'Hey, I've got an article in Discrete Applied Mathematics.' Everyone was quite impressed except Ken Keeler, who said, 'Oh yeah, I had a paper in that journal a couple of months ago.'" With a wry smile on his face, Cohen bemoaned: "What does it mean that I come to write for The Simpsons and I cannot even be the only writer on this show with a paper in Discrete Applied Mathematics?

One can also use compactifications to view the continuous as the limit of the discrete: for instance, it is possible to compactify the sequence / 2,/ 3,/ 4, . . . of cyclic groups in such a way that their limit is the circle group = /.

In this chapter we will look at the entire edifice of QFT. We will see that it is based on three simple principles. We will also list some of its achievements, including some new insights and understandings not previously mentioned. THE FOUNDATION QFT is an axiomatic theory that rests on a few basic assumptions. Everything you have learned so far, from the force of gravity to the spectrum of hydrogen, follows almost inevitably from these three basic principles. (To my knowledge, Julian Schwinger is the only person who has presented QFT in this axiomatic way, at least in the amazing courses he taught at Harvard University in the 1950's.) 1. The field principle. The first pillar is the assumption that nature is made of fields. These fields are embedded in what physicists call flat or Euclidean three-dimensional space-the kind of space that you intuitively believe in. Each field consists of a set of physical properties at every point of space, with equations that describe how these particles or field intensities influence each other and change with time. In QFT there are no particles, no round balls, no sharp edges. You should remember, however, that the idea of fields that permeate space is not intuitive. It eluded Newton, who could not accept action-at-a-distance. It wasn't until 1845 that Faraday, inspired by patterns of iron filings, first conceived of fields. The use of colors is my attempt to make the field picture more palatable. 2. The quantum principle (discetization). The quantum principle is the second pillar, following from Planck's 1900 proposal that EM fields are made up of discrete pieces. In QFT, all physical properties are treated as having discrete values. Even field strengths, whose values are continues, are regarded as the limit of increasingly finer discrete values. The principle of discretization was discovered experimentally in 1922 by Otto Stern and Walther Gerlach. Their experiment (Fig. 7-1) showed that the angular momentum (or spin) of the electron in a given direction can have only two values: +1/2 or -1/2 (Fig. 7-1). The principle of discretization leads to another important difference between quantum and classical fields: the principle of superposition. Because the angular momentum along a certain axis can only have discrete values (Fig. 7-1), this means that atoms whose angular momentum has been determined along a different axis are in a superposition of states defined by the axis of the magnet. This same superposition principle applies to quantum fields: the field intensity at a point can be a superposition of values. And just as interaction of the atom with a magnet "selects" one of the values with corresponding probabilities, so "measurement" of field intensity at a point will select one of the possible values with corresponding probability (see "Field Collapse" in Chapter 8). It is discretization and superposition that lead to Hilbert space as the mathematical language of QFT. 3. The relativity principle. There is one more fundamental assumption-that the field equations must be the same for all uniformly-moving observers. This is known as the Principle of Relativity, famously enunciated by Einstein in 1905 (see Appendix A). Relativistic invariance is built into QFT as the third pillar. QFT is the only theory that combines the relativity and quantum principles.

WHY STUDY DISCRETE MATHEMATICS? There are several important reasons for studying discrete mathematics. First, through this course you can develop your mathematical maturity: that is, your ability to understand and create mathematical arguments. You will not get very far in your studies in the mathematical sciences without these skills. Second, discrete mathematics is the gateway to more advanced courses in all parts of the mathematical sciences. Discrete mathematics provides the mathematical foundations for many computer science courses, including data structures, algorithms, database theory, automata theory, formal languages, compiler theory, computer security, and operating systems. Students find these courses much more difficult when they have not had the appropriate mathematical foundations from discrete mathematics.

To summarize, from simple counting using the God-given integers, we made various extensions of the ideas of numbers to include more things. Sometimes the extensions were made for what amounted to aesthetic reasons, and often we gave up some property of the earlier number system. Thus we came to a number system that is unreasonably effective even in mathematics itself; witness the way we have solved many number theory problems of the original highly discrete counting system by using a complex variable.

What Zeno is forcing us to do is to ask the question of whether space (which is not made of atoms) can be infinitely divvied up. If it can be, the slacker will not reach his goal. If it cannot be, there must be discrete "space atoms," and continuous real-number mathematics is not a proper model for space. We cannot, however be so flippant about asserting that space is discrete and not continuous. The world certainly does not look discrete. Movement has the feel of being continuous. Much of mathematical physics is based on calculus, which assumes that the real world is infinitely divisible. Outside of some quantum theory and Zeno, the continuous real number make a good model for the physical world. We build rockets and bridges using mathematics that assumes that the world is continuous. Let us not be so quick to abandon it.

If we assume the world is discrete, the mathematics needed to build rockets and bridges is far more complicated than calculus. Perhaps calculus is simply an easy approximation of the true mathematics that has to be done to concretely model the discrete world in which we live.

Again, there is a problem abandoning the notion of continuous time for discrete time. Modern physics and engineering are based on the fact that time is continuous. All the equations have a continuous-time variable usually denoted by t. And yet, as Zeno has shown us, the notion of continuous time is illogical.

Any discrete piece of information can be represented by a set of numbers. Systems that compute can represent powerful mappings from one set of numbers to another. Moreover, any program on any computer is equivalent to a number mapping. These mappings can be thought of as statements about the properties of numbers; hence, there is a close connection between computer programs and mathematical proofs. But there are more possible mappings than possible programs; thus, there are some things that simply cannot be computed. The actual process of computing can be defined in terms of a very small number of primitive operations, with recursion and/or iteration comprising the most fundamental pieces of a computing device. Computing devices can also make statements about other computing devices. This leads to a fundamental paradox that ultimately exposes the limitations not just of machine logic, but all of nature as well.

Replace "r" with "h" in Goodreads, that's popular. An unexpected application of Hamming distance from discrete mathematics.

But Yorke had offered more than a mathematical result. He had sent a message to physicists: Chaos is ubiquitous; it is stable; it is structured. He also gave reason to believe that complicated systems, traditionally modeled by hard continuous differential equations, could be understood in terms of easy discrete maps.

Chaos should be taught, he argued. It was time to recognize that the standard education of a scientist gave the wrong impression. No matter how elaborate linear mathematics could get, with its Fourier transforms, its orthogonal functions, its regression techniques, May argued that it inevitably misled scientists about their overwhelmingly nonlinear world. “The mathematical intuition so developed ill equips the student to confront the bizarre behaviour exhibited by the simplest of discrete nonlinear systems,” he wrote. “Not only in research, but also in the everyday world of politics and economics, we would all be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamical properties.

Pedagogically speaking, a good share of physics and mathematics was—and is—writing differential equations on a blackboard and showing students how to solve them. Differential equations represent reality as a continuum, changing smoothly from place to place and from time to time, not broken in discrete grid points or time steps. As every science student knows, solving differential equations is hard. But in two and a half centuries, scientists have built up a tremendous body of knowledge about them: handbooks and catalogues of differential equations, along with various methods for solving them, or “finding a closed-form integral,” as a scientist will say. It is no exaggeration to say that the vast business of calculus made possible most of the practical triumphs of post-medieval science; nor to say that it stands as one of the most ingenious creations of humans trying to model the changeable world around them. So by the time a scientist masters this way of thinking about nature, becoming comfortable with the theory and the hard, hard practice, he is likely to have lost sight of one fact. Most differential equations cannot be solved at all.

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As the sensations of motion and discreteness led to the abstract notions of the calculus, so may sensory experience continue thus to suggest problem for the mathematician, and so may she in turn be free to reduce these to the basic formal logical relationships involved. Thus only may be fully appreciated the twofold aspect of mathematics: as the language of a descriptive interpretation of the relationships discovered in natural phenomena, and as a syllogistic elaboration of arbitrary premise.

I took 17 computer science classes and made an A in 11 of them. 1 point away from an A in 3 of them and the rest of them didn't matter. Math is a tool for physics,chemistry,biology/basic computation and nothing else. CS I(Pascal Vax), CS II(Pascal Vax), Sr. Software Engineering, Sr. Distributed Systems, Sr. Research, Sr. Operating Systems, Sr. Unix Operating Systems, Data Structures, Sr. Object Oriented A&D, CS (perl/linux), Sr. Java Programming, Information Systems Design, Jr. Unix Operating Systems, Microprocessors, Programming Algorithms, Calculus I,II,III, B Differential Equations, TI-89 Mathematical Reasoning, 92 C++ Programming, Assembly 8086, Digital Computer Organization, Discrete Math I,II, B Statistics for the Engineering & Sciences (w/permutations & combinatorics) -- A-American Literature A-United States History 1865 CLEP-full year english CLEP-full year biology A-Psychology A-Environmental Ethics

A few books that I've read.... Pascal, an Introduction to the Art and Science of Programming by Walter Savitch Programming algorithms Introduction to Algorithms, 3rd Edition (The MIT Press) Data Structures and Algorithms in Java Author: Michael T. Goodrich - Roberto Tamassia - Michael H. Goldwasser The Algorithm Design Manual Author: Steven S Skiena Algorithm Design Author: Jon Kleinberg - Éva Tardos Algorithms + Data Structures = Programs Book by Niklaus Wirth Discrete Math Discrete Mathematics and Its Applications Author: Kenneth H Rosen Computer Org Structured Computer Organization Andrew S. Tanenbaum Introduction to Assembly Language Programming: From 8086 to Pentium Processors (Undergraduate Texts in Computer Science) Author: Sivarama P. Dandamudi Distributed Systems Distributed Systems: Concepts and Design Author: George Coulouris - Jean Dollimore - Tim Kindberg - Gordon Blair Distributed Systems: An Algorithmic Approach, Second Edition (Chapman & Hall/CRC Computer and Information Science Series) Author: Sukumar Ghosh Mathematical Reasoning Mathematical Reasoning: Writing and Proof Version 2.1 Author: Ted Sundstrom An Introduction to Mathematical Reasoning: Numbers, Sets and Functions Author: Peter J. Eccles Differential Equations Differential Equations (with DE Tools Printed Access Card) Author: Paul Blanchard - Robert L. Devaney - Glen R. Hall Calculus Calculus: Early Transcendentals Author: James Stewart And more....

Continuity connotes unity; discreteness, plurality.

Planck’s trick amounts to treating light, which physicists thought of as a continuous wave, as coming in discrete chunks, like particles. Planck’s “oscillators” could only emit light in discrete units of brightness. This is a little like imagining a pond where waves can only be one, two, or three centimeters high, never one and a half or two and a quarter. Everyday waves don’t work that way, but that’s what Planck’s mathematical model requires.

The primordial intuition of mathematics and every intellectual activity is the substratum of all observations of change when divested of all quality; a unity of continuity and discreteness.

Is intelligence a formal (or mathematically definable) system? Is life a recursive (or mechanically calculable) function? What happens when you replicate discrete-state microprocessors by the billions and run these questions the other way? (Are formal systems intelligent? Are recursive functions alive?) Life and intelligence have learned to operate on any number of different scales: larger, smaller, slower, and faster than our own. Biology and technology evidence parallel tendencies toward collective, hierarchical processes based on information exchange. As information is distributed, it tends to be represented (encoded) by increasingly economical (meaningful) forms. This evolutionary process, whereby the most economical or meaningful representation wins, leads to a hierarchy of languages, encoding meaning on levels that transcend comprehension by the system’s individual components—whether genes, insects, microprocessors, or human minds.

However weak the position of intuitionism seemed to be after this period of mathematical development, it has recovered by abandoning Kant’s a-priority of space but adhering the more resolutely to the a-priority of time. This neo-intuitionism considers the falling apart of moments of life into qualitatively different parts, to be reunited only while remaining separated by time as the fundamental phenomenon of the human intellect, passing by abstracting from its emotional content into the fundamental phenomenon of mathematical thinking, the intuition of the bare two-oneness. This intuition of two-oneness, the basal intuition of mathematics, creates not only the numbers one and two, but also all finite ordinal numbers, inasmuch as one of the elements of the two-oneness may be thought of as a new two-oneness, which process may be repeated indefinitely; this gives rise still further to the smallest infinite ordinal number ω. Finally this basal intuition of mathematics, in which the connected and the separate, the continuous and the discrete are united, gives rise immediately to the intuition of the linear continuum, i.e., of the “between,“ which is not exhaustible by the interposition of new units and which therefore can never be thought of as a mere collection of units.

[T]his basal intuition of mathematics, in which the connected and the separate, the continuous and the discrete are united, gives rise immediately to the intuition of the linear continuum, i.e., of the "between," which is not exhaustible by the interposition of new units and which therefore can never be thought of as a mere collection of units.

The constructive and operational aspect of mathematics is necessarily linked to its discrete aspect, that is, to discontinuity. Since the operations cannot be carried out simultaneously, each of them takes place in a well-defined segment of time. The indefinite interation of operations creates objects for which it is often difficult, if not impossible, to obtain intuitive representations.