Fourier Transform Quotes

You know what I believe? I remember in college I was taking this math class, this really great math class taught by this tiny old woman. She was talking about fast Fourier transforms and she stopped midsentence and said, ‘Sometimes it seems the universe wants to be noticed.’ “That’s what I believe. I believe the universe wants to be noticed. I think the universe is improbably biased toward consciousness, that it rewards intelligence in part because the universe enjoys its elegance being observed. And who am I, living in the middle of history, to tell the universe that it—or my observation of it—is temporary?

Leibniz’s brilliant monadic system naturally gives rise to calculus (the main tool of mathematics and science). But it was not Leibniz who linked the energy of monads to waves – that was done later following the work of the French genius Jean Baptiste Joseph Fourier on Fourier series and Fourier transforms. Nevertheless, Leibniz’s idea of energy originating from countless mathematical points and flowing across a plenum is indeed the first glimpse in the modern age of “field theory” that now underpins contemporary physics. Leibniz was centuries ahead of his time. Leibniz’s system is entirely mathematical. It brings mathematics to life. The infinite collection of monads constitutes an evolving cosmic organism, unfolding according to mathematical laws.

Neoclassical Assumptions in Contemporary Prescriptive Grammar,” “The Implications of Post-Fourier Transformations for a Holographically Mimetic Cinema,” “The Emergence of Heroic Stasis in Broadcast Entertainment” —’ ‘ “Montague Grammar and the Semantics of Physical Modality”?’ ‘ “A Man Who Began to Suspect He Was Made of Glass”?’ ‘ “Tertiary Symbolism in Justinian Erotica”?

but in areas and with titles, I’m sure you recall quite well, Hal: “Neoclassical Assumptions in Contemporary Prescriptive Grammar,” “The Implications of Post-Fourier Transformations for a Holographically Mimetic Cinema,” “The Emergence of Heroic Stasis in Broadcast Entertainment”—’ ‘ “Montague Grammar and the Semantics of Physical Modality”?’ ‘ “A Man Who Began to Suspect He Was Made of Glass”?’ ‘ “Tertiary Symbolism in Justinian Erotica”?

Roughly speaking what Fourier developed was a mathematical way of converting any pattern, no matter how complex, into a language of simple waves. He also showed how these wave forms could be converted back into the original pattern. In other words, just as a television camera converts an image into electromagnetic frequencies and a television set converts those frequencies back into the original image, Fourier showed how a similar process could be achieved mathematically. The equations he developed to convert images into wave forms and back again are known as Fourier transforms.

Great mathematical geniuses such as Pythagoras, Plato, Descartes, Leibniz, Fourier and Gödel all recognized that mathematics is ontological. ‘Real’ mathematics is ontological mathematics that tells us about reality; it’s not abstract mathematics that has no connection with reality, as most professional mathematicians seem to believe. Reality is 100% mathematical. Above all, Fourier mathematics is the key to the mystery of the universe because it’s the answer to the mystery of mind and matter and how they interact. Mind is the Fourier frequency domain and matter is the inverse Fourier spacetime domain, and the two domains are absolutely tied together in feedback loop. The universe, finally, is a hologram and holography is all about Fourier mathematics.

the simple algebraic equation ω+k3 = 0. This is called the dispersion relation of (1): with the help of the Fourier transform it is not hard to show that every solution is a superposition of solutions of the form ei(kx-ωt), and the dispersion relation tells us how the “wave number” k is related to the “angular frequency” ω in each of these elementary solutions.

Susurrus whispers through the grass and gorse, godling of the Martian wind, gene-spliced tyke of Zephyros and Ares. His story needs no Ovid, tells itself in the rustle of striplings and flowers he loves, the tale that he is: a zygote collaged from: spermatazoa flensed to nuclear caducei; a mathematical transform by the Fréres Fourier, Jean and Charles, flip of an axis changing Y to X; and the egg from which Eros hatched, is always hatching, offered up blithely to a god of war gone broody, Ares a sharper marksman than any brat with bow and arrow, no more to be argued with than the groundling Renart in a frum.

One of the greatest ideas of mathematics is that all time series one is likely to encounter in nature can be described as a superposition of periodic functions.

Distribution theory was one of the two great revolutions in mathematical analysis in the 20th century. It can be thought of as the completion of differential calculus, just as the other great revolution, measure theory (or Lebesgue integration theory), can be thought of as the completion of integral calculus. There are many parallels between the two revolutions. Both were created by young, highly individualistic French mathematicians (Henri Lebesgue and Laurent Schwartz). Both were rapidly assimilated by the mathematical community, and opened up new worlds of mathematical development. Both forced a complete rethinking of all mathematical analysis that had come before, and basically altered the nature of the questions that mathematical analysts asked.

Distribution theory was one of the two great revolutions in mathematical analysis in the 20th century. It can be thought of as the completion of differential calculus, just as the other great revolution, measure theory (or Lebesgue integration theory), can be thought of as the completion of integral calculus. There are many parallels between the two revolutions. Both were created by young, highly individualistic French mathematicians (Henri Lebesgue and Laurent Schwartz). Both were rapidly assimilated by the mathematical community, and opened up new worlds of mathematical development. Both forced a complete rethinking of all mathematical analysis that had come before, and basically altered the nature of the questions that mathematical analysts asked.

... the development of mathematics, for the sciences and for everybody else, does not often come from pure math. It came from the physicists, engineers, and applied mathematicians. The physicists were on to many ideas which couldn’t be proved, but which they knew to be right, long before the pure mathematicians sanctified it with their seal of approval. Fourier series, Laplace transforms, and delta functions are a few examples where waiting for a rigorous proof of procedure would have stifled progress for a hundred years. The quest for rigor too often meant rigor mortis. The physicists used delta functions early on, but this wasn’t really part of mathematics until the theory of distributions was invoked to make it all rigorous and pure. That was a century later! Scientists and engineers don’t wait for that: they develop what they need when they need it. Of necessity, they invent all sorts of approximate, ad hoc methods: perturbation theory, singular perturbation theory, renormalization, numerical calculations and methods, Fourier analysis, etc. The mathematics that went into this all came from the applied side, from the scientists who wanted to understand physical phenomena. [...] So much of mathematics originates from applications and scientific phenomena. But we have nature as the final arbiter. Does a result agree with experiment? If it doesn’t agree with experiment, something is wrong.

The frequency domain of mind (a mind, it must be stressed, is an unextended, massless, immaterial singularity) can produce an extended, spacetime domain of matter via ontological Fourier mathematics, and the two domains interact via inverse and forward Fourier transforms. An inverse Fourier transform converts a frequency (mind) function into a spacetime (material) function, and a forward Fourier transform does the opposite. So, mind can causally affect the material world, and matter can inform mind about its condition, its state. This is thus the long-sought answer to the world-historic problem of Cartesian substance dualism.

Beyond craftsmanship lies invention, and it is here that lean, spare, fast programs are born. Almost always these are the result of strategic breakthrough rather than tactical cleverness. Sometimes the strategic breakthrough will be a new algorithm, such as the Cooley-Tukey Fast Fourier Transform or the substitution of an n log n sort for an n2 set of comparisons. Much more often, strategic breakthrough will come from redoing the representation of the data or tables. This is where the heart of a program lies.

Then there is before us the matter of not the required two but nine separate application essays, some of which of nearly monograph-length, each without exception being—’ different sheet—‘the adjective various evaluators used was quote “stellar”—’ Dir. of Comp.: ‘I made in my assessment deliberate use of lapidary and effete.’ ‘—but in areas and with titles, I’m sure you recall quite well, Hal: “Neoclassical Assumptions in Contemporary Prescriptive Grammar,” “The Implications of Post-Fourier Transformations for a Holographically Mimetic Cinema,” “The Emergence of Heroic Stasis in Broadcast Entertainment”—’ ‘ “Montague Grammar and the Semantics of Physical Modality”?’ ‘ “A Man Who Began to Suspect He Was Made of Glass”?’ ‘ “Tertiary Symbolism in Justinian Erotica”?’ Now showing broad expanses of recessed gum. ‘Suffice to say that there’s some frank and candid concern about the recipient of these unfortunate test scores, though perhaps explainable test scores, being these essays’ sole individual author.

As a result I early asked the question, "Why should I do all the analysis in terms of Fourier integrals? Why are they the natural tools for the problem?" I soon found out, as many of you already know, that the eigenfunctions of translation are the complex exponentials. If you want time invariance, and certainly physicists and engineers do (so that an experiment done today or tomorrow will give the same results), then you are led to these functions. Similarly, if you believe in linearity then they are again the eigenfunctions. In quantum mechanics the quantum states are absolutely additive; they are not just a convenient linear approximation. Thus the trigonometric functions are the eigenfunctions one needs in both digital filter theory and quantum mechanics, to name but two places. Now when you use these eigenfunctions you are naturally led to representing various functions, first as a countable number and then as a non-countable number of them-namely, the Fourier series and the Fourier integral. Well, it is a theorem in the theory of Fourier integrals that the variability of the function multiplied by the variability of its transform exceeds a fixed constant, in one notation l/2pi. This says to me that in any linear, time invariant system you must find an uncertainty principle.

The Fourier transform is one of the most used tools in all of pbysics, engineering, and computer science. It can be found in electronic circuits and lies at the foundation for sending and receiving signals to satellites and back to Earth through electromagnetic waves. And the Fourier transform is essential to understanding how the structure of the universe arises.

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.