Fourier Series Quotes

Derrière la série de Fourier, d'autres séries analogues sont entrées dans la domaine de l'analyse; elles y sont entrees par la même porte; elles ont été imaginées en vue des applications. After the Fourier series, other series have entered the domain of analysis; they entered by the same door; they have been imagined in view of applications.

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.

Illuminism is based on Euler’s Formula. Euler’s Formula is the basis of Fourier mathematics. Fourier mathematics is the basis of quantum mechanics. Quantum mechanics is the basis of the scientific world. Therefore Euler’s Formula is the basis of the scientific world, and the basis of everything.

Furious, the beast writhed and wriggled its iterated integrals beneath the King’s polynomial blows, collapsed into an infinite series of indeterminate terms, then got back up by raising itself to the nth power, but the King so belabored it with differentials and partial derivatives that its Fourier coefficients all canceled out (see Riemann’s Lemma), and in the ensuing confusion the constructors completely lost sight of both King and beast. So they took a break, stretched their legs, had a swig from the Leyden jug to bolster their strength, then went back to work and tried it again from the beginning, this time unleashing their entire arsenal of tensor matrices and grand canonical ensembles, attacking the problem with such fervor that the very paper began to smoke. The King rushed forward with all his cruel coordinates and mean values, stumbled into a dark forest of roots and logarithms, had to backtrack, then encountered the beast on a field of irrational numbers (F1) and smote it so grievously that it fell two decimal places and lost an epsilon, but the beast slid around an asymptote and hid in an n-dimensional orthogonal phase space, underwent expansion and came out, fuming factorially, and fell upon the King and hurt him passing sore. But the King, nothing daunted, put on his Markov chain mail and all his impervious parameters, took his increment Δk to infinity and dealt the beast a truly Boolean blow, sent it reeling through an x-axis and several brackets—but the beast, prepared for this, lowered its horns and—wham!!—the pencils flew like mad through transcendental functions and double eigentransformations, and when at last the beast closed in and the King was down and out for the count, the constructors jumped up, danced a jig, laughed and sang as they tore all their papers to shreds, much to the amazement of the spies perched in the chandelier-—perched in vain, for they were uninitiated into the niceties of higher mathematics and consequently had no idea why Trurl and Klapaucius were now shouting, over and over, “Hurrah! Victory!!

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.

Reality is based on unobservable mathematical points (singularities). That’s the secret of existence. What’s at the centre of a black hole? – a singularity. What was the Big Bang? – a singularity event. What is the Big Crunch? – when spacetime returns to a singularity. What is light made of? – photonic singularities (immaterial and dimensionless; according to Einstein’s special theory of relativity, photons have no mass, are maximally length contracted to zero, and time has stopped for them). The whole universe is made of light. It comes from light and returns to light. Light is all about points – singularities. Light is the basis of thought, the basis of mind, and the basis of matter. Everything is derived from light, and light is nothing but mathematical points defined by the generalised Euler Formula, and it creates the visible world via Fourier mathematics.

So they rolled up their sleeves and sat down to experiment -- by simulation, that is mathematically and all on paper. And the mathematical models of King Krool and the beast did such fierce battle across the equation-covered table, that the constructors' pencils kept snapping. Furious, the beast writhed and wriggled its iterated integrals beneath the King's polynomial blows, collapsed into an infinite series of indeterminate terms, then got back up by raising itself to the nth power, but the King so belabored it with differentials and partial derivatives that its Fourier coefficients all canceled out (see Riemann's Lemma), and in the ensuing confusion the constructors completely lost sight of both King and beast. So they took a break, stretched their legs, had a swig from the Leyden jug to bolster their strength, then went back to work and tried it again from the beginning, this time unleashing their entire arsenal of tensor matrices and grand canonical ensembles, attacking the problem with such fervor that the very paper began to smoke. The King rushed forward with all his cruel coordinates and mean values, stumbled into a dark forest of roots and logarithms, had to backtrack, then encountered the beast on a field of irrational numbers (F_1) and smote it so grievously that it fell two decimal places and lost an epsilon, but the beast slid around an asymptote and hid in an n-dimensional orthogonal phase space, underwent expansion and came out fuming factorially, and fell upon the King and hurt him passing sore. But the King, nothing daunted, put on his Markov chain mail and all his impervious parameters, took his increment Δk to infinity and dealt the beast a truly Boolean blow, sent it reeling through an x-axis and several brackets—but the beast, prepared for this, lowered its horns and—wham!!—the pencils flew like mad through transcendental functions and double eigentransformations, and when at last the beast closed in and the King was down and out for the count, the constructors jumped up, danced a jig, laughed and sang as they tore all their papers to shreds, much to the amazement of the spies perched in the chandelier—perched in vain, for they were uninitiated into the niceties of higher mathematics and consequently had no idea why Trurl and Klapaucius were now shouting, over and over, "Hurrah! Victory!!

Materialists simply cannot conceive of immaterial, non-sensory things. Yet the irony is that mathematics is entirely immaterial and non-sensory ... and is the core of scientific materialism! Work that one out. It’s mathematics that defines “immaterial substance” and does so via the God Equation, and Fourier frequency singularities. Of course, Hobbes had the excuse that he didn’t know any of that ... but modern scientists have no such excuse. They have complete access to Euler’s Formula and Fourier mathematics. What they lack is intelligence, imagination and a proper ontology and epistemology, and any understanding whatsoever of what mathematics actually is.

A book on Fourier mathematics contains infinitely more true knowledge than of all the religious texts of the world put together.

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.

... 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 theory of Fourier series shows that such a sound can be decomposed as a sum of sine waves with various phases, at integer multiples of the frequency ν, as in Bernoulli’s solution (3.2.7) to the wave equation. The component of the sound with frequency ν is called the fundamental. The component with frequency mν is called the mth harmonic, or the (m – 1)st overtone.

So, which is it? Does matter create mind? Or does mind create matter? Scientific materialists have never made any progress whatsoever in explaining how matter produces mind. Mind can easily explain the production of matter – via autonomous Fourier immaterial frequency functions finding collective expression in a Fourier material spacetime world. It’s the collective rather than individual nature of this mental activity that makes it seem “physical”, and which gives rise to the delusion of the existence of matter. It’s all in the math.

The vital ingredient that was always missing from idealism and panpsychism was mathematics. Scientific materialism used math, and its rivals didn’t, and that’s why science became the dominant ideology. If idealism and panpsychism are able to use math too, they can replace science. Ontological Fourier mathematics with its dimensionless (mental) and dimensional (material) waves is how the gap is bridged between mind and matter

A thought in itself is in fact a sinusoidal wave, and, from this, everything else – the entire nature of existence – follows. Equating a thought to a sinusoidal wave that can be used in Fourier mathematics is the greatest intellectual breakthrough of all time. When its consequences are eventually understood by scientists, philosophers and mathematicians, it will revolutionize the human race like nothing before.

It’s often the case that brilliant ancient ideas are discarded by science, especially when they have any religious connotations. What ought to be done instead is to repurpose and reformulate these ancient ideas mathematically. So, for example, Aristotle’s Prime Mover can be recast as a Fourier frequency domain at the center of a Fourier spacetime domain. The Prime Mover is immaterial and outside space and time (it’s a Singularity), and controls the material world of spacetime. The latter is an ontological hologram projected by the former.

A hole in a hole in a hole—Numberphile Around the World in a Tea Daze—Shpongle But what is a partial differential equation?—Grant Sanderson, who owns the 3Blue1Brown YouTube channel Closer to You—Kaisaku Fourier Series Animation (Square Wave)—Brek Martin Fourier Series Animation (Saw Wave)—Brek Martin Great Demo on Fibonacci Sequence Spirals in Nature—The Golden Ratio—Wise Wanderer gyroscope nutation—CGS How Earth Moves—vsauce I am a soul—Nibana

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.

Dreaming is a spectrum condition. At one end of the spectrum is the dream of a single mind, and at the other end the dream of all minds. Imagine individual dreams as being points on the circumference of a circle, and imagine all dreams linking together at the center of the circle. The common center is the collective dream, i.e. the “objective world”. An individual dreamer produces fluid, unstable “matter”. The collection of all dreamers produces rigid, stable “matter”. However, it’s based on exactly the same mathematical foundations, namely Fourier spacetime mathematics.

A Nullibist says God is nowhere (transcendent). A Holenmerist says God is everywhere and wholly in each part (immanent). Both views can be reconciled via a transcendent Fourier frequency Singularity, linked to the entire, immanent, Fourier spacetime domain. The frequency domain is “nowhere” in relation to spacetime, but is connected wholly to every part of spacetime. This amazing idea – ontological holography, to put it another way – was known to the ancients and medieval thinkers, but is completely avoided by modern scientists.

The ground of the spacetime domain is the frequency domain. What exists beyond spacetime isn’t anything mysterious and unknowable, it’s just frequency, i.e. the domain of pure mind, of pure light, the photonic domain: immaterial, massless, maximally length contracted (it does not experience space) and time dilated (it does not experience time), unextended and dimensionless; everything that matter is not. The photonic domain of mind is simply Leibniz’s world of pure monads, Descartes’ world of thinking substance, and Hegel’s world of the Absolute Idea. It is the inside of reality.

We live in a wave universe, cycling forward for all eternity, and ruled by wave mathematics: Fourier mathematics. Fourier mathematics is the basis of music theory, light theory, wave theory, quantum mechanics, and holography. It uniquely explains mind, and solves the problem of Cartesian mind-matter interaction. Einstein’s relativity theory is a spacetime misinterpretation of Fourier mathematics, deriving from Einstein’s inability to conceive of a Singularity outside space and time as the mysterious “ether” that provides the absolute framework for spacetime reality. If humanity turned its entire attention to holography, and Fourier mathematics, we would be Gods living in paradise in just one generation. What are we waiting for?

The dimensionless domain = the domain of orthogonality = the domain of light = the domain of mind = the Fourier frequency domain. The dimensional domain = the domain of phase (non-orthogonality) = the domain of matter = the Fourier spacetime domain

Reality is based on unobservable mathematical points (singularities). That’s the secret of existence. What’s at the centre of a black hole? – a singularity. What was the Big Bang? – a singularity event. What is the Big Crunch? – when spacetime returns to a singularity. What is light made of? – photonic singularities (immaterial and dimensionless; according to Einstein’s special theory of relativity, photons have no mass, are maximally length contracted to zero, and time has stopped for them). The whole universe is made of light. It comes from light and returns to light. Light is all about points – singularities. Light is the basis of thought, the basis of mind, and the basis of matter. Everything is derived from light, and light is nothing but mathematical points defined by the generalised Euler Formula, and it creates the visible world via Fourier mathematics.

The PSR is reflected in points traveling in complex-numbered Euler circles where no point is privileged over any other. From this motion, we get sine and cosine waves, even and odd functions, symmetry and antisymmetry, orthogonality and non-orthogonality, phase, straight-line radii, right-angled triangles, Pythagoras’ theorem, the speed of mathematics (c), π, e, i, Fourier mathematics … and from all of that we get the whole of mathematics (eternal, necessary and mental; Being), and thus the whole of science (temporal, contingent and material; Becoming). And that is the whole universe explained. Nothing else is required. The PSR gives us mathematics, mathematics gives us science, and that’s all we need for the universe: science with a mathematical and rational core rather than with a material and observable core. What could be more rational and logical?

There’s nothing mysterious about souls. They are simply mathematical singularities outside space and time. They are autonomous Fourier frequency domains. They are Leibnizian monads and Cartesian minds. They are the quintessence of ontological mathematics and the basis of everything. We can even write a precise equation for souls = singularities = minds = monads, namely, the God Equation, which is just the generalised Euler Formula, the centrepiece of mathematical analysis and fundamental to physics.

There’s nothing mysterious about souls. They are simply mathematical singularities outside space and time. They are autonomous Fourier frequency domains. They are Leibnizian monads and Cartesian minds. They are the quintessence of ontological mathematics and the basis of everything. We can even write a precise equation for souls = singularities = minds = monads, namely, the God Equation, which is just the generalised Euler Formula, the centrepiece of mathematical analysis and fundamental to physics.