I am interested in reading about possible meaning and implications of the time-frequency duality in Fourier transforms and the uncertainty principle (not necessarily the usually discussed Heisenberg Uncertainty in the philosophy of physics although that might also be included, but especially the time-frequency uncertainty as found in sampling theory, or Gabor Uncertainty - they seem to be all mathematically related, but I am looking for possible theories that discuss about what they really mean or if they have any potential as philosophical foundations) and the possibility of this duality to be anything deeper than just a mathematical abstraction.

So just like Badiou used set theory as a foundation of his philosophy, or Deleuze had the tension between difference and repetition (maybe this comes closer as the repetition part resembles the frequency domain?), I am looking for philosophers who looked into the mathematics of the time-frequency dualism for founding their ideas... Maybe I am being too general, but I could not find anything so far...

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    I don't think you're going to have any luck finding a philosopher that bases their philosophical ideas off of Fourier transforms, continental or otherwise. It seems very specific but also a little bit random. Maybe you can find philosophical writings about Fourier transforms but I doubt you'll find someone who developed their ideas around that tool. Set theory was developed to be a foundational paradigm for mathematics so there's a good reason why Badiou used it for his purposes. – Not_Here May 2 '17 at 20:18
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    Something that could be viewed as philosophical and has to do specifically with quantum Fourier transforms: academia.edu/2552667/… – Not_Here May 2 '17 at 20:27
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    One thing I find fascinating and philosophical is that there are no true band-limited entities in the real world (scientific real world) because every signal has a finite duration which obliges an infinite frequency response. There are plenty of signals where we feel comfortable ignoring the higher harmonics, but they always exist. A thing which is band limited cannot be time limited, and vice versa. – Cort Ammon May 2 '17 at 20:48
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    @CortAmmon One thing I find fascinating is that there are no true time-limited entities in the real world... – Dave May 2 '17 at 20:50
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    Here's my favorite factoid FWIW. Cantor was studying the zeros of trigonometric functions. They arranged themselves into infinite hierarchies of infinite hierarchies that Cantor investigated and clarified as the transfinite ordinal numbers. Thus transfinite set theory is not some magic thing that Cantor pulled out of his religious imagination; but rather a natural phenomenon that arose from physical considerations, namely the transfer of heat as studied by Fourier. – user4894 May 2 '17 at 21:29

A prolific modern Leibnizian, Mike Hockney, might be somewhat like what you are looking for, see e.g. his Mathematical Universe and Smith's review of it, but do not expect the caliber of Badiou or Deleuze. Hockney, following Leibniz, interprets the phenomenal world as derivative from the primary ontology of dimensionless minds, monads (which Leibniz tells are "not parts but foundations of phenomena"). The Fourier transform, or rather its inverse, plays the key role of converting the timeless monadology ("frequency domain") into extensional spacetime. Here is a taste:

"The whole mystery of existence is contained within Fourier mathematics because it's none other than the means by which unextended Cartesian minds (frequency domains) communicate with extended Cartesian bodies (spacetime entities)… Monads are composed of nothing but eternal, immutable sine and cosine waves of every conceivable type (via the generalized Euler Formula)…

Via inverse Fourier transforms, this monadic frequency information can be combined to create any spacetime representation. In other words, mental “ideas” can be converted into physical “bodies” via Fourier mathematics. Mind is the basis of matter, not the other way around (as scientific materialists have always claimed). The phenomenal world is simply a mathematical way of presenting noumenal frequency (mental) data. Fourier mathematics... turned the Cartesian unextended, thinking domain into an eternal monadic frequency domain (the Soul Domain), and the Cartesian extended, material domain into a spacetime domain (the World)…".

Hockney also gives an atemporal interpretation of the Big Bang based on his metaphysics, according to which the Big Bang singularity is not an event in time but rather a metaphysical posit somewhat reminiscent of the "initial state" of Plotinian One or Hegel's Geist. I suppose there is also some affinity to the no-boundary cosmological proposals, a la Hartle–Hawking, where the temporal world is emergent on a timeless substrate in a manner that can be remotely analogized to performing a Fourier transform, see On the Emergence of Time in Quantum Gravity by Isham and Butterfield (especially pp.52-62). Here is more Hockney:

"If there are infinite monads all occupying the same Singularity, they can all be regarded as mental bosons. However, the application of a simple antisymmetry operation converts them into mental fermions. This has the most astounding consequence: it confers unique coordinates on each of the monads and instantly creates an extended Cartesian coordinate grid.

The monads haven't actually moved anywhere – they are still inside the Singularity – but they now have unique identifiers (coordinates) and this produces the effect (illusion) of all monads now being separated from each other. To put it another way, they now have extended (fermionic) relations with each other. A Cartesian extended world has come into being... The infinite Cartesian grid is still wholly contained within the Singularity and is thus a mathematical illusion. Existence can only ever take place within a single point (everything that exists is contained within the Singularity: we live inside the Big Bang Singularity, and the Big Bang itself took place within that Singularity, and remains contained within it). The Big Bang was nothing but an internal mathematical restructuring of the Singularity via an antisymmetry operation and Fourier mathematics."


I am also fascinated by the philosophical aspects of the Fourier transformations. Neuroscience way of modelling how the brain creates consciousness applies Fourier transformations. I published my findings on this topic in The mirror Brain-Mind. You can read it on https://www.scribd.com/document/503405944/The-Mirror-Brain-Mind or the Esoteric story The vibes of the spirits. https://www.scribd.com/document/503429914/The-Vibes-of-the-Spirits-2021

alternatively, try https://archive.org/details/brain-mind-scribd

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