The unreasonable effectiveness of the Fourier transform

Debate over the “unreasonable effectiveness” framing

• Many comments push back on the title pattern (“The unreasonable effectiveness of X”) as overused, silly, or rhetorically manipulative because it hides a claim (“X is unreasonably effective”) inside a noun phrase.
• Others defend it as a well-understood allusion to Wigner and as meaning “surprisingly useful in ways we might not have anticipated,” not literally “unreasonable.”
• There is disagreement over Wigner’s original essay: some see it as profound and historically grounded; others think, in hindsight, math’s effectiveness is obvious and thus not “unreasonable.”
• Related discussion touches on math as “language of science,” its uneven effectiveness across fields (e.g., physics vs. psychology), and the invented-vs-discovered debate.
• Several commenters argue for humility toward past work: what looks “silly” from today may have been genuinely surprising then.

Perceptions of the talk and OFDM angle

• Some dismiss the slides as “FT 101,” similar to a basic signals course.
• The presenter clarifies that the early part is introductory, but the OFDM application at the end is where the “unreasonable effectiveness” feeling comes from.
• There’s interest in practical follow-on, including mention of contributing to open-source LTE modem projects like OpenLTE.

History and culture around Fourier/FFT

• A popular side thread recounts Gauss independently discovering an FFT-like algorithm long before Cooley–Tukey, but never publishing it.
• Discussion expands to Gauss’s habit of keeping results in his desk, his attitude toward his children entering math, and a contrast with more collaborative modern mathematicians.
• This is used to highlight how mathematical culture shifted from solitary “hoarding” to open, student-driven collaboration.

Real-world transforms and applications

• Practitioners note that real systems almost never use the ideal infinite Fourier transform; they use FFTs on windowed, discrete-time data (DTFT), often alongside wavelets and DCT.
• Fourier-like transforms underpin many modern codecs (JPEG, H.264, MP3), though motion prediction is also critical.
• Several examples illustrate “frequency-domain thinking”: extracting heart rate from webcam video, remote PPG, motion magnification, and even reconstructing speech from filmed vibrations.

Fourier, uncertainty, and coordinate systems

• Commenters emphasize the theorem that a signal cannot be both time- and band-limited, tying it directly to the Heisenberg uncertainty principle as a purely mathematical consequence of the Fourier relationship.
• Gaussians are highlighted as optimally trading off time and frequency localization.
• A broader conceptual theme is that transforms like Fourier, Laplace, and Walsh–Hadamard are powerful because they choose a problem-adapted basis; “frequency space” is just one important example of “the right coordinates.”
• Some speculate that understanding models like GPT may similarly require discovering a better internal coordinate system.

Teaching and intuition

• Multiple commenters recount Fourier transforms as a mind-opening moment, especially realizing arbitrary signals can be decomposed into sinusoids.
• Others criticize how DSP is often taught—lots of formulas, little emphasis on the deeper viewpoint of “change of basis” and coordinate choice that makes problems simple.