The next Fourier transform: rewiring, representation, and artificial mathematical creativity
Abstract
What distinguishes the solution of a difficult mathematical problem from a conceptual revolution? Some mathematical innovations do more than establish new results: they reorganize the relations among problems. We propose to describe such events through changes in the transfer structure of mathematics, namely the pattern determining which problems can naturally inform one another, which methods can move between them, and which new questions become accessible. The eighteenth-century controversy over the vibrating string provides our central historical case. D'Alembert found a famous model for wave propagation, Bernoulli proposed a modal interpretation, Euler broadened the class of admissible profiles, and Lagrange constructed a finite-dimensional strategy based on modal decomposition. Later, Fourier analysis gave the trigonometric language in which all the previous descriptions can interact. This motivates our Rewiring Thesis: a conceptual revolution is a persistent, large-scale reorganization of transfer relations among mathematical problems. The productive representation is one mechanism capable of producing such rewiring, but some stress tests (involving Galois, Riemann, Lebesgue-Schwartz and Wiles) show that it is not the only one. The musical notation provides an independent case in which the representation becomes operationally generative without being informationally complete. The resulting framework suggests a stronger criterion for artificial mathematical creativity than problem solving alone. The relevant question is not merely "whether AI can solve a famous conjecture", but "whether it can recognize that an inherited conceptual organization is itself an obstruction and produce a new one that reorganizes several problems at once". In this sense, the decisive achievement may be the invention of "the next Fourier transform".
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BibTeX
@misc{fanelli2026next,
title = {{The next Fourier transform: rewiring, representation, and artificial mathematical creativity}},
author = {Luca Fanelli},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26974},
eprint = {2609.26974},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26974},
}
RIS
TY - UNPB
TI - The next Fourier transform: rewiring, representation, and artificial mathematical creativity
AU - Luca Fanelli
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26974
UR - https://arxiv.org/abs/2609.26974
AB - What distinguishes the solution of a difficult mathematical problem from a conceptual revolution? Some mathematical innovations do more than establish new results: they reorganize the relations among problems. We propose to describe such events through changes in the transfer structure of mathematics, namely the pattern determining which problems can naturally inform one another, which methods can move between them, and which new questions become accessible. The eighteenth-century controversy over the vibrating string provides our central historical case. D'Alembert found a famous model for wave propagation, Bernoulli proposed a modal interpretation, Euler broadened the class of admissible profiles, and Lagrange constructed a finite-dimensional strategy based on modal decomposition. Later, Fourier analysis gave the trigonometric language in which all the previous descriptions can interact. This motivates our Rewiring Thesis: a conceptual revolution is a persistent, large-scale reorganization of transfer relations among mathematical problems. The productive representation is one mechanism capable of producing such rewiring, but some stress tests (involving Galois, Riemann, Lebesgue-Schwartz and Wiles) show that it is not the only one. The musical notation provides an independent case in which the representation becomes operationally generative without being informationally complete. The resulting framework suggests a stronger criterion for artificial mathematical creativity than problem solving alone. The relevant question is not merely "whether AI can solve a famous conjecture", but "whether it can recognize that an inherited conceptual organization is itself an obstruction and produce a new one that reorganizes several problems at once". In this sense, the decisive achievement may be the invention of "the next Fourier transform".
ER -