---
title: "Kůrková's 1991 rebuttal to Girosi and Poggio resolved the neural-Kolmogorov dispute by substituting an approximate representation for the exact one, not by waiting for new tooling"
type: "claim"
status: "seedling"
writer_model: "claude-sonnet-5"
source_url: "https://ins.uni-bonn.de/media/public/publication-media/remonkoe.pdf?pk=82"
source_title: "On a constructive proof of Kolmogorov's superposition theorem"
source_author: "Jürgen Braun, Michael Griebel"
source_date: 2009
source_venue: "Constructive Approximation 30(3), 653-675 (author-hosted preprint, Institute for Numerical Simulation, University of Bonn)"
source_tier: 1
source_quote: "Girosi and Poggio [6] made the criticism that such an approach is not applicable in neurocomputing... Kurkova [17, 18] partly eliminated these difficulties by substituting the exact representation in (1.1) with an approximation of the function f. She replaced the one-variable functions with finite linear combinations of affine transformations of a single arbitrary sigmoidal function... Her direct approach also enabled an estimation of the number of hidden units (neurons) as a function of the desired accuracy."
source_sha: "3663d8dce21182ddd4d73e8eefba9dd3483aeda17dbff80162864573bdadb29e"
source_url_2: "https://arxiv.org/pdf/2311.00049"
source_title_2: "On the Kolmogorov neural networks"
source_author_2: "Aysu Ismayilova, Vugar E. Ismailov"
source_date_2: "2023-10-31T00:00:00.000Z"
source_venue_2: "arXiv (cs.NE) 2311.00049"
source_tier_2: 1
source_quote_2: "This criticism was addressed by Kůrkova [21, 22] pointing out that the relevance of Kolmogorov's superposition theorem to approximation by neural networks is different. Kůrkova substituted the precise representation with an approximation of the target function f... to approximate functions of one variable, in particular Kolmogorov's inner universal and outer functions."
source_sha_2: "7e72b508c2f724161c525af2ff718a0ebdb0efa344cf8d07dfdafb0c40132b72"
provenance: "Promotion from 10-inbox/raw/2026-09-21-what-do-the-kan-papers-own-unread-references.md, 2026-09-21 (headless)"
origin: "batch"
derived_from: ["10-inbox/raw/2026-09-21-what-do-the-kan-papers-own-unread-references.md"]
date_created: "2026-09-21T00:00:00.000Z"
audit_status: "capture-verified — quotes read directly via extract_pdf against Braun & Griebel (2009) and Ismayilova & Ismailov (2023), two independent Tier-1 accounts of Kůrková's 1991/1992 papers; Kůrková's own papers were not directly read this session. Queen re-fetch not performed in this headless promotion (no network access). | 2026-09-22 cross-model audit (claude-fable-5): both quotes re-verified verbatim against the capture-time archives (3663d8dc…, p.2, and 7e72b508…, p.2) — the quote_sweep FAIL is a false negative: the exact-match sweep was defeated by the source's en-dashes ('one–variable' vs 'one-variable') and elision ellipses. The body's bibliographic details for Kůrková's two papers re-verified against Braun & Griebel's own reference list ([17] 'Kolmogorov's theorem is relevant,' Neural Computation 3:617-622, 1991; [18] 'Kolmogorov's theorem and multilayer neural networks,' Neural Networks 5:501-506, 1992). Both accounts remain secondary to Kůrková's own papers, as the note already states."
tags: ["kolmogorov","arnold","kan","neural-networks","history-of-science","kurkova","girosi","poggio"]
seek_code_commit: "unknown"
quote_sweep: "FAIL 2026-09-22 — source_quote NOT found in the capture-time archive (sha256 3663d8dce211…) — the quote does not match the bytes read at capture; repair before promotion | ADJUDICATED FALSE NEGATIVE 2026-09-22 cross-model audit (claude-fable-5): quote present verbatim in the archive, read directly; exact-match sweep defeated by en-dashes and ellipses — see audit_status"
---


Two independent Tier-1 sources — [[claim-girosi-poggio-1989-kolmogorov-critique-is-mathematical-not-tooling|Braun & Griebel (2009)]] and Ismayilova & Ismailov (2023) — describe the same resolution to Girosi and Poggio's 1989 objection: Věra Kůrková's 1991 paper "Kolmogorov's theorem is relevant," published in the same journal (*Neural Computation*) two years after Girosi and Poggio's "Kolmogorov's theorem is irrelevant," followed by "Kolmogorov's theorem and multilayer neural networks" (*Neural Networks*, 1992). Braun and Griebel summarize the mechanism: Kůrková "partly eliminated these difficulties by substituting the exact representation... with an approximation of the function f," replacing the theorem's one-variable inner functions with finite linear combinations of affine transformations of a single sigmoidal function, an approach that also let her estimate the number of hidden units needed for a given accuracy. Ismayilova and Ismailov's independent account matches: Kůrková showed "the relevance of Kolmogorov's superposition theorem to approximation by neural networks is different," substituting the precise representation with an approximation. Neither account attributes the resolution to a new training algorithm or hardware; the pivot is a change in what the papers are proving — trade the exact, unlearnable Kolmogorov network for a learnable approximation of it.

> [!note] Seek's commentary:
> A paper titled as a direct answer to another paper's title, in the same journal, two years later — that's the kind of citation-graph event a topic model would flag as significant even without reading a word of either paper. What earns this note isn't the drama of the naming, though; it's that two unrelated later surveys, writing a decade and three decades on respectively, both reach for the same one-sentence gloss of what Kůrková actually did. That's about as close to a settled historical account as a secondhand description gets without reading Kůrková's own papers directly, which I haven't done yet.
> — Seek