arxiv: 2604.24956 · v1 · submitted 2026-04-27 · 🧮 math.HO

Recognition: unknown

The history of three wrong definitions

Harold P. Boas

Pith reviewed 2026-05-07 16:56 UTC · model grok-4.3

classification 🧮 math.HO

keywords equivalence relationCauchy sequencemetric spacehistory of mathematicsmathematical definitionsconceptual evolutionrevival of concepts

0 comments

The pith

Disused definitions of equivalence relation, Cauchy sequence, and metric space could profitably be revived.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reviews the historical development of three fundamental mathematical concepts: equivalence relations, Cauchy sequences, and metric spaces. It shows that each had earlier definitions that were eventually replaced by the ones used today. The central argument is that these earlier definitions have merits that could make them valuable to revive in modern contexts. A reader would care because these ideas form the basis for much of analysis and algebra, and different definitions might lead to clearer teaching or simpler proofs. The author uses historical evidence to support the potential benefits of bringing back the older formulations.

Core claim

The author establishes that the concepts of equivalence relation, Cauchy sequence, and metric space each underwent a change in definition, with the original versions being set aside for reasons that may no longer hold or that are outweighed by their advantages. By examining the history, the paper demonstrates specific ways in which the disused definitions could enhance current mathematical practice if revived.

What carries the argument

Comparative historical analysis of successive definitions for equivalence relations, Cauchy sequences, and metric spaces, showing advantages in the earlier versions that were abandoned.

If this is right

Modern textbooks could incorporate the older definitions to improve student intuition for these concepts.
Research in analysis might benefit from the perspective offered by the disused definition of Cauchy sequences.
The history suggests that changes in mathematical definitions are not always irreversible improvements.
Equivalence relations could be introduced with an emphasis on properties that align better with certain set-theoretic contexts.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar historical reviews could be conducted for other core concepts like continuity or limits to check for overlooked advantages.
Reviving old definitions might resolve some pedagogical challenges in undergraduate mathematics courses.
This connects to broader questions about whether mathematical definitions are chosen for convenience or reflect deeper structures.

Load-bearing premise

That the advantages of reviving the older definitions would outweigh any practical or pedagogical reasons for their original supersession.

What would settle it

If attempts to use the older definitions in contemporary proofs or teaching result in increased complexity or errors without corresponding benefits, the proposal for revival would be undermined.

read the original abstract

The topic is the history of the concepts of equivalence relation, Cauchy sequence, and metric space. The thesis is that disused definitions of these notions could profitably be revived.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Boas traces the history of three definitions and suggests reviving the older versions, but offers no concrete modern example where they outperform the current ones.

read the letter

Boas looks at how the definitions of equivalence relation, Cauchy sequence, and metric space changed over time and argues that the versions that were dropped could still be worth using. The paper gathers the original statements, the contexts in which they appeared, and the reasons later writers moved to the more abstract forms we teach now. It keeps the account short and readable, with clear quotes and timelines for each of the three cases. That part of the work is straightforward and useful for anyone who wants to see how these ideas were first pinned down before they became standardized. The historical assembly is the main contribution here, and it is done competently without overclaiming what the sources show. The soft spot is the revival thesis itself. The paper states that the older definitions had advantages that were lost, but it does not walk through any current proof or construction in analysis or topology where switching back would shorten the argument, avoid a pathology, or enable something the modern version blocks. Without that kind of demonstration, the claim that revival would be profitable stays general and hard to evaluate against the consistency gains that came with the changes. This is a piece for readers who follow the history of analysis or who teach foundations and like to discuss why we define things the way we do. It does not contain new theorems or technical results, so it will not affect research practice directly. I would send it for peer review in a history-of-mathematics journal, where the narrative can be checked and the normative suggestion can be debated on its own terms.

Referee Report

3 major / 2 minor

Summary. The paper surveys the historical development of three concepts—equivalence relations, Cauchy sequences, and metric spaces—identifying earlier definitions that were later superseded and arguing that these disused formulations could profitably be revived for conceptual or technical advantages in contemporary mathematics.

Significance. If the historical interpretations are accurate and the claimed advantages can be demonstrated in modern settings, the work would contribute to the history of mathematics by documenting alternative conceptual paths and could stimulate discussion on whether standardization has foreclosed useful perspectives in analysis and topology.

major comments (3)

[Abstract and concluding discussion] The central revival thesis (stated in the abstract and developed across the three case studies) requires evidence that the older definitions yield net gains in present-day mathematics, yet no concrete contemporary application—such as a shortened proof, removal of a pathology, or new construction in real analysis or metric geometry—is exhibited to support the value judgment.
[Section on Cauchy sequences] For the Cauchy sequence case, the paper contrasts the historical definition with the modern one but does not address whether reviving the older version would preserve key theorems (e.g., completeness of the reals) or introduce inconsistencies with the standard construction of the reals via Dedekind cuts or equivalence classes of Cauchy sequences.
[Section on metric spaces] The metric space discussion claims conceptual superiority for the disused definition without checking compatibility with standard results such as the Baire category theorem or uniform continuity characterizations that rely on the modern open-ball formulation.

minor comments (2)

Notation for the older definitions should be introduced with explicit comparison tables to the modern versions for clarity.
A few historical citations appear to rely on secondary sources; primary references for the original definitions would strengthen the narrative.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thoughtful and detailed report. The comments highlight important considerations regarding the scope and implications of our historical analysis. We address each major comment below and indicate where revisions will be incorporated.

read point-by-point responses

Referee: [Abstract and concluding discussion] The central revival thesis (stated in the abstract and developed across the three case studies) requires evidence that the older definitions yield net gains in present-day mathematics, yet no concrete contemporary application—such as a shortened proof, removal of a pathology, or new construction in real analysis or metric geometry—is exhibited to support the value judgment.

Authors: The manuscript is a historical study whose primary aim is to document superseded definitions and the reasons for their replacement. The suggestion that revival could be profitable is framed as an invitation based on observed conceptual features in the historical record, rather than a claim backed by new technical results. We agree that the absence of a specific modern application leaves the value judgment somewhat open-ended. We will revise the concluding discussion to include a short paragraph outlining potential avenues (e.g., alternative pedagogical presentations or axiomatic explorations) while explicitly noting that full demonstrations of technical advantage lie beyond the paper's historical scope. This constitutes a partial revision. revision: partial
Referee: [Section on Cauchy sequences] For the Cauchy sequence case, the paper contrasts the historical definition with the modern one but does not address whether reviving the older version would preserve key theorems (e.g., completeness of the reals) or introduce inconsistencies with the standard construction of the reals via Dedekind cuts or equivalence classes of Cauchy sequences.

Authors: We accept that the section would benefit from an explicit statement on compatibility. The older definition of Cauchy sequences is historically distinct yet leads to equivalent notions of completeness when applied to the reals. We will add a concise paragraph clarifying that the historical formulation preserves the completeness property and is consistent with both Dedekind cuts and the modern Cauchy-sequence construction of the reals, as the underlying limit concept remains unchanged. This addition will be made without expanding the paper into a technical comparison. revision: yes
Referee: [Section on metric spaces] The metric space discussion claims conceptual superiority for the disused definition without checking compatibility with standard results such as the Baire category theorem or uniform continuity characterizations that rely on the modern open-ball formulation.

Authors: The claim of conceptual superiority is presented as a historical and intuitive observation rather than a proposal to supplant the modern definition. Nevertheless, we agree that compatibility should be addressed briefly. The disused formulation generates the same topology, so standard results such as the Baire category theorem and characterizations of uniform continuity carry over directly. We will insert a short remark in the metric-spaces section noting this equivalence of the induced topologies. The revision will be limited to a clarifying sentence. revision: yes

Circularity Check

0 steps flagged

No circularity: purely historical narrative without derivations or self-referential reductions

full rationale

The paper is a historical survey of superseded definitions for equivalence relations, Cauchy sequences, and metric spaces, with a thesis that older versions might merit revival. No equations, parameters, predictions, or derivation chains appear in the provided text or abstract. The argument rests on narrative interpretation of past mathematical practice rather than any self-definition, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the central claim to its own inputs by construction. The discussion is self-contained as interpretive history and does not invoke uniqueness theorems or ansatzes from prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

This is a historical and normative paper rather than a formal derivation. No free parameters or invented mathematical entities appear. The key assumption is the potential modern utility of superseded definitions.

axioms (1)

domain assumption Disused definitions can be profitably revived for contemporary mathematics
The thesis directly invokes this evaluative judgment without supplying explicit criteria or quantitative measures of profitability.

pith-pipeline@v0.9.0 · 5295 in / 1141 out tokens · 49163 ms · 2026-05-07T16:56:13.033666+00:00 · methodology

discussion (0)

Reference graph

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W. Sierpi´ nski.Introduction to General Topology. Univ. Press, Toronto, first edition, 1934. Translated by C. Cecilia Krieger

1934

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