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arxiv: 2112.05520 · v1 · submitted 2021-12-07 · 🧮 math.GM

A category-theoretic approach to modeling John Cage's Silent piece

Michael Fowler This is my paper

Pith reviewed 2026-05-24 13:02 UTC · model grok-4.3

classification 🧮 math.GM
keywords category theoryJohn CageSilent piecepushout constructioncategory presentationsspatio-temporal structuresmeta-work modeling
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The pith

John Cage's Silent piece is modeled as the pushout of categories derived from three compositions.

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

The paper tries to establish a category-theoretic model for John Cage's Silent piece by treating it as a meta-work built from three specific compositions. It does this by defining category presentations for each composition and using translations between them, followed by a pushout operation to form a new category. If successful, this would allow formal reasoning about the enduring features of the piece across different realizations. Sympathetic readers would care because it provides a structured way to think about conceptual unity in indeterminate music.

Core claim

By presenting categories A, B, and C corresponding to instances of 4'33'', 0'00'', and One3, and constructing the pushout of B and C along A, the category S is obtained as a model of the Silent piece. From the fiber order of S a semantics is derived that supports reasoning about persistent spatio-temporal structures in the meta-work.

What carries the argument

The pushout construction that combines category presentations B and C along A to produce the category S for the Silent piece.

If this is right

  • The category S provides a specification and fiber order for the meta-work.
  • Semantics from the fiber enable analysis of persistent spatio-temporal structures.
  • The model integrates the three compositions into a unified formal representation.

Where Pith is reading between the lines

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

  • Such categorical models could be used to compare different interpretations or performances systematically.
  • The method suggests a way to formalize other works that exist as collections of related pieces.
  • Extensions might involve enriching the categories with additional data on performance contexts.

Load-bearing premise

The compositions 4'33'', 0'00'', and One3 can be faithfully captured as category presentations whose pushout produces a meaningful model of the Silent piece.

What would settle it

A detailed comparison showing that the derived semantics does not reflect the persistent spatio-temporal structures present in actual realizations of the Silent piece would falsify the central claim.

Figures

Figures reproduced from arXiv: 2112.05520 by Michael Fowler.

Figure 1
Figure 1. Figure 1: Olog of A. produced at, or by the site of the performance. Such sounds were confirmed by Tudor (Gann, 2010, page 4) during the premiere, for which incidental sounds came from confused audience members during the performance. We also declare the olog A as containing the following path equivalences: pfulfillsq » pcontainsq ˝ pis realization ofq (14) 10 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The database DA. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Category of elements A1 “ ş A pIq, for which πI : A1 Ñ A. fiber π ´1 I psq forms a discrete category, that is, a category in which there exists only objects and no morphisms between objects. The pre-image π ´1 I pfq of f : s Ñ s 1 is the set of morphisms between objects π ´1 I psq, and π ´1 I ps 1 q, for which the subcategory π ´1 I pfq Ď I can be described as the function π ´1 I pfq : π ´1 I psq Ñ π ´1 I … view at source ↗
Figure 4
Figure 4. Figure 4: Category presentation of B in which the type φpJq is given the label ‘an actant.’ S. We define C to be the vertex in B labelled φpJq “ φpLq that contains the subset S of φpJq, and φpLq such that: φpJq φpLq C Ω m m1 p (24) Here we assert that both a listener and a performer are a type of actant (mpjq “ c, and m1 plq “ c) and thus S “ tc P C | ppcq “ Trueu. Given that some listeners (audience members) assist… view at source ↗
Figure 5
Figure 5. Figure 5: Category presentation of C in which the morphism ψphq is labelled as the aspect ‘is perceived by,’ and the object ψpAq the type ‘a set of sounds.’ such that f2 ˝ Fpqq “ Gprq ˝ f1u. (27) for which the following diagram commutes: Fpa1q Gpb1q Fpa2q Gpb2q Fpqq Gprq f1 f2 (28) We call the diagram A FÝÑ C GÐÝ B the setup for the comma category pF ÓC Gq. There are two canonical functors called the left projection… view at source ↗
Figure 6
Figure 6. Figure 6: Olog of the pushout category S with additional chained pushouts W “ G Ů T D, and L “ W Ů D T, and morphism ‘spatializes’ (m : Q Ñ L). equivalences regarding the composition of morphisms, but they are also use￾ful when reasoning on schemas with pushouts. Consider the olog of the categorical schema S of the meta-work the Silent piece given in [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Category presentation S. E2 :“ e » pP ˝ wq E3 :“ j » ps ˝ wq E4 :“ t » pa ˝ fq E5 :“ pi1 ˝ uq » pi2 ˝ fq » pi1 ˝ c ˝ jq » pi1 ˝ c ˝ s ˝ wq E6 :“ pi5 ˝ i1 ˝ uq » pi5 ˝ i1 ˝ c ˝ jq » pi5 ˝ i1 ˝ c ˝ s ˝ wq » pi5 ˝ i2 ˝ fq » pi6 ˝ z ˝ fq E7 :“ pi5 ˝ i2 ˝ dq » pi6 ˝ XBq » pi6 ˝ z ˝ dq E8 :“ hC » phB ˝ YBq E9 :“ m » pi6 ˝ pBq » pi6 ˝ XB ˝ pCq » pi6 ˝ z ˝ d ˝ pCq » pi5 ˝ i2 ˝ d ˝ pCq E10 :“ l » phC ˝ pCq » phB ˝ … view at source ↗
Figure 8
Figure 8. Figure 8: Lattice of the fiber order, fbrpSq “ xspecpSq, ěSy of E. E11 :“ pB » pXB ˝ pCq » pz ˝ d ˝ pCq E12 :“ XB » pz ˝ dq E13 :“ pi5 ˝ i2q » pi6 ˝ zq (39) From E we can also consider how particular instances that form the category of elements ş S pIq satisfy (model) an S-fact  P eqnpSq as I |ùS . Consider the population of S depicted in [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
read the original abstract

We derive a schema of John Cage's meta-work the Silent piece from his compositions 4'33'', 0'00''(4'33'' No. 2), and One3, using the mathematics of category theory within Spivak and Kent's (2012) framework of ontological logs for knowledge representation. A category presentation A of a database that describes an instance of 4'33'' from its premiere in 1952 is translated via two functors into the category presentations B (0'00'') and C (One3). A pushout of B and C along A allows for the presentation of the category S (the meta-work the Silent piece), and a discussion of the category's S-specification and fiber order. Finally, we derive a semantics from the fiber in order to reason on persistent spatio-temporal structures of Cage's Silent piece.

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.

Referee Report

3 major / 2 minor

Summary. The paper claims to model John Cage's Silent piece meta-work by presenting categories A, B, and C for the compositions 4'33'' (1952 premiere), 0'00'', and One3 respectively within the Spivak-Kent 2012 ontological logs framework; it translates A via functors into B and C, forms the pushout S of B and C along A, discusses the S-specification and fiber order, and derives semantics from the fiber to reason about persistent spatio-temporal structures.

Significance. If the category presentations and functors are explicitly defined and the pushout construction yields non-trivial, verifiable semantics, the work would offer a formal category-theoretic schema for representing musical meta-works and their persistent structures, extending standard pushout and fiber-order techniques from the cited 2012 framework to an artistic domain. The approach credits the use of established operations (functors, pushouts) without introducing free parameters or ad-hoc axioms beyond the framework itself.

major comments (3)
  1. [Abstract] Abstract: the central claim that the pushout of B and C along A produces a category S whose fiber order supplies independent reasoning about persistent spatio-temporal structures rests on the unverified premise that the compositions are faithfully captured as category presentations A/B/C; no generators, relations, or explicit functor images are supplied, so the construction cannot be checked for non-triviality or faithfulness.
  2. [Abstract] Abstract: the derivation of semantics from the fiber order in S is asserted but not demonstrated; without concrete computation of the pushout or the resulting fiber, it is impossible to assess whether the semantics are derived from the performative content or are artifacts of the chosen presentations.
  3. [Abstract] The paper invokes the Spivak-Kent 2012 framework for ontological logs but provides no verification steps or explicit category definitions, leaving the soundness of the translation via functors and the subsequent pushout unassessable.
minor comments (2)
  1. The manuscript would benefit from an appendix or section explicitly listing the objects, morphisms, generators, and relations for presentations A, B, and C.
  2. Clarify whether the category of presentations is the standard one or a custom ontological-log variant, and cite the precise theorem or construction from Spivak-Kent 2012 used for the pushout.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the pushout of B and C along A produces a category S whose fiber order supplies independent reasoning about persistent spatio-temporal structures rests on the unverified premise that the compositions are faithfully captured as category presentations A/B/C; no generators, relations, or explicit functor images are supplied, so the construction cannot be checked for non-triviality or faithfulness.

    Authors: We agree that the abstract omits explicit generators, relations, and functor images, which limits immediate verifiability. The full manuscript defines the category presentations A, B, and C within the Spivak-Kent framework and specifies the functors, but to strengthen the paper we will add a new subsection that lists the generators and relations for A, B, and C together with the explicit images of each generator under the two functors. This will permit direct checking of faithfulness and non-triviality of the pushout. revision: yes

  2. Referee: [Abstract] Abstract: the derivation of semantics from the fiber order in S is asserted but not demonstrated; without concrete computation of the pushout or the resulting fiber, it is impossible to assess whether the semantics are derived from the performative content or are artifacts of the chosen presentations.

    Authors: The manuscript derives semantics from the fiber order after constructing S, yet we acknowledge that a fully expanded, step-by-step computation of the pushout and the resulting fiber is not supplied. In revision we will insert an explicit computation of the pushout diagram, the colimit category S, and the fiber order, showing how each semantic claim follows from the performative data encoded in the original compositions rather than from arbitrary choices of presentation. revision: yes

  3. Referee: [Abstract] The paper invokes the Spivak-Kent 2012 framework for ontological logs but provides no verification steps or explicit category definitions, leaving the soundness of the translation via functors and the subsequent pushout unassessable.

    Authors: We will expand the manuscript with a verification subsection that recalls the relevant definitions from Spivak-Kent 2012, states the explicit category presentations, confirms that the functors are well-defined on generators and relations, and verifies that the pushout is formed in the category of ontological logs. These additions will make the soundness of the translations and the pushout construction directly assessable. revision: yes

Circularity Check

0 steps flagged

No significant circularity; modeling via external category theory framework

full rationale

The paper applies standard pushout and functor constructions from the cited Spivak-Kent 2012 ontological logs framework to build category presentations A, B, C and their pushout S. This is a direct modeling construction rather than any derivation that reduces by definition or self-citation to its inputs. No fitted parameters are relabeled as predictions, no uniqueness theorems are imported from the author's prior work, and the central step (pushout yielding S) is the explicit application of the external formalism to the chosen generators, not a tautological renaming or self-referential loop. The result is self-contained against the external benchmark of category theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Based on the abstract alone, the claim rests on standard category theory and the applicability of the Spivak-Kent ontological logs framework to musical compositions. The new entity S is introduced without independent evidence outside the construction itself. No free parameters are mentioned.

axioms (2)
  • standard math Standard axioms of category theory (objects, morphisms, functors, pushouts)
    The construction explicitly invokes functors and pushouts to relate the category presentations.
  • domain assumption Spivak and Kent (2012) framework of ontological logs for knowledge representation
    The paper states it operates within this framework to model the compositions as databases and categories.
invented entities (1)
  • Category S (the Silent piece meta-work) no independent evidence
    purpose: To represent the overarching meta-work as the pushout of B and C along A
    S is constructed in the paper as the result of the pushout operation; no external falsifiable evidence is provided.

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Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages · 2 internal anchors

  1. [1]

    S. Awodey. Category Theory. Oxford Logic Guides. OUP Oxford, Oxford, 2010

    work page 2010

  2. [2]

    Spaces speak are you listening? Experiencing Aural Architecture

    Barry Blesser and Linda-Ruth Salter. Spaces speak are you listening? Experiencing Aural Architecture. MIT Press, Boston, 2007

    work page 2007

  3. [3]

    Pragmatics of silence

    William Brooks. Pragmatics of silence. In Nicky Losseff and Jenny Doctor, editors, Silence, Music, Silent Music, pages 97--126. Ashgate, Aldershot, UK, 2007

    work page 2007

  4. [4]

    Cage and L

    J. Cage and L. Kuhn. The Selected Letters of J ohn C age . Wesleyan University Press, Middletown, CT, 2016

    work page 2016

  5. [5]

    John Cage. Silence. Wesleyan University Press, Middletown, CT, 1961

    work page 1961

  6. [6]

    0'00'' (4'33'' No

    John Cage. 0'00'' (4'33'' No. 2). Henmar Press, New York, 1962

    work page 1962

  7. [7]

    A Year from Monday: Lectures and Writings

    John Cage. A Year from Monday: Lectures and Writings. Calder and Boyars, New York, 1968

    work page 1968

  8. [8]

    John Cage. 4'33''. Peters Edition, New York, 1986

    work page 1986

  9. [9]

    John Cage. One ^ 3 . John Cage Trust, New York, 1989. unpublished score, Fax manuscript

    work page 1989

  10. [10]

    Composing under the Skin: The Music-making Body at the Composer's Desk

    Paul Craenen. Composing under the Skin: The Music-making Body at the Composer's Desk. Leuven University Press, Leuven, 2014

    work page 2014

  11. [11]

    Silence and the void: Aesthetics of absence in space and time

    Dieter Daniels. Silence and the void: Aesthetics of absence in space and time. In Y. Kaduri, editor, The Oxford Handbook of Sound and Image in Western Art, Oxford Handbooks. Oxford University Press, Oxford, 2016. https://books.google.de/books?id=29f\_DQAAQBAJ

    work page 2016

  12. [12]

    John C age's 4'33": Is it music? Australasian Journal of Philosophy, 75 0 (4): 0 448--482, 1997

    Stephen Davies. John C age's 4'33": Is it music? Australasian Journal of Philosophy, 75 0 (4): 0 448--482, 1997

    work page 1997

  13. [13]

    Themes in the Philosophy of Music

    Stephen Davies. Themes in the Philosophy of Music. Oxford University Press, Oxford, 2003

    work page 2003

  14. [14]

    What 4'33'' is

    Julian Dodd. What 4'33'' is. Australasian Journal of Philosophy, 96 0 (4): 0 629--641, 2018

    work page 2018

  15. [15]

    John Cage's theatre pieces: notations and performances

    William Fetterman. John Cage's theatre pieces: notations and performances. Harwood Academic Publishers GmbH, Amsterdam, 1996

    work page 1996

  16. [16]

    On the relationship between inside and outside: Conceptualising acoustic space in J ohn C age's V ariations IV

    Michael Fowler. On the relationship between inside and outside: Conceptualising acoustic space in J ohn C age's V ariations IV . Architectural Theory Review, 17 0 (1): 0 158--177, 2012

    work page 2012

  17. [17]

    John C age's S ilent P iece and the J apanese gardening technique of shakkei: Formalizing W hittington's conjecture through conceptual graphs

    Michael Fowler. John C age's S ilent P iece and the J apanese gardening technique of shakkei: Formalizing W hittington's conjecture through conceptual graphs. Journal of Mathematics and Music, 13 0 (1): 0 4--26, 2019

    work page 2019

  18. [18]

    Zero Decibels: The Quest for Absolute Silence

    George Michelsen Foy. Zero Decibels: The Quest for Absolute Silence. Simon and Schuster, New York, 2010

    work page 2010

  19. [19]

    No Such Thing as Silence: J ohn C age's 4'33''

    Kyle Gann. No Such Thing as Silence: J ohn C age's 4'33'' . Yale University Press, New Haven, CT, 2010

    work page 2010

  20. [20]

    The cultural politics of 4'33'': identity and sexuality

    Philip Gentry. The cultural politics of 4'33'': identity and sexuality. In Matthieu Saladin, editor, Tacet \#1: Who is John Cage? Les presses du r \'e el, Dijon, 2018

    work page 2018

  21. [21]

    A. J. Greimas. Structural Semantics: An Attempt at a Method. University of Nebraska Press, Lincoln, 1983

    work page 1983

  22. [22]

    J. Johnson. There is no silence now. In J. Cage and R. Kostelanetz, editors, John C age: An anthology , pages 145--149. Da Capo Press, New York, 1970

    work page 1970

  23. [23]

    Branden W. Joseph. John C age and the architecture of silence. October, 81 0 (Summer): 0 80--104, 1997

    work page 1997

  24. [24]

    John C age: Silence and silencing

    Douglas Kahn. John C age: Silence and silencing. The Musical Quarterly, 81 0 (4): 0 556--598, 1997

    work page 1997

  25. [25]

    Absence, presence and potentiality: J ohn C age's 4'33'' revisited

    Karl Katschthaler. Absence, presence and potentiality: J ohn C age's 4'33'' revisited. In Werner Wolf and Walter Bernhart, editors, Silence and Absence in Literature and Music, pages 166--179. Rodopi, Boston, 2016

    work page 2016

  26. [26]

    In the blink of an ear: towards a non-cochlear sonic art

    Seth Kim-Cohen. In the blink of an ear: towards a non-cochlear sonic art. Contimuum, New York, 2009

    work page 2009

  27. [27]

    Kostelanetz and J

    R. Kostelanetz and J. Cage. Conversing with Cage. Readers' Guides to Essential Criticism. Routledge, London, 2003

    work page 2003

  28. [28]

    Cagean structures

    Liz Kotz. Cagean structures. In Julia Robinson, editor, The Anarchy of Silence: John Cage and experimental art. Musea d' Art C ontemporani de B arcelona, Barcelona, 2009

    work page 2009

  29. [29]

    John C age and nonrepresentational spaces of music

    Robert Kruse. John C age and nonrepresentational spaces of music. Social & Cultural Geography, 2019. 10.1080/14649365.2019.1628291

    work page doi:10.1080/14649365.2019.1628291 2019

  30. [30]

    Background Noise, Second Edition: Perspectives on Sound Art

    Brandon LaBelle. Background Noise, Second Edition: Perspectives on Sound Art. Bloomsbury Publishing, London, 2015

    work page 2015

  31. [31]

    Categories for the Working Mathematician

    Saunders MacLane. Categories for the Working Mathematician. Graduate Texts in Mathematics. Springer, New York, 2013

    work page 2013

  32. [32]

    D. Massey. For space. Sage, London, 2009

    work page 2009

  33. [33]

    Moholy-Nagy and D.M

    L. Moholy-Nagy and D.M. Hoffmann. The New Vision: Fundamentals of Bauhaus Design, Painting, Sculpture, and Architecture. Dover Fine Art, History of Art. Dover Publications, London, 2012

    work page 2012

  34. [34]

    Nicht alles, nicht nichts, etwas

    Michael Pisaro. Nicht alles, nicht nichts, etwas. T he opposing tensions of J ohn C age's 0'00'' and R oaratorio. In Matthieu Saladin, editor, Tact:\#1: Who is John Cage?, pages 110--125. Les presses du r\' e el, Dijon, 2018

    work page 2018

  35. [35]

    The Music of J ohn C age

    James Pritchett. The Music of J ohn C age . Cambridge University Press, Cambridge, 1996

    work page 1996

  36. [36]

    Silence changed: O ne ^ 3 , 2018

    James Pritchett. Silence changed: O ne ^ 3 , 2018. http://rosewhitemusic.com/piano/2018/10/01/silence-changed-one3/

    work page 2018

  37. [37]

    Poethics of a complex realism

    Joan Retallack. Poethics of a complex realism. In Marjorie Perloff and Charles Junkerman, editors, John Cage: Made in America, pages 242--274. University of Chicago Press, Chicago, 2013

    work page 2013

  38. [38]

    Categories

    Horst Schubert. Categories. Springer-Verlag, New York, 1972. Eva Gray, translator

    work page 1972

  39. [39]

    Silencing the Sounded Self: J ohn C age and the A merican E xperimental T radition

    Christopher Shultis. Silencing the Sounded Self: J ohn C age and the A merican E xperimental T radition . University Press of New England, Lebanon, NH, 2013

    work page 2013

  40. [40]

    David I. Spivak. Functorial data migration. CoRR, abs/1009.1166, 2010. http://arxiv.org/abs/1009.1166

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  41. [41]

    Category theory for scientists (Old version)

    David I Spivak. Category theory for scientists (old version). arXiv preprint arXiv:1302.6946 , 2013 a

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  42. [42]

    David I. Spivak. Database queries and constraints via lifting problems. Mathematical Structures in Computer Science, 24 0 (6), 2013 b . ISSN 1469-8072. http://dx.doi.org/10.1017/S0960129513000479

    work page doi:10.1017/s0960129513000479 2013

  43. [43]

    Ologs: a categorical framework for knowledge representation

    David I Spivak and Robert E Kent. Ologs: a categorical framework for knowledge representation. PLoS One, 7 0 (1): 0 e24274, 2012

    work page 2012

  44. [44]

    John C age: S eptember 5, 1912- A ugust 12, 1992

    Mark Swed. John C age: S eptember 5, 1912- A ugust 12, 1992. The Musical Quarterly, 77 0 (1): 0 132--144, 1993

    work page 1912

  45. [45]

    Christian Vandendorpe. Actant. In I.R. Makaryk, editor, Encyclopedia of Contemporary Literary Theory: Approaches, Scholars, Terms, Theory / culture, page 505. University of Toronto Press, Toronto, 1993

    work page 1993

  46. [46]

    Digging in J ohn C age's garden: C age and R y\= o anji

    Stephen Whittington. Digging in J ohn C age's garden: C age and R y\= o anji. Malaysian Music Journal, 2 0 (2): 0 12--21, 2013

    work page 2013