Thoughts on Platonic Discretisation and the Digital

 David M. Berry

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Plato's Timaeus presents a cosmos constructed from mathematical principles, where geometrical forms constitute the key architecture of reality. It has exercised considerable influence on subsequent thought, becoming arguably Plato's most influential dialogue for over a millennium and providing a key philosophical basis for pre-modern cosmology, medicine, and natural philosophy until the Scientific Revolution. In it, Plato claims that mathematical structures underlie physical reality. Yet its assumptions about the relationship between mathematical form and material existence are also relevant to contemporary debates about digital technology. In the dialogue, Plato argues that the world is reducible to proportional relationships, structures, and discrete combinatorial elements showing his reliance on Pythagorean mathematics concerned with "geometrical proportion only" (Cornford, 1937: 45). 

This mathematical foundationalism finds an unexpected echo in computational thinking and ideas of discretisation in thinking about the digital more broadly. The process of discretisation is central to all digital representation in the conversion of continuous phenomena into discrete, quantifiable units. When we digitise an archive, model a text corpus, or render human behaviour as data points, we engage in a kind of Platonic operation in which we impose mathematical form upon the flux of experience.[1] 

This raises the question of whether discretisation necessarily entails a Platonic metaphysics? Does the technical operation of converting analogue to digital carry with it the ontological commitments of the Timaeus? Elsewhere I have suggested that whilst discretisation need not imply such metaphysics, unreflexive computational practice risks recreating exactly these assumptions. As I explain, "computation thought of as a mathematical or logical force comes to be seen as an independent participant in human social relations, it is given 'life' fixed by its own nature and a power to shape social life. This is the reification of a social relation" (Berry, 2023: 127). The danger lies not in discretisation per se, but in mistaking our computational models for the structure of reality itself, therefore, we might add, allowing the Timaeus to become an unconscious metaphysics of the digital age.

The question separating what we might call critical from uncritical discretisation concerns the status of mathematical models. They raise the question of whether they reveal the essential structure of reality (ontology) or provide tools for representing it (epistemology)? In the Timaeus, Plato makes an unambiguous ontological claim. The regular solids are not merely useful descriptions of fire, air, water, and earth, rather they constitute their fundamental nature in some way. As he states in the dialogue, "the first will be the simplest and smallest construction, and its element is that triangle which has its hypotenuse twice the lesser side" (Plato 1998). The triangular surfaces are not models of elements but their actual composition, a claim that reality is essentially mathematical. 

Yet Cornford argues there is something more complex within the dialogue itself which points to the problem of discretisation as a metaphysical claim. Plato "indicates that there is something arbitrary in starting from this assumption" of triangles as irreducible elements (Cornford, 1937: 212). Indeed, the dialogue appears to acknowledge deeper levels (Plato 1997a: 53d). Cornford identifies these as "lines and numbers," noting that "triangles themselves can be constructed of lines, and lines can be expressed as numbers" (Cornford, 1937: 212). The dialogue therefore seems to point towards a hierarchy of foundations with planes constructed from triangles, triangles from lines, lines from numbers. Yet this raises the question of where does the sequence end? If the Timaeus stops at triangles, philosophy might presumably pursue the chain further but do numbers rest on still more fundamental principles (the One, the Dyad, or whatever serves as ultimate metaphysical foundation). This acknowledgement undermines any simple reading of Platonic discretisation as reaching some sort of bedrock reality. Within the Timaeus, mathematical "atoms" turn out to be compound, their apparent simplicity concealing further structure. Cornford suggests that "Plato's reason for stopping short at triangles was perhaps the need to keep his exposition within reasonable bounds" (Cornford, 1937: 213). This epistemic humility is an important point to keep in mind in relation to his ontological claims. 

However, Taylor (2023) argues that this form of mathematical foundationalism threatens a regress of epistemic justification. He explains that if we explain the Forms (including the Form of the Good) through mathematical principles, "then either we have an infinite regress of knowledge or we have some foundations of knowledge, knowledge of which is grounded on nothing but themselves" (Taylor 2023: 445). The Republic suggests the Good is "epistemologically primary," yet understanding goodness as order and proportion seems to require explaining it "in terms of other concepts," making "those concepts now more basic than goodness" (Taylor 2023: 445). This is not the same problem as the ontological sequence from triangles to lines to numbers, but it does concern how we know or justify our claims about Forms. Taylor therefore proposes that "to give an account of a concept is not to explain it in terms of anything more basic but to locate it in a coherent structure of concepts, and specifically to show the explanatory role that each concept plays within that structure" (Taylor 2023: 446). He argues that mathematical principles and the Good mutually constitute each other within a coherent system, neither is simply foundational.

This epistemic problem operates differently from the ontological hierarchy in the Timaeus that reduces triangles to lines to numbers. The ontological reduction claims to move towards more basic constituents, simpler elements from which complex forms are built. But Taylor's epistemic concerns are about how we justify our knowledge of these elements at all. Even if triangles are ontologically posterior to lines, our knowledge of geometrical principles might still face circular justification. We know the Forms through mathematics, but we justify mathematics through the Forms. The two problems intersect in revealing that discretisation cannot be justified through self-evident foundations. Whether we pursue ontological reduction downwards (e.g. seeking simpler building blocks) or epistemic justification upwards (e.g. seeking grounds for our knowledge), we encounter either arbitrary stopping points or circular dependencies.

Whether we face an ontological hierarchy (triangles, lines, numbers, the One...) or an epistemic circle (mathematics explains the Good which grounds mathematics), discretisation emerges as embedded in wider systematic relations rather than resting on self-evident "atoms". As Johansen observes, Timaeus "posits" his geometrical starting points whilst acknowledging that "god and of men he who is a friend to god know the principles still higher than these" (Plato 1997a: 53d6-7, cited in Fine, 2023: 296). The foundations are not discovered but postulated, a fact the dialogue itself admits through its methodology of "likely accounts" (εἰκὼς λόγος) (Plato 1997a, 53d6-7). Similarly, computational specifications are not discoveries of pre-existing structure but documents and protocols that establish that structure. The Unicode Consortium does not discover what characters exist, rather it defines what will count as a character for computation. Yet computation, and software engineering in particular, systematically obscure this definitional moment, treating the specifications as neutral transcriptions of reality rather than theory-laden constructions that rest on nothing but themselves and the coherence they produce.

Contemporary debates about the digital or the computation, by contrast, make strong epistemological claims. When digital humanists discretise an archive or model textual patterns, the stated purpose is representational rather than ontological. Digitisation is understood as a method for managing, analysing, and transmitting cultural materials through computers. Yet there is a persistent tendency for this epistemological modesty to collapse into ontological claims. For example, digital humanities has developed "an instrumentalism revealed in its main two approaches, digital archives and digital tools" which tends to become "means-focussed, allowing other disciplines to define the ends to which their work was oriented" (Berry, 2023: 125). This instrumental orientation paradoxically enables a kind of metaphysical smuggling where technical choices about data structures, metadata schemas, and algorithmic procedures are treated as neutral implementations rather than theory-laden constructions.

Burnyeat (2012) identifies this problem in his analysis of ancient debates about mathematical objects. He argues that the dispute about whether "mathematicals" exist "in sensibles" or "separately from sensibles" reflects a deeper question about truth itself. He states, "the dispute, as [Sextus Empiricus in  Against the Mathematicians] was bound to conclude, is about their manner of existence" (Burnyeat 2012: 151). For both Plato and Aristotle, mathematical theorems must be true of something, as "all parties to the debate agree that mathematics is true. All parties are therefore committed to accepting that mathematicals exist" (Burnyeat, 2012: 151). The crucial difference lies in whether mathematical truth reveals pre-existing structure or constructs helpful representations. As Burnyeat observes, "when the question is put, 'What are the objects of mathematical diánoia?', the Republic replies: That is a problem we must think about" (Burnyeat, 2012: 152). It seems that Plato acknowledges mathematical practice without settling its metaphysical implications, a philosophical modesty largely absent from contemporary debates about digital metaphysics.

The consequences of this conflation are significant. Indeed, computation "can lead to a valorisation of the mathematisation of thought whereby formalisation of knowledge through computation is seen as not just one approach to thinking but the exemplary one, often one that is misplaced" (Berry, 2023: 127). This mirrors the Platonic error identified by Marx in his critique of Hegel of treating abstractions as if they possessed autonomous existence independent of the material conditions of their production. Thus "in order … to find an analogy we must take flight into the misty realm of religion. There the products of the human brain appear as autonomous figures endowed with a life of their own" (Marx cited in Berry, 2023: 127).

This implies that the digital risks becoming a new Platonic realm of Forms, a mathematical heaven to which messy material reality must conform. This is evident in practices where computational models are not tested against reality but rather reality is adjusted to fit the model. The implications of a prior ontological "truth" is that computational systems, once built, exert pressure to remake social relations in their image. The tail wags the dog when "it becomes easier for computationalists to conceive of changing the world, rather than change the computational model" (Berry, 2023: 128). But in fact it is crucial that "one identifies how deeply capitalistic logic is embedded within computational thinking, it becomes clear that markets, individual monads, and transactional relations tend to be paramount" (Berry, 2023: 128).[2] 

A critical approach must therefore maintain careful attention to the ontological-epistemological distinction. Discretisation remains a powerful epistemological tool, as a method for making certain kinds of patterns visible, certain comparisons possible, certain questions answerable. But the moment we forget that digital models are pragmatic, situated, partial constructions and begin to treat them as revelations of underlying structure, we have entered Platonic metaphysics through the back door. It remains helpful to keep a materialist perspective as "materialism, unlike idealism, always understands thinking to be the thinking of a particular people within a particular period of time" (Berry, 2023: 129).

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The Timaeus itself reveals the key problem with ontological mathematisation, which is it cannot account for its own conditions of possibility. Indeed, Plato must introduce the chora or "receptacle" (χώρα), a formless, barely intelligible substrate, to explain how geometrical forms take on material existence. Cornford notes that whilst the receptacle shares the eternal, indestructible character of the Forms, it remains radically different in kind. The receptacle, or what Cornford calls "Space", is described as "everlastingly existent and not admitting destruction," a description that "differs only verbally from that applied to the Form, 'ungenerated and indestructible'" (Cornford, 1937: 192). The Forms are intelligible, grasped through rational thought, whereas the receptacle is "apprehended without the senses by a sort of bastard reasoning" (Plato 1997b: 52b).[3] 

This is not merely difficult knowledge but a different mode of access altogether. The receptacle cannot be known through the same rational procedures that grasp mathematical forms because it is what makes those forms capable of instantiation. It is not an object within the system but the condition for the system having objects at all. As Sallis notes, the receptacle must be such as to be capable of receiving all things whilst itself remaining devoid of all character, he calls it "not only receptacle but also reception" (Sallis, 1999: 99). Its characterlessness is not a deficiency but its very nature as pure receptivity. The moment we attempt to characterise it positively, we have already imposed form upon it, contradicting its function as that which receives form. The Receptacle becomes a precondition for mathematical forms taking material existence that the mathematical system cannot account for. The receptacle can be seen as an analogy of what computation requires but cannot compute, the material substrate that makes discretisation possible whilst escaping its categories.[4]

The receptacle functions not as what remains after discretisation but as what discretisation presupposes yet cannot incorporate. This distinction is crucial. A residue would be merely what the system fails to capture, suggesting that better methods might eventually encompass it. A condition, by contrast, is what makes the system possible in the first instance, and therefore cannot be internalised without paradox. Just as the receptacle must remain formless to receive forms, the material substrate of computation must remain outside computational representation to serve as the condition of possibility for computation and requires a computational ideology to reinforce this (Berry 2014; Chun 2011).

For example in terms of the relationship between an algorithm and its execution, the algorithm specifies a sequence of discrete operations, yet its execution requires continuous electrical flow, thermal dynamics, and quantum mechanical processes in the semiconductors. These physical processes are not themselves discrete, they do not operate through the step-by-step logic of the algorithm. Rather, continuous material processes are harnessed to approximate discrete state transitions. The discreteness exists at the level of abstraction, of logical specification, whilst the substrate remains continuous and material. This is not a gap that better engineering could close, it is the constitutive relationship between form and matter that makes computation possible.

Interestingly, the same structure appears in human labour. For example, when annotators tag training data, interpret ambiguous cases, or correct algorithmic outputs, they exercise capacities that resist formalisation using tacit knowledge, situated judgement, and interpretative skills. Yet these capacities are not failures to be automated away but that which enables the algorithmic system to function at all. The algorithm therefore cannot account for the human capacities that make its application possible. As Irani notes in her ethnography of crowdwork, "human intelligence has become part of network infrastructure" yet remains invisible in computational accounts (Irani, 2015: 724). The receptacle for Plato, similarly, remains invisible to maintain the appearance that Forms alone constitute reality.

Computational models require material instantiation in hardware, electricity, human labour, and institutional structures, and yet these conditions of possibility tend to vanish from view, leaving only an abstraction of the algorithm. Critical theory insists on making these conditions visible, refusing the Platonic gesture that would separate form from matter, model from instantiation, code from its political economy. Yet the question of visibility or invisibility does not capture the receptacle's function. The point is not that computational discourse ignores material conditions, though it often does. Rather, these conditions must be excluded to maintain computation's claims to universality and abstraction. Just as Plato requires the receptacle to be formless, characterless, barely intelligible, computational ideology requires its material substrate to disappear from view. This disappearance is not accidental but structural.

The algorithm appears as pure logical form, implementable anywhere, whilst the specificity of hardware architectures, energy infrastructures, and labour is lost in generic "implementation details." Yet these details are key to what the algorithm is. A machine learning model trained on exploited clickwork labour, running on servers cooled by appropriated water resources, optimised for Nvidia's proprietary Blackwell cores, is not the same entity as the algorithm expressed in Python code. The receptacle reminds us that form without matter is not just incomplete but impossible. There is no algorithm as such, only specific material instantiations that enable certain operations whilst foreclosing others. This helps explain why computational systems resist the totalisation they promise. Precisely because they make computation possible, they retain capacities and characteristics that exceed formal description. Hardware failures, emergent behaviours, unforeseen reactions with their environments, these are symptoms of matter's irreducibility to form, not "bugs" to be resolved. The system cannot close because its closure would destroy the very conditions of its operation. This structural incompleteness is shown most clearly in discretisation's inability to account for continuity and change. 

If the continuous, flowing, qualitative reality is reduced to the rearrangement of discrete units (however defined) this is a reduction that leaves the phenomenon of change unexplained. This problem haunts all discretisation. Digital representation necessarily involves sampling continuous phenomena at discrete intervals, quantising continuous values into discrete bins, and imposing categorial boundaries on fluid experience. One faces the challenge of explaining how one entity becomes another. Do the discrete units (in a discrete metaphysics) share the same fundamental components? This reveals an irreducible remainder within a discrete system itself and even Plato's mathematical ontology cannot achieve the complete reducibility it promises. 

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Discretisation is not a neutral transcription but an interpretative act involving countless micro-decisions about what to include and exclude, how to categorise, where to draw boundaries, what resolution to employ, which features to encode. Each choice involves loss, what information theorists call quantisation error, but also theoretical commitments. The Timaeus ultimately acknowledges the inadequacy of its discrete ontology by introducing elements that escape geometrical reduction. Time, described as "the moving image of eternity," resists incorporation into the system of triangles and solids. Space or the receptacle, as noted above, remains mysterious and incomprehensible, necessary for the system but irreducible to it. Even the soul, especially the mortal soul associated with becoming, desire, and embodiment, cannot be fully rationalised within its mathematical framework. Plato admits that "the truth concerning the soul can only be established by the word of God" and settles for probability and myth rather than geometrical demonstration (Plato, 1998). The qualitative, the contextual, the embodied, the tacit, these dimensions of human experience and cultural meaning are not merely difficult to capture computationally but may be categorically different in kind. Attempting to force them into a discrete form may not just produce a lossy representation but rather a category error. 

This involves not only acknowledging quantisation error but distinguishing between different kinds of remainder that discretisation produces. Firstly, there are practical limitations where finer discretisation could in principle capture more through higher sampling rates, more granular categories, or expanded metadata schemas. This is quantisation error in the technical sense, a problem of resolution rather than kind. When a digital archive samples manuscript images at 300 dpi rather than 600 dpi, information is lost, but this loss is contingent on technical capacity rather than structural impossibility. 

Secondly, there are qualities that resist measurement not contingently but structurally. Heidegger's analysis of "readiness-to-hand" (Zuhandenheit) points to this dimension. The way we know how to use a hammer is not reducible to propositional knowledge about hammers. As Dreyfus developed in his critique of artificial intelligence, skilful coping depends on background understanding that computation cannot formalise precisely because it is constituted through embodied practice rather than explicit representation (Dreyfus, 1992). When a digital humanities project encodes textual variants into TEI markup, what escapes is not merely detail but modes of knowledge, that is, the temporality of reading across folios, the haptic recognition of scribal hands, the peripheral awareness that enables palaeographers to notice patterns. These are not information "in" the manuscript that better encoding could capture but rather competencies constituted through embodied interaction with material artefacts.

Lastly, there are emergent meanings constituted through social practice and historical sedimentation. When computational methods discretise textual patterns through topic modelling or sentiment analysis, they necessarily abstract from the history of reading practices, institutional contexts, and interpretative communities that give texts meaning. This is not information "in" the text that more sophisticated algorithms might extract but rather significance that exists in the relationship between text and interpretative tradition. Literary irony, for instance, depends on recognising multiple incompatible readings simultaneously, a semantic instability that resists encoding not because current methods are inadequate but because the phenomenon itself is constituted through interpretative indeterminacy.

The Timaeus parallel becomes clearer in this discussion. The receptacle is not merely "what's left over" after geometrical analysis but what makes geometrical forms capable of material instantiation at all. It is condition rather than residue. Similarly, the computational remainder should not be understood as what computation fails to capture but as what makes computational capture meaningful in the first place. Human labour, embodied practice, and social relations are not external additions to computational systems but their constitutive conditions. The work of annotation, the interpretative decisions encoded in data structures, the institutional power that determines what gets digitised, these do not supplement computation but make it possible.

Crucially, computation does not simply "miss" this remainder. Rather, discretisation actively produces it through its operations. The act of converting continuous phenomena into discrete units creates the appearance that reality was always already discrete, waiting to be discovered rather than constructed through measurement. What resists discretisation then appears as merely empirical noise, as technical limitation to be overcome through better sensors or finer granularity, rather than as evidence that the categories themselves enact violence upon the phenomenon. This is the ideological effect of discretisation as it naturalises its own abstractions, treating the remainder not as constitutive absence but as contingent incompleteness.

This reification parallels the Platonic gesture where mathematical forms are mistaken for the structure of reality itself. Just as Plato must introduce the receptacle to account for what his geometrical system cannot explain, computational practice must continually confront what exceeds its models. Yet rather than recognising this remainder as revealing the limits of discretisation, there is persistent pressure to expand the domain of the computable, to treat resistance to formalisation as a technical challenge rather than a category error. This involves not only acknowledging quantisation error but questioning the theoretical commitments in discretisation. It requires attending to what exceeds or escapes digital capture, the "remainder" that is crucial to any critical project. Indeed, "material factors are the repressed factor" in idealist or metaphysical theories of computation (Berry, 2023: 129). 

This tension between invariant discrete structure and temporal becoming maps onto what I call "one of the paradoxes of computationalism", namely "the way in which it is simultaneously understood as a logical foundationalism with a demonstrative method combined with a developmental or processual explanation" (Berry, 2023: 128). Indeed, this kind of "foundationalism tends towards an invariant conception of entities and relations frozen at the time of their computational fixation. The world is tethered to the ideal forms of the computational" (Berry, 2023: 128). Computational systems are specified by deterministic rules, through algorithms, data structures, formal specifications, etc. and yet they unfold in time, interact with environments, and produce emergent behaviours. The code is fixed, but crucially its execution is processual. 

Instead we need to insist on the priority of historical process and material conditions. Elsewhere I argue that it is important that "knowledge is seen as a historical and material phenomenon" (Berry, 2023: 129). Computational systems are not eternal Forms instantiated in matter but historical artefacts produced by specific people in specific circumstances to serve specific interests. Their apparent discretisation is an ideological effect that conceals the processual, contingent, political nature of their construction and deployment.

This image of perfected mathematical order remains ever seductive. Digital systems promise similar completeness through comprehensive archives, total surveillance, probabilistic prediction. But such totality is always illusory, always concealing what it excludes. The continuous exceeds the discrete, the qualitative resists the quantitative, the social cannot be reduced to the computational. A critical approach to the digital remains attentive to this remainder, to the non-identical that escapes conceptual capture.

** Headline image generated using Google Gemini Pro. December 2025. The prompt used was: "An ancient philosophical diagram, split screen composition. Upper section: illuminated manuscript style on beige papyrus, displaying the Platonic solids (tetrahedron, octahedron, icosahedron, cube) glowing with elemental magic, deconstructed into geometric triangles, scientific illustrations, clean lines. Lower section: separated by a jagged rift, diving into a dark void of metaphysics, abstract white lines and numbers floating in a dark misty abyss, symbols of ontology, mysterious and deep. Border: Greek key frame. Style: digital fantasy illustration, high detail, 8k resolution, photorealistic textures, chart aesthetic. --ar 4:3" Due to the probabilistic way in which these images are generated, future images generated using this prompt are unlikely to be the same as this version. 

Notes

[1] Not all computational practice involves sampling continuous phenomena, exceptions would include symbolic AI, formal logic systems, and rule-based computation which operate on already-discrete symbolic structures. Indeed, certain representational schemes (vector graphics, parametric curves, functional programming) work with continuous mathematical functions rather than discrete samples. However, I maintain that discretisation remains constitutive of digital computation in a deeper sense, even symbolic computation requires discrete state machines, finite symbol sets, and determinate transitions between states. For example, the computer operates through discrete voltage levels, binary states, and quantised time steps (clock cycles). As Kittler observes, "there is no software" in the sense that all symbolic operations ultimately reduce to discrete physical switching events (Kittler 1997). This is what distinguishes digital from analogue computation and what makes the Platonic parallel interesting. Both claim that reality is constituted by discrete, combinatorial elements subject to mathematical laws. Even computation that works with continuous mathematical functions does so through discrete symbolic manipulation and finite approximation. The question is not whether specific representational formats are discrete but whether computation depends on discretisation at some level.

[2] Due to limits of space, the relationship between discretisation and commodity will be discussed in more detail in a future post. I will just note here that Marx's analysis of exchange value depends on rendering qualitatively different use values commensurable through abstract labour time, a process of discretisation that can be seen to parallel computational reduction of heterogeneous phenomena to quantifiable units. As Sohn-Rethel argues, the commodity abstraction performs a "real abstraction" prior to thought itself, where "the exchange abstraction... is not thought-induced" but rather "arises from spatio-temporal activity" in human action (Sohn-Rethel, 1978: 20). Digital discretisation might therefore be understood not merely as analogous to commodity exchange but as its contemporary technical instantiation, where the computational rendering of social relations into discrete data extends the commodity form into previously resistant domains of experience. The Platonic ontology embedded in computational thinking is crucial to understand as a claim to a timeless metaphysical commitment that obscures the historical dominance of exchange relations that require discrete, quantifiable units.

[3] The Greek term is logismos nothos (λογισμῷ νόθῳ), literally "bastard" or "illegitimate reasoning" or "false logic". The term nothos carries connotations of spuriousness, hybridity, and ontological impurity, something that falls outside proper categories. As Cornford notes, this odd epistemological status marks the receptacle as "barely an object of belief" (Cornford, 1937: 192), accessible neither through direct sensory experience nor through pure rational thought. Plato himself acknowledges that his system requires something that can only be known through an epistemologically suspect, "illegitimate" mode of access.

[4] The analogy I am drawing here between Plato's receptacle and the computational substrate is suggestive but we should recall that Plato's cosmology requires the receptacle as metaphysical necessity within an idealist system. The receptacle is useful to understand within Platonic metaphysics as it points to a symptom of what mathematical foundationalism cannot account for, revealing the necessary failure of any attempt to reduce material reality to formal structures. Where Plato treats matter as derivative and not central to his philosophy, critical materialism instead sees it as the condition and content of all abstraction. The receptacle's appearance in the Timaeus marks the point where idealism is confronted by its own impossibility. 

Bibliography

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