# Actualising the Virtual ## Source Screenshot ![[16-01-25 Difference and Repetition (17).jpg]] ## Metadata - **Source**: Difference and Repetition, screenshot 17 - **Author**: Gilles Deleuze - **Context**: Concrete examples of how Ideas actualize from virtual multiplicities into actual species and parts ## Original Passage > Ideas contain all the varieties of differential relations and all the distributions of singular points coexisting in diverse orders 'perplicated' in one another. When the virtual content of an Idea is actualised, the varieties of relation are incarnated in distinct species while the singular points which correspond to the values of one variety are incarnated in the distinct parts characteristic of this or that species. The Idea of colour, for example, is like white light which perplicates in itself the genetic elements and relations of all the colours, but is actualised in the diverse colours with their respective spaces; or the Idea of sound, which is also like white noise. There is even a white society and a white language, the latter being that which contains in its virtuality all the phonemes and relations destined to be actualised in diverse languages and in the distinctive parts of a given language. Thus, with actualisation, a new type of specific and partitive distinction takes the ## Initial Reflections - The "white" metaphors (light, noise, society, language) beautifully illustrate virtual multiplicities containing all actualized forms - Ideas are "perplicated" - folded together in complex virtual coexistence before actualization - The actualization process involves incarnation into distinct species and parts ## Analysis This passage provides concrete examples of how virtual Ideas actualize into actual multiplicities, giving vivid content to the technical differentiation/differenciation distinction. Ideas exist as virtual multiplicities where "all the varieties of differential relations and all the distributions of singular points coexist in diverse orders 'perplicated' in one another." The beautiful metaphors - white light containing all colors, white noise containing all sounds, white language containing all phonemes - illustrate how virtual multiplicities are genuinely full and complete as virtual rather than being mere potentials waiting for actualization. For your thesis, this suggests mathematical Ideas like category theory exist as "white mathematics" containing all mathematical relations virtually, actualizing differently in algebra, topology, logic, etc. This supports mathematical pluralism and democratic participation - the same virtual mathematical multiplicities can actualize in diverse mathematical species and parts accessible to different mathematical communities rather than requiring single expert interpretation. ## Key Concepts Defined | Concept | Definition/Interpretation | | ------------ | ------------------------- | | [[perplicated Ideas]] | "Ideas contain all the varieties of differential relations...coexisting in diverse orders 'perplicated' in one another" - virtual multiplicities where all differential relations are folded together in complex coexistence. Like [[category theory]] where all mathematical relationships coexist virtually before actualizing in specific mathematical domains. | | [[varieties of relation incarnated]] | "The varieties of relation are incarnated in distinct species" - when Ideas actualize, their virtual differential relations become embodied in actual distinct types or categories. Like how categorical relationships actualize in different mathematical species (algebraic, topological, logical). | | [[singular points in parts]] | "The singular points which correspond to the values of one variety are incarnated in the distinct parts characteristic of this or that species" - virtual singularities become actual distinctive elements within actualized species. Like mathematical singularities actualizing as specific theorems within mathematical domains. | | [[white light metaphor]] | "The Idea of colour...is like white light which perplicates in itself the genetic elements and relations of all the colours, but is actualised in the diverse colours" - white light as virtual multiplicity containing all colors that actualize as distinct colors with their spaces. Perfect model for virtual mathematical multiplicities. | | [[white noise]] | "The Idea of sound, which is also like white noise" - white noise containing all sound frequencies virtually, actualizing as distinct sounds. Like virtual mathematical structures containing all mathematical relationships that actualize as specific mathematical constructions. | | [[white society]] | "There is even a white society" - virtual social multiplicity containing all social relations that actualize in distinct social forms and institutions. Like virtual mathematical communities containing all possible mathematical relationships that actualize in different mathematical practices. | | [[white language]] | "A white language...which contains in its virtuality all the phonemes and relations destined to be actualised in diverse languages and in the distinctive parts of a given language" - virtual linguistic multiplicity actualizing in different actual languages and their parts. | | [[genetic elements]] | White light "perplicates in itself the genetic elements and relations of all the colours" - virtual multiplicities contain the generative/productive elements that will create actual multiplicities. Like [[universal properties]] as genetic elements of categorical constructions. | | [[diverse orders coexisting]] | Virtual differential relations and singular points "coexisting in diverse orders" - multiple organizational structures existing simultaneously in virtual multiplicities before separating in actualization. Like mathematical structures having multiple simultaneous organizational possibilities. | | [[specific and partitive distinction]] | "With actualisation, a new type of specific and partitive distinction takes the..." - actualization creates distinctions between species (types) and parts (elements within types) that don't exist in virtual multiplicities. Like mathematical actualization creating domain distinctions and theorem hierarchies. | ## Philosophical Implications This passage reveals the rich structure of virtual-actual relationships: - **Virtual Richness**: Virtual multiplicities are genuinely full and complete, not impoverished potentials waiting for actualization - **Pluralistic Actualization**: The same virtual multiplicity can actualize in diverse species and parts without losing virtual unity - **Genetic Virtuality**: Virtual multiplicities contain the productive elements that generate actual multiplicities - **Coexistent Orders**: Virtual multiplicities contain multiple organizational possibilities simultaneously - **Non-Reductive Actualization**: Actualization creates new distinctions without reducing or exhausting virtual content For your thesis, this supports mathematical pluralism - virtual mathematical multiplicities can actualize in diverse mathematical forms accessible to different communities. ## Mathematical Resonance **White Mathematics**: - **Category Theory**: Virtual mathematical multiplicity containing all mathematical relationships that actualize in algebra, topology, logic, computer science - **Universal Properties**: Genetic mathematical elements that generate diverse mathematical constructions - **Natural Transformations**: Virtual mathematical relationships actualizing as specific mathematical morphisms - **Foundational Structures**: Virtual mathematical multiplicities actualizing in different foundational approaches (set theory, type theory, category theory) **Mathematical Actualization Examples**: - **Abstract Algebra**: Virtual algebraic relationships actualizing in groups, rings, fields with their distinct parts (elements, operations, theorems) - **Topology**: Virtual spatial relationships actualizing in metric spaces, topological spaces, manifolds with their characteristic parts - **Logic**: Virtual inferential relationships actualizing in classical logic, constructive logic, modal logic with their distinctive inference rules - **Geometry**: Virtual geometric relationships actualizing in Euclidean, non-Euclidean, differential geometry with their specific properties **Perplicated Mathematical Relations**: - **Interdisciplinary Mathematics**: How mathematical Ideas actualize differently in physics, computer science, economics while maintaining virtual unity - **Historical Mathematics**: Same mathematical Ideas actualizing differently in different historical periods and cultural contexts - **Educational Mathematics**: Mathematical Ideas actualizing differently in elementary, secondary, university contexts with different parts and emphases ## Connections ### Internal Connections - Develops [[Definition of Differentiation and Differenciation]] - providing concrete examples of how differenciation works - Links to [[Virtual Opposed to Actual, Not Real]] - showing the genuine richness of virtual reality - Connects to [[Progressive Determination of Adjunct Fields]] - how actualization opens new mathematical territories - Extends [[Two Types of Non-Being]] - virtual coexistence as creative alternative to exclusionary logic ### External Connections - **Optics and Color Theory** - white light as virtual multiplicity actualizing in color spectrum - **Acoustics** - white noise containing all frequencies virtually - **Linguistics** - virtual language structures actualizing in diverse actual languages - **Social Theory** - virtual social structures actualizing in diverse social forms ## Questions & Further Investigation 1. How does [[category theory]]'s [[Yoneda lemma]] formalize the relationship between virtual mathematical "white light" and actual mathematical colors? 2. Can [[topos theory]]'s [[geometric morphisms]] model how "white mathematics" actualizes in different logical contexts? 3. Is there connection to [[Fourier analysis]] - white noise mathematically containing all frequencies that actualize as specific signals? 4. How does [[homotopy type theory]]'s approach to [[mathematical foundations]] relate to "white mathematics" actualizing in different mathematical species? 5. Can [[machine learning]]'s [[representation learning]] be understood as discovering virtual "white" structures that actualize in different domains? 6. How does [[collaborative mathematics]] involve collective exploration of "white mathematical" multiplicities actualizing through diverse community contributions? 7. What would mathematical education look like if it engaged with "white mathematics" - virtual mathematical multiplicities before their actualization in specific domains? ## Notes for Synthesis - Central for [[Essay - White Mathematics: Virtual Mathematical Multiplicities and Pluralistic Actualization]] - Links to [[Essay - From White Light to Category Theory: Virtual Unity and Actual Diversity]] - Develops [[Essay - Mathematical Perplication: How Mathematical Ideas Fold Together Before Actualization]] - Connection to recurring theme of [[virtual unity enabling actual diversity]] - Counter-argument to consider: Doesn't mathematical rigor require precise distinctions rather than virtual multiplicities? - Bridge to [[Essay - Democratic White Mathematics: Collective Participation in Virtual Mathematical Multiplicities]] - Your thesis connection: [[Essay - Category Theory as White Mathematics: Virtual Foundation for Mathematical Democracy]] ## Implications for Democratic Mathematical Foundations This conception of virtual actualization has profound implications for your thesis: - **Mathematical Pluralism**: "White mathematics" - virtual mathematical multiplicities like category theory - can actualize in diverse mathematical species (algebra, topology, logic) accessible to different mathematical communities rather than requiring single expert interpretation - **Inclusive Mathematical Actualization**: The same virtual mathematical relationships can incarnate in different mathematical parts and species appropriate to different educational levels, cultural contexts, and practical applications without losing mathematical validity - **Collective Mathematical Exploration**: Mathematical communities can collectively explore "white mathematical" multiplicities - the virtual coexistence of mathematical relationships before their separation into distinct mathematical domains and hierarchies - **Non-Reductive Mathematical Democracy**: Mathematical democracy doesn't require reducing all mathematics to a single accessible form, but rather involves collective participation in virtual mathematical multiplicities that can actualize in diverse forms - **Genetic Mathematical Elements**: Mathematical communities can engage with the "genetic elements" of mathematical Ideas - the productive virtual relationships that generate diverse mathematical actualization rather than just accessing pre-given mathematical products **Concrete Examples**: - **Category Theory as White Mathematics**: Virtual mathematical multiplicity that actualizes differently in pure mathematics, applied mathematics, computer science, physics - **Mathematical Education**: Same mathematical Ideas actualizing differently in elementary, secondary, university contexts while maintaining virtual unity - **Interdisciplinary Mathematics**: Mathematical communities from different disciplines collectively exploring virtual mathematical multiplicities that actualize in their respective domains - **Open Source Mathematics**: Virtual mathematical relationships actualizing through diverse community contributions and implementations **Democratic Mathematical Vision**: Instead of mathematical democracy as equal access to fixed mathematical content, this suggests mathematical democracy as **collective participation in virtual mathematical multiplicities** that can actualize in forms appropriate to diverse mathematical communities while maintaining genuine mathematical unity. The "white" metaphors suggest mathematical foundations are not exclusive expert domains but **rich virtual multiplicities** accessible to diverse communities through different forms of mathematical actualization - supporting truly democratic mathematical participation without sacrificing mathematical rigor or creativity. --- ## References - Screenshot: [[16-01-25 Difference and Repetition (17).jpg]] - Book: [[Difference and Repetition MOC]] - Related Notes: [[Definition of Differentiation and Differenciation]], [[Virtual Opposed to Actual, Not Real]], [[Progressive Determination of Adjunct Fields]] ## Processing Status - [x] Passage transcribed - [x] Initial analysis complete - [x] Key concepts defined - [x] Connections established - [x] Questions identified - [x] Ready for integration