Introduction
At first glance, Leibniz's philosophy and modern quantum mechanics seem to have little in common. But the idea of monads, as Leibniz describes them in his metaphysics, could be seen as a philosophical counterpart to quantum entanglement. This essay develops the vision that monads not only exist as immaterial, independent substances, but can also be entangled or unentangled - a model that can be formally described using the mathematical structures of Hilbert space.
In addition, this concept offers the possibility of building a bridge between science, philosophy and spirituality and recognizing a deeper unity in human experience. This vision may not only address questions about the metaphysical foundations of the universe, but also provide comfort and inspiration in difficult times. Particularly exciting is the connection to existing approaches, as discussed in the article "Quantum Sociology of Entangled Monads", which sheds light on the unity of state spaces in science, sociology and philosophy.
1. The monads according to Leibniz
Leibniz describes monads as the fundamental, indivisible building blocks of reality. They are immaterial, have no extension and are completely self-contained. Nevertheless, they reflect the entire universe - a concept he calls "pre-established harmony".
1.1. Properties of monads:
Self-sufficiency: Monads do not interact directly with each other.
Mirroring of the universe: Each monad represents the entire universe from its own perspective.
Pre-established harmony: A cosmic order ensures that monads act in harmony with each other, even though they are isolated.
1.2. Quantum entanglement and its parallels
In quantum mechanics, entanglement describes a non-local connection between quantum objects. Two entangled particles share a common state that cannot be reduced to the sum of their individual states. A measurement on one particle influences the state of the other - regardless of the distance between them.
Parallels to monads:
Nonlocality: Exacerbated monads could be synchronized by a nonlocal connection.
Wholeness: Exacerbated states of monads resemble the shared states of entangled quantum objects.
Timeless connection: Like entangled particles, monads also appear to be connected to each other independently of space and time.
1.3. Entangled and unentangled monads
A metaphysical system could describe monads in both entangled and unentangled states:
Entangled monads: These monads are not independent, but connected in a common harmony. Their state is only defined by their relationship to each other.
Unentangled monads: These monads remain autonomous and isolated, but continue to reflect the universe individually.
This model could explain synchronicity, such as the simultaneity of similar thoughts in different people, or why certain decisions are made in harmony without direct communication. The article "Quantum Sociology of Entangled Monads" provides a sociological perspective that interprets these phenomena as the result of entangled state spaces.
1.4. Monads in Hilbert space
Hilbert space provides a mathematical basis for describing monads as states in an abstract space. Here is a suggested formal representation:
1.4.1. Monads as state vectors
Each monad Mi can be represented as a vector in Hilbert space H:
|ψi⟩ ∈ H
1.4.2. Entangled Monads
Entangled monads share a common state, which is described by superposition:
|Ψ⟩ = α |00⟩ + β |11⟩
Here |00⟩ and |11⟩ reflect the coherent connection of the monads, where α and β represent the weights of their states.
1.4.3. Pre-established harmony as operator
The harmony between monads can be represented by an operator Ĥ which guarantees their synchronicity:
Ĥ |ψi⟩ = |ψj⟩ ∀ i, j
This operator ensures that monads remain consistent in an entangled state.
1.4.4. Temporal development
The dynamics of monads could be described by the Schrödinger equation:
iℏ ∂|ψ(t)⟩/∂t = Ĥ |ψ(t)⟩
Here the Hamiltonian operator Ĥ models the pre-established harmony.
1.4.5. Tensor product structure
Hilbert space offers the possibility of describing entangled states by tensor products. If two monads M1 and M2 are entangled, the result is:
HM₁, M₂ = HM₁ ⊗ HM₂
This shows that their conditions are inextricably linked.
1.4.6. Projection operators
To extract the state of a single monad from an entangled system, projection operators can be used:
Pi = |ψi⟩ ⟨ψi|
These operators project the overall state onto the subspace of the monad under consideration.
1.4.7. Experience, residues and collapse:
Experiencing a monad in an entangled state causes the system to collapse. Analogous to complex analysis, the residue of the monad could represent a quantifiable property that is extracted when interacting with the system:
The residue represents the specific contributions or properties of a monad that influence the overall system. This corresponds to the decision about what relationship the monads have with each other and could be viewed analogously to the measurement of qubits.
|Ψ⟩ → |00⟩ oder |11⟩
This corresponds to the decision of what relationship the monads have with each other.
2. Mathematics of Residues and their connection to Monads
Residues are a mathematical concept from complex analysis that is closely related to singularities of functions. In the context of monads, residuals could be understood as quantifiable properties that emerge when they interact with the system. Here are the relevant mathematical principles:
2.1. Definition of a residue
The residue of a function f(z) at an isolated singularity z0 is the coefficient of the term 1/(z - z0) in the Laurent series:
f(z) = ∑ an(z - z0)n, n=-∞ bis ∞
The residue is then given by:
Res(f, z0) = a-1
2.2. The residue theorem
The residue theorem states that the closed curve integral of a function f(z) along a closed curve γ, that includes all singularities can be calculated as:
∮γ f(z) dz = 2πi ∑ Res(f, zk)
Here the contribution of each residue of the included singularities is summed up.
2.3. Connection to monads
In the theory of monads, residues could be understood as specific contributions of individual monads that express their properties or their role in the system. When interacting with other monads or experiencing a monad, this residue is "extracted" and influences the overall system.
2.4. Residues and the collapse of monads
The collapse of a monad could be understood as the "extraction" of its residue, analogous to measurement in quantum mechanics. Formally, this would be expressed by:
Ψ → Res(f, z0)
where Ψ is the state of the monad that is resolved by the residue.
2.5. Extension for entangled monads
For entangled monads, the residuals could quantify their mutual contributions to a common state, similar to coefficients in a superposition:
|Ψ⟩ = α |00⟩ + β |11⟩
The values α and β could be represented here by residuals that express their individual strength and weighting.
3. A metaphysical outlook
In such a system, monads could be viewed as fundamental units of consciousness that are interconnected. This model offers comfort by showing that we are part of a larger, harmonious whole—a universal connection that is not limited by space or time. Even as individual monads evolve and decay, their connection to the universe remains.
Conclusion
The idea of entangled monads unites Leibniz's philosophy with modern quantum mechanics. By embedding these monads in Hilbert space, a model is created that brings together science, philosophy, and spirituality. The connection to the "quantum sociology of entangled monads" shows how sociological and philosophical perspectives can complement this unity. It shows that the universe is not just a mechanical whole, but also a deeply connected and meaningful one.
Literature and sources
Philosophy and metaphysics
- Leibniz, G. W.: Monadologie. Übersetzt und kommentiert von Herbert Herring. Hamburg: Meiner Verlag, 2005.
- Rescher, N.: Leibniz's Metaphysics of Nature: A Group of Essays. Dordrecht: Springer, 1981.
Quantum mechanics and mathematics
- Griffiths, D. J.: Introduction to Quantum Mechanics. Pearson, 2018.
- Shankar, R.: Principles of Quantum Mechanics. Springer, 1994.
- Arfken, G. B., & Weber, H. J.: Mathematical Methods for Physicists. Academic Press, 2013.
Complex Analysis and Residues
- Conway, J. B.: Functions of One Complex Variable. Springer, 1978.
- Lang, S.: Complex Analysis. Springer, 1999.
Connection between philosophy and quantum mechanics
- Bohm, D.: Wholeness and the Implicate Order. Routledge, 1980.
- Penrose, R.: The Road to Reality: A Complete Guide to the Laws of the Universe. Knopf, 2004.
Hilbert spaces and operators
- Sakurai, J. J., & Napolitano, J. J.: Modern Quantum Mechanics. Cambridge University Press, 2020.
- Reed, M., & Simon, B.: Functional Analysis (Methods of Modern Mathematical Physics). Academic Press, 1981.
Quantum Sociology and Interdisciplinary Perspectives
- Barad, K.: Meeting the Universe Halfway: Quantum Physics and the Entanglement of Matter and Meaning. Duke University Press, 2007.
- Tenckhoff, J.: Quantensoziologie verschränkter Monaden. Verfügbar unter tenckhoff.de., 2024.
