The mind-body problem for mathematicians

Mathematicians may be more likely to understand the category errors that are usually made by philosophers in the mind-body problem:

Dimension 1: Type of description

• Physique → objective, material description (e.g. neurobiology, electrophysiology)
• Psyche → subjective, experiential description (e.g. emotion, thought, sensation)

Dimension 2: Perspective of description

• 3. Person → outside perspective, observing (e.g. natural science)
• 1. Person → an inside perspective, experiential (e.g. introspection)

Formalization: The 2×2 model of reference systems

We present the four systems as a Cartesian product:

Reference systems={Physique,Psyche}×{1st person,3rd person}

This results in four combinations, which can be written as a matrix, for example:

The Error in Thinking (Category Error)

What many philosophers (and some scientists) do:

• They confuse these four systems with four ontological entities (i.e., with really separate things).
• Particularly fatal is:
  • Physique (3rd person) is  considered "the real"
  • Psyche (1st person) is  labeled as "the inexplicable", "the emergent", "the secret", etc

For example, they try to explain a transition between:

Physique (3rd P) → Psyche (1st P)

which, however, is not a causal transition, but a change of perspective.

The error in thinking is therefore to explain the functioning within a system (I) between different reference systems – which is logically impossible.

What is correct instead?

All four systems describe one and the same thing:

The I – once from the outside, once from the inside; once as matter, once as experience.

From a formal point of view:

There is an entity, let's call it I, whose properties are encoded in four types of statements:

I = functional unit ∈ R^4

with four coordinates:

I=(P3,P1,Ψ3,Ψ1)

whereby:

• P3: Physique, 3rd person (e.g. EEG, hormones)
• P1: Physicality, 1st person (e.g. hunger, fatigue)
• Ψ3: Psyche, 3rd person (e.g. behavioural tests)
• Ψ1: Psyche, 1st person (e.g. feelings, consciousness)

Fallacy:

Attempts to postulate causalities between these components (e.g., P3→Ψ1  as the "hard problem of consciousness") misunderstand that these are just coordinates of the same entity.

Ultimately, these fallacies lead to religion and esotericism.

Mathematical Transformation Approach

We start from the already defined 2×2 model:

Reference systems = {physique, psyche} × {1st person, 3rd person}

This results in four coordinates of a unified System I:

• P₃: Physical, 3rd person (objective physical description)
• P₁: Physique, 1st person (subjective physical description)
• Ψ₃: Psyche, 3rd person (objective psychological description)
• Ψ₁: Psyche, 1st person (subjective description of experience)

Transformations between the reference systems

We define transformations T as mappings between these reference frames:

1. Transformations between objective and subjective physique

• T₁: P₃ → P₁ (How are objective physical states subjectively experienced?)
  • Example: Neuronal pain stimuli → pain perception
  • Mathematical properties: Non-injective, contextual
  • Philosophical Position: Phenomenal Reductionism
• T₂: P₁ → P₃ (How do subjective physical sensations become objectively measurable?)
  • Example: Subjective feeling of hunger → measurement of ghrelin levels
  • Mathematical properties: Incomplete, probabilistic
  • Philosophical position: Psychophysics

2. Transformations between objective physique and objective psyche

• T₃: P₃ → Ψ₃ (How do physical states correlate with observable behavior?)
  • Example: Neuronal activity patterns → behavioral reactions
  • Mathematical properties: complex, non-linear, emergent
  • Philosophical Position: Biological Naturalism
• T₄: Ψ₃ → P₃ (How does behavior manifest itself in physical states?)
  • Example: Observable behavior → neuronal correlates
  • Mathematical properties: Underdetermined, multiple feasibility
  • Philosophical Position: Functionalism

3. Transformations between the objective and subjective psyche

• T₅: Ψ₃ → Ψ₁ (How are observable psychological states subjectively experienced?)
  • Example: Observable emotional reactions → subjective emotional experience
  • Mathematical properties: privacy, incompleteness
  • Philosophical Position: Phenomenology
• T₆: Ψ₁ → Ψ₃ (How do subjective experiences manifest themselves in observable behavior?)
  • Example: Inner thoughts → verbal utterances
  • Mathematical properties: expressive, selective, in need of interpretation
  • Philosophical Position: Expressionivism

4. Transformations between Subjective Physique and Subjective Psyche

• T₇: P₁ → Ψ₁ (correlations between subjective physical sensations and subjective experience)
  • Example: Physical fatigue and emotional upset as parallel descriptions
  • Mathematical properties: Correlative, non-causal
  • Philosophical position: Dual aspect theory (not to be confused with "embodied cognition", which problematically mixes determinant and determinandum)
• T₈: Ψ₁ → P₁ (correlations between subjective experiences and body perception)
  • Example: Parallel description of anxiety and physical tension
  • Mathematical properties: Correlative, non-reductive
  • Philosophical Position: Psychophysical Parallelism

5. Diagonal transformations

• T₉: P₃ → Ψ₁ (The "hard problem of consciousness")
  • Example: Neural activity → qualia/consciousness
  • Mathematical properties: explanatory gap, categorical difference
  • Philosophical position: Non-reductive physicalism
• T₁₀: Ψ₁ → P₃ (The Problem of Mental Causation)
  • Example: Voluntary decision → neuronal activity
  • Mathematical properties: causally underdetermined
  • Philosophical Position: Libertarian Compatibilism
• T₁₁: P₁ → Ψ₃ (correlations between subjective physical sensation and observable social behavior)
  • Example: Correlation of pain sensations with observable behavioral changes
  • Mathematical properties: correlative, probabilistic
  • Philosophical position: behavioral analysis (without the mixing of categories of "embodied sociality")
• T₁₂: Ψ₃ → P₁ (correlations between social signals and subjective body perception)
  • Example: Correlation between social interaction and subjective physical sensations
  • Mathematical properties: associative, non-causal
  • Philosophical Position: Descriptive Phenomenology

Mathematical properties of transformations

1. Formal characterization

• Completeness: Some transformations are incomplete in principle (e.g. T₉: P₃ → Ψ₁)
• Injectivity/Surjectivity: Many transformations are neither injective nor surjective
• Context dependency: Transformations are dependent on the state and history of the system I

2. Uncertainty Principle

For complementary description systems, a kind of "uncertainty principle" applies:

The more precisely a state is described in reference system A, the less precise its description becomes in reference system B.

Formal: For the accuracy Δ of the descriptions, the following applies: ΔP₃ · ΔΨ₁ ≥ k (where k is a constant)

3. Non-commutativity

The order of the transformations is decisive: T₁ ∘ T₃ ≠ T₃ ∘ T₁

4. Emergence

Some transformations create qualitatively new properties: T₃(P₃) ⊃ {properties that are not explicitly contained in P₃}

Philosophical implications

1. Reinterpretation of classical positions

• Reductionism: Claims T₉: P₃ → Ψ₁ is completely possible
• Dualism: Claims T₉: P₃ → Ψ₁ is in principle impossible
• Emergence: Describes T₃: P₃ → Ψ₃ as a qualitatively new property level
• Functionalism: Focused on T₃ and T₄ as a sufficient explanation
• Phenomenology: Prioritizes Ψ₁ as an independent access route to reality

2. The "hard problem of consciousness"

The "hard problem" appears as an explanatory gap in the transformation T₉: P₃ → Ψ₁ because:

• The transformation transcends categorical boundaries
• The target domain (Ψ₁) has private, non-public properties
• The qualia properties of Ψ₁ in P₃ have no structural equivalents

3. Complementarity instead of reduction

The four frames of reference offer complementary, irreducible perspectives on the same reality:

• They are all equivalent and necessary for a complete description
• No perspective can completely replace the others
• Together, they form a coherent overall picture of Entity I

Applications of the transformation model

1. Neuroscience research

• Clarification of the research questions by clarifying which reference systems are being investigated
• Recognition of the Limits of Principle of Certain Claims to Declaration
• Avoidance of categorical errors in the interpretation of neuroscientific data

2. Clinical Applications

• Clearer distinction between different levels of description (e.g. physiological vs. phenomenological)
• More precise language that prevents category confusion
• Recognition of the different, irreducible description systems

3. Artificial intelligence

• Clarification of conceptual confusions in the AI discussion
• Precise distinction between functional description (Ψ₃) and phenomenal consciousness (Ψ₁)
• Avoidance of categorical errors such as the "extended mind" concept, which mistakenly equates external tools (P₃) with mental processes (Ψ₁/Ψ₃)

Summary

The transformation approach:

1. Formalizes the relationships between the four reference systems
2. Identifies the mathematical properties and limits of these transformations
3. Explains why certain philosophical problems arise
4. Provides a framework for the integration of different research approaches
5. Avoids the category error of confusing perspectives with ontological entities

By considering I as a unified entity with four complementary systems of description and analyzing the transformations between these systems, we can reconceptualize the mind-body problem – not as a metaphysical puzzle, but as a consequence of the complementarity of different systems of description of the same reality.