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Notes

A personal research notebook. Emphasis on explanation over formalism.

• Definition

  💡Explanatory Power

  An explanation has explanatory power when it accounts for observations not by fitting parameters to data, but by providing a hard-to-vary account of the underlying reality.

  For instance, consider the equation $E = mc^2$. This is not merely a formula that fits experimental data; it reveals a deep relationship between energy $E$, mass $m$, and the speed of light $c$.

  The explanatory power lies in the fact that this relationship is constrained by the structure of spacetime itself, not by arbitrary parameters.

• Note

  λSK Combinators and Computational Universality

  Schönfinkel and Curry discovered that all computable functions can be expressed using just two primitive combinators: $S$ and $K$.

  $$S = \lambda x. \lambda y. \lambda z. (x z) (y z)$$

  $$K = \lambda x. \lambda y. x$$

  The $S$ combinator distributes application, while $K$ discards its second argument. Together, they form a complete basis for computation; any λ-term can be translated into SK combinator terms.

  This reveals something profound: computational universality does not require many primitives. Complex behavior emerges from the composition of extremely simple rules. Wolfram's computational equivalence principle generalizes this: simple programs can be as powerful as any computation.

• Conjecture

  ◈Wolfram Physics Project and Hypergraphs

  The Wolfram Physics Project proposes that spacetime emerges from the evolution of hypergraphs: abstract networks where nodes can be connected by edges linking more than two nodes simultaneously.

  Starting from simple rewrite rules applied to these hypergraphs, the project aims to derive fundamental physics: general relativity, quantum mechanics, and potentially quantum gravity. The computational irreducibility of these systems means we cannot predict their behavior analytically; we must simulate.

  A key insight is causal invariance: if different sequences of rule applications lead to equivalent states, this corresponds to reference frame independence in physics. The observer's perception of space and time emerges from their computational perspective within the hypergraph.

  This suggests our universe is fundamentally discrete and computational at the Planck scale, with continuous spacetime as an emergent approximation. If true, it would unify computation and physics: the universe does not compute. It is computation.

• Note

  ⇌Isomorphism and Structural Identity

  Two structures $\mathcal{A}$ and $\mathcal{B}$ are isomorphic if there exists a bijection $f: A \to B$ preserving all relations and operations:

  $$R^{\mathcal{A}}(a_1, \ldots, a_n) \iff R^{\mathcal{B}}(f(a_1), \ldots, f(a_n))$$

  This definition captures structural identity: isomorphic structures are indistinguishable from the perspective of their internal relations. But what does this mean for mathematical truth?

  If we accept the invariance principle (that mathematical properties should be invariant under isomorphism) then statements distinguishing isomorphic structures are not truly mathematical. This motivates category theory: objects matter only up to isomorphism, and morphisms (structure-preserving maps) become primary.

  Consider the natural numbers. We cannot ask “is 3 a set?” in a mathematically meaningful way, because any isomorphic copy of $\mathbb{N}$ has equal claim to being “the” natural numbers. Structure, not substance, determines mathematical identity.