Compression is all you need: Modeling Mathematics

Human mathematics (HM), the mathematics humans discover and value, is a vanishingly small subset of formal mathematics (FM), the totality of all valid deductions. We argue that HM is distinguished by its compressibility through hierarchically nested definitions, lemmas, and theorems. We model this with monoids. A mathematical deduction is a string of primitive symbols; a definition or theorem is a named substring or macro whose use compresses the string. In the free abelian monoid $A_n$, a logarithmically sparse macro set achieves exponential expansion of expressivity. In the free non-abelian monoid $F_n$, even a polynomially-dense macro set only yields linear expansion; superlinear expansion requires near-maximal density. We test these models against MathLib, a large Lean~4 library of mathematics that we take as a proxy for HM. Each element has a depth (layers of definitional nesting), a wrapped length (tokens in its definition), and an unwrapped length (primitive symbols after fully expanding all references). We find unwrapped length grows exponentially with both depth and wrapped length; wrapped length is approximately constant across all depths. These results are consistent with $A_n$ and inconsistent with $F_n$, supporting the thesis that HM occupies a polynomially-growing subset of the exponentially growing space FM. We discuss how compression, measured on the MathLib dependency graph, and a PageRank-style analysis of that graph can quantify mathematical interest and help direct automated reasoning toward the compressible regions where human mathematics lives.

Adversarial KA

Regarding the representation theorem of Kolmogorov and Arnold (KA) as an algorithm for representing or {\guillemotleft}expressing{\guillemotright} functions, we test its robustness by analyzing its ability to withstand adversarial attacks. We find KA to be robust to countable collections of continuous adversaries, but unearth a question about the equi-continuity of the outer functions that, so far, obstructs taking limits and defeating continuous groups of adversaries. This question on the regularity of the outer functions is relevant to the debate over the applicability of KA to the general theory of NNs.

The Proof of Kolmogorov-Arnold May Illuminate Neural Network Learning

Kolmogorov and Arnold, in answering Hilbert's 13th problem (in the context of continuous functions), laid the foundations for the modern theory of Neural Networks (NNs). Their proof divides the representation of a multivariate function into two steps: The first (non-linear) inter-layer map gives a universal embedding of the data manifold into a single hidden layer whose image is patterned in such a way that a subsequent dynamic can then be defined to solve for the second inter-layer map. I interpret this pattern as "minor concentration" of the almost everywhere defined Jacobians of the interlayer map. Minor concentration amounts to sparsity for higher exterior powers of the Jacobians. We present a conceptual argument for how such sparsity may set the stage for the emergence of successively higher order concepts in today's deep NNs and suggest two classes of experiments to test this hypothesis.