The Splendors and Miseries of Heavisidisation

TL;DR

Addresses whether scientific answers can be reformulated as an ${\cal G}: X\longrightarrow Z$ expressed by iterated Heaviside functions and recovered via a smoothed steepest-descent lift with gauge fixing. Develops a toolkit of elementary blocks (logic, arithmetic, zero-detection, sector functions) to build Heaviside networks, and extends the approach toward algebraic numbers by examining quadratic and Cardano-type representations; it also analyzes gauge freedom and the role of activation smoothing. The paper demonstrates both potential and limitations by formalizing gradient dynamics on simple examples and by a practical TensorFlow discussion that highlights human-guided training as essential. It argues that Heavisidisation offers a principled route to reformulate parts of science in constructive terms, while identifying key theoretical and practical challenges—especially for higher-degree problems and precise algebraic representations. Overall, it lays a framework for exploring algebraic structures and scientific knowledge through iterated Heaviside constructions, pointing to future work in formalizing gauge fixing, extensibility to complex problems, and connections to deep mathematical structures.

Abstract

Machine Learning (ML) is applicable to scientific problems, i.e. to those which have a well defined answer, only if this answer can be brought to a peculiar form ${\cal G}: X\longrightarrow Z$ with ${\cal G}(\vec x)$ expressed as a combination of iterated Heaviside functions. At present it is far from obvious, if and when such representations exist, what are the obstacles and, if they are absent, what are the ways to convert the known formulas into this form. This gives rise to a program of reformulation of ordinary science in such terms -- which sounds like a strong enhancement of the constructive mathematics approach, only this time it concerns all natural sciences. We describe the first steps on this long way.

Figures (4)

• Figure 1: The ( node number, bias value) plot of descent of initial bias values (blue) to the line (orange) expected in (\ref{['eval_map']}) via TensorFlow training with ’sigmoid’ activation function (smoothed version of $\theta$).
• Figure 2: The ( node number, bias value) plot as a result of TensoFlow model.fit() calculation (orange) with the same initial state (blue) of $\{b_i,\ i=1\dots 10\}$ using training sequences [2, 5, 5, 5, 5] and [2, 2, 2, 2, 5] correspondingly.
• Figure 3: ( sample number, $f$-value) plot for 20 samples of testing (blue) vs. prediction (orange) values produced by a 2-layer network trained on the map: $f(b,c)=b^2+c$
• Figure 4: ( sample number, $f$-value) plot for 20 samples of network output, trained for the map $f(b,c)=b^2+c$ with sigmoid activation (blue), compared to the output computed for the same weights and biases, but with Heaviside activation (orange).