Isaac Newton and Gottfried Leibniz built calculus using intuitive infinitesimals that nineteenth-century mathematicians banished as logically inconsistent; the 'chunk and permeate' framework constructs a rigorous, contradiction-tolerant field of infinitesimals.
Calculus was born from the intuitive idea of 'infinitesimals'—quantities smaller than any positive number yet greater than zero. However, treating infinitesimals as simultaneously zero and non-zero led nineteenth-century purists like Weierstrass to banish them in favor of epsilon-delta limits.
Standard Non-Standard Analysis recovered infinitesimals using complex hyperreal ultrafilters, but non-constructive model-theoretic machinery obscures the simple, intuitive calculations used by practicing physicists.
This mathematical work applies paraconsistent logic's 'chunk and permeate' strategy to calculus, partitioning mathematical proofs into consistent chunks and controlling the flow of infinitesimal information across boundaries to prevent logical explosion.
This constructive field of infinitesimals rehabilitates intuitive infinitesimal reasoning for computer algebra systems, automatic differentiation in AI, and pedagogical physics education.
A Constructive Field of Infinitesimals: Chunk and Permeate Approach
While intuitive, naïve infinitesimal reasoning is classically inconsistent, and rigorous nonstandard analysis relies on non-constructive machinery. We resolve this tension by constructing an explicit, totally ordered field using only real sequences and Cauchy convolution. We model the combined real and hyperreal axioms via the Chunk and Permeate strategy, a paraconsistent technique that isolates contradictions without global collapse. Equipping with a two-tier topology, we develop a calculus where infinitesimal derivatives and integrals permeate cleanly to their classical counterparts. We further introduce a -continuity hierarchy capturing infinitesimal smoothness invisible to standard or transfer-based models. Finally, yields a direct algebraic consistency proof for Sergeyev's Grossone arithmetic, and we establish strict computability bounds on field operations. By guaranteeing infinitesimal contradictions never reach the classical chunk, this work bridges paraconsistent logic, constructive mathematics, and nonstandard analysis into a transparent, computationally tractable framework for infinitesimal reasoning.
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Probabilists assumed that the higher-order mathematical moments of normal bell curves were always universally positive; mathematicians discovered an elegant low-dimensional counterexample where the math unexpectedly turns negative. By using computer-assisted optimization to shatter the Gaussian Moments Conjecture, probability theorists have corrected a fundamental assumption in random matrix theory and quantum information science.
Statistical learning theory assumed that accumulating infinite data guarantees convergence to underlying empirical truths; this mathematical proof reveals rigorous topological boundaries where specific hypotheses remain fundamentally unlearnable.
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