Peano as a Limit: Discrete Arithmetic and the Resolution of Multiplicity

Peano's axioms define the natural numbers through the successor function, whichrequires uniqueness: each number has exactly one successor. This paper argues thatPeano arithmetic is applicable only in domains where multiplicity has been resolved.Drawing on the Landau-Stuart equation--which describes systems capable of multiplestable states (multistability)--we show that discrete arithmetic presupposes a priorcollapse of multiplicity. Quantum mechanics provides a parallel: superposition statescannot be counted; measurement collapses multiplicity, enabling enumeration. Wepropose that Peano arithmetic is not a universal foundation but a special case: the limitof Landau-Stuart dynamics when multiplicity approaches zero. This reframingaddresses Wigner's puzzle regarding the unreasonable effectiveness of mathematicsin physics: mathematics applies precisely where physical processes have resolved themultiplicity that mathematics cannot represent.Keywords: Peano axioms, Landau-Stuart equation, multiplicity, quantummeasurement, foundations of mathematics, emergence