The δ Foundation: Distinction, Arithmetic, and the Limits of Abstraction

Starting with a distinction that can be repeated, this book builds arithmetic and asks which conclusions follow from the construction and which require additional logical principles. Eight parts develop counting, signed differences, ratios and the line; proof rules and their logical resources; construction, iteration and the continuum; compactness, linear algebra and certified computation; finite physical records; the cost of a comparison and the forcing chain built on it; classical physics on the line; and the library, learning and open problems. Each stage of the number tower, from the tallies through the ratios to the line, is the unique solution of its task; for the line this uses countable choice. Each additional logical principle receives an exact price, proved in both directions. With excluded middle the book's rules are Peano arithmetic, and with no added principle they are Heyting arithmetic. This fifth edition adds three parts. On the line built from ratios, the composition law and its calibration force the cost J(x) = (x + 1/x)/2 − 1 with no omniscience principle and no choice, and the book prices every step of the forcing chain of Recognition Science as it restates it: of 130 rows, 119 are free, seven have an exact price, four are bracketed and none is open. Three is the only dimension in which a circle can be linked, again with no omniscience and no choice. The mechanics of Kepler and Newton, the perihelion advance of relativistic test orbits to second order and the second law hold with explicit constants, and on transverse-traceless strains the long-wave form of J is the Fierz–Pauli form of linearized gravity. The conventions of a classical library and the learning of laws from records receive exact prices, and a final chapter collects conjectures, refutations and open problems. Each result states what it rests on: a proof on the page or in an appendix that its proof names, a cited standard fact, or a stated input. This 1222-page monograph has thirty chapters and sixteen appendices, including exercise solutions. It is the mathematical companion to The Structure of Reality. Version 5.1 adds 104 numbered results with their proofs, a price table of the mathematics that physics uses, and a rebuilt list of open problems. It corrects 46 sentences of version 5.0 that the new results change. The book now has 1,475 pages. Every theorem, section, table and appendix number of version 5.0 is unchanged: the new material is in new sections at the ends of chapters and in a new last appendix.