Professor Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization
. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers.
The essays have in common a relatively unexplored aspect of the intellectual formation of nineteenth-century Americans: arithmetization
. Alphabetization itself has only recently been welcomed into literary studies from the margins of cultural history thanks to scholars like Patricia Crain, whose work has thoroughly demonstrated how this low designation belies the alphabet's significant role in the operations of institutional power and its production of lettered subjects.
The synthesis between episteme and techne, the systemic application of the Leibnizian project of mathesis universalis as calculation, and the modern development of productive forces, determined not only the progressive technicization of knowledge--perceived here as a continuous dialectic between prometheia and epimetheia (2)--but also a systemic arithmetization
of life in the enframing of biological functions by computational, homogeneous, and productive systems of algebraic laws.
Specifically, it can be the case that the low-degree part of the arithmetization
of P vanishes or becomes negative for some inputs where the linear/quadratic polynomial is positive (i.e.
The rows of standardized human figures are immediately reminiscent of the factory discipline under which the real workers worked on the assembly line as well as Carnap's contemporary arithmetization
The nineteenth century arithmetization
of analysis initiated by Cauchy led to a separation of mathematics from such physicalistic foundations.
In the field of analysis, Gray distinguishes between early foundational efforts (Cauchy's arithmetization
, Weierstrass's rigorization) and later more abstract developments in analyzing the nature and meaning of numbers (Dedekind on real numbers and natural numbers, Cantor on transfinite ordinal and cardinal numbers).
They continue to present material in a two-semester format, the first on computability theory (enumerability, diagonalization, Turing compatibility, uncomputability, abacus computability, recursive functions, recursive sets and relations, equivalent definitions of computability) and basic metalogic (syntax, semantics, the undecidability of first-order logic, models and their existence, proofs and completeness, arithmetization
, representability of recursive functions, indefinability, undecidability, incompleteness and the unprobability of inconsistency).
In 1833 Hamilton read a paper expressing complex numbers as algebraic couples, and in 1837 presented an article on the arithmetization
of analysis .
Feferman, S., 1960, "Arithmetization
of Metamathematics in a General Setting", Fundamenta Mathematicae, vol.
After a brief but engaging account of Frege's life and career, Noonan's introductory chapter provides a helpful sketch of the origins and development of his leading ideas in their philosophical and mathematical context--Kant's thesis that mathematics, while a priori, must be synthetic and his associated insistence on the role of intuition; the emergence of non-Euclidean geometries; and the drive for rigor and the arithmetization
of analysis by Augustine Cauchy, Karl Weierstrass, and others--followed by a concise overview of Frege's main contributions which serves as a useful background to their more detailed discussion in the chapters that follow.