The second was the twentieth-century arithmetization
of logic and computation.
The essays have in common a relatively unexplored aspect of the intellectual formation of nineteenth-century Americans: arithmetization
This fact concerns the Liar sentence, independently of its formulation as 'this sentence is false' or 'this sentence is not true', both recorded via arithmetization
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.
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 .
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.
Different methods of curve generation have been suggested: a byte-oriented technique [Butz 1971], an arithmetization
of the curve [Sagan 1992], L-system formalization with turtle interpretation [Prusinkiewicz et al.