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.
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.
Gregory, the calculus of operations, and the Cambridge Mathematical Journal; developments in the theory of algebras over number fields; Emmy Noether's 1932 ICM lecture on non-commutative methods in algebraic number theory; and the arithmetization
of algebraic geometry.
For undergraduate math students and teachers, this introduction to the history of the subject and influential mathematicians encompasses the Greeks, Indian arithmetic, integral and differential calculus, the analytic geometry of Rene Descartes, non-commutative algebra, the arithmetization
of analysis, and the beginnings of algebra as well as figures such as Newton, Hamilton, Kepler, Fibonacci, and Euclid.