function-theory: * see EndNote<A>

aka real analysis (Theory of functions of a real variable) branch of mathematical analysis dealing with real numbers & real-valued functions of a real variable; studies behaviour of real numbers, sequences & series of real numbers, and real functions; it includes convergence, limits, continuity, smoothness, differentiability & integrability.

 

Theory of Groups (functions):

groups are defined as a set with an operator that combines any 2 elements to form a 3rd element; it is associative & has an identity & inverse elements.  These group axioms hold for number systems & many other mathematical structures.  An example of a group are all integers together with the addition operation.  Group theory has 3 main historical sources all dating from the 19th century; these strands were unified around 1880.  Since then, the impact of group theory has been considerable.

 

Theory of Aggregates (values of the variables):

see above page 426- aggregates

 

algebra of logic: * see EndNote<B>

explicit algebraic system showing the underlying mathematical structure of logic, introduced by Boole (1815–1864) in “The Mathematical Analysis of Logic” (1847). His methodology was successfully continued in the 19th century in the work of Jevons (1835–1882), Peirce (1839–1914), Schröder (1841–1902), establishing a tradition in mathematical logic.

 

geometrical axioms & epistemology:

For 2000 years Euclid’s axioms were accepted as intuitively & metaphysically true; in the 19th century it was realized these axioms did not prove his theorems.  Starting with Pasch in 1882, many improved axiomatic systems emerged, the best known being those of Hilbert, Birkhoff & Tarski.  In addition to this revolution, earlier in the century, non-Euclidean geometries were discovery of by Lobachevsky (1792–1856), Bolyai (1802–1860) & Gauss (1777–1855).  In the early 20th century Einstein's theory of special relativity involved a 4-dimensional space-time, the Minkowski space, which is non-Euclidean.

 

unity of science:

see above page 425

 

Faustian symbol of Infinite space:

Spengler identifies this as the Prime Symbol of Faustian culture, analogous to the naked sculpture of the Apollonian

 

anthropomorphic:

ascribing human form or attributes to a being not human, especially to a deity

 

Skepsis:

see above page 418

 

morphological relationships:

relationships between the parts of a system

 

Dynamics:

branch of physics developed in classical mechanics, concerned with forces & their effects on motion; Newton was the first to formulate the physical laws governing dynamics, especially his second law of motion.  Spengler is treating this conceptual entity as a symbol, equivalent as Faustian symbols to the Gothic cathedral or Romanesque sculpture.

 

Analysis:

branch of mathematics which studies infinite processes, it includes calculus & is applicable to real numbers, complex variables, trigonometric functions & algorithms, as well as non-classical concepts (constructivism, harmonics, infinity, vectors).  Spengler is treating this conceptual entity as a symbol, equivalent as Faustian symbols to the Gothic cathedral or Romanesque sculpture.

 

Romanesque ornament:

see above page 380, 396

 

Christian-German dogma:

imprecise reference to Reformation of the initial Summer period of Faustian Culture, the age of Luther, Karlstadt, Calvin

 

the dynastic state:

reference to the Faustian political entity during the Baroque period (1500-1800) or Faustian Summer, a time when the state reached its maturity & fullest expression under the ruler ship of the Bourbons, Hapsburgs & Stuarts