Classical science (200 BC):

The Apollonian Winter, the Hellenistic age (323-31 BC)  witnessed the flowering of science, the completion of the mathematical form-world, their concluding thoughts.  It was an age of great cities but little spiritual creativity.  Science benefited from the cross-fertilization of Greek ideas within the larger Hellenistic world & was supported by Alexander & later royal patrons of the Diadochi kingdoms.  A wealthy urban population seeking luxury & comfort demanded practical knowledge to solve the problems of daily urban life.  It thrived in the great urban centres throughout the Mediterranean, Syracuse, Pergamum & especially 3rd century BC Alexandria.  Astronomy, mathematics, geography, medicine & physics were the main areas which advanced in this age.  Most famous of the astronomers was Aristarchus (310-230 BC), the first to grasp the enormous size of the universe, who posited the immobility of the "fixed" stars is due to their vast distance from the earth & proposed a heliocentric theory (which was never accepted, it conflicted with Aristotle & did not sit well with the Jews & other Orientals, a significant proportion of the Hellenistic population).  Euclid (325-265 BC) developed proofs for the Pythagorean Theorem, the infinitude of primes, worked on the 5 Platonic solids; his Elements became the foundation text of geometry.  Archimedes (287–212 BC) discovered displacement or specific gravity, & scientific formulations for the lever, pulley & screw.  Herophilos (335–280 BC) performed the first dissections of humans & animals & provide accurate descriptions of the nervous system, liver & other organs.  The Empiric school of Alexandria (emerged 250 BC) based their medical theories purely on observation, rejecting unseen causes

 

Faustian science (Gauss, Cauchy and Riemann):

These 3 mathematicians worked during the Winter period (mid to late 19th century) & completed work done by an earlier generation.  In 1827 Gauss (1777- 1855) defined curved surfaces in a manner similar to the modern topological understanding.  The investigation of the validity of infinite series also begins with Gauss, who established simpler criteria of convergence, and the questions of remainders & range of convergence.  In 1821, the Frenchman Cauchy (1789-1857) put calculus on a firm logical foundation by formulating calculus in terms of geometric ideas & infinitesimals.  He also introduced the concept of the Cauchy sequence, and started the formal theory of complex analysis.  While Newton & Leibniz provided a systematic approach to integration they lacked rigour; the development of limits was needed.  In the mid-19th century Riemann (1826- 1866)  introduced limits with his theory of integration, providing a rigorous formalization.

 

Skepsis:

archaic, from Greek skeptikos, one who reflects upon; member of one of the ancient Greek schools of philosophy (in particular Pyrrho) who believed that real knowledge of things is impossible

 

enumerative:

to ascertain the number of; count

 

probability (enumerative & statistical methods):

Einstein was the first to create a purely quantum explanation in his fundamental paper “On the Quantum Theory of Radiation” (1916).  It changed the nature of the evolving quantum theory by introducing the fundamental role of random chance.  The paper discusses the exchange of momentum between atoms and radiation by making use of the theory of Brownian motion.  He shows that in every elementary process of radiation, and in particular in spontaneous emission, an amount of momentum (ho/c) is emitted in a random direction.  The spontaneous emission has on the average no preferred direction & therefore does not in the average transmit momentum to the atom.  It leaves the time and direction of elementary processes to ‘chance’;

 

motion-problem: * see EndNote<A>

This refers to the dispute between absolute conceptions of space, time and motion, and relational conceptions.  It was addressed by Aristotle and his Physics which dominated Western thinking until the Enlightenment.  Galileo disputed Aristotle & proved him wrong.  Descartes began eroding Aristotle with his Dualism which questioned the teleological approach.  Newton cleared the deck & dominated until the late 19th century, until Einstein.

and see above pages 416 and 388

 

gravitation (Newton’s hypothesis limited): * see EndNote<B>

General relativity was a theory of gravitation developed by Einstein between 1907 & 1915; it states that the observed gravitational effect between masses results from their warping of spacetime, and not force (as posited by Newton).  Experiments & observations show that this description accounts for several effects unexplained by Newton's law, notably the minute anomalies in the orbits of Mercury & other planets.  It also predicts novel effects of gravity, such as gravitational waves, gravitational lensing and an effect of gravity on time known as gravitational time dilation.  Many of these predictions have been confirmed by experiment or observation.

 

Conservation of Energy (validity): * see Endnote <C>

a law of Newtonian physics, stating that physical measurable quantities (energy, momentum, angular momentum, mass & electric charge) do not change in the course of time within an isolated physical system.  The law requires the system to be isolated, meaning it is so far removed from other systems that it does not interact with them, or (if a thermodynamic system) is enclosed by rigid immovable walls through which neither mass nor energy can pass.  Spengler is drawing attention to this clause: “a system of bodies self-contained and not externally influenced”.  Holistically, an infinite universe cannot be self- contained by definition.  In an infinite universe light emissions (energy) from stars travel outward, into infinity, never to return; thereby depleting the energy of the system.  Spengler is thus claiming the infinite universe (see below) invalidates Newtonian law

 

an infinite space:* See EndNote<D>

The size of the universe & its motion came under scrutiny in the first decades of the 20th century.  The static & finite universe of Newton was gradually losing the debate; astronomy came to accept that the Milky Way was but 1 among millions of galaxies (the islandunvierse), vast distances away & moving away at a high rate of speed (the expanding universe).  Photographic & spectroscopic analysis expanded the field of study.  Astronomers like H Curtis & VM Slipher made ground breaking discoveries.  Observational evidence was further supported in 1922 when Alexander Friedmann (using Einstein field equations) provided theoretical evidence that the universe was expanding.

 

Non-Euclidean geometries:

2 geometries based on axioms related to Euclidean geometry; they arises by either relaxing the metric requirement (which results in kinematic geometries), or replacing the parallel postulate with an alternative (which gives rise to hyperbolic geometry & elliptic geometry).

see Chapter II page 67, Chapter V page 170, Chapter XI page 385

 

“finite and yet unbounded”:

quote from Einstein’s Relativity: The Special and General Theory (1920), the title of Chapter 31 (The Possibility of a “Finite” and Yet “Unbounded” Universe).  He suggests a “spherical” universe, one in which we can venture out in a straight line, and circumnavigate back to our starting position.  A finite yet unbounded universe is one that has a finite volume, a specific number of galaxies, but has no 3D “edge”.  Such a universe is called spherical, or “closed”.  No matter what direction you travel in the universe, you will always end up back where you started.

 

luminiferous aether:

aka aether; postulated medium for the propagation of light, invoked to explain the ability of wave-based light to propagate through empty space.  As it was believed waves could not travel through a vacuum, such propagation needed a medium, an invisible & infinite material with no interaction with physical objects.

 

Lord Kelvin:

(1824-1907) aka William Thomson, Irish-British mathematical physicist & engineer born in Belfast, Professor of Natural Philosophy at the University of Glasgow; made important contributions to the mathematical analysis of electricity & helped with the formulation of the 1st & 2nd laws of thermodynamics;

 

Lord Kelvin (proved mathematically no aether):

In 1901 Lord Kelvin wrote “On Ether and Gravitational Matter through Infinite Space.”, an amplification of a lecture given in 1884 at Baltimore.  It summarizes his theoretical musings on the nature of the aether, an attempt to deduce properties of the aetherial matter based solely on a general understanding of the properties of wave motion.  He attempts to determine the density of the aether by broad analogy with other types of waves.  His conclusions make for a very strange material: aether is a solid, but incredibly less dense than hydrogen gas; it has inertial mass, but not gravitational mass; it is a massive solid that does not interact with ordinary matter.  Yet despite all the evidence piling up against aether, Kelvin never doubts its existence.  He simply highlights how perplexing it was.

 

Fresnel:

(1788- 1827) French civil engineer & physicist, his research in optics led to the almost unanimous acceptance of the wave theory of light (which excluded Newton's corpuscular theory of light); he was famous for his catadioptric (reflective/refractive) Fresnel lens and for pioneering the use of "stepped" lenses to extend the visibility of lighthouses, saving countless lives at sea.

 

Fresnel (experiments with light-waves): * see EndNote<E>

Between 1816-21 Fresnel experimented with various devices for producing interference fringes & diffraction; he became convinced that the wave theory of light was correct.  His mathematical description used Huygens’s principle that every point on a wave front can be considered a secondary source of spherical wavelets.  In 1816 he studied the laws of the interference of polarized light & in 1817 he was the first to obtain circularly polarized light, a discovery which led him to the conclusion that light was not a longitudinal wave as previously supposed but a transverse wave.  By supposing that light waves are purely transverse he explained the nature of polarization.  However pure transverse waves, implied that the aether was an elastic solid, but, unlike other elastic solids, incapable of transmitting longitudinal waves.  This was difficult to accept so long as the luminiferous aether was accepted as a fact.

 

transversal (light waves):* see EndNotes<F>

wave whose oscillations are perpendicular to the direction of the wave's advance (in contrast to a longitudinal wave which travels in the direction of its oscillations); such waves commonly occur in elastic solids due to the shear stress generated;

 

rigid body (quaint properties):

Although Maxwell seemed to be moving away from the aether concept, its influence as a legacy paradigm was strong.  He retained the concept even as he was disproving it!  His aether had to be "still" universally.  He proposed several mechanical models of aether based on wheels & gears.  GF FitzGerald went so far as to construct a working model of one of them.  These models had to agree with the fact that the electromagnetic waves are transverse but never longitudinal (see below)

 

longitudinal (light waves): * see EndNotes<G>

aka compression or pressure waves; waves in which the displacement of the medium is in the same (or opposite) direction of the wave transmission; they produce compression & rarefaction when traveling through a medium,  they also produce increases and decreases in pressure.  Sound travels through longitudinal waves.

 

Maxwell-Hertz equations (Electromagnetic theory of light & aether):

four equations produced by Maxwell, giving a description of the production & interrelation of electric & magnetic fields.  Between 1845 & 1847 Faraday investigated light and magnetism; he proposed light was a high-frequency electromagnetic vibration, able to propagate even in the absence of a medium such as the aether.  This inspired Maxwell to study electromagnetic radiation & light.  He discovered self-propagating electromagnetic waves would travel through space at a constant speed (the speed of light) & concluded light was a form of electromagnetic radiation (1862).  In 1873 he published a full mathematical description of the behaviour of electric & magnetic fields (Maxwell's equations).  Between 1886 & 1889 Hertz confirmed Maxwell's theory experimentally by generating & detecting radio waves in the lab & demonstrating they behaved exactly like light, exhibiting properties such as reflection, refraction, diffraction and interference.  Maxwell's theory & Hertz's experiments led to the development of radio, radar, television, electromagnetic imaging & wireless communications.