math.ca   100

« earlier    

Isaac Newton: the first physicist.
[...] More fundamentally, Newton's mathematical approach has become so basic to all of physics that he is generally regarded as _the father of the clockwork universe_: the first, and perhaps the greatest, physicist.

The Alchemist

In fact, Newton was deeply opposed to the mechanistic conception of the world. A secretive alchemist [...]. His written work on the subject ran to more than a million words, far more than he ever produced on calculus or mechanics [21]. Obsessively religious, he spent years correlating biblical prophecy with historical events [319ff]. He became deeply convinced that Christian doctrine had been deliberately corrupted by _the false notion of the trinity_, and developed a vicious contempt for conventional (trinitarian) Christianity and for Roman Catholicism in particular [324]. [...] He believed that God mediated the gravitational force [511](353), and opposed any attempt to give a mechanistic explanation of chemistry or gravity, since that would diminish the role of God [646]. He consequently conceived such _a hatred of Descartes_, on whose foundations so many of his achievements were built, that at times _he refused even to write his name_ [399,401].

The Man

Newton was rigorously puritanical: when one of his few friends told him "a loose story about a nun", he ended their friendship (267). [...] He thought of himself as the sole inventor of the calculus, and hence the greatest mathematician since the ancients, and left behind a huge corpus of unpublished work, mostly alchemy and biblical exegesis, that he believed future generations would appreciate more than his own (199,511).

[...] Even though these unattractive qualities caused him to waste huge amounts of time and energy in ruthless vendettas against colleagues who in many cases had helped him (see below), they also drove him to the extraordinary achievements for which he is still remembered. And for all his arrogance, Newton's own summary of his life (574) was beautifully humble:

"I do not know how I may appear to the world, but to myself I seem to have been only like a boy, playing on the sea-shore, and diverting myself, in now and then finding a smoother pebble or prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."

Before Newton

...

1. Calculus. Descartes, in 1637, pioneered the use of coordinates to turn geometric problems into algebraic ones, a method that Newton was never to accept [399]. Descartes, Fermat, and others investigated methods of calculating the tangents to arbitrary curves [28-30]. Kepler, Cavalieri, and others used infinitesimal slices to calculate volumes and areas enclosed by curves [30], but no unified treatment of these problems had yet been found.
2. Mechanics & Planetary motion. The elliptical orbits of the planets having been established by Kepler, Descartes proposed the idea of a purely mechanical heliocentric universe, following deterministic laws, and with no need of any divine agency [15], another anathema to Newton. _No one imagined, however, that a single law might explain both falling bodies and planetary motion_. Galileo invented the concept of inertia, anticipating Newton's first and second laws of motion (293), and Huygens used it to analyze collisions and circular motion [11]. Again, these pieces of progress had not been synthesized into a general method for analyzing forces and motion.
3. Light. Descartes claimed that light was a pressure wave, Gassendi that it was a stream of particles (corpuscles) [13]. As might be guessed, Newton vigorously supported the corpuscular theory. _White light was universally believed to be the pure form_, and colors were some added property bequeathed to it upon reflection from matter (150). Descartes had discovered the sine law of refraction (94), but it was not known that some colors were refracted more than others. The pattern was the familiar one: many pieces of the puzzle were in place, but the overall picture was still unclear.

The Natural Philosopher

Between 1671 and 1690, Newton was to supply definitive treatments of most of these problems. By assiduous experimentation with prisms he established that colored light was actually fundamental, and that it could be recombined to create white light. He did not publish the result for 6 years, by which time it seemed so obvious to him that he found great difficulty in responding patiently to the many misunderstandings and objections with which it met [239ff].

He invented differential and integral calculus in 1665-6, but failed to publish it. Leibniz invented it independently 10 years later, and published it first [718]. This resulted in a priority dispute which degenerated into a feud characterized by extraordinary dishonesty and venom on both sides (542).

In discovering gravitation, Newton was also _barely ahead of the rest of the pack_. Hooke was the first to realize that orbital motion was produced by a centripetal force (268), and in 1679 _he suggested an inverse square law to Newton_ [387]. Halley and Wren came to the same conclusion, and turned to Newton for a proof, which he duly supplied [402]. Newton did not stop there, however. From 1684 to 1687 he worked continuously on a grand synthesis of the whole of mechanics, the "Philosophiae Naturalis Principia Mathematica," in which he developed his three laws of motion and showed in detail that the universal force of gravitation could explain the fall of an apple as well as the precise motions of planets and comets.

The "Principia" crystallized the new conceptions of force and inertia that had gradually been emerging, and marks the beginning of theoretical physics as the mathematical field that we know today. It is not an easy read: Newton had developed the idea that geometry and equations should never be combined [399], and therefore _refused to use simple analytical techniques in his proofs_, requiring classical geometric constructions instead [428]. He even made his Principia _deliberately abstruse in order to discourage amateurs from feeling qualified to criticize it_ [459].

[...] most of the rest of his life was spent in administrative work as Master of the Mint and as President of the Royal Society, _a position he ruthlessly exploited in the pursuit of vendettas_ against Hooke (300ff,500), Leibniz (510ff), and Flamsteed (490,500), among others. He kept secret his disbelief in Christ's divinity right up until his dying moment, at which point he refused the last rites, at last openly defying the church (576). [...]
org:junk  people  old-anglo  giants  physics  mechanics  gravity  books  religion  christianity  theos  science  the-trenches  britain  history  early-modern  the-great-west-whale  stories  math  math.CA  nibble  discovery 
11 weeks ago by nhaliday
Subgradients - S. Boyd and L. Vandenberghe
If f is convex and x ∈ int dom f, then ∂f(x) is nonempty and bounded. To establish that ∂f(x) ≠ ∅, we apply the supporting hyperplane theorem to the convex set epi f at the boundary point (x, f(x)), ...
pdf  nibble  lecture-notes  acm  optimization  curvature  math.CA  estimate  linearity  differential  existence  proofs  exposition  atoms  math  marginal  convexity-curvature 
august 2017 by nhaliday
Archimedes Palimpsest - Wikipedia
Using this method, Archimedes was able to solve several problems now treated by integral calculus, which was given its modern form in the seventeenth century by Isaac Newton and Gottfried Leibniz. Among those problems were that of calculating the center of gravity of a solid hemisphere, the center of gravity of a frustum of a circular paraboloid, and the area of a region bounded by a parabola and one of its secant lines. (For explicit details, see Archimedes' use of infinitesimals.)

When rigorously proving theorems, Archimedes often used what are now called Riemann sums. In "On the Sphere and Cylinder," he gives upper and lower bounds for the surface area of a sphere by cutting the sphere into sections of equal width. He then bounds the area of each section by the area of an inscribed and circumscribed cone, which he proves have a larger and smaller area correspondingly. He adds the areas of the cones, which is a type of Riemann sum for the area of the sphere considered as a surface of revolution.

But there are two essential differences between Archimedes' method and 19th-century methods:

1. Archimedes did not know about differentiation, so he could not calculate any integrals other than those that came from center-of-mass considerations, by symmetry. While he had a notion of linearity, to find the volume of a sphere he had to balance two figures at the same time; he never figured out how to change variables or integrate by parts.

2. When calculating approximating sums, he imposed the further constraint that the sums provide rigorous upper and lower bounds. This was required because the Greeks lacked algebraic methods that could establish that error terms in an approximation are small.
big-peeps  history  iron-age  mediterranean  the-classics  innovation  discovery  knowledge  math  math.CA  finiteness  the-trenches  wiki  trivia  cocktail  stories  nibble  canon  differential 
may 2017 by nhaliday

« earlier    

related tags

245a_-_real_analysis  accretion  acm  acmtariat  additive-combo  additive  alg-combo  algebra  algebraic-complexity  amt  announcement  aphorism  applications  approximation  arrows  article  atoms  better-explained  big-list  big-peeps  big-picture  big-surf  binomial  book  books  boolean-analysis  britain  brunn-minkowski  calculation  caltech  canon  cartoons  characterization  cheatsheet  christianity  clever-rats  closure  coarse-fine  cocktail  comparison  composition-decomposition  concentration-of-measure  concept  conference  convergence  convexity-curvature  cool  counterexample  course  curiosity  curvature  cycles  definition  differential  dimensionality  direction  discovery  distribution  draft  duality  dynamic  dynamical  early-modern  economics  electromag  embeddings  encyclopedic  entropy-like  erdos  ergodic  essay  estimate  examples  existence  expert-experience  expert  explanation  exposition  expository  extrema  fedja  feynman  film  finance  finiteness  fixed-point  flexibility  fluid  fourier  geometry  giants  gowers  gradient-descent  gravity  ground-up  harvard  hi-order-bits  hierarchy  high-dimension  higher-ed  history  homepage  ideas  identity  ieee  information-theory  init  inner-product  innovation  insight  integral  interdisciplinary  intersection  intuition  invariance  inverse_function_theorem  iron-age  israel  jean_saint_raymond  knowledge  learning-theory  lecture-notes  lectures  lens  levers  lifts-projections  limits  linear-algebra  linearity  liner-notes  links  list  local-global  logic  magnitude  manifolds  marginal  markov  martingale  math.at  math.co  math.cv  math.ds  math.fa  math.gn  math.gr  math.mg  math.nt  math.rt  math  math_overflow  mathtariat  measure  measure_theory  mechanics  mediterranean  meta:math  metabuch  metameta  methodology  metric-space  metrics  micro  mit  moments  motivation  multi  multiplicative  nibble  nitty-gritty  norms  novelty  objektbuch  ocw  old-anglo  oly  online-learning  open-problems  optimization  orders  orfe  org:bleg  org:edu  org:inst  org:junk  oscillation  overflow  p:**  p:*  p:someday  p:whenever  papers  pdf  people  philosophy  physics  plots  polynomials  positivity  preprint  princeton  probabilistic-method  probability  problem-solving  prof  proofs  properties  puzzles  q-n-a  qra  quantifiers-sums  quantum  questions  quotes  rand-approx  random-matrices  random  recommendations  reduction  reference  reflection  regularity  regularization  relaxation  religion  research  rigidity  rigor  roadmap  ryan-odonnell  s:**  s:*  s:null  science  sdp  sequential  series  simplex  simulation  singularity  slides  smoothness  soft-question  space  spatial  stat-mech  stirling  stochastic-processes  store  stories  street-fighting  structure  sum-of-squares  summary  supply-demand  survey  symmetry  synchrony  synthesis  tcs  tcstariat  teaching  techtariat  temperature  tensors  the-classics  the-great-west-whale  the-trenches  theos  thermo  thinking  tidbits  tightness  time  top-n  topics  topology  tricki  tricks  trivia  uniqueness  unit  update  visual-understanding  visualization  washington  waves  wiki  wild-ideas  wisdom  wormholes  yoga  zooming  👳 

Copy this bookmark:



description:


tags: