The Structure of Scientific Revolutions

Chaos theory A double rod pendulum animation showing chaotic behavior. Starting the pendulum from a slightly different initial condition would result in a completely different trajectory. The double rod pendulum is one of the simplest dynamical systems that has chaotic solutions. Chaos: When the present determines the future, but the approximate present does not approximately determine the future. Chaotic behavior can be observed in many natural systems, such as weather and climate.[6][7] This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Introduction[edit] Chaos theory concerns deterministic systems whose behavior can in principle be predicted. Chaotic dynamics[edit] The map defined by x → 4 x (1 – x) and y → x + y mod 1 displays sensitivity to initial conditions. In common usage, "chaos" means "a state of disorder".[9] However, in chaos theory, the term is defined more precisely. where , and , is: .

Occam's razor The sun, moon and other solar system planets can be described as revolving around the Earth. However that explanation's ideological and complex assumptions are completely unfounded compared to the modern consensus that all solar system planets revolve around the Sun. Ockham's razor (also written as Occam's razor and in Latin lex parsimoniae) is a principle of parsimony, economy, or succinctness used in problem-solving devised by William of Ockham (c. 1287 - 1347). Solomonoff's theory of inductive inference is a mathematically formalized Occam's Razor:[2][3][4][5][6][7] shorter computable theories have more weight when calculating the probability of the next observation, using all computable theories which perfectly describe previous observations. History[edit] Formulations before Ockham[edit] Part of a page from Duns Scotus' book Ordinatio: "Pluralitas non est ponenda sine necessitate", i.e., "Plurality is not to be posited without necessity" Ockham[edit] Later formulations[edit]

Axiomatic Theories of Truth First published Mon Dec 26, 2005; substantive revision Mon Nov 4, 2013 An axiomatic theory of truth is a deductive theory of truth as a primitive undefined predicate. Because of the liar and other paradoxes, the axioms and rules have to be chosen carefully in order to avoid inconsistency. 1. There have been many attempts to define truth in terms of correspondence, coherence or other notions. In semantic theories of truth (e.g., Tarski 1935, Kripke 1975), in contrast, a truth predicate is defined for a language, the so-called object language. As with other formal deductive systems, axiomatic theories of truth can be presented within very weak logical frameworks. Formal work on axiomatic theories of truth has helped to shed some light on semantic theories of truth. This entry outlines the most popular axiomatic theories of truth and mentions some of the formal results that have been obtained concerning them. 1.1 Truth, properties and sets 1.2 Truth and reflection ∀x(BewS(x) → Tx) 2. 3.

Phase diagram Overview[edit] Common components of a phase diagram are lines of equilibrium or phase boundaries, which refer to lines that mark conditions under which multiple phases can coexist at equilibrium. Phase transitions occur along lines of equilibrium. Triple points are points on phase diagrams where lines of equilibrium intersect. Types of phase diagrams[edit] 2D phase diagrams[edit] The curves on the phase diagram show the points where the free energy (and other derived properties) becomes non-analytic: their derivatives with respect to the coordinates (temperature and pressure in this example) change discontinuously (abruptly). The existence of the liquid–gas critical point reveals a slight ambiguity in labelling the single phase regions. Other thermodynamic properties In addition to just temperature or pressure, other thermodynamic properties may be graphed in phase diagrams. In a two-dimensional graph, two of the thermodynamic quantities may be shown on the horizontal and vertical axes.

Euclid "Euclid" is the anglicized version of the Greek name Εὐκλείδης, meaning "Good Glory".[4] Life Little is known about Euclid's life, as there are only a handful of references to him. The date and place of Euclid's birth and the date and circumstances of his death are unknown, and only roughly estimated in proximity to contemporary figures mentioned in references. Euclid is rarely, if ever, referred to by name by other Greek mathematicians from Archimedes onward, who instead call him "ό στοιχειώτης" ("the author of Elements").[5] The few historical references to Euclid were written centuries after he lived, by Proclus ca. 450 AD and Pappus of Alexandria ca. 320 AD.[6] Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclid's Elements, "Euclid replied there is no royal road to geometry A detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. Elements Other works See also Notes References

Sense data In the philosophy of perception, the theory of sense data was a popular view held in the early 20th century by philosophers such as Bertrand Russell, C. D. Broad, H. H. Talk of sense-data has since been largely replaced by talk of the closely related qualia. Examples[edit] Bertrand Russell heard the sound of his knuckles rapping his writing table, felt the table's hardness and saw its apparent colour (which he knew 'really' to be the brown of wood) change significantly under shifting lighting conditions. H. When we twist a coin it 'appears' to us as elliptical. Consider a reflection which appears to us in a mirror. The nature of sense data[edit] The idea that our perceptions are based on sense data is supported by a number of arguments. Abstract sense data[edit] Abstract sense data is sense data without human judgment, sense data without human conception and yet evident to the senses, found in aesthetic experience. Criticisms[edit] See also[edit] References[edit]

Phase space Phase space of a dynamic system with focal instability, showing one phase space trajectory A plot of position and momentum variables as a function of time is sometimes called a phase plot or a phase diagram. Phase diagram, however, is more usually reserved in the physical sciences for a diagram showing the various regions of stability of the thermodynamic phases of a chemical system, which consists of pressure, temperature, and composition. In classical mechanics, any choice of generalized coordinates q i for the position (i.e. coordinates on configuration space) defines conjugate generalized momenta pi which together define co-ordinates on phase space. More abstractly, in classical mechanics phase space is the cotangent space of configuration space, and in this interpretation the procedure above expresses that a choice of local coordinates on configuration space induces a choice of natural local Darboux coordinates for the standard symplectic structure on a cotangent space.

Euclid's Elements Euclid's Elements (Ancient Greek: Στοιχεῖα Stoicheia) is a mathematical and geometric treatise consisting of 13 books written by the ancient Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books cover Euclidean geometry and the ancient Greek version of elementary number theory. The work also includes an algebraic system that has become known as geometric algebra, which is powerful enough to solve many algebraic problems,[1] including the problem of finding the square root of a number.[2] With the exception of Autolycus' On the Moving Sphere, the Elements is one of the oldest extant Greek mathematical treatises,[3] and it is the oldest extant axiomatic deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science. The name 'Elements' comes from the plural of 'element'.

Lokayata/Carvaka – Indian Materialism | Internet Encyclopedia of Philosophy Lokayata/Carvaka—Indian Materialism In its most generic sense, "Indian Materialism" refers to the school of thought within Indian philosophy that rejects supernaturalism. It is regarded as the most radical of the Indian philosophical systems. It rejects the existence of other worldly entities such an immaterial soul or god and the after-life. The terms Lokāyata and Cārvāka have historically been used to denote the philosophical school of Indian Materialism. Table of Contents 1. Traces of materialism appear in the earliest recordings of Indian thought. a. Vedic thought, in the most comprehensive sense, refers to the ideas contained within the Samhitas and the Brāhamaṇas, including the Upaniṣads. The Vedic period marked the weakest stage of the development of Indian Materialism. Once upon a time Bṛhaspati struck the goddess Gāyatrī on the head. The term "Svabhava" in Sanskrit can be translated to "essence" or "nature." b. 2. a. b. 3. a. b. c. 4. 5. a.

Complementarity (physics) In physics, complementarity is a fundamental principle of quantum mechanics, closely associated with the Copenhagen interpretation. It holds that objects governed by quantum mechanics, when measured, give results that depend inherently upon the type of measuring device used, and must necessarily be described in classical mechanical terms. Further, a full description of a particular type of phenomenon can only be achieved through measurements made in each of the various possible bases — which are thus complementary. The complementarity principle was formulated by Niels Bohr, the developer of the Bohr model of the atom, and a leading founder of quantum mechanics.[1] Bohr summarized the principle as follows: ...however far the [quantum physical] phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms. For example, the particle and wave aspects of physical objects are such complementary phenomena. Physicists F.A.M. Dr.

The Compendious Book on Calculation by Completion and Balancing A page from the book Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala (Arabic: الكتاب المختصر في حساب الجبر والمقابلة‎, "The Compendious Book on Calculation by Completion and Balancing"), also known under a shorter name spelled as Hisab al-jabr w’al-muqabala, Kitab al-Jabr wa-l-Muqabala and other transliterations) is a mathematical book written in Arabic language in approximately AD 820 by the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī in Baghdad, the capital of the Abbasid Caliphate at the time. The book was translated into Latin in the mid 12th century under the title Liber Algebrae et Almucabola (with algebrae and almucabola being simply Latinized corruptions of the words in the Arabic title). Today's term algebra is derived from the term الجبر al-ğabr in the title of this book. The al-ğabr provided an exhaustive account of solving for the positive roots of polynomial equations up to the second degree.[1] Legacy[edit] R. J. The book[edit] References[edit] Barnabas B.

45 ways to avoid using the word 'very' Three Telling Quotes About ‘Very’ Substitute ‘damn’ every time you’re inclined to write ‘very;’ your editor will delete it and the writing will be just as it should be. ~Mark Twain‘Very’ is the most useless word in the English language and can always come out. More than useless, it is treacherous because it invariably weakens what it is intended to strengthen. ~Florence KingSo avoid using the word ‘very’ because it’s lazy. If you enjoyed this, you will love: If you want to learn how to write a book, join our Writers Write course. by Amanda Patterson © Amanda Patterson