Entropy (information theory) 2 bits of entropy. A single toss of a fair coin has an entropy of one bit. A series of two fair coin tosses has an entropy of two bits. This definition of "entropy" was introduced by Claude E. Entropy is a measure of unpredictability of information content. Now consider the example of a coin toss. English text has fairly low entropy. If a compression scheme is lossless—that is, you can always recover the entire original message by decompressing—then a compressed message has the same quantity of information as the original, but communicated in fewer characters. Shannon's theorem also implies that no lossless compression scheme can compress all messages. Named after Boltzmann's H-theorem, Shannon defined the entropy H (Greek letter Eta) of a discrete random variable X with possible values {x1, ..., xn} and probability mass function P(X) as: Here E is the expected value operator, and I is the information content of X.[8][9] I(X) is itself a random variable. . The average uncertainty , with
Principia Mathematica ✸54.43: "From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2." —Volume I, 1st edition, page 379 (page 362 in 2nd edition; page 360 in abridged version). (The proof is actually completed in Volume II, 1st edition, page 86, accompanied by the comment, "The above proposition is occasionally useful." Τhey go on to say "It is used at least three times, in ✸113.66 and ✸120.123.472.") The title page of the shortened Principia Mathematica to ✸56 I can remember Bertrand Russell telling me of a horrible dream. Hardy, G. He [Russell] said once, after some contact with the Chinese language, that he was horrified to find that the language of Principia Mathematica was an Indo-European one Littlewood, J. The Principia Mathematica (often abbreviated PM) is a three-volume work on the foundations of mathematics written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. PM has long been known for its typographical complexity.
Quantum Aspects of Life Quantum Aspects of Life is a 2008 science text, with a foreword by Sir Roger Penrose, which explores the open question of the role of quantum mechanics at molecular scales of relevance to biology. The book adopts a debate-like style and contains chapters written by various world-experts; giving rise to a mix of both sceptical and sympathetic viewpoints. The book addresses questions of quantum physics, biophysics, nanoscience, quantum chemistry, mathematical biology, complexity theory, and philosophy that are inspired by the 1944 seminal book What Is Life? by Erwin Schrödinger. Contents[edit] Foreword by Sir Roger Penrose Section 1: Emergence and Complexity Chapter 1: "A Quantum Origin of Life?" Section 2: Quantum Mechanisms in Biology Chapter 3: "Quantum Coherence and the Search for the First Replicator" by Jim Al-Khalili and Johnjoe McFaddenChapter 4: "Ultrafast Quantum Dynamics in Photosynthesis" by Alexandra Olaya-Castro, Francesca Fassioli Olsen, Chiu Fan Lee, and Neil F. See also[edit]
Ada Lovelace Augusta Ada King, Countess of Lovelace (10 December 1815 – 27 November 1852), born Augusta Ada Byron and now commonly known as Ada Lovelace, was an English mathematician and writer chiefly known for her work on Charles Babbage's early mechanical general-purpose computer, the Analytical Engine. Her notes on the engine include what is recognised as the first algorithm intended to be carried out by a machine. Because of this, she is often described as the world's first computer programmer.[1][2][3] Ada described her approach as "poetical science" and herself as an "Analyst (& Metaphysician)". As a young adult, her mathematical talents led her to an ongoing working relationship and friendship with fellow British mathematician Charles Babbage, and in particular Babbage's work on the Analytical Engine. Biography[edit] Childhood[edit] Ada, aged four On 16 January 1816, Annabella, at George's behest, left for her parents' home at Kirkby Mallory taking one-month-old Ada with her. Adult years[edit]
The Notation in Principia Mathematica 1. Why Learn the Symbolism in Principia Mathematica? Principia Mathematica [PM] was written jointly by Alfred North Whitehead and Bertrand Russell over several years, and published in three volumes, which appeared between 1910 and 1913. This entry is intended to assist the student of PM in reading the symbolic portion of the work. 2. Below the reader will find, in the order in which they are introduced in PM, the following symbols, which are briefly described. 3. An immediate obstacle to reading PM is the unfamiliar use of dots for punctuation, instead of the more common parentheses and brackets. The use of dots. 3.1 Some Basic Examples Consider the following series of extended examples, in which we examine propositions in PM and then discuss how to translate them step by step into modern notation. Example 1 ⊢:p∨p.⊃.pPp This is the second assertion of “star” 1. ⊢[p∨p.⊃.p] So the brackets “[” and “]” represent the colon in ∗1·2. ⊢(p∨p)⊃p Example 2 p.q.=. (p&q)=df[∼(∼p∨∼q)] p&q=df∼(∼p∨∼q) ⊢:∼p.∨.
The Emperor's New Mind The Emperor's New Mind: Concerning Computers, Minds and The Laws of Physics is a 1989 book by mathematical physicist Sir Roger Penrose. Penrose argues that human consciousness is non-algorithmic, and thus is not capable of being modeled by a conventional Turing machine-type of digital computer. Penrose hypothesizes that quantum mechanics plays an essential role in the understanding of human consciousness. The collapse of the quantum wavefunction is seen as playing an important role in brain function. The majority of the book is spent reviewing, for the scientifically minded layreader, a plethora of interrelated subjects such as Newtonian physics, special and general relativity, the philosophy and limitations of mathematics, quantum physics, cosmology, and the nature of time. Penrose states that his ideas on the nature of consciousness are speculative, and his thesis is considered erroneous by experts in the fields of philosophy, computer science, and robotics.[1][2][3] See also[edit]
Top 50 Free Open Source Classes on Computer Science : Comtechtor Computer science is an interesting field to go into. There are a number of opportunities in computer science that you can take advantage of. With computers increasingly becoming a regular part of life, those who can work with computers have good opportunities. You can find a good salary with a program in computer science, and as long as you are careful to keep up your skills. Introduction to Computer Science Learn the basics of computer science, and get a foundation in how computer science works. Introduction to Computer Science: Learn about the history of computing, as well as the development of computer languages. Comprehensive Computer Science Collections If you are interested in courses that are a little more comprehensive in nature, you can get a good feel for computer science from the following collections: Programming and Languages Get a handle on computer programming, and learn about different computer languages used in programming. Computer Software Computer Processes and Data
On the Space-Theory of Matter From Wikisource Riemann has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says, although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space.
Orchestrated objective reduction Orchestrated objective reduction (Orch-OR) is a controversial 20-year-old theory of consciousness conceptualized by the theoretical physicist Sir Roger Penrose and anesthesiologist Stuart Hameroff, which claims that consciousness derives from deeper level, finer scale quantum activities inside the cells, most prevalent in the brain neurons. It combines approaches from the radically different angles of molecular biology, neuroscience, quantum physics, pharmacology, philosophy, quantum information theory, and aspects of quantum gravity.[1] The Penrose–Lucas argument[edit] The Penrose–Lucas argument states that, because humans are capable of knowing the truth of Gödel-unprovable statements, human thought is necessarily non-computable.[23] In 1931, mathematician and logician Kurt Gödel proved that any effectively generated theory capable of proving basic arithmetic cannot be both consistent and complete. Criticism of the Penrose–Lucas argument[edit] Objective reduction[edit] Motivation[edit]
Image evolution What is this? A simulated annealing like optimization algorithm, a reimplementation of Roger Alsing's excellent idea. The goal is to get an image represented as a collection of overlapping polygons of various colors and transparencies. We start from random 50 polygons that are invisible. In each optimization step we randomly modify one parameter (like color components or polygon vertices) and check whether such new variant looks more like the original image. If it is, we keep it, and continue to mutate this one instead. Fitness is a sum of pixel-by-pixel differences from the original image. This implementation is based on Roger Alsing's description, though not on his code. How does it look after some time? 50 polygons (4-vertex) ~15 minutes 644 benefitial mutations 6,120 candidates 88.74% fitness 50 polygons (6-vertex) ~15 minutes 646 benefitial mutations 6,024 candidates 89.04% fitness 50 polygons (10-vertex) ~15 minutes 645 benefitial mutations 5,367 candidates 87.01% fitness Requirements