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  • feedwordpress 09:01:26 on 2019/01/14 Permalink
    Tags: , Elements, Euclid, , , , incompleteness theorems, logic, , ,   

    “The laws of nature are but the mathematical thoughts of God”*… 


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    euclid

    2,300 years ago, Euclid of Alexandria sat with a reed pen–a humble, sliced stalk of grass–and wrote down the foundational laws that we’ve come to call geometry. Now his beautiful work is available for the first time as an interactive website.

    Euclid’s Elements was first published in 300 B.C. as a compilation of the foundational geometrical proofs established by the ancient Greek. It became the world’s oldest, continuously used mathematical textbook. Then in 1847, mathematician Oliver Byrne rereleased the text with a new, watershed use of graphics. While Euclid’s version had basic sketches, Byrne reimagined the proofs in a modernist, graphic language based upon the three primary colors to keep it all straight. Byrne’s use of color made his book expensive to reproduce and therefore scarce, but Byrne’s edition has been recognized as an important piece of data visualization history all the same…

    Explore elemental beauty at “A masterpiece of ancient data viz, reinvented as a gorgeous website.”

    * Euclid, Elements

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    As we appreciate the angles, we might spare a thought for Kurt Friedrich Gödel; he died on this date in 1978.  A  logician, mathematician, and philosopher, he is considered (along with Aristotle, Alfred Tarski— whose birthday this also is– and Gottlob Frege) to be one of the most important logicians in history.  Gödel had an immense impact upon scientific and philosophical thinking in the 20th century.  He is, perhaps, best remembered for his Incompleteness Theorems, which led to (among other important results) Alan Turing’s insights into computational theory.

    Kurt Gödel’s achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a landmark which will remain visible far in space and time. … The subject of logic has certainly completely changed its nature and possibilities with Gödel’s achievement.                  — John von Neumann

    kurt_gödel source

     

     
  • feedwordpress 09:01:10 on 2017/01/17 Permalink
    Tags: American Enlightenment, , , logic, , , , ,   

    “For, after all, how do we know that two and two make four?”*… 


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    “The teaching of Logic or Dialetics” from a collection of scientific, philosophical and poetic writings, French, 13th century; Bibliotheque Sainte-Genevieve, Paris, France. Photo by Bridgeman

    Is logical thinking a way to discover or to debate? The answers from philosophy and mathematics define human knowledge..

    The history of logic should be of interest to anyone with aspirations to thinking that is correct, or at least reasonable. This story illustrates different approaches to intellectual enquiry and human cognition more generally. Reflecting on the history of logic forces us to reflect on what it means to be a reasonable cognitive agent, to think properly. Is it to engage in discussions with others? Is it to think for ourselves? Is it to perform calculations?…

    The rise and fall and rise of logic: “What is logic?

    * George Orwell, 1984

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    As we ruminate on reason, we might send enlightened birthday greetings to Benjamin Franklin; he was born on this date in 1706.  One of the Founding Fathers of the United States, Franklin was a renowned polymath: a leading author, printer, political theorist, politician, freemason, postmaster, scientist, inventor, civic activist, statesman, and diplomat. As a scientist, he was a major figure in the American Enlightenment and the history of physics for his discoveries and theories regarding electricity. As an inventor, he is known for the lightning rod, bifocals, and the Franklin stove, among other innovations.  And as a social entrepreneur (who grasped the fact that by united effort a community could have amenities which only the wealthy few can afford for themselves), he helped establish several institutions people now take for granted: a fire company (1736), a library (1731), an insurance company (1752), an academy (the University of Pennsylvania, 1751), a hospital (1751), and the U.S. Postal Service (starting as postmaster of the Colonies in 1753, then becoming U.S. Postmaster during the Revolution).  In most cases these foundations were the first of their kind in North America.

    In a Franklin could be merged the virtues of Puritanism without its defects, the illumination of the Enlightenment without its heat.

    – Henry Steele Commager

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  • feedwordpress 09:01:27 on 2016/12/03 Permalink
    Tags: , , , John Wallis, logic, logical fallacy, ,   

    “All opinions are not equal. Some are a very great deal more robust, sophisticated and well supported in logic and argument than others”*… 


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    Now more than ever:  Get a free logical fallacy poster.

    * Douglas Adams, The Salmon of Doubt

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    As we dedicate ourselves to discipline, we might send carefully-calculated birthday greetings to John Wallis; he was born on this date in 1616.  An English mathematician who served as chief cryptographer for Parliament and, later, the royal court, he helped develop infinitesimal calculus and is credited with introducing the symbol ∞ for infinity.

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  • feedwordpress 09:01:21 on 2016/02/15 Permalink
    Tags: Alfred North Whitehead, , , , , , logic, , , ,   

    “Gravity is a habit that is hard to shake off”*… 


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     click here for zoomable version

    Last week, scientists at The Laser Interferometer Gravitational-Wave Observatory, or LIGO, announced that they had confirmed Einstein’s century-old theoretical prediction of “gravitational waves,” a feature of his theory of general relativity.

    Our friends at PhD Comics explain why that matters:

    email readers click here for video

    * Terry Pratchett

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    As we go with the flow, we might send carefully-calculated birthday greetings to Einstein’s rough contemporary Alfred North Whitehead; he was born on this date in 1861.  Whitehead began his career as a mathematician and logician, perhaps most famously co-authoring (with his former student, Bertrand Russell), the three-volume Principia Mathematica (1910–13), one of the twentieth century’s most important works in mathematical logic.

    But in the late teens and early 20s, Whitehead shifted his focus to philosophy, the central result of which was a new field called process philosophy, which has found application in a wide variety of disciplines (e.g., ecology, theology, education, physics, biology, economics, and psychology).

    “There is urgency in coming to see the world as a web of interrelated processes of which we are integral parts, so that all of our choices and actions have consequences for the world around us.”

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  • feedwordpress 09:01:51 on 2015/11/25 Permalink
    Tags: , field equations, General Theory of Relativity, logic, , , , , Smulyan,   

    “If it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.”*… 


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    Your correspondent is headed into the chilly wilds for the Thanksgiving holiday, so this will be the last post until after the passing of the tryptophan haze.  By way of keeping readers amused in the meantime, the puzzle above…

    Find a step-by-step guide to its answer at “How to Solve the Hardest Logic Puzzle Ever.”

    * Tweedledee, in Lewis Caroll’s Through the Looking-Glass, and What Alice Found There

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    As we muddle in the excluded middle, we might recall that it was on this date in 1915 that Albert Einstein presented the Einstein Field Equations to the Prussian Academy of Sciences.  Einstein developed what was elaborated into a set of 10 equations to account for gravitation in the curved spacetime described in his General Theory of Relativity; they are used to determine spacetime geometry.

    (German mathematician David Hilbert reached the same conclusion, and actually published the equation before Einstein– though Hilbert, who was a correspondent of Einstein’s, never suggested that Einstein’s credit was inappropriate.)

    On the right side of the equal sign, the distribution of matter and energy in space; on the left, the geometry of the space, the so-called metric, a prescription for how to compute the distance between two points.

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