Posts Tagged ‘logic’
“The laws of nature are but the mathematical thoughts of God”*…
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
###
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
“For, after all, how do we know that two and two make four?”*…
“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
###
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
“All opinions are not equal. Some are a very great deal more robust, sophisticated and well supported in logic and argument than others”*…
Now more than ever: Get a free logical fallacy poster.
* Douglas Adams, The Salmon of Doubt
###
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.
“Gravity is a habit that is hard to shake off”*…
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
###
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.”
“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.”*…
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
###
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.
“Use what language you will, you can never say anything but what you are”*…
Lars Yenken‘s “The Great Language Game” is an interactive game, being played worldwide, that challenges users to distinguish among (currently) 87 languages based on their sound alone. As Lars explains,
There are perhaps six or seven thousand languages in the world. Even so-called hyperpolyglots, people who learn to speak six or more fluently, barely scratch the surface. You and I will never be able to communicate in all these languages without machine aids, but learning to identify what’s being spoken near us, that’s within our reach…
Besides, it’s fun!
[TotH to reddit]
* Ralph Waldo Emerson
###
As we prick up our ears, we might send thoughtful birthday greetings to 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.”
Time travel…
Chino Otsuka, 1980 and 2009, Nagayama, Japan
In the photo series “Imagine Finding Me,” photographer Chino Otsuka revisits her childhood by digitally inserting herself in old photos of her as a child. Otsuka likens her double self-portraits to a kind of time travel:
“The digital process becomes a tool, almost like a time machine, as I’m embarking on the journey to where I once belonged and at the same time becoming a tourist in my own history…”
Chino Otsuka, 1975 and 2005, Spain
See (and read) more at Laughing Squid and at AGO.
###
As we revisit Memory Lane, we might spare a thought for Kurt Friedrich Gödel; he died on this date in 1978. Considered (with Aristotle and Frege) one of the most important logicians in history, Gödel published the work for which he is probably most widely remembered– his two incompleteness theorems— in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. He demonstrated that:
- If a system is consistent, it cannot be complete.
- The consistency of a systems axioms cannot be proven within the system.
Gödel’s theorems ended a half-century of attempts, beginning with the work of Frege and culminating in Russell and Whitehead’s Principia Mathematica and Hilbert’s formalism, to find a set of axioms sufficient for all mathematics.
As the Anschluss swept Austria, Gödel fled to the U.S., landing at Princeton, where he joined Albert Einstein at the Institute for Advanced Studies. In 1951, as a 70th birthday present for Einstein, Gödel demonstrated the existence of paradoxical solutions to Einstein’s field equations in general relativity (they became known as the Gödel metric)– which allowed for “rotating universes” and time travel… and which caused Einstein to have doubts about his own theory.
