## Posts Tagged ‘**Gödel**’

## “Nothing happens until something moves”*…

What determines our fate? To the Stoic Greek philosophers, fate is the external product of divine will, ‘the thread of your destiny’. To transcendentalists such as Henry David Thoreau, it is an inward matter of self-determination, of ‘what a man thinks of himself’. To modern cosmologists, fate is something else entirely: a sweeping, impersonal physical process that can be boiled down into a single, momentous number known as the Hubble Constant.

The Hubble Constant can be defined simply as the rate at which the Universe is expanding, a measure of how quickly the space between galaxies is stretching apart. The slightest interpretation exposes a web of complexity encased within that seeming simplicity, however. Extrapolating the expansion process backward implies that all the galaxies we can observe originated together at some point in the past – emerging from a Big Bang – and that the Universe has a finite age. Extrapolating forward presents two starkly opposed futures, either an endless era of expansion and dissipation or an eventual turnabout that will wipe out the current order and begin the process anew.

That’s a lot of emotional and intellectual weight resting on one small number…

How scientists pinned a single number on all of existence: “Fate of the Universe.”

[Readers might remember that the Big Bang wasn’t always an accepted paradigm— and that on-going research continues to surface challenges.]

* Albert Einstein

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**As we center ourselves,** 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

## “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

Elementswas 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

## “Mystery has its own mysteries”*…

Finally, an answer to a question that puzzled Cantor and Hilbert (proprietor of The Infinite Hotel) and challenged Cohen and Gödel…

In a breakthrough that disproves decades of conventional wisdom [and confounds common sense], two mathematicians have shown that two different variants of infinity are actually the same size. The advance touches on one of the most famous and intractable problems in mathematics: whether there exist infinities between the infinite size of the natural numbers and the larger infinite size of the real numbers…

Connecting the sizes of infinities and the complexity of mathematical theories: “Mathematicians Measure Infinities and Find They’re Equal.”

* “Mystery has its own mysteries, and there are gods above gods. We have ours, they have theirs. That is what’s known as infinity.” – Jean Cocteau

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**As we go big,** we might spare a thought for Paul Erdős; he died on this date in 1996. One of the most prolific mathematicians of the 20th century (he published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed), he is remembered both for his “social practice” of mathematics (he engaged more than 500 collaborators) and for his eccentric lifestyle (he spent his waking hours virtually entirely on math; he would typically show up at a colleague’s doorstep and announce “my brain is open”, staying long enough to collaborate on a few papers before moving on a few days later).

Erdős’s prolific output with co-authors prompted the creation of the Erdős number, the number of steps in the shortest path between a mathematician and Erdős in terms of co-authorships. Low numbers are a badge of pride– and a usual marker of accomplishment: As of 2016, all Fields Medalists have a finite Erdős number, with values that range between 2 and 6, and a median of 3. Physics Nobelists Einstein and Sheldon Glashow have an Erdős number of 2. Baseball Hall of Famer Hank Aaron can be considered to have an Erdős number of 1 because they both autographed the same baseball (for number theorist Carl Pomerance). Natalie Portman’s undergraduate collaboration with a Harvard professor earned her an Erdős number of 5; Danica McKellar (“Winnie Cooper” in *The Wonder Years*) has an Erdős number of 4, for a mathematics paper coauthored while an undergraduate at UCLA.

## Time travel…

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…”

See (and read) more at Laughing Squid and at AGO.

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**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.