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Posts Tagged ‘Hilbert

“Sometimes the only way to move forward is to revisit the things in your past that were holding you back”*…

Adobe Flash was the language of choice for a generation of game developers, helping kickstart an indie revolution on the still-young web of the 1990s and 2000s. But it withered on the proprietary and insecure vine, and both web browsers and Adobe have now canned it, threatening countless games and interactive presentations with the memory hole. The Internet Archive comes to the rescue, not only archiving the flash files but emulating the player itself, allowing history to live on.

“The Internet Archive has begun emulating Flash Animations, Games and Toys in a new collection,” wrote archivist Jason Scott on Twitter. “It’s at https://archive.org/details/softwarelibrary_flash and it’s going to be past 1,000 items in 24 hours. You can add your own and get them running, and the animations have never ran smoother or better.”…

From our friends at Boing Boing: “Internet Archive turns on Flash emulation, already has 1000 items to check out.”

* “Barry Allen” (The Flash)


As we celebrate that what’s old is new again, we might recall that it was on this date in 1915 that Albert Einstein presented the Einstein Field Equation to the Prussian Academy of Sciences.  Einstein soon after elaborated it into the set of 10 equations that account for gravitation in the curved spacetime that he 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.


“I am incapable of conceiving infinity, and yet I do not accept finity”*…



Suppose you’re working at a hotel with infinitely many rooms in it, numbered 1, 2, 3, 4, 5, … all the way up forever and ever. (This is known as a Hilbert Hotel.) One evening when every single room is occupied, a traveler arrives and requests to be accommodated too. You’re the manager. What do you do to help the traveler?

Simple. You just ask each occupant to one room forward. 1 goes to 2, and 2 goes to 3, and so on. Every previous occupant gets a new room. And the first room is now open for the traveler.

The procedure above is characterized by an infinite number of actions or tasks to be carried out in a finite amount of time. Procedures with this character are known as supertasks…

More on the ins and outs of infinities at “Introducing Supertasks.” (More fun musings on infinity here and here; and more on Hilbert’s Hotel here.)

* Simone de Beauvoir, La Vieillesse


As we muse on many, we might spare a thought for Hermann Hankel; he died on this date in 1873.  A mathematician who worked with Möbius, Riemann, Weierstrass,  and Kronecker (among others), he made important contributions to the understanding of complex numbers and quaternions… and to work begun by Bernard Bolzano on infinite series.

220px-Hankel source


Written by LW

August 29, 2018 at 1:01 am

“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


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.



Written by LW

September 20, 2017 at 1:01 am

“I may be going nowhere, but what a ride”*…


Nine salvaged bikes were reassembled into a carousel formation. The bike is modular and can be dismantled, transported and reassembled. It is normally left in public places where it can attract a variety of riders and spectators.

From artist Robert Wechsler, the Circular Bike.

* Shaun Hick


As we return to where we started, we might send carefully-calculated birthday greetings to Stephen Smale; he was born on this date in 1930.  A winner of both the Fields Medal and the Wolf Prize, the highest honors in mathematics, he first gained recognition with a proof of the Poincaré conjecture for all dimensions greater than or equal to 5, published in 1961.  He then moved to dynamic systems, developing an understanding of strange attractors which lead to chaos, and contributing to mathematical economics.  His most recent work is in theoretical computer science.

In 1998, in the spirit of Hilbert’s famous list of problems produced in 1900, he created a list of 18 unanswered challenges– known as Smale’s problems– to be solved in the 21st century.  (In fact, Smale’s list contains some of the original Hilbert problems, including the Riemann hypothesis and the second half of Hilbert’s sixteenth problem, both of which are still unsolved.)



Written by LW

July 15, 2015 at 1:01 am

No reservation? No problem!…


Jeff Dekofsky explains Hilbert’s paradox of the Grand Hotel, a thought experiment proposed in the 1920s by German mathematician David Hilbert to illustrate some surprising properties of infinite sets, in this TED-Ed animated lecture

email readers click here for video

As a special bonus, another amusing video (via Kottke)– an explanation of why it is that the sum of all positive integers (1 + 2 + 3 + 4 + 5 + …) = -1/12…  Euler actually proved this result in 1735, but the result was only made rigorous later; and now physicists have been seeing this result actually show up in nature.  (Spoiler alert: the answer turns on what one means by “sum” mathematically…)


As we pray for more fingers and toes, we might spare a thought for Harald August Bohr; he died on this date in 1951.  While materially less well-known than his brother Niels, Harald was a formidable mathematician (founder of the field of almost periodic functions), a gifted athlete (an accomplished footballer who won a silver medal at the 1908 Summer Olympics as a member of Denmark’s team), an inspirational teacher (the annual award for outstanding teaching at the University of Copenhagen is called “the Harald” in his honor), and an out-spoken critic of the anti-Semitic policies that took root in the German mathematical establishment in the 1930s.


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