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Posts Tagged ‘Hilbert

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

 

hilberts_hotel

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

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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

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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.

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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

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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.)

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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…)

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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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From the Not-Sure-I-Really-Want-To-Know Department…

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As readers know, some physicists believe that the universe as we know it is actually a giant hologram, giving us the illusion of three-dimensions, while in fact all the action is occurring on a two-dimensional boundary region (see here, here, and here)… shadows on the walls of a cave, indeed.

But lest one mistake that for the frontier of freakiness, others (c.f., e.g., here and here) believe that the existence we experience is nothing more (or less) than a Matrix-like simulation…

A common theme of science fiction movies and books is the idea that we’re all living in a simulated universe—that nothing is actually real. This is no trivial pursuit: some of the greatest minds in history, from Plato, to Descartes, have pondered the possibility. Though, none were able to offer proof that such an idea is even possible. Now, a team of physicists working at the University of Bonn have come up with a possible means for providing us with the evidence we are looking for; namely, a measurable way to show that our universe is indeed simulated. They have written a paper describing their idea and have uploaded it to the preprint server arXiv…

Phys.Org has the whole story at “Is it real? Physicists propose method to determine if the universe is a simulation“; the paper mentioned above can be downloaded here.

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As we reach for the “reset” button, we might send carefully-calculated birthday greetings to Paul Isaac Bernays; he was born on this date in 1888.  A close associate of David Hilbert (of “Hilbert’s Hotel” fame), Bernays was one the foremost philosophers of mathematics of the Twentieth Century, who made important contributions to mathematical logic and axiomatic set theory.  Bernays is perhaps best remembered for his revision and improvement of the (early, incomplete) set theory advanced by John von Neumann in the 1920s; Bernays’s work, with some subsequent modifications by Kurt Gödel, is now known as the Von Neumann–Bernays–Gödel set theory.

Lest, per the simulation speculation above suggest that cosmology has a hammerlock on weirdness:  Set theory is used, among other purposes, to describe the symmetries inherent in families of elementary particles and in crystals. Materials such as a liquid or a gas in equilibrium, made of uniformly distributed particles, exhibit perfect spatial symmetry—they look the same everywhere and in every direction… a condition that “breaks” at very low temperature, when the particles form crystals (which have some symmetry, but less)…  Now Nobel Laureate Frank Wilczek has suggested that there may exist “Time Crystals“– whose structure would repeat periodically, as with an ordinary crystal, but in time rather than in space… a kind of “perpetual motion ‘machine'” (weirder yet, one that doesn’t violate the laws of thermodynamics).

Paul Bernays

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Written by LW

October 17, 2012 at 1:01 am

Infinitely cool…

 click here for video

How to Count to Infinity (or “Yes, Virginia, some infinities are bigger than others…”)

Many more sixty-second epiphanies at MinutePhysics’ You Tube channel (or via New Scientist TV)

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As we check in to Hilbert’s Hotel, we might spare a thought for Joesph Fourier; the French mathematician, physicist, Egyptologist and administrator who died on this date in 1830. Fourier introduced Jean-Francois Champollion to the Rosetta Stone, which Champollion subsequently decoded/translated.  And after calculating that a body the size of earth, at earth’s distance form the sun, should be cooler than our world is, discovered what we now call “the greenhouse effect.”  But Fourier is best remembered for his contributions to mathematical physics through his Théorie analytique de la chaleur (1822; The Analytical Theory of Heat), which introduced an infinite mathematical series to aid in solving conduction equations. (The technique allowed the function of any variable to be expanded into a series of sines of multiples of the variable– now known as “the fourier series.”)

True greatness is when your name is like ampere, watt, and fourier—when it’s spelled with a lower case letter.

– Richard Hamming (in a 1986 Bell Labs Colloquium)

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Written by LW

May 16, 2012 at 1:01 am

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