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  • feedwordpress 08:01:51 on 2018/08/29 Permalink
    Tags: Hermann Hankel, Hilbert, , , , , , supertasks, , Zeno   

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


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    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

     

     
  • feedwordpress 08:01:49 on 2017/09/20 Permalink
    Tags: Cantor, , Erdős number, , Hilbert, , , ,   

    “Mystery has its own mysteries”*… 


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    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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  • feedwordpress 08:01:21 on 2015/07/15 Permalink
    Tags: bicycle, circular, Hilbert, , , , Riemann, Robert Wechsler, Smale, strange attractor,   

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


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    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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