Archive for July, 2020

Berry’s Paradox and Gödel’s Incompleteness Theorem

 

Chaitin-Boolos

Left: Argentine-American mathematician
Gregory Chaitin [image from here]. Right: American philosopher and logician
George Boolos [image Wikimedia Commons].

A young librarian at the Bodleian Library in Oxford devised an intriguing paradox. He defined a number by means of a statement of the form

THE SMALLEST NATURAL NUMBER THAT CANNOT BE
DEFINED IN FEWER THAN TWENTY WORDS.

Continue reading ‘Berry’s Paradox and Gödel’s Incompleteness Theorem’

Does Numerical Integration Reflect the Truth?

Many problems in applied mathematics involve the solution of a differential equation. Simple differential equations can be solved analytically: we can find a formula expressing the solution for any value of the independent variable. But most equations are nonlinear and this approach does not work; we must solve the equation by approximate numerical means. The big question is:

Does the numerical solution resemble the true solution of the equation?

The answer is: “Not necessarily”.

There are often specific criteria that must be satisfied to ensure that the answer `crunched out’ by the computer is a reasonable approximation to reality. Although the principles of numerical stability are quite general, they are best illustrated by simple examples. We will look at some of these below.

CFL-Criterion-1

Smooth curve: True solution. Black dots: stable solution. Red dots: unstable solution (time step too large).

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Cornelius Lanczos – Inspired by Hamilton’s Quaternions

Lanczos240In May 1954, Cornelius Lanczos took up a position as senior professor in the School of Theoretical Physics at the Dublin Institute for Advanced Studies (DIAS). The institute had been established in 1940 by Eamon de Valera, with a School of Theoretical Physics and a School of Celtic Studies, reflecting de Valera’s keen interest in mathematics and in the Irish language. Later, a School of Cosmic Physics was added. DIAS remains a significant international centre of research today [TM191 or search for “thatsmaths” at irishtimes.com].

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Buridan’s Ass

Jean-Buridan

Jean Buridan (c. 1300-1360).

“Buridan’s Ass” is a paradox in philosophy, in which a hungry donkey, located at the mid-point between two bales of hay, is frozen in indecision about which way to go and faces starvation — he is unable to move one way or the other.

Jean Buridan was a French philosopher who lived in the fourteenth century. He was not interested in donkeys, but in human morality. He wrote that if two courses of action are judged to be morally equal, we must suspend a decision until the right course of action becomes clear. The idea of the paradox can be found in the writings of the ancients, including Aristotle.

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The Ever-growing Goals of Googology

In 1920, a kindergarten class was asked to describe the biggest number that they could imagine. One child proposed to “write down digits until you get tired”. A more concrete idea was to write a one followed by 100 zeros. This number, which scientists would express as ten to the power 100, was given the name “googol” by its inventor [TM190; or search for “thatsmaths” at irishtimes.com ].

OneGoogol

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