Tag: mathematics

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Definition of a system

There are many definitions of what a system is, I like Mario Bunge’s definition:

“A system is a complex object every part or component of which is connected with other parts of the same object in such a manner that the whole possesses some features that its components lack – that is emergent properties” (Bunge 1996: 20).

Examples of emergent properties of a system: structure or history of social system. Examples of emergent properties of a component (i.e., part would not possess property if were independent or isolated): role, civil right, scarcity, price (Bunge 1996: 20).

quotes from: Bunge, Mario Augusto. Finding Philosophy in Social Science. New Haven and London: Yale University Press, 1996.

In my understanding of open systems, its components are continuously changing with their environment. Open systems have the emergent property of continuous exchange (input/output) of their components with the environment. Ludwig von Bertalanffy argues that open systems reach states of equilibrium within and through this interaction with their environment. Closed systems are isolated from their environment. Is mathematics a closed system?

[Photomedia Forum post by T.Neugebauer from Jan 10, 2007]

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Optical Art and Mathematics

The connection between mathematics and art has always been of interest to me, perhaps because I see a lot of potential in this mix that has only been explored on a surface-level with movements such as Optical Art

The department of mathematics at the National University of Singapore has an interesting page from a course called, Mathematics in Art and Architecture

Michael Bach has this collection of optical illusions & visual phenomena.

In discussions of the intersection of mathematics and art, fractal art will inevitably come up. A fractal in mathematics is defined as a geometric shape with a Hausdorff dimension (1) greater than its Lebesgue covering dimension (2).

 

“A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales. The object need not exhibit exactly the same structure at all scales, but the same “type” of structures must appear on all scales. A plot of the quantity on a log-log graph versus scale then gives a straight line, whose slope is said to be the fractal dimension. The prototypical example for a fractal is the length of a coastline measured with different length rulers. The shorter the ruler, the longer the length measured, a paradox known as the coastline paradox.

source:(Mathworld – Fractal)
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Examples of visual representations of fractals are plentiful, see for example, Jack Cooper’s Fractal Recursions gallery

[Edited from Photomedia Forum post by T.Neugebauer from 2005-2006  ]

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Gödel’s Proof

Douglas R. Hofstadter, the author of Godel, Escher, Bach: An Eternal Golden Braid edited a new edition of a wonderful 1958 book by Ernest Nagel and James R. Newman called Gödel’s Proof.

The original 1931 article by Godel is available in English translation by Martin Hirzel (On formally undecidable propositions of Principia Mathematica and related systems); I recommend reading the Nagel book first.

see also: Gödel’s Incompleteness Theorem (MathWorld), Gödel number (MathWorld)

[Photomedia Forum post by T.Neugebauer from May 22, 2007]

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Grigory Perelman

Grigory Perelman, mathematician that recently offered a solution to Poincaré’s conjecture has an impressive list of prizes that he did not accept: the Fields Medal, the $1 million Millennium Prize, the prizes from the International Congress of Mathematicians in Madrid, a prize from the European Mathematical Society.

“Perelman has said: “If everyone is honest, it is natural to share ideas.”
“[…]Of course, there are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest.” He has also said that “It is not people who break ethical standards who are regarded as aliens. It is people like me who are isolated.”

source: The New Yorker – MANIFOLD DESTINY
A legendary problem and the battle over who solved it.
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“
“If anybody is interested in my way of solving the problem, it’s all there — let them go and read about it,” he told The Telegraph. “I have published all my calculations. This is what I can offer the public.”

source: New York Times – The Math Was Complex, the Intentions, Strikingly Simple
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[Edited from Photomedia Forum post by T.Neugebauer from Nov 12, 2006]

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constructive mathematics and the Markov principle

Someone asked me recently about constructivism in mathematics, which led me to the Douglas Bridges’ article in Stanford Encyclopedia of Philosophy on constructive mathematics. What is constructivism? It is a philsophy of mathematics that “asserts that it is necessary to find (or “construct”) a mathematical object to prove that it exists” (wikipedia), as opposed to classical mathematics where it is enough to assume that the object does not exist and derive a contradiction.

In the Bridges’ article, I found a priceless illustrative example, the Markov Principle (MP):

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For each binary sequence (an), if it is contradictory that all the terms an equal 0, then there exists a term equal to 1.

[…]

Markov’s Principle represents an unbounded search: if you have a proof that all terms an being 0 leads to a contradiction, then, by testing the terms a1,a2,a3, in turn, you are guaranteed to come across a term equal to 1; but this guarantee does not extend to an assurance that you will find the desired term before the end of the universe. ”

[Photomedia Forum post by T.Neugebauer from  May 6, 2006 ]