Some delights are embodied in objects such as a slice of baked New York cheese cake or a glass of Cape Chamonix Reserve Chardonnay. Other delights (although still of course embodied) are more abstract. Abstract delights are most pleasingly evident when you do or think about consilience, evolutionary biology, socio-biology, provability logic, proof theory, model theory - (which can be thought of as universal algebra plus logic or algebraic geometry minus fields), set theory, topology, astronomy, cosmology, earth systems sciences or music.
These pages are about such abstract delights...
The bunny, for example, although she looks so solid to us is really quite delightfully abstract. She is a conformal structure bunny invented by professors X. Gu and S.T. Yau; an image from an algorithmic method for tackling fundamental problems in computer aided geometry design, computer graphics or computer vision.
The image is used on these pages by kind permission of professors X. Gu and S.T. Yau. The conformal structure bunny appeared in X. Gu and S.T. Yau, "Computing Conformal Structures of Surfaces", Communications in Information and Systems, Vol.2, No. 2, pp. 121-146, December 2002. International Press & professors Gu and Yau, are the copyright holders.
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When thinking changes your mind, that's philosophy. When God changes your mind, that's faith. When facts change your mind, that's science. from www.edge.com
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It would be neat if the universe were a set of mathematical structures
Below are some quotations from various authors expressing this grand idea
Pythagoras and his followers believed that in some sense it must be true that 'All is Number'. This was their way of saying that the universe is a mathematical structure. The Pythagoreans recognised the mathematical nature of music and believed in some sort of mystical numerical harmony of the universe.
To Pythagoras and his followers we must attribute the deep insight, "that space can be a mathematical abstraction, and, just as important, that the abstraction can apply to many different circumstances." Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace, Leonard Mlodinow, Allen Lane Penguin, 2001. So in some interpretation, for the Pythagoreans, the stuff of the universe was numbers and geometry.
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Galileo Galilei wrote; "Philosophy is written in this grand book - I mean the Universe - which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles and other geometrical figures, without which it is humanly impossible to understand a single word of it."
"Mathematics is the language with which God has written the universe."
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Gödel believed that what makes mathematics true is that it's descriptive of an abstract reality. Mathematical intuition is like sense perception. In the abstract Platonic world we can see mathematical and logical truths which must be true. See Gödel's paper: "What Is Cantor's Continuum Hypothesis?" for an elaboration of what we can deduce through pure reason. Gödel's Platonism is the view that there exists "a non-sensual reality, which exists independently both of the acts and the dispositions of the human mind and is only perceived, and probably perceived very incompletely, by the human mind" ("Some basic theorems on the foundations of mathematics and their philosophical implications." p.323. (1951) Reproduced in Gödel. Collected Works, vol. 3. Feferman, S., Dawson, J., Goldfarb, W., Parsons, C., Solovay, R., and van Heijenoort, J. (eds.). 1995 Oxford: Oxford University Press.
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Max Tegmark continues the grand tradition of universal abstraction. Max Tegmark is a precision cosmologist, that is someone who combines theoretical work with new measurements to place sharp constraints on cosmological models and their free parameters.
In The Mathematical Universe Max Tegmark argues that given the physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans; and assuming a broad definition of mathematics, ERH implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure.
An overview map of the delightfully abstract landscape of maths by Max Tegmark
Relationships between various basic mathematical structures. The arrows generally indicate addition of new symbols or axioms. Arrows that meet indicate the combination of structures - for instance, an algebra is a vector space that is also a ring, and a Lie group is a group that is also a manifold.
The map is used with permission from prof. Max Tegmark. It was used in M. Tegmark, Is 'the theory of everything' merely the ultimate ensemble theory? arxiv:gr-qc/9704009 and subsequently in Which Mathematical Structure is Isomorphic to our Universe? in the print edition of the New Scientist. http://space.mit.edu/home/tegmark/toe_frames.html
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An interesting attitude to travel
Horace's philosophy of travel.
Horace was no traveler. "For him the tranquil delights of his farm near Licenza in the Sabine hills, his poetry, his library, his excellent wine cellar, his friends and the occasional dancing girl are all that a man could possibly desire. His one trip abroad to study in Athens, was not a success. He was lured by Brutus into military action, fought ingloriously at the battle of Philippi and was glad to get back to Rome."
Lone Traveller; one woman, two wheels and the world by Anne Mustoe, p. 209. Virgin, London 2000.
Those who rush across the sea change their skies, not their souls.
Horace (65 - 8 BC. Latin poet and satirist)
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So you want to understand the abstract delights of quantum physics but you have almost no math?
Someone who can write clearly and engagingly should write a small paperback which could put the beginning thinker on the right path of understanding quantum physics - a paperback which could sit beside the Greene books on the popular science shelves in bookshops. If this happened the beginning thinker would be shocked; since according to Bohr, "Anyone who is not shocked by quantum theory has not understood it".
At present (2008) in Cape Town there is only Lee Smolin The Trouble with Physics in paperback shelved next to the Greene books. And Smolin is not trying in his book to be an introduction to quantum physics.
Popular Books which presuppose only simple high-school math:
a. Herbert, Nick. Quantum Reality. Anchor Press/Doubleday, Garden City, New York, 1985.
b. Feynman, Richard P. QED: the strange theory of light and matter. Princeton U. Press, Princeton, 1985.
Popular Books which presuppose introductory university or advanced high-school math:
c. Chester, Marvin. Primer of Quantum Mechanics. John Wiley & Sons, New York, 1987.
d. Mattuck, Richard D. A Guide to Feynman Diagrams in the Many-Body Problem (2nd. Ed.). Dover Publications, New York, 1976.
Finding the right books from which to learn quantum physics from a start position of no knowledge is hard.
And there are many false steps one could take.
The books listed above are rare exaamples of excellence in didactic presentation.
The Feynman and Herbert books have different approaches to the material but beautifully complement each other.
The more advanced books are both conceptual (philosophical) and mathematical.
Such a combination is not often found.
Chester presents the standard mathematical objects of quantum theory and Mattuck explains the advanced concepts.
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There's a new Darwin. His name is Edward O. Wilson
Tom Wolfe in Sorry but your soul just died 1996
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The Fundamental property of Venice
Henry James wrote:
"almost every one interesting, appealing, melancholy, memorable, odd, seems at one time or another, after many days and much life, to have gravitated to Venice by a happy instinct, settling in it and treating it, cherishing it, as a sort of repository of consolations; all of which today, for the conscious mind, is mixed with its air and constitutes its unwritten history.
The deposed, defeated, the disenchanted, or even only the bored, have seemed to find there something that no other place could give." (Italian Hours, 1909)
Marcel Proust ("When I went to Venice, I discovered that my dream had become - incredibly, but quite simply - my address") (1925) and Thomas Mann (Death in Venice, 1912) found the same atmosphere of transience and decay which caused them to be enchanted by it. So of course did Palladio, Tiepolo, Titian, Tintoretto, Renoir, Turner, Signac, Delacroix...the list goes on and on...
Its something that a visitor to Venice picks up on during her/his first progress down the Grand Canal, or never understands...