Metaphor's  Algebra   ^{© 2000} 

Lines in blue are "Links".
Lines in white are "quotes".
Lines beginning with a yellow (g) are news or updates.
Black lines are my text, you are allowed to copy from the text, providing the copyright is properly attributed to [© Ben Tamari ( www.ecometry.biz )].
Update 15-November-2007

This page is dedicated to my friend  Jacob Lavie, who died of cancer on 20-03-2004 at the age of  75. He was a Socratic type.

Metaphors Quotes

"The unconscious helps by communicating things to us, or making figurative allusions."

  Jung C. G. (1961) "Memories, Dreams, Reflections", p. 302, Vintage Books (April 1989) NY.

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"We reveal ourselves in the metaphors we choose for depicting the cosmos in miniature."

  Gould S. J. (1996) "Full House", p. 7, Harmony Books, NY.

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"Humans are pattern-seeking animals."

Graham B. (2006) "The Intelligent Investor", p. 220, Revised Edition, Updated with New Commentary by Jason Zweig, Collins.

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 "What distinguishes a mathematical model from, say, a poem, a song, a portrait, (a metaphor, my addition BT) or any other kind of "model," is that the mathematical model is an image or picture of reality painted with logical symbols instead of with words, sounds or watercolors.    These symbols are then strung together in accordance with a set of rules expressed in a special language, the language of mathematics."

Casti L. Johan (1992) "REALITY RULES: Picturing the World in Mathematics", Preface vii, Johan Wily & Sons, inc.

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 "Geometrical perception, which is extremely rich and elaborate, is directly tied up with the visual areas of the brain. Using these areas you can immediately contemplate a picture and perceive the beauty of it."

Alain Connes  in  Karl Sabbagh (2002) "The Riemann  Hypothesis", p. 242, Farrar, Straus and Giroux.

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Fractalization of the Language   (25-August-2002)

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'A word is worth a thousand ...?...' 

As the old saying goes:  " A picture is worth a thousand words "

[ But sometimes: One (good) sentence is worth a thousand pictures.for example: "A picture is worth a..., " (To follow suit Mobius Strip). (5-April-2007).]

'A metaphor is worth a thousand pictures'

'An allegory is worth a thousand metaphors'

'A.....?..... is worth a thousand allegories'

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Pythagoras Theorem a² + b² = c²  is the  Seed - Einstein Formula E = MC² is the Tree. (2000)
To follow suit Friedrichs K.O., (1965) "From Pythagoras to Einstein", MAA.

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Mathematics are the Metaphors of Realities. (15-Januar-2007).

The Rules:

* New idea, the result of the combination of two (or three) metaphors, pictures or tools, must give birth to yet a new insight, or solve a certain problem, instead.
  (e.g., Columbus).

* Each metaphor must come from a different discipline or technology (e.g., Kepler).

* At least one of the drawings must be an original work of the ‘inventor’ (e.g., Sturgeon).

* It is not mandatory to be the first originator, but must be the first to analyses (e.g., Marshall).

* Simply describing a dream is not sufficient by itself (e.g., the snake of Kekule).

Columbus - The Way to India

Christopher Columbus (1451 - 1506).

In order to achieve his goal, Columbus made several assumptions: The world is round (a theory that was already well known before Columbus’s time); it is only 2,300 miles in circumference; and therefore, to reach Cathay, he needed no more than two months, in those days the maximum time a ship could be at sea without putting into port. The question was how to get the wind at his back. As a sailor who traveled between Spain and Africa, he noticed that the wind in Spain comes from the west (blows eastward from the sea), and the wind on the coast of Africa blows  to the sea (westward).

To persuade the committee to allow him to make his voyage, he depicted the earth as pear-shaped. The Direction of the Wind - That was Columbus’s great secret: From the African coast to the coast of India (which turned out to be America), the wind blows westward. From the Indian (American) coast to the European coast, the wind blows eastward. Columbus began his first voyage in the Canary Islands and returned via the Azores.

The Earth                                    Winds                                Columbus's first voyage, Morison (1942) p. 222.
 

   Books:

Morison (1942) "Admiral of The Ocean Sea", Little  Brown & Com, Boston.

  Links to sites about Christopher Columbus: 

http://www1.minn.net/~keithp

http://muweb.millersv.edu/~columbus

http://www.win.tue.nl/cs/fm/engels/discovery/columbus.html

http://www.neta.com/~1stbooks/colony4a.htm

Kepler - The Distances Between the Planets in the Solar System

  Johannes Kepler (1571 - 1630).

Building upon the theories of Copernicus, Kepler envisioned the planets arranged around the sun as platonic solids (polyhedrons), one inside the other, in a three-dimensional universe. His great achievement was the shift from Copernicus’s two-dimensional map to three dimensions. The distances between the moving planets could be estimated by using the platonic solids nested one inside the other as a starting point.

The heavens as a Disk, as envisioned by Copernicus  he five platonic Solids  The heavens as envisioned by Kepler

  Links to sites about Johannes Kepler: 

http://www.kepler.arc.nasa.gov/johannes.html

www.kepler-museum.de

http://es.rice.edu/ES/humsoc/Galileo/People/kepler.html

http://antwrp.gsfc.nasa.gov/apod/ap960831.html

http://home.cvc.org/science/kepler.htm

Sturgeon -  The Inventor of the ElectroMagnet device

  William Sturgeon (1783 - 1850).

William Sturgeon was an English electrical engineer.

In 1825 he built the first practical electromagnet, which led to the invention of the telegraph, the electric motor, and numerous other devices basic to modern technology.

 

  Links to sites about William Sturgeon:

http://chem.ch.huji.ac.il/~eugeniik/history/sturgeon.html

http://www.acmi.net.au/AIC/STURGEON_BIO.html

http://inventors.about.com/library/inventors/blelectromagnet.htm

http://physics.kenyon.edu/EarlyApparatus/Electricity/Electromagnet/Electromagnet.html

http://www.sparkmuseum.com/MAGNET.HTM

Probably, Fleeming Jenkin (1833 - 1885) was the first to introduce the demand and supply together in one diagram, ("The Graphic Representation of the Laws of Supply and Demand, and their application to labour" Edinburgh, 1870, Rep. Political Economy, London 1931). But "It was left to Marshall to synthesize for general use the ideas of Jevons and others, respecting demand, with those of Ricardo and Johan Stuart Mill, on cost of production and supply, giving to the English-speaking world a broader foundation for value theory than had been furnished by either of antecedent schools." (My bold B.T.). Edmund Whittaker (1940) "A History of Economic Ideas" Longmans NY, p. 453.

Marshall (1890) "Principles Of Economics", p. 346, n. 1., MacMillan 8ed. (1949). "To represent the equilibrium of demand and supply geometrically we may draw the demand and supply curves together as in Fig. 19.".

 

The Demand,  p. 96.                            The Supply,  p. 344.                     The Demand and Supply together, p.  346. 

This metaphor is dedicated to the memory of Don Patinkin.

  Links to sites about Alfred Marshall:

http://cepa.newschool.edu/het/profiles/marshall.htm

http://www.econlib.org/library/Enc/bios/Marshall.html

http://staff-www.uni-marburg.de/~multimed/theorie/economics/marshall/bio/Marshall.html

http://en.wikipedia.org/wiki/Alfred_Marshall

http://www.economyprofessor.com/theorists/alfredmarshall.php

Benoit B. Mandelbrot (1924 - ).

The first picture features Julia figures which, together with Fatu, was among the firsts to study the Iterations Functions. Julia discovered a whole new class of strange and beautiful shapes, called today 'Julia sets'.

The second picture is for the computer as a concept. Computer as an Extended Mind - Mandelbrot mind in this case. Inside display frame one can find one of Mandelbrot's experiments in which he trails in his way to discover the M set.

The third picture is the Mandelbrot set (or M set). Mandelbrot discovered the M set when he was trying to make a map of Julia sets. M set is defined as The set of all points C for which the iteration Z_t+1 = Z_t² + C, with Z₀=0, stays bounded as t →∞.

Julia (1918) p. 18.                                 Mandelbrot (2004)  p. 162.             Mandelbrot Set (or M set)

This is what Mandelbrot wrote on the discovery of the M-set:

"Not only did I have access to a computer in 1979, but I was familiar with its power (My bold, B.T.). The fact that no one knew what was going to emerge was enough to make these calculations worth trying. A fishing expedition led to a primitive form of Mandelbrot Set." Lesmoir-Gordon (2004). p. 59.

"To approximate the IFS (iterated function systems or schemes) limit involves an easy mechanical process, and that should be undertaken first. Next, one should inspect pictures of a sufficient number of variants of the IFS limit and check for structures that may be suggested by the eye. A deep thinking mode should come later. Of course, this is precisely the strategy whose best-known success was the discovery, then about to be announced, of the Mandelbrot set." Mandelbrot (2004) p. 177.

  Books:

Nigel Lesmoir-Gordon (2004) "The Colours of Infinity: The Beauty, The Power and the Sense of Fractals" include CD, Clear Books.

  Links to sites about Benoit B. Mandelbrot:

Ben Tamari  (1942 - ).
The speculative index of the stock markets behaves like water flowing through a sleeve (pipe).

Assumption: The mathematical root of the speculative stock markets is the “sleeve (pipe) model”.

see Tamari B., Foundations, p. 47, Fig. 3 and Stocks.

        

The Sleeve (Pipe, Torus) The speculative index of Tel-Aviv stock market (1.1.1980-31.10.2007) The Sleeve (Pipe) Model  for the stock markets.

Conjecture: The mathematical foundations of the speculative stock markets is Pappus's and Desargues's Theorems:

Pappus's Theorems

Eric W. Weisstein. "Pappus's Hexagon Theorem.

Note 1, 04-October-2005   A Note On .."Psychology Of Invention"

See: Jacques Hadamard (1945) "The Psychology Of Invention In The Mathematical Field", Dover, p. 142.

A letter from Albert Einstein. (My bold B.T.)■

Note 2, 25-Merch-2006   A Note On Metaphors

See: George Lakoff and Mark Johnson (1980) "METAPHORS We Live By", The University of Chicago Press.

"The essence of metaphor is understanding and experiencing one kind of thing in terms of another." p. 5.

"We claim that most of our normal conceptual system is metaphorically structured; that is, most concepts are partially understood in terms of other concepts. This raises an important question about the grounding of our conceptual system. Are there any concepts at all that are understood directly, without metaphor?  If not, how can we understand anything at all ?" p. 56.

"The primary function of metaphor is to provide a partial understanding of one kind of experience in terms of another kind of experience. This may involve preexisting isolated similarities, the creation of new similarities, and more." p. 154.

"The reason we have focused so much on metaphor is that it unites reason and imagination." p. 193.

"The natural question to ask, then, is whether people actually think and act in terms of consistent sets of metaphors. A special case where they do is in the formulation of scientific theories, say, in biology, psychology, or linguistics. Formal scientific theories are attempts to consistently extend a set of ontological and structural metaphors." p. 220.

"New metaphors are capable of creating new understandings and, therefore, new realities." p. 235.

"But metaphor is not merely a matter of language. It is a matter of conceptual structure." p. 235.

"..., what stands out most in our minds are the metaphors themselves and the insights they have given us into our own daily experiences." p. 239.

It is my hope that my arguments (accompanied by examples) support theirs. I have used the term 'metaphor' in a more generalized way, that is, the metaphor as a kind of a 'image', 'picture', 'vision' or a 'megametaphor'. From my point of view, the pictures in my mind's eye appeared first and then their symbols followed suit - Letters, Words, Numbers, and other symbols or signs. See also Note 2 - Form.