Published 2000, Update 9-June-2012
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 QuotesJosé Ortega y Gasset (1925) "The Dehumanization of Art", Tr. Helene Weyl , 2 ed., Princeton. p. 32, 33, respectively. (added 10-10-2010)
"Poetry has become the higher algebra of metaphors."
"The metaphor is perhaps one of mans most fruitful potentialities. Its efficacy verges on magic, and it seems a tool for creation which God forgot inside one of His creature when He made him."
Ernst Mach (1943) "Popular Scientific Lectures", Tr. McCormack, 5 ed. The Open Court Pub, p. 275.
" The movement of our thoughts obeys the law of association."
Carl G. Jung (1961) "Memories, Dreams, Reflections", Vintage Books (April 1989) NY, p. 302.
"The unconscious helps by communicating things to us, or making figurative allusions."
Rudy Rucker (1987) "MIND TOOLS: The Five Levels of Mathematical Reality", Houghton Mifflin Company, Boston, p. 247.
" Church's theorem is a metaphor for the fact that no simple test can give Yes or No answers for important questions."
" Godel's theorem is a metaphor for the fact that no logical program can hope, even in the limit, to answer all the questions."
Montague Ullman (October 1992) in Steven M. Rosen "Science, Paradox, and the Moebius Principle", State University of NY Press, Foreword, p. xi.
"The poets ends her poem out into the world.
The dreamer is communicating to himself.
Both use the medium of metaphor, The poet by rearranging words,
The dreamer by rearranging images."
Stephen Jay Gould(1996) "Full House", Harmony Books, NY, p. 7.
"We reveal ourselves in the metaphors we choose for depicting the cosmos in miniature."
Jorge Luis Borges (2000) "This Craft of Verse".
"Now we are led to the two obvious and major conclusions of this lecture. The first is, of course, that though there are hundreds and indeed thousands of metaphors to be found, they may all be traced back to a few simple patterns. But this need not trouble us, since each metaphor is different: every time the pattern is used, the variations are different."
John L. Casti and Werner DePauli (2000) "Gödel: A Life of Logic", Basic Books, p. 122.
"...a material object - the human brain - gives rise somehow to the mind, the mind itself seems to have no material composition. Rather, it appears to consist solely of patterns of information existing in some realm beyond ordinary space and time."
Connes Alain in Karl Sabbagh (2002) "The Riemann Hypothesis", Farrar, Straus and Giroux, p. 242.
"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. "
Marcel Danesi (2004) "Poetic Logic: The Role of Metaphor in Thought, Language, and Culture", Atwood Pub., p. 55, 125 respectively.
"It is impossible to study the mind without recourse to metaphor."
"Metaphor is the default form of thought, providing many angles from which to literally "see" the world."
Freeman Dyson (2007) "Mathematics as Metaphor: Selected Essays of Yuri I. Manin.", American Mathematical Society, Foreword p. vii.
"Coordinates, fluxions, symbolic logic and Riemann surfaces
are all metaphors,
extending the meanings of words from familiar to unfamiliar contexts."
Terry Marks-Tarlow (2008) "Psyche's Veil: Psychotherapy, fractals and complexity", Routledge, p. 92.
"The art and science of psychotherapy merge within its metaphors. .....
Metaphor is a primary cognitive tool by which we make sense of the world."
Fractalization of the Language I (April 2002)
A word is worth a thousand ...?... .
"A picture is worth a thousand words", as the old saying goes.
But sometimes: One (good) sentence is worth a thousand pictures.
for example: "A picture is worth a thousand words" (following Mobius strip). (added: 5-April-2007). A metaphor is worth a thousand pictures. An allegory is worth a thousand metaphors. A .....?..... is worth a thousand allegories.
Fractalization of the Language II (15 November-2009)
"Between any two 'points' in a set"
of: points, numbers,
words, sentences, paragraphs, thoughts, etc.,
"we can put another 'point' " and if the set is not big enough, we can add another 'point' .
Wislawa Szymborska (2005) "COLON" "The nightmare of the poet" (translate Deena Land, 15-July-201)
"An allocated number of words.
Never a needless one
Meaning - no poetry
And no philosophy and no religion."
Pythagoras theorem - a2 + b2 = c2 - is the seed. Einstein Formula - E = MC2 - is the Tree. (added: 10-July-2000)
following Friedrichs (1965) From Pythagoras to Einstein, MAA.
A metaphor is a bridge from the known to the unknown.
(following Ariston 7-March -2009)
or more exactly: A metaphor is a harpoon thrown from the known into the unknown. (7-August -2010)
A good metaphor is the cornerstone of a new theory. (20-August -2010)
The Rules to Metaphor's Algebra:
1. 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).
2. Each metaphor must come from a different discipline or technology (e.g., Kepler).
3. At least one of the drawings must be an original work of the ‘inventor’ (e.g., Sturgeon, Volta).
4. It is not mandatory to be the first originator, but must be the first to analyses (e.g., Marshall).
5. 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.
Morison (1942) "Admiral of The Ocean Sea", Little Brown & Com, Boston.
Copernicus - The Universe
Nicolaus Copernicus (1473 - 1543). Aristotle supposed that the Earth was motionless in the center of the universe. Aristotle's cosmology and Christian theology together were accepted as the correct picture of the Universe. Observations since Aristotle's times were not compatible with this cosmology, in which the Planets, the Sun and the Moon move in perfect circles around the Earth at constant speed. The difficulties were that sometimes the Planets moved too fast or too slow, and sometimes they stopped or moved backward. After a long process of corrections Aristotle's cosmology became Ptolemy's cosmology, which included different centers for the movement of the Planets. The planets moved in an epicycle around the point on the circle of the planets and the Sun's orbit was in an angel with the orbits of the other planets. Copernicus changed Ptolemy's cosmology to fit observations with geometric considerations. Mainly by changing the position of the Earth with the position of the Sun. Now the Sun was in the center and standing, and the Earth was in an orbit and moved as the other planets.
Copernicus created the new cosmology by combining two disciplines - Cosmology and Geometry (mathematics in the terminology of his time) or in his words:
"We find, then, in this arrangement the marvelous symmetry of the universe, and a sure linking together in harmony of the motion and size of the spheres, such as could be perceived in no other way." Copernicus, p. 50.
Aristotle's cosmology Geometry Considerations Copernicus's cosmology
Copernicus (1976) Trans. A.M. Duncan "Copernicus: On the Revolutions of the Heavenly Spheres", David Charles.
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 The five platonic Solids The heavens as envisioned by Kepler
N Volta - The Inventor of the Voltaic Pile - the Battery
Allessandro Volta (1745 - 1827). Galvani's 'animal electricity' is the result of contact of two different metals in a conductive environment. He announced (at 1800) a new electrical device - the Voltaic Pile - and described it as an 'electric organ', which is like the torpedo fish.
It was the first direct current battery.
" I have the pleasure of sending you some striking results at which I have arrived in pursuing my experiments on the electricity produced by the simple mutual contact of different metals. The chief result is the construction of apparatus having the properties (such as ability to give shocks) of Leyden jars which operate continuously, or whose charge is restored automatically after each discharge. The apparatus of which I speak, and which doubtless will astonish you, is nothing but the assembly of a number of good conductors of different kinds, arranged in a certain manner, 30, 40, 60 pieces, or more, of copper or silver, each placed next to a piece of tin or, better, zinc, and an equal number of layers of water, or some other conducting liquid such as salt water, or pieces of leather or card soaked in these liquids, are placed between each pair of different metals. Behold, this is all that makes up my novel instrument, which imitates, as I have said, the effects of a Leyden jar." http://www.electric-history.com/~zero/004-Battery.htm
The frog legs (Galvani) + Torpedo fish (Volta) Volta torpedo fish pattern - The artificial one The Voltaic Pile - Battery
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.
Marshall - The Economic's Equilibrium, the demand and supply together.
Alfred Marshall (1842 - 1924). Probably, Fleeming Jenkin (1833 - 1885) was the first to introduce the demand and supply together in one diagram. 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." Edmund Whittaker (1940) "A History of Economic Ideas", Longmans NY, p. 453. (My underline B.T.).
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 Marshall metaphor is dedicated to the memory of Don Patinkin.
Mandelbrot - The Discovery of the Mandelbrot Set (M-set)
Benoit B. Mandelbrot (1924 - 2010). The first picture feature 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 Zt+1 = Zt2 + C, with Z0=0, stays bounded as t → ∞.
Julia sets (1918) p. 18. Mandelbrot screen (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. 59i.
"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.
M set is a virtual entity which lived only in a computer. The computer is an extension of the human mind, extension which creates a new culture - CompuMind. The computer to the mind is more then ships to legs (Columbus) or telescopes to eyes (Kepler). Maybe, one day, we will find that our brain was designed by God when He was playing with M set.
Julia (1918) "Memoir on Iterations of Rational Functions", Translated in English by Alessandro Rosa (2001).
Nigel Lesmoir-Gordon (2004) "The Colours of Infinity: The Beauty, The Power and the Sense of Fractals", include CD, Clear Books.
Mandelbrot (2004) "Fractals and Chaos" Springer.
I do hope Mandelbrot will forgive me for the simplicity of the representation.
Tamari - The Sleeve (Pipe) and the Stock Market
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 Ben Tamari (1990) Foundations of Economics (Hebrew) p. 47, Fig. 3 and Stocks.
The Sleeve (Torus) The speculative index of Tel-Aviv stock market (1980-2007) The Sleeve (Pipe) Model for the stock markets.
Conjecture The mathematical foundations of the speculative stock markets is Pappus's (or Pascal's Theorem) and Desargues's Theorems:
Pappus'sTheorem, Eric W.
Weisstein "Pappus's Hexagon Theorem."
Application of Pappus (Hexagon) Theorem on The Speculative Cycle of Tel Aviv Stock Exchange
Note 1, 04-October-2005 Note On ...Psychology Of InventionJacques Hadamard (1945) "The Psychology Of Invention In The Mathematical Field", Dover, p. 142. A letter from Albert Einstein (My bold B.T.)
Note 2, 25-03-2006 A Note On Metaphors
Lakoff and 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 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.
Or in the words of Marcel Danesi (2004) "Poetic Logic: The Role of Metaphor in Thought, Language, and Culture" Atwood Pub, p. 97.
"...Metaphor is evidence of the human ability to visualize the universe as a coherent organism. Proof of our capacity, not just to see one thing in another but to change the very nature of things. When a metaphor is accepted as fact, it enters groupthink, taking on an existence in the real world."
Koestler A. (1964) "The Act Of Creation", Pan Books ltd.
José Ortega Y Gasset (1925) "The Dehumanization of Art", , Tr. Helene Weyl , 2 ed. Princeton.
Lakoff and a href="http://www.cog.brown.edu/~mj/"Johnson (1980) "METAPHORS We Live By", The University of Chicago Press.
Merton and Barber (2004) "The Travel and Adventures of Serendipity" Princeton UP.
Ruelle D. (2007) "The Mathematician's Brain" Princeton UP.
Danesi M. (2004) "Poetic Logic: The Role of Metaphor in Thought, Language, and Culture" Atwood Pub.
Manin U. i. (2007) "Mathematics as Metaphor: Selected Essays", American Mathematical Society.
Ortony A, Editor (1993) "Metaphor and Thought" Cambridge UP.
Rucker R. (1987) "MIND TOOLS: The Five Levels of Mathematical Reality", p. 247, Houghton Mifflin Company, Boston.
Ramachandran V. S. and Hubbard E. M. (2003) "Hearing Colors, Tasting Shapes", Scientific American. Synesthesia and Metaphors.