Lapidarium RSS

Amira's favorite quotes

"Everything you can imagine is real."— Pablo Picasso

Lapidarium notes


Age of information
Artificial intelligence
Cognition, relativity
Cognitive science
Collective intelligence
Greek & Latin
Human being
Mind & Brain
Science & Art
Self improvement
The other


A Box Of Stories



Lipman Bers, American mathematician born in Riga who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups, (1914-1993), cited in D Albers, G Alexanderson, C Reid, More Mathematical People, Harcourt Brace Jovanovich, 1990. (tnx therx)
Beauty in mathematics is seeing the truth without effort.
George Pólya, Hungarian Jewish mathematician. He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory (1888-1985), cited in The study of education at Stanford: report to the university, Tom 8, Stanford University, 1969, p. 48.
Mathematics is the only infinite human activity. It is conceivable that humanity could eventually learn everything in physics or biology. But humanity certainly won’t ever be able to find out everything in mathematics, because the subject is infinite. Numbers themselves are infinite.
Paul Erdős, Hungarian mathematician (1913-1996) cited in Paul Hoffman’s, The man who loved only numbers: the story of Paul Erdös and the search for mathematical truth, Hyperion, 1998, p. 56.

Inside a mathematical proof lies literature. Some of the greatest mathematicians were also some of classical history’s most poetic storytellers

“Like novelists, mathematicians are creative authors. With diagrams, symbolism, metaphor, double entendre and elements of surprise, a good proof reads like a good story. (…) [Reviel] Netz reveals the stunning stylistic similarities between Hellenistic poetry and mathematical texts from the same era. (…) In the very layout, in the use of a particular formulaic language, in the structuring of the text (…) its success or failure depends entirely on features residing in the text itself. It is really an activity very powerfully concentrated around the manipulation of written documents, more perhaps than anywhere else in science, and comparable, then, to modern poetry. (…)

Metaphor is fairly standard in mathematics. Mathematics can only become truly interesting and original when it involves the operation of seeing something as something else – a pair of similarly looking triangles, say, as a site for an abstract proportion; a diagonal crossing through the set of all real numbers.” “
Reviel Netz, Professor of Classics and Philosophy at Stanford University, Inside a mathematical proof lies literature, says Stanford’s Reviel Netz, Stanford University Report, May 7, 2012. See also: 
Oulipo - a group of writers interested in exploring the application of mathematical structures, patterns and algorithms to writing
The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.
G. H. Hardy, was a prominent English mathematician, known for his achievements in number theory and mathematical analysis (1877-1947), A Mathematician’s Apology (1941)
George Dyson: Unravelling the digital code

“We now live in a world (…) increasingly run by self-replicating strings of code. Everything we love and use today is, in a lot of ways, self-reproducing exactly as Turing, von Neumann, and Barricelli prescribed. It’s a very symbiotic relationship: the same way life found a way to use the self-replicating qualities of these polynucleotide molecules to the great benefit of life as a whole, there’s no reason life won’t use the self-replicating abilities of digital code, and that’s what’s happening. (…)

In 1945 we actually did create a new universe. This is a universe of numbers with a life of their own, that we only see in terms of what those numbers can do for us. (…) And that’s not just a metaphor for something else. It actually is. It’s a physical reality. (…)

But it was Turing who developed the one-dimensional model, and von Neumann who developed the two-dimensional implementation, for this increasingly three-dimensional digital universe in which everything we do is immersed. And so, the next breakthrough in understanding will also I think come from some oddball. It won’t be one of our great, known scientists. It’ll be some 22-year-old kid somewhere who makes more sense of this. (…)

We’re seeing a fraction of one percent of it, and there’s this other 99.99 percent that people just aren’t looking at. (…)

I think they [Turing & von Neumann] would be immediately fascinated by the way biological code and digital code are now intertwined. Von Neumann’s consuming passion at the end was self-reproducing automata. And Alan Turing was interested in the question of how molecules could self-organize to produce organisms. (…) They would be amazed by the direct connection between the code running on computers and the code running in biology—that all these biotech companies are directly reading and writing nucleotide sequences in and out of electronic memory, with almost no human intervention. That’s more or less completely mechanized now, so there’s direct translation, and once you translate to nucleotides, it’s a small step, a difficult step, but, an inevitable step to translate directly to proteins. And that’s Craig Venter’s world, and it’s a very, very different world when we get there.” “
George Dyson, author and historian of technology whose publications broadly cover the evolution of technology in relation to the physical environment and the direction of society, ☞ Science historian George Dyson: Unravelling the digital codeEdge, Mar 26, 2012 

“The most beautiful and satisfying experiences open to humankind are not derived from the outside, but are bound up with the development of the individual’s own feeling, thinking and acting. The genuine artists, investigators and thinkers have always been persons of this kind. However inconspicuously the life of these individuals runs its course, none the less the fruits of their endeavors are the most valuable contributions which one generation can make to its successors. (…)

Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships.” “
Albert Einstein, German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics, Nobel Prize laureate (1879-1955), Professor Einstein Writes in Appreciation of a Fellow-Mathematician. To the Editor of The New York Times, Princeton University, May 1, 1935.
(Illustration: Albert Einstein sailing his boat on Saranac Lake (Courtesy: The Fantova Collection, Princeton University)

“There is still a difference between something and nothing, but it is purely geometrical and there is nothing behind the geometry.” “
Martin Gardner, American mathematics and science writer specializing in recreational mathematics (1914-2010), Mathematical magic show: more puzzles, games, diversions, illusions & other mathematical sleight-of-mind from Scientific American, Vintage Books, 1978, p. 21. (Illustration: Agnes Denes, Metropolitan Museum NYC)
Tim Love on Science and the Arts

“Maths and music make claims to be universal languages of sorts. Science’s base metaphors are increasingly mathematical - building conceptual models from billiard balls is a thing of the past. Some theorists (for example Wimsatt) consider metaphor central to poetry. Colin Turbayne [1] thinks that that science is metaphor-laden too, the metaphors dead. Waismann [quoted in 2] argues that scientific concepts are only closed in specific contexts and that they are not different in kind to the metaphors of poetry. (…)

Quantum Theory - In Quantum Theory, probabilities can be calculated but only when an observation is made can any certainty be established. Observation is said to ‘collapse the probability function.’ This has been used for an analogy to the way that a text is interpreted (dis-ambiguated) by the act of reading [3]. (…)

Relativity - Connections are made between Einstein’s Special Relativity and analytic cubism. Awareness of the equal importance of world viewpoints, the impossibility of absolute motion and time perhaps permeated via the Zeitgeist to artists; the link came from no deep mutual understanding.

Gödel - Gödel’s findings have helped soften artists’ views on science and has removed an aim of classical science. They have only made maths more obviously like the other sciences. The gap between science and the arts hasn’t thereby been reduced.

Geometry - Mondrian is heavily geometric and minimalist. This doesn’t make him more appealing to mathematicians. Equally, the 4-colour problem in maths isn’t appealing to artists.” “
Tim Love, poet, computer officer at Cambridge University, Science and the Arts, PhysLink, Sept 1995 [1] C. Turbayne, 'The Myth of Metaphor', Univ of South Carolina Press, 1970, [2] P.L. Hagen, Peter Lang, ‘Metaphor’s Way of Knowing’, 1995, [3] Ian Mills, 'The Quantum Uncertainty of the Narrator', in ‘Poetry Review’ V85.1, Spring 1995.
If we go back to our checker game, the fundamental laws are rules by which the checkers move. Mathematics may be applied in the complex situation to figure out what in given circumstances is a good move to make. But very little mathematics is needed for the simple fundamental character of the basic laws. They can be simple stated in English for checkers.
The platonist metaphor assimilates mathematical enquiry to the investigations of the astronomer: mathematical structures, like galaxies, exist, independently of us, in a realm of reality which we do not inhabit but which those of us who have the skill are capable of observing and reporting on. The constructivist metaphor assimilates mathematical activity to that of the artificer fashioning objects in accordance with the creative power of the imagination.
Michael Dummett, British philosopher. He was, until 1992, Wykeham Professor of Logic at the University of Oxford, Truth and Other Enigmas, Harvard University Press, 1978, p.225.
Freeman Dyson: ‘Some mathematicians are birds, others are frogs’

Joris Hoefnagel, Allegorie für Abraham Ortelius, c.1563

“Some mathematicians are birds, others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. They delight in concepts that unify our thinking and bring together diverse problems from different parts of the landscape. Frogs live in the mud below and see only the flowers that grow nearby. They delight in the details of particular objects, and they solve problems one at a time. (…)

Mathematics needs both birds and frogs. Mathematics is rich and beautiful because birds give it broad visions and frogs give it intricate details. Mathematics is both great art and important science, because it combines generality of concepts with depth of structures. (…)

At the beginning of the seventeenth century, two great philosophers, Francis Bacon in England and René Descartes in France, proclaimed the birth of modern science. Descartes was a bird, and Bacon was a frog. Each of them described his vision of the future. Their visions were very different. Bacon said, “All depends on keeping the eye steadily fixed on the facts of nature.” Descartes said, “I think, therefore I am.” According to Bacon, scientists should travel over the earth collecting facts, until the accumulated facts reveal how Nature works. The scientists will then induce from the facts the laws that Nature obeys. According to Descartes, scientists should stay at home and deduce the laws of Nature by pure thought. (…)

For four hundred years since Bacon and Descartes led the way, science has raced ahead by following both paths simultaneously. Neither Baconian empiricism nor Cartesian dogmatism has the power to elucidate Nature’s secrets by itself, but both together have been amazingly successful.

For four hundred years English scientists have tended to be Baconian and French scientists Cartesian. Faraday and Darwin and Rutherford were Baconians; Pascal and Laplace and Poincaré; were Cartesians. Science was greatly enriched by the cross-fertilization of the two contrasting cultures. Both cultures were always at work in both countries. Newton was at heart a Cartesian, using pure thought as Descartes intended, and using it to demolish the Cartesian dogma of vortices. Marie Curie was at heart a Baconian, boiling tons of crude uranium ore to demolish the dogma of the indestructibility of atoms.”

Freeman Dyson, British-born American theoretical physicist and mathematician, famous for his work in quantum field theory, solid-state physics, astronomy and nuclear engineering, Birds and Frogs (pdf), Notices of the AMS, Vol 56, Nr 2, 2009

Michael Gove speaks to the Royal Society on maths and science

Euclid in Raphael's detail from The School of Athens

History is driven, above all, by mathematics and the power it gives us to understand, predict and control the world.

The emergence of the first, truly great, Western civilization, in the scattered city states of Ancient Greece, was intimately connected with the first systematic thinking about reason, logic and number.
Although Pythagoras himself is a figure shrouded by myth, the Pythagorean revolution he and his disciples set in motion was the prelude to the astonishing flowering of classical philosophy which laid the foundations of the Western world.

On those first foundations men such as Euclid and Archimedes devised a means of making sense of the world which enabled their contemporaries, and successors, to master it. Greece bequeathed her mathematical heritage to Rome and the achievements of the Caesars, their imperial highways, feats of engineering and centralised accounts, were all the fruits of mathematical knowledge.

Rome’s fall was the prelude to Islam’s rise and again mathematical innovation was the leading indicator of historical progress. While Western Europe was sunk in a Dark Age of dynastic squabbling, pagan aggression and superstitious poverty the Islamic world flourished, advanced and subdued its foes while also nurturing a series of mathematical thinkers responsible for transmitting wisdom and generating great historic breakthroughs. Whether it was the establishment of Arabic numerals as the principal method of mathematical notation or the invention of algebra, Arabic and Islamic culture was the world’s forcing-house of progress for centuries.

Europe only caught up again in the sixteenth century, but when we did it was with a burst of mathematical innovation which once more moved the world on its axis. Galileo and Descartes authored advances in mechanics and geometry which were hugely ground-breaking. They were followed by the arguably even greater geniuses of Newton and Leibniz.

Newton, the greatest President this society has had - so far - was the godfather of the Enlightenment, mankind’s great period of intellectual flowering, the liberation from ignorance on which our current freedoms rest.

In the nineteenth century, the greatest mathematicians were Germans - like Karl Friedrich Gauss and Bernhard Riemann - reflecting the shift of intellectual innovation, and economic power, to central Europe.

In the twentieth century, the flight of mathematicians like Kurt Gödel from a fascist Europe sunk in a new barbarism to a new world of liberty and promise again presaged a fundamental shift in economic, political and intellectual power. (…)

Richard Feynman has described the precision of quantum mechanics as like being able to measure the distance from New York to L.A to the nearest hair’s breadth. And for those of us navigating journeys even more fraught and perilous than an odyssey across America - such as driving from West London to Westminster without hitting roadworks - the precision of GPS satellite technology can guide us - and all thanks to the extraordinary precision of relativity’s equations.”

Michael Gove, British Conservative politician, journalist and author, Secretary of State for Education, Michael Gove speaks to the Royal Society on maths and science (full speech), 29 June 2011
" "And once you take on board the idea that mathematics itself can, through its inherent structure, embody any and all aspects of reality—sentient minds, heavy rocks, vigorous kicks, stubbed toes—you’re led to envision that our reality is nothing but math. In this way of thinking, everything you’re aware of—the sensation of holding this book, the thoughts you’re now having, the plans you’re making for dinner—is the experience of mathematics. Reality is how math feels. To be sure, this perspective requires a conceptual leap not everyone will be persuaded to take; personally, it’s a leap I’ve not taken. But for those who do, the worldview sees math as not just “out there,” but as the only thing that’s “out there.” “
Brian Greene, American theoretical physicist and string theorist, professor at Columbia University, The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos, Knopf, 2011 (tnx johnsparker)
The same mathematics of networks that governs the interactions of molecules in a cell, neurons in a brain, and species in an ecosystem can be used to understand the complex interconnections between people, the emergence of group identity, and the paths along which information, norms, and behavior spread from person to person to person.
James Fowler is a political scientist at the University of California, San Diego, answering the question "If you only had a single statement to pass on to others summarizing the most vital lesson to be drawn from your work, what would it be?" in Starting Over, SEED, Aprill 22, 2011.