Beginning from these very beauties,
for the sake of that highest beauty he ascends eternally,
just as if employing the rungs of a ladder, from one to two,
and from two to all beautiful bodies;
from beautiful bodies he proceeds to beautiful pursuits,
from pursuits to beautiful sciences,
and from these sciences arrives at that science which is
concerned with the beautiful itself and nothing else,
so that finally he comes to know what the beautiful is.
Socrates, on Diotima’s wisdom, on Plato’s Symposium
My views on Plato’s Archetypes
I am opposed to the idea of Platonism as an independent and abstract dimension concept that split Man from the Archetypes, (Theory of forms) as the result of the separation of the Physical from the Spiritual, as to deny the subjective integral world of ideas, as unreal on the basis it’s is a mind product, and not an objective reality. Mathematicians, at least some of them have seeing the intimate reality of conceptual ideas on the way the Universe works; Not only is the Platonism under discussion not Plato’s, Platonism as characterized above is a purely metaphysical view, it should be distinguished from other views that have substantive epistemological content. Many older characterizations of Platonism add strong epistemological claims to the effect that we have some immediate grasp of, or insight into, the realm of abstract objects.
I believe Semantics is at the root of the problem, and the long but now obsolete definitions of charged words like Metaphysics, it is really conceptual ideas, subjective insights that define something beyond our physical dimension, really something beyond ourselves the creators of the ideas, the definers of concrete objects, and subjective images, and understanding?
The Universe can exist without Man as a witness? No doubt, considering the relative young age of Man, but it’s not Man the essential key to explain it? No Man, no consciousness, no consciousness, not even conceptual existence, since the conceptual is totally dependent on the conscious observer…Of course it could be argued that consciousness it’s in everything even in the mineral kingdom, to which I agree, the Universe/s it’s consciousness.
“I was a Treasure unknown then I desired to be known so I created a creation to which I made Myself known; then they knew Me.”
For those interested on the Platonic Archetypes you can read my post of THE WORLD WITHOUT DUST GEOGRAPHICAL ARCHETYPES OF THE SOUL, posted on March 2012
Love and Math: The Heart of a Hidden Reality
Here are long excerpts of the New York Review of Books article by Jim Holt on Edward Frenkel’s book Love and Math: The Heart of a Hidden Reality, a very interesting article that deal in a quasi mystic idea of the existence of the Independence of mathematics in relation to our understanding that Mathematical objects are independent of intelligent agents and their language, thought, and practices.
“For those who have learned something of higher mathematics, nothing could be more natural than to use the word “beautiful” in connection with it. Mathematical beauty, like the beauty of, say, a late Beethoven quartet, arises from a combination of strangeness and inevitability. Simply defined abstractions disclose hidden quirks and complexities. Seemingly unrelated structures turn out to have mysterious correspondences. Uncanny patterns emerge, and they remain uncanny even after being underwritten by the rigor of logic.
So powerful are these aesthetic impressions that one great mathematician, G.H. Hardy, declared that beauty, not usefulness, is the true justification for mathematics. To Hardy, mathematics was first and foremost a creative art. “The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful,” he wrote in his classic 1940 book, A Mathematician’s Apology. “Beauty is the first test: there is no permanent place in the world for ugly mathematics.”
And what is the appropriate reaction when one is confronted by mathematical beauty? Pleasure, certainly; awe, perhaps. Thomas Jefferson wrote in his seventy-sixth year that contemplating the truths of mathematics helped him to “beguile the wearisomeness of declining life.” To Bertrand Russell—who rather melodramatically claimed, in his autobiography, that it was his desire to know more of mathematics that kept him from committing suicide—the beauty of mathematics was “cold and austere, like that of sculpture…sublimely pure, and capable of a stern perfection.” For others, mathematical beauty may evoke a distinctly warmer sensation. They might take their cue from Plato’s Symposium. In that dialogue, Socrates tells the guests assembled at a banquet how a priestess named Diotima initiated him into the mysteries of Eros—the Greek name for desire in all its forms.
One form of Eros is the sexual desire aroused by the physical beauty of a particular beloved person. That, according to Diotima, is the lowest form. With philosophical refinement, however, Eros can be made to ascend toward loftier and loftier objects. The penultimate of these—just short of the Platonic idea of beauty itself—is the perfect and timeless beauty discovered by the mathematical sciences. Such beauty evokes in those able to grasp it a desire to reproduce—not biologically, but intellectually, by begetting additional “gloriously beautiful ideas and theories.” For Diotima, and presumably for Plato as well, the fitting response to mathematical beauty is the form of Eros we call love.”
As with Galois theory, the Langlands Program had its origins in a letter. It was written in 1967 by Robert Langlands (then in his early thirties) to one of his colleagues at the Institute for Advance Study, André Weil. In his letter, Langlands proposed the possibility of a deep analogy between two theories that seemed to lie at opposite ends of the mathematical cosmos: the theory of Galois groups, which concerns symmetries in the realm of numbers; and “harmonic analysis,” which concerns how complicated waves (e.g., the sound of a symphony) are built up from simple harmonics (e.g., the individual
instruments). Certain structures in the harmonic world, called automorphic forms, somehow “knew” about mysterious patterns in the world of numbers. Thus it might be possible to use the methods of one world to reveal hidden harmonies in the other—so Langlands conjectured. If Weil did not find the intuitions in the letter persuasive, Langlands added, “I am sure you have a waste basket handy.”
But Weil, a magisterial figure in twentieth-century mathematics (he died in 1998 at the age of ninety-two), was a receptive audience. In a letter that he had written in 1940 to his sister, Simone Weil, he had described in vivid terms the importance of analogy in mathematics. Alluding to the Bhagavad-Gita (he was also a Sanskrit scholar), André explained to Simone that, just as the Hindu deity Vishnu had ten different avatars, a seemingly simple mathematical equation could manifest itself in dramatically different abstract structures. The subtle analogies between such structures were like “illicit
liaisons,” he wrote; “nothing gives more pleasure to the connoisseur.”
As it happens, Weil was writing to his sister from prison in France, where he had been temporarily confined for desertion from the army (after nearly being executed as a spy in Finland).
The Langlands Program is a scheme of conjectures that would turn such hypothetical analogies into sturdy logical bridges, linking up diverse mathematical islands across the surrounding sea of ignorance. Or it can be seen as a Rosetta stone that would allow the mathematical tribes on these various islands—number theorists, topologists, algebraic geometers—to talk to one another and pool their conceptual resources. The Langlands conjectures are largely unproved so far. Are they even true? There is an almost Platonic confidence among mathematicians that they must be. As Ian Stewart has remarked, the Langlands Program is “the sort of mathematics that ought to be true because it was so beautiful.”
Quantum Physycs Connection
The next move was to extend the Langlands Program beyond the borders of mathematics itself. In the 1970s, it had been noticed that one of its key ingredients—the “Langlands dual group”—also crops up in quantum physics. This came as a surprise. Could the same patterns that can be dimly glimpsed in the worlds of number and geometry also have counterparts in the theory that describes the basic forces of nature?
Frenkel was struck by the potential link between quantum physics and the Langlands Program, and set about to investigate it—aided by a multimillion-dollar grant that he and some colleagues received in 2004 from the Department of Defense, the largest grant to date for research in pure mathematics. (In addition to being clean and gentle, pure mathematics is cheap: all its practitioners need is chalk and a little travel money. It is also open and transparent, since there are no inventions to patent.)
This brought him into a collaboration with Edward Witten, widely regarded as the greatest living mathematical physicist (and, like Langlands himself, a member of the Institute for Advanced Study in Princeton). Witten is a virtuoso of string theory, an ongoing effort by physicists to unite all the forces of nature, including gravity, in one neat mathematical package. He awed Frenkel with his “unbreakable logic” and his “great taste.” It was Witten who saw how the “branes” (short for “membranes”) postulated by string theorists might be analogous to the “sheaves” invented by mathematicians. Thus opened a rich dialogue between the Langlands Program, which aims to unify mathematics, and string theory, which aims to unify physics. Although optimism about string theory has faded somewhat with its failure (thus far) to deliver an effective description of our universe, the Langlands connection has yielded deep insights into the workings of particle physics.
This is not the first time that mathematical concepts studied for their pure beauty have later turned out to illumine the physical world. “How can it be,” Einstein asked in wonderment, “that mathematics, being after all a product of human thought independent of experience, is so admirably appropriate to the objects of reality?” Frenkel’s take on this is very different from Einstein’s. For Frenkel, mathematical structures are among the “objects of reality”; they are every bit as real as anything in the physical or mental world.
Moreover, they are not the product of human thought; rather, they exist timelessly, in a Platonic realm of their own, waiting to be discovered by mathematicians. The conviction that mathematics has a reality that transcends the human mind is not uncommon among its practitioners, especially great ones like Frenkel and Langlands, Sir Roger Penrose and Kurt Gödel. It derives from the way that strange patterns and correspondences unexpectedly emerge, hinting at something hidden and mysterious. Who put those patterns there? They certainly don’t seem to be of our making.
The problem with this Platonist view of mathematics—one that Frenkel, going on in a misterioso vein, never quite recognizes as such—is that it makes mathematica lknowledge a miracle. If the objects of mathematics exist apart from us, living in a Platonic heaven that transcends the physical world of space and time, then how does the human mind “get in touch” with them and learn about their properties and relations? Do mathematicians have ESP? The trouble with Platonism, as the philosopher Hilary Putnam has observed, “is that it seems flatly incompatible with the simple fact that we think with
our brains, and not with immaterial souls.”
Perhaps Frenkel should be allowed his Platonic fantasy. After all, every lover harbors romantic delusions about his beloved. In 2009, while Frenkel was in Paris as the occupant of the Chaire d’ Excellence of the Fondation Sciences Mathématiques, he decided to make a short film expressing his passion for mathematics. Inspired by Yukio Mishima’s Rite of Love and Death, he titled it Rites of Love and Math. In this silent Noh-style allegory, Frenkel plays a mathematician who creates a formula of love. To keep the formula from falling into evil hands, he hides it away from the world by tattooing it with a bamboo stick on the body of the woman he loves, and then prepares to sacrifice himself for its
Upon the premiere of Rites of Love and Math in Paris in 2010, Le Monde called it “a stunning short film” that “offers an unusual romantic vision of mathematicians.” The “formula of love” used in the film was one that Frenkel himself discovered (in the course of investigating the mathematical underpinnings of quantum field theory). It is beautiful, yet forbidding. The only numbers in it are zero, one, and infinity. Isn’t love like that?