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Showing posts with label Shing-tung Yau. Show all posts
Showing posts with label Shing-tung Yau. Show all posts

Sunday, January 15, 2012

WHAT IS YOUR FAVORITE DEEP, ELEGANT, OR BEAUTIFUL EXPLANATION?

Scientists' greatest pleasure comes from theories that derive the solution to some deep puzzle from a small set of simple principles in a surprising way. These explanations are called "beautiful" or "elegant". Historical examples are Kepler's explanation of complex planetary motions as simple ellipses, Bohr's explanation of the periodic table of the elements in terms of electron shells, and Watson and Crick's double helix. Einstein famously said that he did not need experimental confirmation of his general theory of relativity because it "was so beautiful it had to be true." See:2012 : WHAT IS YOUR FAVORITE DEEP, ELEGANT, OR BEAUTIFUL EXPLANATION?
See which comments resonate with you. Some of my picks as I go through was by :

Raphael Bousso
Professor of Theoretical Physics, Berkeley



My Favorite Annoying Elegant Explanation: Quantum Theory .......General Relativity, in turn, is only a classical theory. It rests on a demonstrably false premise: that position and momentum can be known simultaneously. This may a good approximation for apples, planets, and galaxies: large objects, for which gravitational interactions tend to be much more important than for the tiny particles of the quantum world. But as a matter of principle, the theory is wrong. The seed is there. General Relativity cannot be the final word; it can only be an approximation to a more general Quantum Theory of Gravity.

But what about Quantum Mechanics itself? Where is its seed of destruction? Amazingly, it is not obvious that there is one. The very name of the great quest of theoretical physics—"quantizing General Relativity"—betrays an expectation that quantum theory will remain untouched by the unification we seek. String theory—in my view, by far the most successful, if incomplete, result of this quest—is strictly quantum mechanical, with no modifications whatsoever to the framework that was completed by Heisenberg, Schrödinger, and Dirac. In fact, the mathematical rigidity of Quantum Mechanics makes it difficult to conceive of any modifications, whether or not they are called for by observation.

Yet, there are subtle hints that Quantum Mechanics, too, will suffer the fate of its predecessors. The most intriguing, in my mind, is the role of time. In Quantum Mechanics, time is an essential evolution parameter. But in General Relativity, time is just one aspect of spacetime, a concept that we know breaks down at singularities deep inside black holes. Where time no longer makes sense, it is hard to see how Quantum Mechanics could still reign. As Quantum Mechanics surely spells trouble for General Relativity, the existence of singularities suggests that General Relativity may also spell trouble for Quantum Mechanics. It will be fascinating to watch this battle play out.



President, The Royal Society; Professor of Cosmology & Astrophysics; Master, Trinity...

Physical Reality Could Be Hugely More Extensive Than the Patch of Space and Time Traditionally Called 'The Universe' .....As an analogy (which I owe to Paul Davies) consider the form of snowflakes. Their ubiquitous six-fold symmetry is a direct consequence of the properties and shape of water molecules. But snowflakes display an immense variety of patterns because each is molded by its distinctive history and micro-environment: how each flake grows is sensitive to the fortuitous temperature and humidity changes during its growth.

If physicists achieved a fundamental theory, it would tell us which aspects of nature were direct consequences of the bedrock theory (just as the symmetrical template of snowflakes is due to the basic structure of a water molecule) and which cosmic numbers are (like the distinctive pattern of a particular snowflake) the outcome of environmental contingencies. .


Theoretical physicist

An Explanation of Fundamental Particle Physics That Doesn't Exist Yet.....What is tetrahedral symmetry doing in the masses of neutrinos?! Nobody knows. But you can bet there will be a good explanation. It is likely that this explanation will come from mathematicians and physicists working closely with Lie groups. The most important lesson from the great success of Einstein's theory of General Relativity is that our universe is fundamentally geometric, and this idea has extended to the geometric description of known forces and particles using group theory. It seems natural that a complete explanation of the Standard Model, including why there are three generations of fermions and why they have the masses they do, will come from the geometry of group theory. This explanation does not yet exist, but when it does it will be deep, elegant, and beautiful—and it will be my favorite.


Mathematician, Harvard; Co-author, The Shape of Inner Space

A Sphere....Most scientific facts are based on things that we cannot see with the naked eye or hear by our ears or feel by our hands. Many of them are described and guided by mathematical theory. In the end, it becomes difficult to distinguish a mathematical object from objects in nature.

One example is the concept of a sphere. Is the sphere part of nature or it is a mathematical artifact? That is difficult for a mathematician to say. Perhaps the abstract mathematical concept is actually a part of nature. And it is not surprising that this abstract concept actually describes nature quite accurately.



theoretical physicist; Professor, Department of Physics, University of California,...
 Gravity Is Curvature Of Spacetime … Or Is It?......We do not yet know the full shape of the quantum theory providing a complete accounting for gravity. We do have many clues, from studying the early quantum phase of cosmology, and ultrahigh energy collisions that produce black holes and their subsequent disintegrations into more elementary particles. We have hints that the theory draws on powerful principles of quantum information theory. And, we expect that in the end it has a simple beauty, mirroring the explanation of gravity-as-curvature, from an even more profound depth.



Albert Einstein Professor in Science, Departments of Physics and Astrophysical...
Quasi-elegance....As a young student first reading Weyl's book, crystallography seemed like the "ideal" of what one should be aiming for in science: elegant mathematics that provides a complete understanding of all physical possibilities. Ironically, many years later, I played a role in showing that my "ideal" was seriously flawed. In 1984, Dan Shechtman, Ilan Blech, Denis Gratias and John Cahn reported the discovery of a puzzling manmade alloy of aluminumand manganese with icosahedral symmetry. Icosahedral symmetry, with its six five-fold symmetry axes, is the most famous forbidden crystal symmetry. As luck would have it, Dov Levine (Technion) and I had been developing a hypothetical idea of a new form of solid that we dubbed quasicrystals, short for quasiperiodic crystals. (A quasiperiodic atomic arrangement means the atomic positions can be described by a sum of oscillatory functions whose frequencies have an irrational ratio.) We were inspired by a two-dimensional tiling invented by Sir Roger Penrose known as the Penrose tiling, comprised of two tiles arranged in a five-fold symmetric pattern. We showed that quasicrystals could exist in three dimensions and were not subject to the rules of crystallography. In fact, they could have any of the symmetries forbidden to crystals. Furthermore, we showed that the diffraction patterns predicted for icosahedral quasicrystals matched the Shechtman et al. observations. Since 1984, quasicrystals with other forbidden symmetries have been synthesized in the laboratory. The 2011 Nobel Prize in Chemistry was awarded to Dan Shechtman for his experimental breakthrough that changed our thinking about possible forms of matter. More recently, colleagues and I have found evidence that quasicrystals may have been among the first minerals to have formed in the solar system.

The crystallography I first encountered in Weyl's book, thought to be complete and immutable, turned out to be woefully incomplete, missing literally an uncountable number of possible symmetries for matter. Perhaps there is a lesson to be learned: While elegance and simplicity are often useful criteria for judging theories, they can sometimes mislead us into thinking we are right, when we are actually infinitely wrong.




Physicist, Harvard University; Author, Warped Passages; Knocking On Heaven's Door

The Higgs Mechanism......Fortunately that time has now come for the Higgs mechanism, or at least the simplest implementation which involves a particle called the Higgs boson. The Large Hadron Collider at CERN near Geneva should have a definitive result on whether this particle exists within this coming year. The Higgs boson is one possible (and many think the most likely) consequence of the Higgs mechanism. Evidence last December pointed to a possible discovery, though more data is needed to know for sure. If confirmed, it will demonstrate that the Higgs mechanism is correct and furthermore tell us what is the underlying structure responsible for spontaneous symmetry breaking and spreading "charge" throughout the vacuum. The Higgs boson would furthermore be a new type of particle (a fundamental boson for those versed in physics terminology) and would be in some sense a new type of force. Admittedly, this is all pretty subtle and esoteric. Yet I (and much of the theoretical physics community) find it beautiful, deep, and elegant.

Symmetry is great. But so is symmetry breaking. Over the years many aspects of particle physics were first considered ugly and then considered elegant. Subjectivity in science goes beyond communities to individual scientists. And even those scientists change their minds over time. That's why experiments are critical. As difficult as they are, results are much easier to pin down than the nature of beauty. A discovery of the Higgs boson will tell us how that is done when particles acquire their masses.



Professor of Quantum Mechanical Engineering, MIT; Author, Programming the Universe
 The True Rotational Symmetry of Space.....Although this excercise might seem no more than some fancy and painful basketball move, the fact that the true symmetry of space is rotation not once but twice has profound consequences for the nature of the physical world at its most microscopic level. It implies that 'balls' such as electrons, attached to a distant point by a flexible and deformable 'strings,' such as magnetic field lines, must be rotated around twice to return to their original configuration. Digging deeper, the two-fold rotational nature of spherical symmetry implies that two electrons, both spinning in the same direction, cannot be placed in the same place at the same time. This exclusion principle in turn underlies the stability of matter. If the true symmetry of space were rotating around only once, then all the atoms of your body would collapse into nothingness in a tiny fraction of a second. Fortunately, however, the true symmetry of space consists of rotating around twice, and your atoms are stable, a fact that should console you as you ice your shoulder.

Remember even though I pick some of these explanations does not mean I discount all others. It's just that some are picked for what they are saying in highlighted quotations. Lisi's statement on string theory is of course in my opinion far from the truth, yet,  he captures a geometrical truth that I feel exists.:) You sort of get the jest of where I am coming from in the summation of Paul Steinhardt

Friday, August 08, 2008

William Thurston

Xianfeng David Gu and Shing-Tung Yau
To a topologist, a rabbit is the same as a sphere. Neither has a hole. Longitude and latitude lines on the rabbit allow mathematicians to map it onto different forms while preserving information.


William Thurston of Cornell, the author of a deeper conjecture that includes Poincaré’s and that is now apparently proved, said, “Math is really about the human mind, about how people can think effectively, and why curiosity is quite a good guide,” explaining that curiosity is tied in some way with intuition.

“You don’t see what you’re seeing until you see it,” Dr. Thurston said, “but when you do see it, it lets you see many other things.”
Elusive Proof, Elusive Prover: A New Mathematical Mystery

Some of us are of course interested in how we can assign the relevance to perceptions the deeper recognition of the processes of nature. How we get there and where we believe they come from. As a layman I am always interested in this process, and of course, life's mysteries can indeed be a motivating factor. Motivating my interest about the nature of things that go unanswered and how we get there.


William Paul Thurston
(born October 30, 1946) is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields medal for the depth and originality of his contributions to mathematics. He is currently a professor of mathematics and computer science at Cornell University (since 2003).


There are reasons with which I present this biography, as I did in relation to Poincaré and Klein. The basis of the question remains a philosophical one for me that I question the basis of proof and intuition while considering the mathematics.

Mathematical Induction

Mathematical Induction at a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one.

The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science.

Mathematical induction should not be misconstrued as a form of inductive reasoning, which is considered non-rigorous in mathematics (see Problem of induction for more information). In fact, mathematical induction is a form of deductive reasoning and is fully rigorous
.


Deductive reasoning

Deductive reasoning is reasoning which uses deductive arguments to move from given statements (premises), which are assumed to be true, to conclusions, which must be true if the premises are true.[1]

The classic example of deductive reasoning, given by Aristotle, is

* All men are mortal. (major premise)
* Socrates is a man. (minor premise)
* Socrates is mortal. (conclusion)

For a detailed treatment of deduction as it is understood in philosophy, see Logic. For a technical treatment of deduction as it is understood in mathematics, see mathematical logic.

Deductive reasoning is often contrasted with inductive reasoning, which reasons from a large number of particular examples to a general rule.

Alternative to deductive reasoning is inductive reasoning. The basic difference between the two can be summarized in the deductive dynamic of logically progressing from general evidence to a particular truth or conclusion; whereas with induction the logical dynamic is precisely the reverse. Inductive reasoning starts with a particular observation that is believed to be a demonstrative model for a truth or principle that is assumed to apply generally.

Deductive reasoning applies general principles to reach specific conclusions, whereas inductive reasoning examines specific information, perhaps many pieces of specific information, to impute a general principle. By thinking about phenomena such as how apples fall and how the planets move, Isaac Newton induced his theory of gravity. In the 19th century, Adams and LeVerrier applied Newton's theory (general principle) to deduce the existence, mass, position, and orbit of Neptune (specific conclusions) from perturbations in the observed orbit of Uranus (specific data).


Deduction and Induction



Our attempt to justify our beliefs logically by giving reasons results in the "regress of reasons." Since any reason can be further challenged, the regress of reasons threatens to be an infinite regress. However, since this is impossible, there must be reasons for which there do not need to be further reasons: reasons which do not need to be proven. By definition, these are "first principles." The "Problem of First Principles" arises when we ask Why such reasons would not need to be proven. Aristotle's answer was that first principles do not need to be proven because they are self-evident, i.e. they are known to be true simply by understanding them.


Back to the lumping in of theology alongside of Atlantis. Rebel dreams, it is hard to remove one's colour once they work from a certain premise. Atheistic, or not.

Seeking such clarity would be the attempt for me, with which to approach a point of limitation in our knowledge, as we may try to explain the process of the current state of the universe, and it's shape. Such warnings are indeed appropriate to me about what we are offering for views from a theoretical standpoint.

The basis presented here is from a layman standpoint while in context of Plato's work, brings some perspective to Raphael's painting, "The School of Athens." It is a central theme for me about what the basis of Inductive and deductive processes reveals about the "infinite regress of mathematics to the point of proof."

Such clarity seeking would in my mind contrast a theoretical technician with a philosopher who had such a background. Raises the philosophical question about where such information is derived from. If ,from a Platonic standpoint, then all knowledge already exists. We just have to become aware of this knowledge? How so?

Lawrence Crowell:
The ball on the Mexican hat peak will under the smallest perturbation or fluctuation begin to fall off the peak, roll into the trough and the universe tunnels out of the vacuum or nothing to become a “something.”


Whether I attach a indication of God to this knowledge does not in any way relegate the process to such a contention of theological significance. The question remains a inductive/deductive process?

I would think philosophers should weight in on the point of inductive/deductive processes as it relates to the search for new mathematics?

Allegory of the Cave

For me this was a difficult task with which to cypher the greater contextual meaning of where such mathematics arose from. That I should implore such methods would seem to be, to me, in standing with the problems and ultimates searches for meaning about our place in the universe. Whether I believe in the "God nature of that light" should hold no atheistic interpretation to my quest for the explanations about the talk on the origins of the universe.

See:

  • The Sound of Billiard Balls

  • Mathematical Structure of the Universe