The Riemann function depends on the complex variable , and also in a deep way, on the nature of natural numbers. Every natural number can be written as a product of primes, but no prime can be written as a product of more than one natural. This deep asymmetry is at the bottom of Riemann's hypothesis.
The zeroes of this function, away from the real axis, have the property of a unique real part:
Wikipedia
Primes play the role of elementary particles in Physics. All matter (except dark), is composed of quarks and leptons, but as far as we know, these particles are not composite.
Primes then, seem like natural mental constructs, for a mathematical theory of Physics.
Neither Riemann's Conjecture, nor this Physical theory are understood today.
One connection with Physics, is the spectral decomposition of an operator. Operator Zeta Function. One can see the nature of the problem this way: There cannot be a formula which gives us all prime numbers. We will always be looking for the next prime number. This may be related to mathematical logic problems, like the no-halting problem.
Read Schumayer & Hutchinson, and Menezes, Svaiter, & Svaiter.
$$\LaTeX$$ Formula Editor for Blogger
Sunday, March 31, 2013
Sunday, March 24, 2013
Power
I am reading Moisés Naím's book, "The End of Power", at the same time that Cyprus is having a very bad night. From Paul Krugman's piece below, I conclude, that Germany has power over Cyprus, contrary to Naím's concept of power decay. It may be true that old ways to look at power do not work anymore, but as long as there are human beings, power relations are unavoidable.
My conclusion tonight is that I shouldn't expect an ATM when I visit Cyprus, and expect to get my dollars, just as if I was in Warrenville.
Going back to my childhood, when a dollar was worth 12.50 Mexican pesos, and everybody wanted to have real dollars, not pesos :(
My conclusion tonight is that I shouldn't expect an ATM when I visit Cyprus, and expect to get my dollars, just as if I was in Warrenville.
Going back to my childhood, when a dollar was worth 12.50 Mexican pesos, and everybody wanted to have real dollars, not pesos :(
Hot Money Blues
By PAUL KRUGMAN
Published: March 24, 2013 11 Comments
- GOOGLE+
- SAVE
- SHARE
- REPRINTS
Whatever the final outcome in the Cyprus crisis — we know it’s going to be ugly; we just don’t know exactly what form the ugliness will take — one thing seems certain: for the time being, and probably for years to come, the island nation will have to maintain fairly draconian controls on the movement of capital in and out of the country. In fact, controls may well be in place by the time you read this. And that’s not all: Depending on exactly how this plays out, Cypriot capital controls may well have the blessing of the International Monetary Fund, which has already supported suchcontrols in Iceland.
Fred R. Conrad/The New York Times
Related
As Deadline Nears, Cyprus Scrambles to Devise a Bailout(March 25, 2013)
Connect With Us on Twitter
For Op-Ed, follow@nytopinion and to hear from the editorial page editor, Andrew Rosenthal, follow@andyrNYT.
Readers’ Comments
Share your thoughts.
That’s quite a remarkable development. It will mark the end of an era for Cyprus, which has in effect spent the past decade advertising itself as a place where wealthy individuals who want to avoid taxes and scrutiny can safely park their money, no questions asked. But it may also mark at least the beginning of the end for something much bigger: the era when unrestricted movement of capital was taken as a desirable norm around the world.
It wasn’t always thus. In the first couple of decades after World War II, limits on cross-border money flows were widely considered good policy; they were more or less universal in poorer nations, and present in a majority of richer countries too. Britain, for example, limited overseas investments by its residents until 1979; other advanced countries maintained restrictions into the 1980s. Even the United States briefly limited capital outflows during the 1960s.
Over time, however, these restrictions fell out of fashion. To some extent this reflected the fact that capital controls have potential costs: they impose extra burdens of paperwork, they make business operations more difficult, and conventional economic analysis says that they should have a negative impact on growth (although this effect is hard to find in the numbers). But it also reflected the rise of free-market ideology, the assumption that if financial markets want to move money across borders, there must be a good reason, and bureaucrats shouldn’t stand in their way.
As a result, countries that did step in to limit capital flows — like Malaysia, which imposed what amounted to a curfew on capital flight in 1998 — were treated almost as pariahs. Surely they would be punished for defying the gods of the market!
But the truth, hard as it may be for ideologues to accept, is that unrestricted movement of capital is looking more and more like a failed experiment.
It’s hard to imagine now, but for more than three decades after World War II financial crises of the kind we’ve lately become so familiar with hardly ever happened. Since 1980, however, the roster has been impressive: Mexico, Brazil, Argentina and Chile in 1982. Sweden and Finland in 1991. Mexico again in 1995. Thailand, Malaysia, Indonesia and Korea in 1998. Argentina again in 2002. And, of course, the more recent run of disasters: Iceland, Ireland, Greece, Portugal, Spain, Italy, Cyprus.
What’s the common theme in these episodes? Conventional wisdom blames fiscal profligacy — but in this whole list, that story fits only one country, Greece. Runaway bankers are a better story; they played a role in a number of these crises, from Chile to Sweden to Cyprus. But the best predictor of crisis is large inflows of foreign money: in all but a couple of the cases I just mentioned, the foundation for crisis was laid by a rush of foreign investors into a country, followed by a sudden rush out.
I am, of course, not the first person to notice the correlation between the freeing up of global capital and the proliferation of financial crises; Harvard’s Dani Rodrik began banging this drum back in the 1990s. Until recently, however, it was possible to argue that the crisis problem was restricted to poorer nations, that wealthy economies were somehow immune to being whipsawed by love-’em-and-leave-’em global investors. That was a comforting thought — but Europe’s travails demonstrate that it was wishful thinking.
And it’s not just Europe. In the last decade America, too, experienced a huge housing bubble fed by foreign money, followed by a nasty hangover after the bubble burst. The damage was mitigated by the fact that we borrowed in our own currency, but it’s still our worst crisis since the 1930s.
Now what? I don’t expect to see a wholesale, sudden rejection of the idea that money should be free to go wherever it wants, whenever it wants. There may well, however, be a process of erosion, as governments intervene to limit both the pace at which money comes in and the rate at which it goes out. Global capitalism is, arguably, on track to become substantially less global.
And that’s O.K. Right now, the bad old days when it wasn’t that easy to move lots of money across borders are looking pretty good.
A version of this op-ed appeared in print on March 25, 2013, on page A23 of the New York edition with the headline: Hot Money Blues.
Monday, March 18, 2013
Selfnome?
Today I read in Scientific American, on the effort to map the human brain. Also I am in the middle of Antonio Damasio's book: "Self Comes to Mind".
The article has a pessimistic note, not I!
All you need is a correct theory of what Consciousness is, more to the point: What is the Self?
I agree with Dr. Damasio, who himself got his cues from Baruch Spinoza. You need the Heart, without the emotional component to the mind, we will never find The Self.
Now we need a well directed, and funded project to accomplish this goal.
The article has a pessimistic note, not I!
All you need is a correct theory of what Consciousness is, more to the point: What is the Self?
I agree with Dr. Damasio, who himself got his cues from Baruch Spinoza. You need the Heart, without the emotional component to the mind, we will never find The Self.
Now we need a well directed, and funded project to accomplish this goal.
The Human Sense of Self as a Program for the Biosphere
Antonio Damasio tells us in his book: Self Comes to Mind, that our brain makes a map mainly of our body, but also of the world at large. Nevertheless unlike other maps, this one can change the world.
We can program, with Consciousness, i.e., the Sense of Self, a better biosphere.
Neat.
We can program, with Consciousness, i.e., the Sense of Self, a better biosphere.
Neat.
Sunday, March 17, 2013
Universe?
Either this name describes the place we live in, or not. Recentley Fan et al., proposed a parallel universe made of Dark Matter, which exists alongside our known Universe. The idea of Dark Matter is old, it was Fritz Zwicky who proposed that matter affected by gravity, but otherwise invisible to us, could explain the high velocity of stars in regions where there was not enough visible light to account for their motion. Also Hugh Everett III, proposed that we live in parallel universes, and only perceive one of them. Obviously if the Universe is not one, the name does not agree with the thing. I should clarify that Fan et al. do not use Everett's ideas.
I feel that Fan et al., proposal, pertains to a single Universe, and then, we should assign some solution of Einstein Equations, to their new objects, or start anew. Maybe we can have a huge capacitor with their Disk, and our Disk, say one positive, and the other negative. That would be a huge particle accelerator, maybe thus explaining the highest energy cosmic rays reaching Earth.
One Universe!
I feel that Fan et al., proposal, pertains to a single Universe, and then, we should assign some solution of Einstein Equations, to their new objects, or start anew. Maybe we can have a huge capacitor with their Disk, and our Disk, say one positive, and the other negative. That would be a huge particle accelerator, maybe thus explaining the highest energy cosmic rays reaching Earth.
One Universe!
Friday, March 8, 2013
Monday, March 4, 2013
Was Wittgenstein Right?
The Stone is a forum for contemporary philosophers on issues both timely and timeless.
The singular achievement of the controversial early 20th century philosopher Ludwig Wittgenstein was to have discerned the true nature of Western philosophy — what is special about its problems, where they come from, how they should and should not be addressed, and what can and cannot be accomplished by grappling with them. The uniquely insightful answers provided to these meta-questions are what give his treatments of specific issues within the subject — concerning language, experience, knowledge, mathematics, art and religion among them — a power of illumination that cannot be found in the work of others.
A reminder of philosophy’s embarrassing failure, after over 2000 years, to settle any of its central issues.
Admittedly, few would agree with this rosy assessment — certainly not many professional philosophers. Apart from a small and ignored clique of hard-core supporters the usual view these days is that his writing is self-indulgently obscure and that behind the catchy slogans there is little of intellectual value. But this dismissal disguises what is pretty clearly the real cause of Wittgenstein’s unpopularity within departments of philosophy: namely, his thoroughgoing rejection of the subject as traditionally and currently practiced; his insistence that it can’t give us the kind of knowledge generally regarded as its raison d’être.
Wittgenstein claims that there are no realms of phenomena whose study is the special business of a philosopher, and about which he or she should devise profound a priori theories and sophisticated supporting arguments. There are no startling discoveries to be made of facts, not open to the methods of science, yet accessible “from the armchair” through some blend of intuition, pure reason and conceptual analysis. Indeed the whole idea of a subject that could yield such results is based on confusion and wishful thinking.
Free Press
This attitude is in stark opposition to the traditional view, which continues to prevail. Philosophy is respected, even exalted, for its promise to provide fundamental insights into the human condition and the ultimate character of the universe, leading to vital conclusions about how we are to arrange our lives. It’s taken for granted that there is deep understanding to be obtained of the nature of consciousness, of how knowledge of the external world is possible, of whether our decisions can be truly free, of the structure of any just society, and so on — and that philosophy’s job is to provide such understanding. Isn’t that why we are so fascinated by it?
If so, then we are duped and bound to be disappointed, says Wittgenstein. For these are mere pseudo-problems, the misbegotten products of linguistic illusion and muddled thinking. So it should be entirely unsurprising that the “philosophy” aiming to solve them has been marked by perennial controversy and lack of decisive progress — by an embarrassing failure, after over 2000 years, to settle any of its central issues. Therefore traditional philosophical theorizing must give way to a painstaking identification of its tempting but misguided presuppositions and an understanding of how we ever came to regard them as legitimate. But in that case, he asks, “[w]here does [our] investigation get its importance from, since it seems only to destroy everything interesting, that is, all that is great and important? (As it were all the buildings, leaving behind only bits of stone and rubble)” — and answers that “(w)hat we are destroying is nothing but houses of cards and we are clearing up the ground of language on which they stand.”
Associated Press
Given this extreme pessimism about the potential of philosophy — perhaps tantamount to a denial that there is such a subject — it is hardly surprising that “Wittgenstein” is uttered with a curl of the lip in most philosophical circles. For who likes to be told that his or her life’s work is confused and pointless? Thus, even Bertrand Russell, his early teacher and enthusiastic supporter, was eventually led to complain peevishly that Wittgenstein seems to have “grown tired of serious thinking and invented a doctrine which would make such an activity unnecessary.”
But what is that notorious doctrine, and can it be defended? We might boil it down to four related claims.
— The first is that traditional philosophy is scientistic: its primary goals, which are to arrive at simple, general principles, to uncover profound explanations, and to correct naïve opinions, are taken from the sciences. And this is undoubtedly the case.
—The second is that the non-empirical (“armchair”) character of philosophical investigation — its focus on conceptual truth — is in tension with those goals. That’s because our concepts exhibit a highly theory-resistant complexity and variability. They evolved, not for the sake of science and its objectives, but rather in order to cater to the interacting contingencies of our nature, our culture, our environment, our communicative needs and our other purposes. As a consequence the commitments defining individual concepts are rarely simple or determinate, and differ dramatically from one concept to another. Moreover, it is not possible (as it is within empirical domains) to accommodate superficial complexity by means of simple principles at a more basic (e.g. microscopic) level.
— The third main claim of Wittgenstein’s metaphilosophy — an immediate consequence of the first two — is that traditional philosophy is necessarily pervaded with oversimplification; analogies are unreasonably inflated; exceptions to simple regularities are wrongly dismissed.
— Therefore — the fourth claim — a decent approach to the subject must avoid theory-construction and instead be merely “therapeutic,” confined to exposing the irrational assumptions on which theory-oriented investigations are based and the irrational conclusions to which they lead.
Consider, for instance, the paradigmatically philosophical question: “What is truth?”. This provokes perplexity because, on the one hand, it demands an answer of the form, “Truth is such–and-such,” but on the other hand, despite hundreds of years of looking, no acceptable answer of that kind has ever been found. We’ve tried truth as “correspondence with the facts,” as “provability,” as “practical utility,” and as “stable consensus”; but all turned out to be defective in one way or another — either circular or subject to counterexamples. Reactions to this impasse have included a variety of theoretical proposals. Some philosophers have been led to deny that there is such a thing as absolute truth. Some have maintained (insisting on one of the above definitions) that although truth exists, it lacks certain features that are ordinarily attributed to it — for example, that the truth may sometimes be impossible to discover. Some have inferred that truth is intrinsically paradoxical and essentially incomprehensible. And others persist in the attempt to devise a definition that will fit all the intuitive data.
But from Wittgenstein’s perspective each of the first three of these strategies rides roughshod over our fundamental convictions about truth, and the fourth is highly unlikely to succeed. Instead we should begin, he thinks, by recognizing (as mentioned above) that our various concepts play very different roles in our cognitive economy and (correspondingly) are governed by defining principles of very different kinds. Therefore, it was always a mistake to extrapolate from the fact that empirical concepts, such as red or magnetic oralive stand for properties with specifiable underlying natures to the presumption that the notion of truth must stand for some such property as well.
Wittgenstein’s conceptual pluralism positions us to recognize that notion’s idiosyncratic function, and to infer that truth itself will not be reducible to anything more basic. More specifically, we can see that the concept’s function in our cognitive economy is merely to serve as a device of generalization. It enables us to say such things as “Einstein’s last words were true,” and not be stuck with “If Einstein’s last words were that E=mc2, then E=mc2; and if his last words were that nuclear weapons should be banned, then nuclear weapons should be banned; … and so on,” which has the disadvantage of being infinitely long! Similarly we can use it to say: “We should want our beliefs to be true” (instead of struggling with “We should want that if we believe that E=mc2, then E=mc2; and that if we believe … etc.”). We can see, also, that this sort of utility depends upon nothing more than the fact that the attribution of truth to a statement is obviously equivalent to the statement itself — for example, “It’s true that E=mc2” is equivalent to “E=mc2”. Thus possession of the concept of truth appears to consist in an appreciation of that triviality, rather than a mastery of any explicit definition. The traditional search for such an account (or for some other form of reductive analysis) was a wild-goose chase, a pseudo-problem. Truth emerges as exceptionally unprofound and as exceptionally unmysterious.
This example illustrates the key components of Wittgenstein’s metaphilosophy, and suggests how to flesh them out a little further. Philosophical problems typically arise from the clash between the inevitably idiosyncratic features of special-purpose concepts —true, good, object, person, now, necessary — and the scientistically driven insistence upon uniformity. Moreover, the various kinds of theoretical move designed to resolve such conflicts (forms of skepticism, revisionism, mysterianism and conservative systematization) are not only irrational, but unmotivated.The paradoxes to which they respond should instead be resolved merely by coming to appreciate the mistakes of perverse overgeneralization from which they arose. And the fundamental source of this irrationality is scientism.
As Wittgenstein put it in the “The Blue Book”:
Our craving for generality has [as one] source … our preoccupation with the method of science. I mean the method of reducing the explanation of natural phenomena to the smallest possible number of primitive natural laws; and, in mathematics, of unifying the treatment of different topics by using a generalization. Philosophers constantly see the method of science before their eyes, and are irresistibly tempted to ask and answer in the way science does. This tendency is the real source of metaphysics, and leads the philosopher into complete darkness. I want to say here that it can never be our job to reduce anything to anything, or to explain anything. Philosophy really is “purely descriptive.
These radical ideas are not obviously correct, and may on close scrutiny turn out to be wrong. But they deserve to receive that scrutiny — to be taken much more seriously than they are. Yes, most of us have been interested in philosophy only because of its promise to deliver precisely the sort of theoretical insights that Wittgenstein argues are illusory. But such hopes are no defense against his critique. Besides, if he turns out to be right, satisfaction enough may surely be found in what we still can get — clarity, demystification and truth.
NOTE: A response to this post by Michael P. Lynch will be published in The Stone later this week.
NOTE: A response to this post by Michael P. Lynch will be published in The Stone later this week.
Paul Horwich is a professor of philosophy at New York University. He is the author of several books, including “Reflections on Meaning,” “Truth-Meaning-Reality,” and most recently, “Wittgenstein’s Metaphilosophy.”
Saturday, March 2, 2013
From Auger to Fermi LAT
The Pierre Auger Observatory is the result of many years of hard work. The purpose of the installation in Argentina, is to use the Universe as a big High Energy Physics Laboratory.
The Fermi LAT is a Space Based Telescope to study Gamma Rays.
This note is autobiographical, I was invited by Professor Arnulfo Zepeda Dominguez, to join the Auger effort in 1995. Eventually the University of Puebla, where I was working at the time, joined this effort. I met Professor Don Hooper at Fermilab in 2007 on the occasion of a public lecture on Dark Matter, where I bought his book, "Dark Cosmos". Now Hooper has evidence of Dark Matter. This is my take.
Fermi Bubbles were discovered by Fermi LAT. These huge, tens of thousands of light years, structures are perpendicular to the Milky Way Galactic Plane. Now Hooper claims evidence in these bubbles, near the immense black hole in the center of our Galaxy, of new particles in the tens of GeV mass range. This is amazing!
When Arnulfo was my professor at the Centro de Investigacion y Estudios Avanzados, in Mexico, I decided to study at Earth based high energy laboratories, in my youthful mind, that was easier than Astronomy. Not intellectually, but practically. At Fermilab one controls everything, the source, the path, and the target; those conditions would leave, I thought, to more success. I was WRONG.
I hope to have Professor Hooper explain my students, why that is so.
In my mind, the reason is that, what one looses in control, is gained in size. The Galaxy is bigger than Fermilab.
The Fermi LAT is a Space Based Telescope to study Gamma Rays.
This note is autobiographical, I was invited by Professor Arnulfo Zepeda Dominguez, to join the Auger effort in 1995. Eventually the University of Puebla, where I was working at the time, joined this effort. I met Professor Don Hooper at Fermilab in 2007 on the occasion of a public lecture on Dark Matter, where I bought his book, "Dark Cosmos". Now Hooper has evidence of Dark Matter. This is my take.
Fermi Bubbles were discovered by Fermi LAT. These huge, tens of thousands of light years, structures are perpendicular to the Milky Way Galactic Plane. Now Hooper claims evidence in these bubbles, near the immense black hole in the center of our Galaxy, of new particles in the tens of GeV mass range. This is amazing!
When Arnulfo was my professor at the Centro de Investigacion y Estudios Avanzados, in Mexico, I decided to study at Earth based high energy laboratories, in my youthful mind, that was easier than Astronomy. Not intellectually, but practically. At Fermilab one controls everything, the source, the path, and the target; those conditions would leave, I thought, to more success. I was WRONG.
I hope to have Professor Hooper explain my students, why that is so.
In my mind, the reason is that, what one looses in control, is gained in size. The Galaxy is bigger than Fermilab.
Subscribe to:
Posts (Atom)