Exploring the Mountain

Moving over

2011-06-18T04:50:00.000-07:00

I've started a blog on wordpress.com that besides being a repository of my current line of research will also
follow the editorial line of Exploring the Mountain.
To follow up on this move I'll post some of the posts of Exploring the Mountain on my new blog address.

So without further delays here it is: ateixeira.

Watchmath script

2011-05-05T14:02:00.000-07:00

I don't know what's wrong with it but the previous JavaScript I used is down. Thus I had to find another way to display the equations you are so fond of.

The solution was provided by this post in the blog Dysfunctional. It uses the MathJax JavaScript and the equations do look mighty fine.

Novo blog

2011-04-29T05:37:00.000-07:00

Um grupo de amigos decidiu preencher um vácuo da blogosfera em português: Mar de Dirac.

O objectivo do blog é produzir nova física através da interacção dos seus membros e ser também uma plataforma de ensino e discussão da Física (e áreas directamente relacionadas) em português sem ter medo de se recorrer a equações.

Visitem, comentem e divulguem.

Dennis Overbye Interview

2011-04-14T10:25:00.000-07:00

The original is here.

To many scientists, Dennis Overbye needs no introduction. As science writer for the New York Times, he’s been a main bolt at the junction where science meets society. His well-read and well-written articles on physics and astronomy have helped launch many of our favorite projects and some of our careers as well.
Overbye graduated from M.I.T. with a physics degree, failed to finish a novel and then worked as a writer and editor at Sky and Telescope and Discover magazines. He has now written two books: Lonely Hearts of the Cosmos: The Scientific Search for the Secret of the Universe and Einstein in Love: A Scientific Romance.
Dennis and I played phone tag for about six weeks. But finally on Martin Luther King Day we found time to chat about the state of science and science journalism.
MK: How bad are things for science right now, in the US and elsewhere?
DO: How bad are things for science in the US? You should ask me this question in the summer after the new budget has come out. As far as the world is concerned—-Europe has risen from the ashes and it’s pretty much equal to us now, and Asia is coming on strong. Overall I think there’s more research going on, more doctors being trained. But most of them won’t be in the U.S. anymore.
I guess I think that in this country, Science seems to have survived the Bush years, barely. I think you probably have to take it field by field. Some people have done well. Some people haven’t done so well. Climate science people certainly did not prosper during the last administration. I don’t know what’s going to happen to the physicists in the next few years. They are going to have to move to Europe.
Though from where I sit, there seems to be an awful lot of stuff going on. There’s lots of news. There are all sorts of results tumbling in from satellites and experiments.   Of course my editors have gotten tired of listening to me promise these things. I think it’s time to discover Dark Matter already. I think it’s time to discover an Earth-like planet. I think it’s time to say yes or no to super-symmetry.
MK: How would you describe the state of science journalism?
DO: Science journalism is in a very interesting, very turbulent state I think. We still have newspapers. Some newspapers still have science reporters, like the Times. I feel like the blogs have risen up to become huge force in the coverage of science. I think the readership now is very fragmented. I think a lot of people get their information from blogs, where people can be more casual or more arcane if they want to be. I think even at my newspaper there’s a difference between people who read the science times and the font page. There are a lot of these different layers of coverage going on.
The bloggosphere is very fast. The kerfluffle about the Arsenic bacteria is a perfect example. In the blogs it was just a prarie fire. It seemed like that whole story just happened on the web not in the print medium. We eventually caught up, but our press just takes longer. On the other hand you can’t count on anything being covered on the web. They are all volunteers so they might decide to cover something or they might not.
Science journalism has undergone a revolution as a whole in the last 20 years. You have to be much more skeptical and look more carefully at claims and budgets and practices. In some ways it’s less fun.
But certainly if you’re in the biomedical/pharmaceutical end of things, you have to pay a lot of attention to money, who owns the patents and who s paying for a particular discovery. The business of that part of science has become a story in itself.
I’m kind of a dinosaur because I spend most of my time writing about pure research of no immediate use to anybody. I don’t have to ask an astrophysicist if he owns a company that’s just patented his recipe for dark matter.
MK: It some ways it sounds like journalists used to work with scientists, but now you have to separate yourselves from them.
DO: When journalists move to writing about science they are so happy to be there, because for the most part people don’t lie to them in science. They understand that scientists are their friends.
But it’s not really our job to teach anybody science or promote the fortunes of NASA or NSF. We’re really just there for the reader to try to illuminate another sphere of human activity. I’m writing about this stuff because I think it’s realy interesting and cool. But there’s a limit to how much I can enthuse about it in print, especially in a place like the Times. I think there’s a misconception that benefits science journalism that they are there to benefit science. I think that’ how most of us feel personally, but it’s relay not quite our job.
A couple of years ago I wrote about this lawsuit in Honolulu to prevent the Supercollider from turning on.   The federal judge in Honolulu has nothing to do with what CERN does in Geneva. It ended up on the front page because it’s a kind of quirky thing—-the editors thought it was comical. It was kind of a grim day otherwise. They thought they were brightening up the front page.
But scientists reacted as though I’d betrayed them. Neither CERN nor Fermilab responded to my requests for over a week for some kind of comment on it.   Finally they referred me to the justice department. I freaked out and figured I guess I better write this thing.
MK: I guess people are sensitive about their 300 million dollar projects.
DO: (laughing) You mean three billion dollar projects!   I think the science community in the US has felt that it was under siege during the last decade under the Bush administration as a result of reactions to things like evolution and the big bang theory of the universe. I think people have been very sensitive.
MK: Do you think that things are better for science since President Bush left office?
DO: If you’d asked me two years ago I would have said yes. But the current mood, which has swung towards reducing the deficit instead of reducing unemployment, doesn’t look good for science.
I heard a talk by the undersecretary of science last week saying that we’re looking at twenty percent cuts.   Pick your favorite agency and imagine it getting a twenty percent cut.
Congress reauthorized the America COMPETES act a month ago [December 22, 2010], which authorized increases in research funding for NSF, NIH DOE, not NASA—-NASA somehow wasn’t included in the package.   But the new Republicans coming to town have vowed they would cut 100 billion dollars from discretionary spending. Fermilab already threw in towel and said we are not going to run the Tevatron anymore.
MK: What do you think scientists should do about all these cuts?
DO: I think they have to make the case that this is an important thing this nation des. It’s one of the collective endeavors for the good that we engage in. You can quote Robert Wilson, the founder of Fermilab. He was asked during some congressional testimony if Fermilab had anything to do with defense of the country. He said no, it was just one of the things that made the country worth defending. Scientsts should be more forthcoming with what they are doing and they should bring people along with them.
MK: How can scientists directly influence politicians?
DO: My impression is you can go and talk to your congressman. You can give talks, you can write articles, you can write books. You can give intelligent press conferences. You can describe why your work is important without necessarily hyping it as discovering black holes again for the first time—-as NASA did every week after the Hubble Space Telescope was launched.
Brian Greene’s op ed essay about Dark Energy is number 2 on the most emailed list [of New York Times articles] today. People actually are interested in this stuff if the way that it’s presented to them is a way that they can understand but don’t feel talked down to. It’s easier said than done. Brian is a master at it
MK: Since you’ve brought up NASA press releases, what are some of your pet pevees about things you see in press releases?
DO: Press releases tend to generally overstate the importance of things. They tend to kind of leave out qualifications. They are too breathless. Of curse they are competing for my attention. But when they overstate the case I immediately lose interest.
In general I think there’s too many of them. I don’t think every Nature paper deserves to be the subject of a press conference. One suspects that most of it is less motivated by desire to inform the public than by a desire to promote the institution.
Today my attention was drawn to some quantum mechanics paper “Extraction of Timelike Entanglement form the Quantum Vacuum” the idea is that you can have entangled particles going in opposite directions in time as well as in space. Somehow you could detect this correlation by measuring things at different times. It sounds bizarre.   I’m probably not going to write about it. All kinds of weird intriguing little nuggets come though. If two important things happen on the same day I can’t do them both.
MK: What would you like to see us scientists do to make your life better?
DO: One thing is to stop cowering in fear of your editors and funding agencies. Just tell me when something happens and I’ll come and write about it. I’m not fond of covering press conference announcements. It’s not interesting from a literary point of view. It’s much more fun to actually see work being done.   Like I’d love to be there when some dark matter experiment decides they’ve really got it.
MK: So we should skip the press release process and just call you directly?
DO: Yes, but my competitors would be very unhappy about that.
MK: Have any scientists actually done that?
DO: These days people are just too aware of mistakes. I think there might be a time when people have done that in the past.   Sometimes people email me and tell me about something they are doing that might not make the news. I think the more abstruse areas of string theory and cosmology don’t get the press release, press conference treatment.
One pet peeve is press releases about papers that show that string theory is about to be experimentally tested. When you read the fine print that’s never true. There was a press release that the large hadron collider was going to test string theory. It was kind of embarassing for them.
Scientists and science journalists just take these shortcuts And I think they become enshrined as truth in the public mind.
MK:   So you’re telling us: don’t write press releases claiming to discover the same thing over and over.
DO: It’s not going to get you a headline. I mean, the clearest example of this is exoplanets. It used to be a big deal that they found an exoplanet. But the bar for an exoplanet getting into the news now is really high. I’m not interested now unless it’s smaller that the one they just announced.
MK: One thing I’m trying to wrap my head around is this idea of manipulating the press. How does that work?
DO: We’re totally manipulable. We don’t pay our sources so, as [Blanche] said in A Streetcar Named Desire, we depend on the kindness of strangers.
The easiest way to manipulate the press is to embargo some result and then send a press release about it to a thousand different news organizations. They will cover it because they are afraid everyone else will cover it.   It’s a kind of artificial competition that’s stirred up.
It does two things. By embargoing the information it makes it harder to get an informed opinion on the paper. It put you at the mercy of time.   And you whip up competition between news organizations.    You have to have your story ready to go online the instant the embargo ends.
You will see that every story has a little note after it with the time that the story came out so you can see who was first, who was a few minutes late with it. For some people this constitutes bragging rights—-in terms of business news it’s not so silly. So there’s a deadline—-you’ve got to have something to say. Your access to informed opinion may be limited.
In the case of the arsenic bacteria I spent several days sending it to a whole bunch of people. But it’s hard because when the whole community reads the paper….
MK: What can you do when you’re manipulated this way, to register that you don’t like it?
DO: You mean besides telling you that I don’t like it? The easiest way to register your disapproval would be to simply write the story on the spot and print it. Then Science or Nature would cut you off from their press releases, which I think would be wonderful. It would be like a vacation.
I was tempted during the arsenic bacteria thing.   As I was writing the story—-there are certain stories that you know the embargo’s going to break, because it’s just too good. I had written my arsenic story the day before in case the story broke on the web. I was tempted to say lets just publish the sucker. And say good bye to Nature for six months.
I think that if you talk to a lot of science journalists you’ll find that they don’t like this embargo game that the journals play. The astrophysicists are much saner about this—you just put your stuff on the archive.
MK: How do you know when a story is ripe for the New York Times?
DO: It’s a mixture of what I think is interesting, what I think is gong to be significant, what I think people are going to be interested in. There are tragic decisions that get made.
If you think it’s important to people beyond your area of specialty. If it’s arcane but it still has an effect on something important.   If it’s something you think you might want to tell your neighbor about.
MK: You’ve written wonderful books; do you have any advice for me about writing a book for the first time?
DO: Yes. It will be over someday!   When I writing my first book I did three complete drafts, and it was like swimming across the ocean each time.   And remember, what you’re writing is for the readers; it’s not for you.

Mathematical trick in Statistical Physics

2011-04-13T11:43:00.000-07:00

The other day I was looking into some Statistical Physics and ended up doing an exercise that was kinda cute. The exercise in question is on Mandl's Statistical Physics, Second Edition page 66.

The first time I did the exercise I did it the normal way, but then I noticed something that ended up simplifying my calculations and here I am posting it.

A system possesses three energy levels $ {E_1= \epsilon } $, $ {E_2=2 \epsilon } $ and $ {E_3=3 \epsilon } $ with degeneracies $ {g(E_1)=g(E_3)=1} $, $ {g(E_2)=2} $. Find the heat capacity of the system.

Don't worry if you don't understand all the terms in here since the gist of this post is not Statistical Physics but the mathematical trick I ended up using.

To solve this problem we need to calculate the partition function $ {Z} $.

$ \displaystyle \begin{array}{rcl} Z &=& \displaystyle \sum_{E_r}g(E_r)e^{-\beta E_r} \\ &=& 1+2e^{-\beta \epsilon}+e^{-\beta 2\epsilon} \\ &=& e^{\beta \epsilon}+2+2e^{-\beta \epsilon} \\ &=& 2(1+ \cosh (\beta \epsilon)) \end{array} $

After having calculated the partition function we have to calculate the average energy, $ {\bar{E}} $, of this system. By definition it is:

$ \displaystyle \begin{array}{rcl} \bar{E} &=& -\dfrac{\partial}{\partial \beta} \log Z \\ &=& -\dfrac{\partial}{\partial \beta} \log 2(1+ \cosh (\beta \epsilon)) \\ &=& -\dfrac{2 \epsilon \sinh (\beta \epsilon)}{2(1+\cosh (\beta \epsilon))} \\ &=& -\dfrac{\epsilon \sinh (\beta \epsilon)}{1+\cosh (\beta \epsilon)} \\ \end{array} $

Keeping in mind that in Statistical Physics it is $ {\beta = 1/(kT)} $ the heat capacity, $ {c} $, is (this is where I started using the trick):

$ \displaystyle \begin{array}{rcl} c &=& \dfrac{\partial \bar{E}}{\partial T} \\ &=& k \dfrac{\partial \bar{E}}{\partial (kT)} \\ &=& k \dfrac{\partial \bar{E}}{\partial (1/\beta)} \\ &=& -\beta ^2 k \dfrac{\partial \bar{E}}{\partial \beta}\\ &=& \beta ^2 k \dfrac{\partial}{\partial \beta}\left( \dfrac{\epsilon \sinh (\beta \epsilon)}{1+\cosh (\beta \epsilon)}\right) \end{array} $

Notice that up until now we haven't calculated a thing at all. All that we have done is just to change variables in order to ease the difficulty in the derivative we'll have to calculate.

Taking the derivative with respect to $ {T} $ in the last expression isn't that hard, but it sure is boring and if one isn't careful errors are expected to creep in.

On the other hand the last expression is much easier to differentiate, but since we are caught up in the moment we'll just do one more change of variable.

$ \displaystyle \begin{array}{rcl} c &=& \beta ^2 k \dfrac{\partial}{\partial \beta}\left( \dfrac{\epsilon \sinh (\beta \epsilon)}{1+\cosh (\beta \epsilon)}\right) \\ &=& \beta ^2 \epsilon ^2 k \dfrac{\partial}{\partial (\beta \epsilon)}\left( \dfrac{ \sinh (\beta \epsilon)}{1+\cosh (\beta \epsilon)}\right) \\ &=& x ^2 k \dfrac{\partial}{\partial x}\left( \dfrac{ \sinh x}{1+\cosh x}\right) \\ &=& x ^2 k \dfrac{\cosh x + \cosh ^2 x - \sinh ^2 x}{(1+ \cosh x)^2} \\ &=& x ^2 k \dfrac{\cosh x + 1}{(1+ \cosh x)^2} \\ &=& \dfrac{x ^2 k}{1+ \cosh x} \end{array} $

Yes, in this case the simplification wasn't that great but I think that one shouldn't lose sight of the fact that this type of reasoning can greatly simplify some other physical derivations.

As an afterthought just let me point out the fact that we cold get a much simpler life if we just remembered a very easy algebraic identity:

$ \displaystyle \begin{array}{rcl} Z &=& 1+2e^{-\beta \epsilon}+e^{-\beta 2\epsilon} \\ &=& (1+e^{-\beta \epsilon})^2 \\ \end{array} $

With this expression the calculation of $ {\bar{E}} $ and $ {c} $ is a lot easier and the reader is urged to try it out.

The joys of teaching

2010-12-03T04:11:00.000-08:00

The joys of learning.

More talk about numbers

2010-12-01T07:24:00.000-08:00

At the end of this post we had arrived at the rational numbers and were able to derive four mathematical operations. At first sight this might look like an impressive achieving, taking into account the tools we set ourselves to use, but due to some shortcomings that plague our construction we think that things can get better.

— 1. Problems —

Our operations are binary
By that we mean that our mathematical operations only make sense when applied to two numbers. So for instance $ {2+3+4} $ has no answer in our current state of affairs. But worry not dear reader because things can be formalized in a fast and straightforward way. And there's nothing too fancy, or technical in this formalization: it is nothing more than common sense put to action.
How do we add $ {2+3+4} $? We just do $ {2+3=5} $ and $ {5+4=9} $. So the correct formalization of $ {a_1 + a_2 + a_3 + \ldots + a_n} $ just is:
$ \displaystyle ((((a_1 + a_2) + a_3) + \ldots )+ a_n) $
That is to say that we define addition of $ {n} $ numbers to be done two by two (this means that addition is still binary but can be done repeatedly).
Of course that all of this is valid for the three remaining operations even though we haven't explicitly said so. Thus this is one possible problem that was taken care of.
Another problem is that we have two inverse operations and due to the existence of them we could expand the set of the available numbers. But we never said nothing about what happens to the properties of the operations while we continuously do that. And for that matter we never said anything at all about the properties of the operations in the first place!
Another problem that we have is that our number system is rather incomplete.

— 2. Properties of the mathematical operations —

To solve our second problem it's time for us to talk about the properties of addition, multiplication, subtraction, and division.

Associativity
$ \displaystyle (m+n)+p = m+(n+p) $
$ \displaystyle (m \times n) \times p = m\times (n \times p) $
Commutativity
$ \displaystyle m+n=n+m $
$ \displaystyle m \times n = n \times m $
Distributivity
$ \displaystyle m \times (n+p)= m \times n + m \times p $
Neutral Element
For every natural number it is:
$ \displaystyle m+0 = m $
and
$ \displaystyle m \times 1 = m $

If we think about addition and multiplication in the simple terms of the first post in this series it is easy to see why these properties hold. But remember that those simple terms apply to the natural numbers. As we went along in enlarging the set of numbers that we were working with we never cared about what happened to those properties.

Do they still hold in $ { \mathbb{Z} } $ and $ { \mathbb{Q} } $?

We can take the lazy out and just define those sets as being sets in which the properties hold or we can check how things really are. Even though I'll take the lazy way out I want readers of this blog to know that things can be done in an intellectually satisfying way.

— 3. Powers —

To solve our third problem we'll just continue to use our ambition and see where it leads us.

Just like we introduced multiplication as a device of allowing us to write more succinctly a sum of $ {n} $ equal numbers we will now introduce the power operation as a more succinct way of writing a product of equal factors.

So for instance we'll write $ {2^5} $ for $ {2 \times 2 \times 2 \times 2 \times 2} $. In general if we have a number $ {a} $, this number is called the base, multiplying with itself $ {n} $ times, this number is called the exponent, we'll write $ {a^n} $ for the end result.

Now that we have introduced powers as a shorthand notation for multiplication of equal factors we have to know how this new entity behaves with the previous four operations.

When we are summing (subtracting) powers we have to first calculate the result of the powers and after this we sum (subtract) those same results. For example:

$ \displaystyle 2^3+ 3^2 = 8+9=17 $

$ \displaystyle 4^2 + 3^2 = 16+9=25 $

$ \displaystyle 5^3-10^2=125-100=25 $

But when we are multiplying or dividing there are some rules that can be used in order to get the right result in a faster way.

— 3.1. Multiplying powers —

Unlike addition (subtraction) of powers when one is multiplying powers we can do it in a more cleaner way in some number of cases:

Same base
For instance:
$ \displaystyle 2^3 \times 2^4=2\times 2 \times 2 \times 2 \times 2 \times 2 \times 2=2^7 $
Thus the final power has the same base as the initial ones and its exponent is just the sum of the exponents of the initial power.
Of course that the numbers $ {2} $, $ {3} $ and $ {4} $ that we choose initially aren't in any way special and as a general rule it always is:
$ \displaystyle a^n\times a^m = a^{n+m} $
Same exponent
For instance:
$ \displaystyle 3^3 \times 4^3= 3 \times 3 \times 3 \times 4 \times 4 \times 4= (3\times 4) \times (3\times 4) \times (3\times 4)= 12^3 $
Once again the numbers we initially choose have nothing special about them so we can state in full generality that
$ \displaystyle a^n\times b^n = (a\times b)^n $

Yes we have stated these two properties for binary multiplication but they can be extended to any number of factors without any problems whatsoever.

— 3.2. Dividing powers —

In the case of division we'll just state the theorems and won't bother to present any example since this can be done very easily by an interested reader once he grasps the idea behind the examples used for multiplication.

Same base
$ \displaystyle a^n/a^m=a^{n-m} $
Same exponent
$ \displaystyle a^n/b^n=(a/b)^n $

— 4. Here comes trouble! —

Since we haven't abandoned our principle of ambition thus far the questions that are being begged to be answered are if we can introduce the inverse operation of a power? and where such course of action might take us?

The answer to the first question is an emphatic yes. The answer to second question is that the inverse operation of a power takes us to wonderful places and open the floodgates of some wonderful mathematics.

The answer of these two questions will be the subject of our next, and possibly final, installment and in these series of posts.

A first and fast look into Kuhn

2010-11-17T17:39:00.001-08:00

Thomas Kuhn, born in 1922 and passed away in 1996, got interested about the history of Physics during his Ph.D. studies

He was in charge of a course, for humanities students, that was about the episodes in the history of science . To prepare for his lectures he decided to read the original texts as much has he could. Once he started to read Physics by Aristotle he was totally surprised with the new Universe that unfolded before him: the ideas expressed in that book seemed to come from outer space!

But they just had come from another time and space. Kuhn realized that taking into account the historical context and the remarkable internal consistency of Aristotle's Physics, it wasn't bad Physics: it was just another kind of Physics. From that moment on Kuhn was more and more drawn to reading the original sources while always stressing that original sources need to be read and analyzed in context.

Essential to Kuhn's analysis of science is the notion of paradigm. A paradigm is a highly ample concept in Kuhn's usage: it is simultaneously all of the laws, all of the theories, all methodological rules, all applications and extensions, all the models, metaphysical suppositions, conceptual frameworks and even the vague and imprecise scheme of the way Nature works.

Of course that such a broad concept had to face its fair share of scrutiny. To respond to his critics Kuhn dropped the notion of paradigm and focused on the much less vague notion of "disciplinary matrix". A disciplinary matrix has four subdivisions:

Symbolic generalizations: laws and definitions of symbols contained in the laws.
Metaphysical assumptions: shared beliefs of a scientific community during a certain time interval.
Values: the criteria of simplicity, internal coherence, accuracy, elegance, plausibility shared by the scientific community.
Exemplars: the designated body of knowledge contained in lab activities, exercises, textbooks, etc, that will help the student acquire his/her scientific formation.

After knowing what Kuhn's notion of a disciplinary matrix is we can summarize his view of how scientific development occurs.

At first we have the period of pre-paradigmatic science, a period of competition between multiple schools of thought where the explanation of certain natural phenomena cause divergences on what is to be studied and how it is to be studied (this occurrence of competing views of Nature is what Kuhn called incommensurability - a rather controversial, bum rapped and poorly understood concept).

After this period a school of thought has won preponderance and one can speak of an existing scientific community. This community has now a set of procedures, problems and a world view that are more or less broadly shared and is engaged in what mostly is puzzle solving activities. This is the period of normal science and as Kuhn rightly pointed out scientists in this stage have a kind of aversion to novelties on their fields of expertise.

Anomalies in the reigning paradigm are seen as minor nuisances that have to be solved in the context of the scientific zeitgeist. If the anomaly resists numerous attempts of a solution for a long enough period of time it will cause an erosion of the rules that are present in normal science and a period of renewed uncertainty about the disciplinary matrix is sure to follow (this convoluted period that has a lot of similarities with the pre-paradigmatic stage of science is sometimes called multi-paradigmatic science).

Following this either a new theory emerges where the discrepancies are mostly taken care of, or the old theory, with some little modifications, is able to accommodate the results that seemed to be out of its range. The first possibility is what is called a scientific revolution and after it a new period of normal science will follow in which the breadth and depth of the new disciplinary matrix will increase.

Two things are worth noticing in order to finish this very short introduction:

Even though the period of normal science is adverse to scientific novelties it, nevertheless, is a catalyst for them given its constant poking at the limits of its own domains of application.
On Kuhn's approach to the development of science, progress is achieved not in a cumulative way: science consists of long stretches of time where scientists engage in what Kuhn called normal science that are interrupted whenever a sufficient amount of anomalous results are too big to be ignored. In Kuhn's terminology this sets off a period that is called a crisis. After the resolution of this crisis by means of a scientific revolution a new world view emerges from which the old world view results can be derived (for instance Newtonian Dynamics results can be derived from Special Relativity by formally taking the limit $ {c \rightarrow \infty} $), but the underlying metaphysical assumptions of the two theories are profoundly at odds with each other.

Ladies and Gentleman, prepare your bets!

2010-07-16T04:32:00.000-07:00

This is a very interesting bet: “By 2020, no one will have won a Nobel Prize for work on superstring theory, membrane theory, or some other unified theory describing all the forces of nature.”

Superstring theory is a hot topic of debate among the Physics community and it has a very rich history considering its relatively young age.

Physicists from all subdivisions argue if this is a worthy subject of study, if it can be considered Physics at all (or if it is just a part of Mathematics) and the debate still rages to this day.

As interesting as it may be I have to say that I don't have much interest in superstring theory. I won't feign a deep knowledge of it nor a deep understanding, but for the life of me I can't be really bothered with it.
Maybe I'm explaining myself wrong though. I do find interesting to debate the merits and failings of superstring theory and trying to understand its basics on a level that is enough for any self-respecting physicist, but I don't really see myself having the time, nor the energy, to devote myself to a serious study of this.

Two physicists that know what they're talking about have decided to make a bet on the near future outcome of superstring theory: the question is if a Nobel prize will be awarded to some kind of work related to superstring theory (or any other unified theory that describes all known interactions in Nature).

My partially educated guess is that such a thing won't happen and I just hope I'm here to see the rest of this history.

Horgan’s Argument

In purely intellectual terms, a unified theory of physics would be the greatest of all scientific achievements. It would culminate the ancient human quest for knowledge, which began when the first of our ancestors asked, "Why?" It would yield the basic rules governing the entire universe, from the smallest to the largest scales. It would tell us how the universe came into being and why it took this particular form, which permitted our existence. It might even reveal our ultimate cosmic fate. At least, that's what seekers of a unified theory hope, and what I used to believe. In the early 1990s, I came to suspect that the quest for a unified theory is religious rather than scientific. Physicists want to show that all things came from one thing: a force, or essence, or membrane wriggling in eleven dimensions, or something that manifests perfect mathematical symmetry. In their search for this primordial symmetry, however, physicists have gone off the deep end, postulating particles and energies and dimensions whose existence can never be experimentally verified. The Superconducting Supercollider, the monstrous particle accelerator that Congress canceled in 1993, would have been 54-miles in circumference. Gaining access to the infinitesimal microscales where superstrings supposedly wriggle would require an accelerator 1,000 light years around. (The entire solar system is only one light day around.) It is this problem that makes me confident I will win this bet. The Nobel prize judges have always been sticklers for experimental proof. The dream of a unified theory, which some evangelists call a "theory of everything," will never be entirely abandoned. But I predict that over the next twenty years, fewer smart young physicists will be attracted to an endeavor that has vanishingly little hope of an empirical payoff. Most physicists will come to accept that nature might not share our passion for unity. Physicists have already produced theories-Newtonian mechanics, quantum mechanics, general relativity, nonlinear dynamics--that work extraordinarily well in certain domains, and there is no reason why there should be a single theory that accounts for all the forces of nature. The quest for a unified theory will come to be seen not as a branch of science, which tells us about the real world, but as a kind of mathematical theology. By the way, I would be delighted to lose this bet.

Kaku’s Argument

It is often forgotten that physics is mainly done indirectly. Thus, we know that the sun is made of hydrogen gas, yet no one has ever visited the sun. We know that black holes exist in space, yet they are invisible by definition. We know that the Big Bang took place approximately 15 billion years ago, yet no one was there to witness it. We know these things, because we have indirect evidence or "echoes", such as sunlight and characteristic radiation from black holes and Creation. Likewise, you do not need to build an atom smasher the size of the galaxy to prove string theory or M-theory (the leading and, in fact, only candidate for a "theory of everything). Instead, we need to look for echoes from the 10th and 11th dimensions as follows: a) Within a few years, the Large Hadron Collider, the largest atom smasher on earth, will be turned on outside Geneva, Switzerland. It might be able to find "sparticles" or super particles, i.e. higher vibrations or octaves of the superstring. b) Invisible dark matter, which makes up 90% of the matter in the universe, might be shown to consist of sparticles like the photino. This might also verify string theory. c) In this decade, gravity wave detectors should be able to record shock waves from colliding black holes, which might reveal the first quantum corrrection to Einstein's original theory of 1915. These quantum corrections can be compared to those predicted by string theory. d) Within 20 years, NASA plans to send three gravity wave detectors into outer space. They should be sensitive enough to pick up the shock waves from the Big Bang itself created a fraction of a second after the instant of creation. This should be able to prove or disprove string theory. Personally, I feel no need to prove the theory experimentally, since I believe it can be proven using pure mathematics. A theory of everything is also a theory of everyday energies, where we find familar electrons, protons, and atoms. If we can solve the theory mathematically, then we should be able to calculate the properties of electrons, protons, and atoms from pure mathematics. If the results disagree with known data, then string theory will be shown to be a "theory of nothing." However, if the numbers agree, then it will be heralded as the greatest achievement of the human mind. We will have "read the mind of God." So what prevents us from simply solving the theory and comparing the results with nature? The problem is that the theory is smarter than we are. No one on this planet is smart enough to solve this theory. The smartest people on earth are working on this problem, and have so far failed. (This is because the theory was discovered purely by accident in 1968. We were never supposed to see this theory in the 20th century. The mathematics necessary to solve the theory have not yet been discovered.) Because string theory has near-miraculous breakthroughs every 8 to 10 years, we can expect 2 more breakthroughs in the theory before 2020, and hence might be able to solve this theory by then. Perhaps someone reading this bet will be inspired to mathematically solve this theory completely. Maybe that person will then receive a telephone call from Sweden.

Lisa Randall II

2010-07-14T15:09:00.001-07:00

This time is serious. If you have the time watch her full interview.

Lisa Randall

2010-07-14T15:02:00.001-07:00

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Michio Kaku

2010-07-14T14:57:00.001-07:00

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Arnold is gone

2010-07-14T14:45:00.000-07:00

And I only knew about it today...

Vladimir Igorevich Arnold has left us... Out of the blue and so untimely...
Arnold is a token figure of the demimonde of the present-day academic community of Russia since he held records in allpopular citation indices. Much more important is the fact thatArnold is a real teacher of the dozens if not hundreds of thousands of modern scientists who mastered the art of solving ordinary differentialequations with his textbook at hand.

Differentiation discovers trends, and integration forecasts the future fromtrends.Differential equations express connections between the laws we have grasped in the small. Qualitative analysis and solution of the equations underlie thescientific prognosis of future events. Invented three centuries ago as the toolkitof classical mechanics, the differential and integral calculuslays grounds for the theory of dynamical systems which incorporatescelestial mechanics, astronautics, and hydrodynamics.Arnold's Mathematical Principles of Classical Mechanics occupies the place of Newton's Mathematical Principles of Natural Philosophyat the desks of all specialists in these fields.

Arnold, once a student of Kolmogorov, is the most brilliant representativeof the romantic trend in the mathematics of the twentieth century,a successor of the synthetic traditions of Diophantus, Newton, Gauss, Chebychev,Lobachevsky, and Poincaré.During many decades ahead, mathematicians will recall the solution of the thirteenth Hilbert problem Arnold gave as a third-year student of Lomonosov State University,as well as his deep lectures on the history of mathematics,the sharp criticism of the notorious “Bourbakism,”and the bewildering phantasmagorias in the wake of the emerald tablet.

Arnold had entered the history of the world mathematics many years ago andwill reside there forever. Mathematics is a matter of a rather slight rather than paramount importance. To be a great mathematician is quite an honor and, seemingly, this is a pleasure. It is not shameful to be a mathematicianbut it is shameful to be only a mathematician, since specialization suppressespersonality. Arnold was a man much greater that his mathematics and mechanics.It is not excluded that only a few professionals in Russia would recall the mathematical contributions of Arnold after the lapse of a century. However, the name of Arnold will never vanish from the Russian culture while our Russian language survives for Arnold made a discovery of an extraordinarybrightness and talent in Pushkinistics.

All Russians know about the “uncle, man of firm convictions,” from the beginning of the first chapter ofEugene Onegin. Fewer readers pay any attention to the dedicationand the epigraph in French marked as“tire d'une lettre particuliere.”In the first edition of 1825 this epigraph precededthe first chapter, whereas in 1833Pushkin chose it to be the common epigraph to the whole of his novel in verses.At the times of Pushkin all cultivated persons had seemed to understand the epigraph from a half-word andneeded no explanations. Culture changed since then and the intrinsic meaning ofthe epigraph became enigmatic for about one hundred and fifty years.The common hypothesis reads that the epigraph was some mystification belonging to Pushkin's pen.

In the late 1990s Arnold published a short note “About the epigraph to Eugene Onegin” in Proceedings of the Russian Academy of Sciences (Series in Literature and Language).Professor and International Grandmaster Larisa Volpert, anoutstanding expert in the creative heritage of Pushkin, briefly evaluatedthis note as follows:“In 1998 V. I. Arnold, an amateur-pushkinist,happened to find the source of the epigraph.He suggested and in our opinion suggested quite convincinglythat the epigraph is an inexact citationfrom Les Liaisons Dangereuses by Pierre Choderlos de Laclos.A serious finding, even a small discovery, since this concerns the epigraph to the whole novel.It is a pity that Arnold did not proceed further from the concise statement and did not attempt to connect his finding with the poetics of Eugene Onegin.”

This rather unknown episode characterizesthe extraordinary gifts and treasures of Arnold's personality we all enjoyedin Russia as his compatriots and contemporaries.Yesterday Arnold had moved to the pantheon of the world culture.His memories will remain exuberant and enduring.

The always wise Terry Tao

2010-07-14T13:48:00.000-07:00

It is always a great pleasure to read Terry Tao be it on Mathematics, be it on simple life affairs: he's always striving for simplicity and clarity and usually he gets the job done with extreme efficiency.
Here's a recent interview:

America's 300 million-plus people are declaring their identity in the 2010 census. This piece is part of a special series on CNN.com in which people describe how they see their own identity. Terence Tao won a Fields Medal in 2006 and was named a MacArthur fellow the same year. He is a professor of mathematics at UCLA, where he holds the James and Carol Collins Chair in the College of Letters and Science.

When I was a child in Adelaide, Australia, I loved games with clear, unambiguous rules; puzzles that were tough but fair; and the clean, abstract, simplicity of numbers and symbols. So it is perhaps not surprising that I have been drawn to mathematics for as long as I can remember.

For much of my childhood, playing with mathematical problems, equations and facts was one of my favorite activities; I even went so far as to compete in international high school mathematics competitions, racing to solve tricky sets of problems in a few hours.

I still remember the realization in college at Flinders University in Australia that mathematics was not just an abstract game of symbols, but could be used as a tool to analyze and understand the modern world.

Why are some statistics trustworthy and some not? Why are some investment strategies sound, and others risky? How come a computer can search the entire internet for you in a matter of seconds, but cannot read a printed word if it is even just slightly distorted or blurred? How come our modern array of satellites can tell millions of drivers their location with amazing accuracy, but cannot correctly predict the weather a fortnight into the future?

Knowledge of mathematics can answer these questions, and make the world comprehensible and orderly rather than mysterious and capricious.

Once I saw the power of this knowledge, and the satisfying feeling when everything "clicks" and one sees a confusing problem resolve itself into a clear solution, I was hooked for life. I wanted to use mathematics to explore and understand as much of the world as I could.

In 1992, when I was 16, I moved to the United States to start working on my Ph.D. at Princeton University in New Jersey.

I alternated my time between Australia and the United States for many years, but by 1999 I had a permanent job in the United States, on the faculty of UCLA's Department of Mathematics. By 2002, I was married to an American -- we were married in the United States -- and last year I became a U.S. citizen (while retaining Australian citizenship).

Of course, my life is more than just my work. I am a husband and a father and a proud citizen of two countries; my homeland of Australia and my adopted country here in the United States. I identify with them both.

I enjoy a good meal, a good vacation or a good movie, much as anyone else would. But mathematics is both my profession and my hobby, and the place where it seems I am best able to make a contribution; so if I had to answer the question "Who am I?" I would have to say, "a mathematician."

Are you a disgruntled Physicist?

2010-07-13T16:50:00.000-07:00

So are you?

Physics is a fascinating subject that has its degree of difficulty and to really understand some parts of it there is no denying that a lot of hard work is needed.
Some people have no fear of putting their hands and brain up to some work, but somehow along the way they just become discouraged and disillusioned with Physics. People that started with the highest hopes gradually become bored to death with a subject they deeply cared for only a few years ago.
One thing that may cause this break in interest is the sudden awareness that to know Physics one really needs to know Mathematics, and for some students the sudden appearance of so much difficult Mathematics is really off-putting and seems to be highly unreasonable.
One other reason explored by Warren Siegel is that the Physics part of most curricula is just boring as Hell and can suck the life out of any party going Physics enthusiast!
A fascinating subject is transformed at lessons of History (99.99999% of the time it is really poor history with a faint resemblance to actual facts) and there is no sense of unity and elegance in the way things are normally put forth.

Anyway, here are Siegel's original musings and you'd do a lot of good reading it:

Have you ever gotten the impression that the introductory physics textbook you are forced to use is a charlatan,that maybe it was designed by physicians and engineers purposely to make physics look boring,so that anyone with any imagination would be driven away?Generally such a book looks much more like a history book, andapparently the author thinks that the only way to learn from the mistakes of history is to repeat them.(Let's hope he didn't write the lab manual, too.)Or maybe he is just a disgruntled physicist, who thinks that, well, he had to learn it that way,so, by golly, that's the way you're going to have to learn it, too.This type of approach has several important drawbacks, even fornon-physicists:
Such books give no hint of how a scientist really thinks.If you learn anything in school, it should be how to think.Memorization is less useful: You can always look it up.
The science of tomorrow is not the science of yesterday.Even biologists should know quantum mechanics and special relativity,which are relevant to things likemolecular biology and particle-beam cancer treatment.The new ideas of tomorrow, in any branch of science, willnot come from just old physics.
College degrees have breadth requirements for a good reason.Breadth does not mean teaching outside topics as if they were inside.
Here is a list of some of the things I have found in such books given to me as the "TA" of such courses:
The introductory chapter of the book includes a discussion of units,but nowhere is mentioned the fact that the whole point of units isthat you can choose whatever units are most convenient, such as usingthe (reduced) mass of the electron as a unit of mass in atomic physics.
The book begins with "handyman physics": fulcrums, pulleys, etc. Who are we trying to attract to physics, carpenters?(This is why I didn't get interested in physics until I was 8, instead of at 5.By then I learned from Crackerjacks and cereal that the good stuff is at the back, not in the front.But you can't teach a course that way.)
No hint of particle physics.Special relativity appears only at the end, if at all, and after electromagnetism and optics.(Ever hear of nonrelativistic light?)
Newton's laws are discussed before conservation of energy and momentum.Ironically, conservation of energy is used before Newton's lawsfor the special case of constant acceleration (i.e., gravity),but only as a subsidiary condition and not by name.
Electrodynamics is apparently about circuitry.(What are these stupid constants "ε₀" and "μ₀"?)
No complex numbers: sin's, cos's, and complicated trigonometric identities abound.
Ampere's law is given first in its original incorrect form(with little hint given that it is wrong),then corrected much later.
Maxwell's equations appear in full only in integral form, not differential,even though a partial version of the differential equations is applied to radiation.The gradient appears only in a footnote; the "curl" is never defined.
Refraction of light is claimed to have no explanation in terms of particles,quoting Newton's failure. Of course it does: As we know from quantum mechanics,Newton's mistake was to confuse phase and group velocities.
For special relativity, Lorentz transformations come before 4-vectors.Minkowski space isn't even mentioned.
Planck's constant is introduced in its original form, as "h";" ħ" does not even appear.
Vectors are done with unit vectors "i, j, k" (even in upper-division undergraduate courses). Does anybody really use them anymore? If so, what is the unit vector for time in 4-vectors? (In this old quaternion notation it would be 1, but I'm sure nobody has used that since Maxwell.) Obviously this notation was already out of date in 1905, so why do we still teach 19th-century notation in the 21st century?
Here's some more stuff I found in a similar textbook for pre-med students:
Although forces are often given as vectors, only the magnitude of the force is given in the equation for Coulomb's law: The radial unit vector never appears; it is only described in words.
Derivatives & integrals are repeatedly used, but almost never by name. "Δ" is always used instead of "d", and "∑" instead of "∫". Solutions to differential equations are given (e.g., for circuits) without the equations.
Dot products and cross products are frequently used, but never by name. In specific applications, dot products are written either with cos θ or with the projection of one vector on the other, but never with both, nor is the equivalence indicated. The right-hand rule appears to work differently in different cases. No indication is given how these concepts relate to rotational invariance.
Maxwell's equations are called "Maxwell's predictions", and written only in words. Ampere's law is never corrected.
A relativistic cube is said to appear "rectangular". If they had even only watched a little popular science on TV, they would know it appears as a rotated cube.
I almost forgot the most important point: Every problem gives numerical values to all quantities, and answers must be given in numerical values. So every problem is not only a test on units, but also on how well one can use a calculator, which on an exam means a hand-held calculator. This teaches every student:

Arithmetic is more important than algebra. Plug in numbers as soon as possible so you can lose sight of what you're doing.
Use only 1970's technology. 1960's technology (slide rules) or worse yet, doing arithmetic in your head, is obsolete. But 1980's technology (personal computers) is cheating.

PhD allegory

2010-07-06T03:22:00.000-07:00

I received this by email today.

The Lord of the Rings: an allegory of the PhD?
The story starts with Frodo: a young hobbit, quite bright, a bit dissatisfied with what he's learnt sofar and with his mates back home who just seem to want to get jobs and settle down and drink beer.He's also very much in awe of his tutor and mentor, the very senior professor Gandalf, so whenGandalf suggests he take on a short project for him (carrying the Ring to Rivendell), he agrees.

Frodo very quickly encounters the shadowy forces of fear and despair which will haunt the rest ofhis journey and leave permanent scars on his psyche, but he also makes some useful friends. Inparticular, he spends an evening down the pub with Aragorn, who has been wandering the world formany years as Gandalf's postdoc and becomes his adviser when Gandalf isn't around.

After Frodo has completed his first project, Gandalf (along with head of department Elrond)proposes that the work should be extended. He assembles a large research group, including visitingstudents Gimli and Legolas, the foreign postdoc Boromir, and several of Frodo's own friends fromhis undergraduate days. Frodo agrees to tackle this larger project, though he has mixed feelingsabout it. ("'I will take the Ring', he said, 'although I do not know why.'")

Very rapidly, things go wrong. First, Gandalf disappears and has no more interaction with Frodountil everything is over. (Frodo assumes his supervisor is dead: in fact, he's simply found a moreinteresting topic and is working on that instead.) At his first international conference in Lorien,Frodo is cross-questioned terrifyingly by Galadriel, and betrayed by Boromir, who is anxious to getthe credit for the work himself. Frodo cuts himself off from the rest of his team: from now on, hewill only discuss his work with Sam, an old friend who doesn't really understand what it's all about,but in any case is prepared to give Frodo credit for being rather cleverer than he is. Then he sets outtowards Mordor.

The last and darkest period of Frodo's journey clearly represents the writing-up stage, as hestruggles towards Mount Doom (submission), finding his burden growing heavier and heavier yetmore and more a part of himself; more and more terrified of failure; plagued by the figure ofGollum, the student who carried the Ring before him but never wrote up and still hangs around as aburnt-out, jealous shadow; talking less and less even to Sam. When he submits the Ring to the fire,it is in desperate confusion rather than with confidence, and for a while the world seems empty.

Eventually it is over: the Ring is gone, everyone congratulates him, and for a few days he canconvince himself that his troubles are over. But there is one more obstacle to overcome: monthslater, back in the Shire, he must confront the external examiner Saruman, an old enemy of Gandalf,who seeks to humiliate and destroy his rival's protege. With the help of his friends and colleagues,Frodo passes through this ordeal, but discovers at the end that victory has no value left for him.While his friends return to settling down and finding jobs and starting families, Frodo remains in limbo; finally, along with Gandalf, Elrond and many others, he joins the brain drain across theWestern ocean to the new land beyond.

Written by Dave Pritchard

Thanks Ana.

Science in Islam

2010-07-03T15:33:00.000-07:00

One of the goals of me in putting this blog up was to do a semi-professional account of the history of Physics and Mathematics. In particular I intend to dispel a few myths about Science and Islam (namely the role of Science in Islam during Europe's Dark Ages) and this three part documentary will be an appetizer for what will follow.

Despair not dear readers this blog is alive and I will follow through with my promises it's just a matter of me having some free time (and I think it is about to come) and/or some different organization on my part.

Anyway, enough with the small talk and here you go:

From Eternity to Here - an interview with Stephen Colbert

2010-07-03T11:38:00.000-07:00

Sean Carrol, from Cosmic Variance fame has written a new book, From the Eternity to Here, which I haven't read (and most certainly will not read in a near future), and is doing his promotion.

Sean is a theoretical physicist at the California Institute of Technology. He received his Ph.D. from Harvard in 1993, and worked at MIT, the Institute for Theoretical Physics at UC Santa Barbara, and the University of Chicago before moving to Caltech.
His research involves theoretical physics and astrophysics, focusing on issues in cosmology, field theory, and gravitation.

Enjoy this interview where, even though Stephen Colbert hit us all with his usual style, Sean had the time to really provide some deep questions and the reason of why they are deep questions:

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Video interview with David Bohm

2010-07-03T11:30:00.000-07:00

David Bohm is one of my favorite physicists (even though his book on Quantum Mechanics was a disappointment) due to his constant search for originality and consistency.

His critique/analysis of orthodox Quantum Mechanics was a much needed one and it helped pave the way for serious alternatives to the orthodoxy.

Here's a little interview for the ones that want to know this man, his thoughts and his influence on Physics:

He isn't a golpher, but she is

2010-07-02T05:56:00.000-07:00

In order to pretend that there is life in this blog I'll take the initiative to provide a few interviews, videos and whatnot's while giving a few initial commentaries and possibly a final commentary.

The content of this TED talk isn't directly related to our subject matter, but nevertheless it is interesting and useful.

Too much articles

2010-07-02T05:43:00.000-07:00

Everybody in Academia knows that the amount of articles published in any given field is just too much to be read in life time (let's not even talk about understanding all those articles).

In the following article the authors discuss the fact that just by having a bigger number of articles published doesn't mean that there are more important discoveries being made and not even more knowledge is being created.
The authors note the extreme level of redundancy and uselessness (due to poor research) that this kind of attitude causes.

The article is a pretty interesting one and I urge you all to read it in its entirety.

By Mark Bauerlein, Mohamed Gad-el-Hak, Wayne Grody, Bill McKelvey, and Stanley W. Trimble:

Everybody agrees that scientific research is indispensable to thenation's health, prosperity, and security. In the many discussions ofthe value of research, however, one rarely hears any mention of howmuch publication of the results is best. Indeed, for all the regretsone hears in these hard times of research suffering from financingproblems, we shouldn't forget the fact that the last few decades haveseen astounding growth in the sheer output of research findings andconclusions. Just consider the raw increase in the number of journals.Using Ulrich's Periodicals Directory, Michael Mabe shows thatthe number of "refereed academic/scholarly" publications grows at arate of 3.26 percent per year (i.e., doubles about every 20 years). Themain cause: the growth in the number of researchers.

Many people regard this upsurge as a sign of health. They emphasizethe remarkable discoveries and breakthroughs of scientific researchover the years; they note that in the Times Higher Education's rankingof research universities around the world, campuses in the UnitedStates fill six of the top 10 spots. More published output means morediscovery, more knowledge, ever-improving enterprise.
If only that were true.

While brilliant and progressive research continues apace here andthere, the amount of redundant, inconsequential, and outright poorresearch has swelled in recent decades, filling countless pages injournals and monographs. Consider this tally from Science twodecades ago: Only 45 percent of the articles published in the 4,500 topscientific journals were cited within the first five years afterpublication. In recent years, the figure seems to have dropped further.In a 2009 article in Online Information Review, Péter Jacsófound that 40.6 percent of the articles published in the top scienceand social-science journals (the figures do not include the humanities)were cited in the period 2002 to 2006.

As a result, instead of contributing to knowledge in variousdisciplines, the increasing number of low-cited publications only addsto the bulk of words and numbers to be reviewed. Even if read, manyarticles that are not cited by anyone would seem to contain littleuseful information. The avalanche of ignored research has a profoundlydamaging effect on the enterprise as a whole. Not only does the uncitedwork itself require years of field and library or laboratory research.It also requires colleagues to read it and provide feedback, as well asreviewers to evaluate it formally for publication. Then, once it ispublished, it joins the multitudes of other, related publications thatresearchers must read and evaluate for relevance to their own work.Reviewer time and energy requirements multiply by the year. The impactstrikes at the heart of academe.

Among the primary effects:

Too much publication raises the refereeing load on leadingpractitioners—often beyond their capacity to cope. Recognized figuresare besieged by journal and press editors who need authoritativejudgments to take to their editorial boards. Foundations and governmentagencies need more and more people to serve on panels to review grantapplications whose cumulative page counts keep rising. Departments needdistinguished figures in a field to evaluate candidates for promotionwhose research files have likewise swelled.

The productivity climate raises the demand on younger researchers.Once one graduate student in the sciences publishes three first-authorpapers before filing a dissertation, the bar rises for all the othergraduate students.

The pace of publication accelerates, encouraging projects that don'trequire extensive, time-consuming inquiry and evidence gathering. Forexample, instead of efficiently combining multiple results into onepaper, professors often put all their students' names on multiplepapers, each of which contains part of the findings of just one of thestudents. One famous physicist has some 450 articles using such astrategy.

In addition, as more and more journals are initiated, especially themany new "international" journals created to serve the rapidlyincreasing number of English-language articles produced by academics inChina, India, and Eastern Europe, libraries struggle to pay thenotoriously high subscription costs. The financial strain has reached acritical point. From 1978 to 2001, libraries at the University ofCalifornia at Los Angeles, for example, saw their subscription costsalone climb by 1,300 percent.

The amount of material one must read to conduct a reasonable reviewof a topic keeps growing. Younger scholars can't ignore any of it—theynever know when a reviewer or an interviewer might have writtensomething disregarded—and so they waste precious months reviewing apool of articles that may lead nowhere.

Finally, the output of hard copy, not only print journals but alsoarticles in electronic format downloaded and printed, requires enormousamounts of paper, energy, and space to produce, transport, handle, andstore—an environmentally irresponsible practice.
Let us go on.

Experts asked to evaluate manuscripts, results, and promotion filesgive them less-careful scrutiny or pass the burden along to other,less-competent peers. We all know busy professors who ask Ph.D.students to do their reviewing for them. Questionable work finds itsway more easily through the review process and enters into the domainof knowledge. Because of the accelerated pace, the impression spreadsthat anything more than a few years old is obsolete. Older literatureisn't properly appreciated, or is needlessly rehashed in a newer,publishable version. Aspiring researchers are turned intopublish-or-perish entrepreneurs, often becoming more or less cynicalabout the higher ideals of the pursuit of knowledge. They fashionpathways to speedier publication, cutting corners on methodology andturning to politicking and fawning strategies for acceptance.

Such outcomes run squarely against the goals of scientific inquiry.The surest guarantee of integrity, peer review, falls under adebilitating crush of findings, for peer review can handle only so muchmaterial without breaking down. More isn't better. At some point,quality gives way to quantity.

Academic publication has passed that point in most, if not all, disciplines—in some fields by a long shot. For example, Physica Apublishes some 3,000 pages each year. Why? Senior physics professorshave well-financed labs with five to 10 Ph.D.-student researchers.Since the latter increasingly need more publications to compete foracademic jobs, the number of published pages keeps climbing. Whilepublication rates are going up throughout academe, with unfortunateconsequences, the productivity mandate hits especially hard in thesciences.

Only if the system of rewards is changed will the avalanche stop. Weneed policy makers and grant makers to focus not on money for currentlevels of publication, but rather on finding ways to increasehigh-quality work and curtail publication of low-quality work. If onlysome forward-looking university administrators initiated changes inhiring and promotion criteria and ordered their libraries to stoppaying for low-cited journals, they would perform a national service.We need to get rid of administrators who reward faculty members onprinted pages and downloads alone, deans and provosts "who can't readbut can count," as the saying goes. Most of all, we need to understandthat there is such a thing as overpublication, and that pushingthousands of researchers to issue mediocre, forgettable arguments andfindings is a terrible misuse of human, as well as fiscal, capital.

Several fixes come to mind:

First, limit the number of papers to the best three, four, or fivethat a job or promotion candidate can submit. That would encourage morecomprehensive and focused publishing.

Second, make more use of citation and journal "impact factors," fromThomson ISI. The scores measure the citation visibility of establishedjournals and of researchers who publish in them. By that index, Nature and Sciencescore about 30. Most major disciplinary journals, though, score 1 to 2,the vast majority score below 1, and some are hardly visible at all. Ifwe add those scores to a researcher's publication record, thepublications on a CV might look considerably different than a mere listdoes.

Third, change the length of papers published in print: Limit manuscripts to five to six journal-length pages, as Nature and Sciencedo, and put a longer version up on a journal's Web site. The twoversions would work as a package. That approach could be enhanced ifuniversity and other research libraries formed buying consortia, whichwould pressure publishers of journals more quickly and aggressively topursue this third route. Some are already beginning to do so, but anationally coordinated effort is needed.

There may well be other solutions, but what we surely need is achange in the academic culture that has given rise to the oversupply ofjournals. For the fact is that one article with a high citation ratingshould count more than 10 articles with negligible ratings. Moving tothe model that Nature and Science use would have far-reaching and enormously beneficial effects.

Our suggestions would change evaluation practices in committeerooms, editorial offices, and library purchasing meetings. Hiringcommittees would favor candidates with high citation scores, not bulkypublications. Libraries would drop journals that don't register impact.Journals would change practices so that the materials they publishwould make meaningful contributions and have the needed, detailedbackup available online. Finally, researchers themselves would devotemore attention to fewer and better papers actually published, and morejournals might be more discriminating.

Best of all, our suggested changes would allow academe to revert toits proper focus on quality research and rededicate itself to the soberpursuit of knowledge. And it would end the dispiriting paper chase thatturns fledgling inquirers into careerists and established figures intooverburdened grouches.

Ps: It is, perhaps, amusing to note that certainly not everybody will read the 176 comments (at the time of this post) that the article elicited.

Why does Captain Kirk climb the mountain?

2010-04-14T03:30:00.000-07:00

Because he's in love.

Mea Culpa

2010-03-27T18:21:00.000-07:00

Ao contrário do que aqui ficou implícito não é nada a primeira vez que se publica a obra Sidereus Nuncius em português. Em 1987 foi publicada no Brasil o livro "A Mensagem das Estrelas" que é precisamente a primeira tradução da obra de Galileu para português.

A short debunk of Money as Debt

2010-03-22T13:11:00.000-07:00

Right off the bat, let me just say that I'm not a fan of the current financial system. I am a center-left fan of social democracy and so I'm definitely far from the stereotypical libertarian "economist". I may not be on the exact opposite of the spectrum, but I'm definitely on the other side.

That said, this video (and its clones that circulate around the internet) is the Zeitgeist: Addendum equivalent for the monetary system. It feeds you some basic history and when you are starting to fall asleep, it slides in an unsupported false assertion. Before you know what happened, you're already nodding in agreement or wondering what the hell just happened. This goes from conspiracies in worldwide education ("they don't want you to learn about it") to things like "without debt there would be no money". They may give you some quote of a guy also asserting it (bonus points if he's a founding father), but will never really explain you why.

But on to the video itself. The central [premise of the video is that all money is debt. If all money is debt and interest needs to be repaid, then (they conclude) more debt needs to be created to avoid large foreclosures. I'll make just a few comments about it.

The first problem is how they define debt. For them, deposits are also debt. Although it's true that from an accounting perspective deposits are a liability from the side of banks, it's also true that what are liabilities from the side of the general public are also assets from the side of banks. By jumping between sides to claim everything is debt/liability is disingenuous. I'll then talk about IOUs as opposed to debt.

Secondly, only in a barter economy or a fiat money monetary system is some money not involved in a IOU of some sort. If money is redeemable in gold, then money is also basically a IOU. Each paper note is a promise to be repaid in x units of gold. Is that any better? I don't see why. Remember, most of the value of gold comes also solely from the fact that other people believe it is quite valuable and are ready to accept it. If anything, fiat currency allows the government to control the amount of money that has the potential to be debt free. Any other non-barter system is then even "worse" than a fiat money system.

Thirdly, money being a sort of quasi-debt is actually not a big problem. Think about it. Money basically exists to facilitate exchanges (the other functions are side products). You only accept money, because you know that you'll be able to exchange it later for something you find useful. Money is then valuable because it is a unique form of IOU, where you have great flexibility in deciding how you'll be redeemed. You don't even need to know what you'll be exchanging it for when you accept it. It is then only natural that money and debt are related.

The value of the financial system (this is contentious, so it's a more personal view) is simply one: to channel savings into more productive investments. This will necessarily involve debt, but is in the vast majority of cases win-win. The entrepreneur is able to finance a long-term project that involves large initial costs and the saver gets a future return. One could argue that the financial system has grown well beyond this basic function, and I would agree, but this is fundamentally not a problem of the monetary system or commercial banking regulations. It's mostly a problem of investment banking absorbing too much of the value added and too little taxation. But that is another conversation.

So let's get to the core of the video, their 4 questions:

Why borrow from private banks when the government could create money?

Because historically, hyperinflation was the result. This led to the separation between monetary authorities and fiscal authorities. The ability to create money to pay government costs and investment meant that politicians do not internalize the inflation cost of money creation. This meant that the government overspent and needed to create perpetually more money because it was devaluing the currency and people want more money for the same services. Eventually inflation explodes. By forcing governments to pay debt (not just from private banks, bonds are also held by households) this internalizes the cost of money creation and reduces excessive spending. The real question is why do we not care when the government borrows so much. If we want it to spend, then we must accept higher taxes.

Why not create money that circulates permanently?

To be fair, I don't understand what they mean. If they mean having a fixed supply of money, then growth would automatically imply deflation. Deflation is a problem because of deflationary spirals. In short, prices fall, production falls because it is less profitable leading to unemployment and/or lower wages. This depresses demand and may lead to further price decreases. Deflationary episodes are usually quite bad for any country.

Are "perpetually accelerating growth" and sustainability compatible?

First of all, growth is an exponential concept. To talk about "perpetually accelerating growth" is wrong. Constant growth is what we are talking about, but of course it doesn't sound that bad and so it doesn't serve their purpose. But back to the point, which is a typical fallacy:

Growth is not just about multiplication of what already exists. We didn't grow from the industrial revolution to the present just by multiplying the number of factories. Most of it actually came from increases in efficiency, especially for countries at the frontier of technology. The idea that a sustainable economy cannot be compatible with growth is false. I share the sentiment that governments need to do more to ensure sustainability and I agree that this means growing at a lower pace, but this is completely different from asserting the incompatibility between growth and sustainability.

What needs to be changed?

Many things need to be changed, but not really anything related to the monetary system. The issue of sustainability is indeed an important one in my view, but the link to the monetary system is tenuous at best.

And with this, forgive me if I retire from commenting on economic conspiracy theories. I don't think this fits with what this blog should be about...

Starting things off in Economics

2010-03-20T19:18:00.001-07:00

Galileu em português

2010-03-17T17:39:00.000-07:00

Sidereus Nuncius foi publicado em 1610 tendo como autor Galileo Galilei e tornou-se um dos muitos vislumbres do Universo novo que estava por vir. Nesse pequeno livro Galileo relata os frutos das suas observações dos Céus ao utilizar uma luneta: Galileo observa a Lua e vê que ela não é uma esfera perfeira como até então se supunha. Galileo vê muitas mais estrelas que as que foram catalogadas. E relata também a descoberta de quatro corpos celestes nas proximidades de Júpiter.

A importância de todas estas observações é que ajudaram a minar os pressupostos e conclusões do edifício da Física Aristotélica e tornaram ainda mais prementes as necessidades de uma nova Física que pudesse explicar o novo Universo que estava para vir.

Agora, com a tradução de Henrique Leitão, poderemos ter o prazer de ler esta pequena pérola em português.

Dangerous knowledge

2010-03-15T15:39:00.000-07:00

I haven't seen this documentary so I can't say anything about it, but the main theme seems to be interesting.

Google please

2010-03-09T14:24:00.003-08:00

No Scientology adds. Seriously!

On the value of a contextual History of Science

2010-03-07T05:42:00.000-08:00

"We shall recognize also that not only a knowledge of the ideas that have been accepted and cultivated by subsequent teachers is necessary for the historical understanding of a science, but also that the rejected and transient thoughts of the inquirers, nay even apparently erroneous notions, may be very important and very instructive. The historical investigation of the development of a science is most needful, lest the principles treasured up in it become a system of half-understood prescripts, or worse, a system of prejudices. Historical investigation ... brings new possibilities before us, by showing that which exists to be in great measure conventional and accidental. From the higher point of view at which different paths of thought converge we may look about us with freer vision and discover routes before unknown."

Ernst Mach, Science of Mechanics

"We've been scooped"

2010-03-06T18:22:00.000-08:00

Two ways to look at the history of science.

2010-03-04T15:20:00.000-08:00

One of the focus of this blog will be the story/history of how the ideas in Physics (allow me to use this anachronistic term) came to be.
We have two ways in which we can do this: either we can look at history with the eyes of the present and judge things accordingly, or we can look into the past with contextual eyes. That is: we should look into old theories, and ideas not with what we know today in our minds but with what was known at the time.

With the first choice the only insight that we get when looking into the past is that everything is poor and ridiculous science, while with the second choice we get to understand how the ideas that we now have came to be and why they were accepted or rejected in the end.

It is pretty obvious how partial I am to this question and I won't make any feeble attempt to disguise it.
Thus, expect me to take the second road at all times: the analysis and criticism that you'll see will be made, with, what may seem to be, a naive point of a view.

A fun way to teach a Theorem

2010-03-01T15:58:00.000-08:00

In Climbing The Mountain I had already talked about the Bolzano-Weierstrass Theorem for bounded sequences but I haven't provided a full proof of the theorem. Due to some laziness on my part I only posted a sketch of a proof and let it like that.

Anyway if the reader really wants to know the full proof of that theorem it isn't hard to find on the web.
As you see there are thousand of hits on the google engine but the most fun proof that you can find on the web in the one given by Steve Sawin on a youtube video. It is a very well known proof of the theorem but the novelty in this case is the way in which Sawin chose to display the proof. He's rapping all the steps of the proof and I think he does an excellent job in explaining why that method works.

Come on you math majors if you want to be free

From Corporate America you listen to me.

You've got a sequence that you built from your approximating tweakins

And you really need to find a convergent subsequence,

So you ask my man Bolzano and his homie Weierstrass,

Who've found you a solution with a trick that's really boss.

Well are you down with that?

We're down with that!

Well are you down with that?

We're down with that!

Well you haven't got much hope unless your sequence is bounded,

So let's say some interval has got your numbers surrounded.

They're all greater than a, they're all less than some b,

And it's right there with those points, that you thoughts have to be.

So to your right is b, and to your left is a,

And in between your sequence tries to wind its way,

An infinity of ex-ens, this interval has in it,

Still you don't know where to look, to try to find the limit.

So you stand in the middle, halfway in between,

a plus b over two, if you know what I mean,

To your left a line segment, half the big one's size,

To your right the other half, in the same way lies.

You see every ex-en lives in the other or the one,

But kid you'll never believe what this division has done,

Because if every ex-en lies in one of these, dude,

Then in one or the other an infinitude!

Well are you down with that?

We’re down with that!

Well are you down with that?

We're down with that!

So you slide to the side where this infinity lies,

To the middle of an interval of half the size,

This new interval (I say it's half as long),

Contains an infinite subsequence if my logic ain't wrong.

Now you do it again, divide the line in two,

And if you payed attention, you'll know just what to do.

You count up all the ex-ens, on the left and right,

There's infinity in one, though the space is getting tight.

So you do this k times, now we're really getting small,

One half to the k, is our interval.

Yet in this little space, within this little bound,

A whole subsequence can still be found.

Well are you down with that?

We're down with that!

Well are you down with that?

We're down with that!

Well you can do this forever, until Tishebuv,

Cuz infinite recursion is the thing that we love.

A chain of nested intervals, each inside the last,

Like little Russian dolls, and they're getting smaller fast.

But what you have to believe, because then we're nearly done,

Is there's exactly one point that lives in every one!

See all those left endpoints, they have to have a supremum,

The same way that the right ones have to have an infimum.

Well this sup and this inf, they live in each of these sets,

So the distance that’s between them is as small as it gets.

They are both the same point, so I say what the hell,

I think that its our limit so let's call it L!

Well are you down with that?

We're down with that!

Well are you down with that?

We're down with that!

Well I promised a subsequence and I never tell a lie,

To distinguish it from ex I'll call this sequence why.

Recall the kayth interval, and all the points in its span,

Well I only need one, that's just how bad I am.

Why-kay's my name for this point, it lives in interval kay,

Which makes it quite close to L, you see I planned it that way!

Now you can pick epsilon as small as it wants to be,

Cuz I've got nested quantifiers and their working for me.

I will come back with an M, so big I'm sure it will do,

Which of your epsilon is one minus the log base two.

Well the thing about M, is that I picked it so good,

That after it the why-kays lie inside of L's `hood,

L's `hood is epsilon sized, so all those intervals lie in it,

QED, you've got a sequence that converges to a limit!

Well are you down with that?

We're down with that!

Well are you down with that?

We're down with that!

Well I am outta here now, because my rap is at its end,

But I’ll leave you with this exercise: to prove it in Are-en!

The only question I have about this is: What the Hell the reference to "Tishebuv" means?!

Current reading

2010-02-26T14:48:00.000-08:00

Apart from the other 300 books I'm reading now this is the one that I'll study with a little bit of more care from now on:

In this book Alexandre Koyré narrates the beautiful story/history that allowed us (by us I mean people in the western hemisphere - since I'm total ignorant of how it was like in other parts of the world) to step from viewing the world around us as a closed, hierarchical, and finite Universe (Cosmos in short) to the view of a world that is infinite and unified.

During my college years I've already read a little bit on it, but now it's time for to really brush up my knowledge. Why?, some of you might ask. The simple answer is because I want to and I find the History of Ideas in Physics to be an extremely fascinating subject.

For those of you that speak English here is the English version of the book:

On top of that I'll also start reviewing some of the stuff I learned during my History of Physics course and will pour them in here.

So, if any of this is your cup of coffee you better stick around.

And don't forget about the new member Palynka. He´s a dismal Science guy and so his take on this issues will certainly be helpful. On top of that if he gives us some commentary about the Economics/Finance world it will only be an added bonus.

Let us hope that this time this blog will really start.

A montanha pariu um rato

2010-02-26T09:15:00.000-08:00

Is this it? Is this moment when the mountain wins? Even Sisyphus would be proud at how far up Armando has managed to climb while pushing such a giant boulder. But the time is not yet right for his Minsky moment, so that's where I come in.

So how can an economist be helpful here? In fact, when was any economist ever helpful? It's not just an accident that people call it the dismal science. What's that? You want a good answer? So do I, so if you can find a good reason, just let me know. In the meantime, I have some exploring to do...

A great interview by Penrose

2010-01-23T15:45:00.000-08:00

The original is here.

One of the greatest thinkers in physics says the human brain—and the universe itself—must function according to some theory we haven't yet discovered.

by Susan Kruglinski; photography by Oliver Chanarin

Roger Penrose could easily be excused for having a big ego. A theorist whose name will be forever linked with such giants as Hawking and Einstein, Penrose has made fundamental contributions to physics, mathematics, and geometry. He reinterpreted general relativity to prove that black holes can form from dying stars. He invented twistor theory—a novel way to look at the structure of space-time—and so led us to a deeper understanding of the nature of gravity. He discovered a remarkable family of geometric forms that came to be known as Penrose tiles. He even moonlighted as a brain researcher, coming up with a provocative theory that consciousness arises from quantum-mechanical processes. And he wrote a series of incredibly readable, best-selling science books to boot.

And yet the 78-year-old Penrose—now an emeritus professor at the Mathematical Institute, University of Oxford—seems to live the humble life of a researcher just getting started in his career. His small office is cramped with the belongings of the six other professors with whom he shares it, and at the end of the day you might find him rushing off to pick up his 9-year-old son from school. With the curiosity of a man still trying to make a name for himself, he cranks away on fundamental, wide-ranging questions: How did the universe begin? Are there higher dimensions of space and time? Does the current front-running theory in theoretical physics, string theory, actually make sense?

Because he has lived a lifetime of complicated calculations, though, Penrose has quite a bit more perspective than the average starting scientist. To get to the bottom of it all, he insists, physicists must force themselves to grapple with the greatest riddle of them all: the relationship between the rules that govern fundamental particles and the rules that govern the big things—like us—that those particles make up. In his powwow with DISCOVER contributing editor Susan Kruglinksi, Penrose did not flinch from questioning the central tenets of modern physics, including string theory and quantum mechanics. Physicists will never come to grips with the grand theories of the universe, Penrose holds, until they see past the blinding distractions of today’s half-baked theories to the deepest layer of the reality in which we live.

You come from a colorful family of overachievers, don’t you?
My older brother is a distinguished theoretical physicist, a fellow of the Royal Society. My younger brother ended up the British chess champion 10 times, a record. My father came from a Quaker family. His father was a professional artist who did portraits—very traditional, a lot of religious subjects. The family was very strict. I don’t think we were even allowed to read novels, certainly not on Sundays. My father was one of four brothers, all of whom were very good artists. One of them became well known in the art world, Sir Roland. He was cofounder of the Institute of Contemporary Arts in London. My father himself was a human geneticist who was recognized for demonstrating that older mothers tend to get more Down syndrome children, but he had lots of scientific interests.

How did your father influence your thinking?
The important thing about my father was that there wasn’t any boundary between his work and what he did for fun. That rubbed off on me. He would make puzzles and toys for his children and grandchildren. He used to have a little shed out back where he cut things from wood with his little pedal saw. I remember he once made a slide rule with about 12 different slides, with various characters that we could combine in complicated ways. Later in his life he spent a lot of time making wooden models that reproduced themselves—what people now refer to as artificial life. These were simple devices that, when linked together, would cause other bits to link together in the same way. He sat in his woodshed and cut these things out of wood in great, huge numbers.

So I assume your father helped spark your discovery of Penrose tiles, repeating shapes that fit together to form a solid surface with pentagonal symmetry.
It was silly in a way. I remember asking him—I was around 9 years old—about whether you could fit regular hexagons together and make it round like a sphere. And he said, “No, no, you can’t do that, but you can do it with pentagons,” which was a surprise to me. He showed me how to make polyhedra, and so I got started on that.

Are Penrose tiles useful or just beautiful?
My interest in the tiles has to do with the idea of a universe controlled by very simple forces, even though we see complications all over the place. The tilings follow conventional rules to make complicated patterns. It was an attempt to see how the complicated could be satisfied by very simple rules that reflect what we see in the world.

The artist M. C. Escher was influenced by your geometric inventions. What was the story there?
In my second year as a graduate student at Cambridge, I attended the International Congress of Mathematicians in Amsterdam. I remember seeing one of the lecturers there I knew quite well, and he had this catalog. On the front of it was the Escher picture Day and Night, the one with birds going in opposite directions. The scenery is nighttime on one side and daytime on the other. I remember being intrigued by this, and I asked him where he got it. He said, “Oh, well, there’s an exhibition you might be interested in of some artist called Escher.” So I went and was very taken by these very weird and wonderful things that I’d never seen anything like. I decided to try and draw some impossible scenes myself and came up with this thing that’s referred to as a tri-bar. It’s a triangle that looks like a three-dimensional object, but actually it’s impossible for it to be three-dimensional. I showed it to my father and he worked out some impossible buildings and things. Then we published an article in the British Journal of Psychology on this stuff and acknowledged Escher.

Escher saw the article and was inspired by it?
He used two things from the article. One was the tri-bar, used in his lithograph called Waterfall. Another was the impossible staircase, which my father had worked on and designed. Escher used it in Ascending and Descending, with monks going round and round the stairs. I met Escher once, and I gave him some tiles that will make a repeating pattern, but not until you’ve got 12 of them fitted together. He did this, and then he wrote to me and asked me how it was done—what was it based on? So I showed him a kind of bird shape that did this, and he incorporated it into what I believe is the last picture he ever produced, called Ghosts.

Is it true that you were bad at math as a kid?
I was unbelievably slow. I lived in Canada for a while, for about six years, during the war. When I was 8, sitting in class, we had to do this mental arithmetic very fast, or what seemed to me very fast. I always got lost. And the teacher, who didn’t like me very much, moved me down a class. There was one rather insightful teacher who decided, after I’d done so badly on these tests, that he would have timeless tests. You could just take as long as you’d like. We all had the same test. I was allowed to take the entire next period to continue, which was a play period. Everyone was always out and enjoying themselves, and I was struggling away to do these tests. And even then sometimes it would stretch into the period beyond that. So I was at least twice as slow as anybody else. Eventually I would do very well. You see, if I could do it that way, I would get very high marks.

You have called the real-world implications of quantum physics nonsensical. What is your objection?
Quantum mechanics is an incredible theory that explains all sorts of things that couldn’t be explained before, starting with the stability of atoms. But when you accept the weirdness of quantum mechanics [in the macro world], you have to give up the idea of space-time as we know it from Einstein. The greatest weirdness here is that it doesn’t make sense. If you follow the rules, you come up with something that just isn’t right.

In quantum mechanics an object can exist in many states at once, which sounds crazy. The quantum description of the world seems completely contrary to the world as we experience it.
It doesn’t make any sense, and there is a simple reason. You see, the mathematics of quantum mechanics has two parts to it. One is the evolution of a quantum system, which is described extremely precisely and accurately by the Schrödinger equation. That equation tells you this: If you know what the state of the system is now, you can calculate what it will be doing 10 minutes from now. However, there is the second part of quantum mechanics—the thing that happens when you want to make a measurement. Instead of getting a single answer, you use the equation to work out the probabilities of certain outcomes. The results don’t say, “This is what the world is doing.” Instead, they just describe the probability of its doing any one thing. The equation should describe the world in a completely deterministic way, but it doesn’t.

Erwin Schrödinger, who created that equation, was considered a genius. Surely he appreciated that conflict.
Schrödinger was as aware of this as anybody. He talks about his hypothetical cat and says, more or less, “Okay, if you believe what my equation says, you must believe that this cat is dead and alive at the same time.” He says, “That’s obviously nonsense, because it’s not like that. Therefore, my equation can’t be right for a cat. So there must be some other factor involved.”

So Schrödinger himself never believed that the cat analogy reflected the nature of reality?
Oh yes, I think he was pointing this out. I mean, look at three of the biggest figures in quantum mechanics, Schrödinger, Einstein, and Paul Dirac. They were all quantum skeptics in a sense. Dirac is the one whom people find most surprising, because he set up the whole foundation, the general framework of quantum mechanics. People think of him as this hard-liner, but he was very cautious in what he said. When he was asked, “What’s the answer to the measurement problem?” his response was, “Quantum mechanics is a provisional theory. Why should I look for an answer in quantum mechanics?” He didn’t believe that it was true. But he didn’t say this out loud much.

Yet the analogy of Schrödinger’s cat is always presented as a strange reality that we have to accept. Doesn’t the concept drive many of today’s ideas about theoretical physics?
That’s right. People don’t want to change the Schrödinger equation, leading them to what’s called the “many worlds” interpretation of quantum mechanics.

That interpretation says that all probabilities are playing out somewhere in parallel universes?
It says OK, the cat is somehow alive and dead at the same time. To look at that cat, you must become a superposition [two states existing at the same time] of you seeing the live cat and you seeing the dead cat. Of course, we don’t seem to experience that, so the physicists have to say, well, somehow your consciousness takes one route or the other route without your knowing it. You’re led to a completely crazy point of view. You’re led into this “many worlds” stuff, which has no relationship to what we actually perceive.

The idea of parallel universes—many worlds—is a very human-centered idea, as if everything has to be understood from the perspective of what we can detect with our five senses.
The trouble is, what can you do with it? Nothing. You want a physical theory that describes the world that we see around us. That’s what physics has always been: Explain what the world that we see does, and why or how it does it. Many worlds quantum mechanics doesn’t do that. Either you accept it and try to make sense of it, which is what a lot of people do, or, like me, you say no—that’s beyond the limits of what quantum mechanics can tell us. Which is, surprisingly, a very uncommon position to take. My own view is that quantum mechanics is not exactly right, and I think there’s a lot of evidence for that. It’s just not direct experimental evidence within the scope of current experiments.

In general, the ideas in theoretical physics seem increasingly fantastical. Take string theory. All that talk about 11 dimensions or our universe’s existing on a giant membrane seems surreal.
You’re absolutely right. And in a certain sense, I blame quantum mechanics, because people say, “Well, quantum mechanics is so nonintuitive; if you believe that, you can believe anything that’s nonintuitive.” But, you see, quantum mechanics has a lot of experimental support, so you’ve got to go along with a lot of it. Whereas string theory has no experimental support.

I understand you are setting out this critique of quantum mechanics in your new book.
The book is called Fashion, Faith and Fantasy in the New Physics of the Universe. Each of those words stands for a major theoretical physics idea. The fashion is string theory; the fantasy has to do with various cosmological schemes, mainly inflationary cosmology [which suggests that the universe inflated exponentially within a small fraction of a second after the Big Bang]. Big fish, those things are. It’s almost sacrilegious to attack them. And the other one, even more sacrilegious, is quantum mechanics at all levels—so that’s the faith. People somehow got the view that you really can’t question it.

A few years ago you suggested that gravity is what separates the classical world from the quantum one. Are there enough people out there putting quantum mechanics to this kind of test?
No, although it’s sort of encouraging that there are people working on it at all. It used to be thought of as a sort of crackpot, fringe activity that people could do when they were old and retired. Well, I am old and retired! But it’s not regarded as a central, as a mainstream activity, which is a shame.

After Newton, and again after Einstein, the way people thought about the world shifted. When the puzzle of quantum mechanics is solved, will there be another revolution in thinking?
It’s hard to make predictions. Ernest Rutherford said his model of the atom [which led to nuclear physics and the atomic bomb] would never be of any use. But yes, I would be pretty sure that it will have a huge influence. There are things like how quantum mechanics could be used in biology. It will eventually make a huge difference, probably in all sorts of unimaginable ways.

In your book The Emperor’s New Mind, you posited that consciousness emerges from quantum physical actions within the cells of the brain. Two decades later, do you stand by that?
In my view the conscious brain does not act according to classical physics. It doesn’t even act according to conventional quantum mechanics. It acts according to a theory we don’t yet have. This is being a bit big-headed, but I think it’s a little bit like William Harvey’s discovery of the circulation of blood. He worked out that it had to circulate, but the veins and arteries just peter out, so how could the blood get through from one to the other? And he said, “Well, it must be tiny little tubes there, and we can’t see them, but they must be there.” Nobody believed it for some time. So I’m still hoping to find something like that—some structure that preserves coherence, because I believe it ought to be there.

When physicists finally understand the core of quantum physics, what do you think the theory will look like
I think it will be beautiful.

Two Solved Exercises

2010-01-01T07:55:00.000-08:00

In my other blog I received a three challenges regarding the solution of two exercises. Since I think the comments section of my blog isn't really the place for me to post a solution I decided to post the resolutions in here.

As a matter of fact I'm kind of opening an exception that I'm quite sure I won't open many times in the future. My other blog isn't about people sending problems for me to solve (neither is this one by the way). It is about me posting the contents of my physics degree online so that other people can see what physics is about when done from the bottom up. But since I'm sure that the person that send me those exercises is perfectly able to solving them himself I'm doing this this time.

Anyway, enough with the explanations and let's get our hands dirty,because that's the only way for us to learn in Physics and Mathematics.

— 1. Exercise —

$ \displaystyle \lim \left( \dfrac{a}{1+|a|} \right)^n $

— 1.1. First Resolution —

$ {\left| \dfrac{a}{1+|a|} \right|= \dfrac{|a|}{|1+|a||}=\dfrac{|a|}{1+|a|} }$

Since $ {1+|a|>0} $ for all values of $ {a} $

Now $ \dfrac{|a|}{1+|a|} < 1 $ since the numerator is less than the denominator. From this it follows that:

$ \displaystyle \lim \left( \dfrac{a}{1+|a|} \right)^n=0 $

— 1.2. Second Resolution —

First let us suppose that it is $ a = 0 $. In this case it is $ { \lim \left( \dfrac{a}{1+|a|} \right)^n=0 }$ trivially.

Now let us suppose that it is $ a>0 $.

$ { \begin{array}{llc} \lim \left( \dfrac{a}{1+|a|}\right)^n &=& \\ \lim \left( \dfrac{1+|a|}{a}\right)^{-n} &=& \\ \lim \left( \dfrac{1+a}{a}\right)^{-n} &=& \\ \lim \left( \dfrac{1}{a}+ 1 \right)^{-n} &=& 0 \end{array} }$

Where the last equality follows from the fact that $ { \dfrac{1}{a} + 1 > 1 }$.

Finally let us suppose that $ a < 0 $ (and so we can write $ a = -|a| $)

$ { \begin{array}{llc} \lim \left( \dfrac{1-a}{a}\right)^{-n} &=& \\ \lim\left( \dfrac{1}{a} -1 \right)^{-n} &=& \\ - \lim \left( \dfrac{1}{|a|} +1 \right)^{-n} &=& 0\end{array} }$

Where the last equality follows from the previous result.

Thus it always is $ {\lim \left( \dfrac{a}{1+|a|} \right)^n = 0 }$

— 2. Exercise —

$ \displaystyle \lim \sqrt[n]{ \dfrac{(3n)!}{(n!)^3} } $

$ { \begin{array}{rcr} \lim \sqrt[n]{ \dfrac{(3n)!}{(n!)^3} } &=& \\ \lim \exp \left[\log \left(\sqrt[n]{ \dfrac{(3n)!}{(n!)^3} }\right) \right] &=& \\ \lim \exp \left[ \dfrac{1}{n}\log \left( \dfrac{(3n)!}{(n!)^3}\right) \right] &=& \\ \lim \exp \left[ \dfrac{1}{n}(\log (3n)!-3 \log n!) \right] &=& \\ \exp \left[ \lim \dfrac{1}{n}(\log (3n)!-3 \log n!) \right] \end{array} }$

Now we'll calculate $ { \lim \dfrac{1}{n}( \log (3n)! -3 \log n! ) }$ using Stirling's approximation and neglecting the $ { O (\log n) }$ terms since $ {\lim O (\log n)/n=0}$

$ { \begin{array}{rcr} \lim \dfrac{1}{n} (\log (3n)! -3 \log n!) &=& \\ \lim \dfrac{1}{n} (3n \log 3n -3n -3 ( n \log n -n ) ) &=& \\ \lim \dfrac{1}{n} (3n \log 3 + 3n \log n -3n -3n \log n +3n) &=& \\ \lim \dfrac{3n \log 3}{n} &=& \\ 3 \log 3 \end{array} }$

Thus $ {\lim \sqrt[n]{ \dfrac{(3n)!}{(n!)^3} } = \exp (3 \log 3 ) = 3^3=27 }$

Polymath Experience

2009-12-31T09:21:00.000-08:00

This blog post is more of an experience but I promise the readers of this blog that I will write a serious post about the polymath experience and how it went.

There's already a lot of stuff on the web about it, but I also want to give my tow cents worth about a project that was a very valuable experience (I wasn't an actor on that project, just a spectator).

So just stay tuned and news will follow soon enough.

Vitaly Ginzburg: a life in physics

2009-11-06T04:45:00.000-08:00

A great physicist provides a great interview. Enjoy.

Vitaly Ginzburg, who turned 93 last month, is one of the most prominent Russian theoretical physicists of the 20th century and shared the 2003 Nobel Prize for Physics with Alexei Abrikosov and Tony Leggett for their work on the theory of superconductors and superfluids. In an exclusive interview with physicsworld.com, Ginzburg talks to Matin Durrani about how his interest in physics developed, why he distrusts the church's growing role in Russian society, and how his role in developing a hydrogen bomb for the Soviet Union saved his life.

Physics World: How did you first become interested in physics?

Vitaly Ginzburg: I was sent to school in 1927 when I was 11. Education was not compulsory at the time, which meant that I started in the fourth form. But in 1931, after I had been at school for only four years, the Soviet Union imposed another round of idiotic school reforms, abolishing all upper secondary schools, which used to be for 16–18-year-olds. It was recommended that all pupils who had been at school for seven years should transfer to the factory school (the school for "working people") and that only then should they try to find a way to enrol in a university.
But I did not like this path and went to work as a technician in the X-ray laboratory of one of the higher-education technical institutes, where I was first introduced to physics. I was especially impressed with a popular-science book by the Russian physicist Orest Danilovich Hvolvson entitled Physics in Our Day. It was then that I decided to become a physicist and I began to familiarize myself with the knowledge necessary for admission to university, which had just been made a competitive process. Not without difficulties, I became a student of the physics department of Moscow State University (MSU) in 1933 on the second course.

Who or what had the biggest influence on you in the way you think about and approach physics?

I think that my biggest achievement in physics is connected with the theory of superconductivity.

Hvolvson's book deserves another mention, though I never actually met the author in person. Then there were university textbooks such as The Fundamentals of the Theory of Electricity by Igor Tamm (the famous course of Lev Landau and Evgeny Lifschitz had not yet been written). I was also influenced by Leonid I Mandelstam, who was a very prominent figure at that time. Now we have many good books on physics and the latest results are described in review journals, such as Uspekhi Fizicheskikh Nauk [Physics–Uspekhi], which has been published since 1918 and of which I have been editor-in-chief for over 11 years.

Looking back on your scientific career, what do you think has been your single biggest achievement in physics, and why?

To me, the special charm and specific feature of theoretical physics is that you can quickly change what you are studying. Typically, you do not need many years to build new equipment, as experimentalists often do. In this spirit, I worked in many areas of physics and astrophysics. The main highlights of my work are described in my recently published book On Superconductivity and Superfluidity (Springer, 2008) in the chapter called "Scientific Autobiography. An attempt." Having said all that, I think that my biggest achievement in physics is connected with the theory of superconductivity.

Do you have any regrets about your part in developing the hydrogen bomb for the Soviet Union?

We thought at the time that we were working to prevent a monopoly on the atomic bomb – Hitler's monopoly if he got the bomb before Stalin. The thought of what would happen if Stalin had a monopoly on atomic weapons somehow never entered my head. Scary thought. Stalin would seek to subjugate the entire world. I admit this may betray stupidity, but this stupidity was, back then, a common way of thinking in the Soviet Union.
I wish to use this opportunity to clarify my input into the creation of the Soviet Union's hydrogen bomb. In 1948 it was already clear that the Soviet Union would acquire nuclear weapons and would be able to compete with the US. In addition, secret sources revealed that the Americans were starting to think about building an H-bomb and so a group of Soviet theorists were instructed to address this issue, although not in a particularly urgent way as the problem was not yet being treated as a high priority. Before 1948 I had no connection with atomic weapons work, but once it had been decided to race the US for the H-bomb, Tamm was brought in and asked to suggest people to join such a team. I was his deputy in the theory department of the Lebedev Physical Institute so, naturally, Tamm proposed including me. However, as I learned many years later, my candidature was initially rejected; it was approved only some time later.

Why were you initially turned down from being part of the H-bomb team?

The main point was that I was not treated as a "reliable person" by the Soviet authorities: for example, my wife was in exile. But when the problem began to be treated seriously, I was included in the team due to the fact that I had, as one would say nowadays, a high scientific rating. Incidentally, Andrei Sakharov was included in the team at the request of our director, Sergey Vavilov, because Sakharov and his wife had a small child but did not have anywhere to live; after joining the group the Sakharovs were assigned a single small room in a communal flat.

One way or another, we began to get acquainted with the available materials and concluded that they offered no real guidance for the design of the H-bomb. At this stage Sakharov and I advanced two ideas that could solve the problem. In his memoirs, Sakharov called these ideas "the first" and "the second", because they were still classified. His "first" idea was to use alternating layers of uranium and fuel, while my "second" was to use lithium-6 as a fuel, which would, when hit by neutrons, create tritium and helium nuclei and liberate 4 MeV of energy per nucleus. I do not think that either of these ideas was terribly bright, but together they made it possible to create the H-bomb.

What happened next?

Once the authorities had decided to build the H-bomb based on these two ideas, the real work was transferred to Arzamas-16 (the city of Sarov) where the main atomic bomb laboratories were situated. But at this stage I was once again refused clearance for the work and stayed in Moscow. I started working on the thermonuclear energy problem but at the end of 1951 I lost clearance for that work too. That was the end of my "top secret work", period. It was a time of terror. Stalin personally signed orders for 40,000 people to be executed by firing squad. I have no idea why I missed one of those execution lists. I think that I was a very good candidate for this fate: my wife was in exile and I was branded by some as Bezrodny Kosmopolit ["homeless, stateless cosmopolitan"]. Of course, my Jewish origin could also be an "argument". Suffice to say that I was expelled from the scientific council of the Lebedev Institute, seemingly in an attempt to make it stronger. But why I was not on the firing squad list at that time I still do not know. Perhaps it was sheer luck. Maybe an unknown somebody had expressed a kind of gratitude for the "second idea". The bottom line is that I am a very rare – and possibly unique – person in that I was saved by the H-bomb!

You have long been a staunch atheist: what are your concerns about the growing influence of the church in Russian public life?

I am, and always have been, an atheist, but I think that to be – or not to be – religious is a fundamental human right. It is, however, a different matter if the church interferes with secular education, offering creationism as a foundation of science (in other words, the idea that God created everything and that everything then evolved obeying laws that were also established by the divine powers). I fully reject this approach and therefore keep seeking to expose creationism as false. This struggle is not easy because church leaders try to equate belief in God with a struggle for high moral values and, among other things, a struggle against alcoholism. It is clear that when the church is fighting alcoholism, or drug abuse, or immoral behaviour, I am certainly not against it. But I am convinced that the bright future of mankind is connected with the progress of science and I believe it is inevitable that one day religions (at least those existing now) will drop in status to no higher than that of astrology. When that will happen is, of course, is difficult to say.

What do you think are the three most important unsolved problems in physics?

Regarding the current state of physics as a whole, it is possible to divide it into two parts: "modern physics" (ordinary physics) and "the physics of the future" (extraordinary physics). Modern physics covers everything that we know and also what we do not yet know (but, in principle, can learn) in close vicinity to the data that can be obtained at the Large Hadron Collider at CERN; this is partly true for cosmology too. I also include condensed-matter physics and plasma physics as part of modern physics, the main task of which is to create substances with special properties that we wish to have. Currently, not all possible combinations can be calculated, let alone implemented. One example of such a task is to create room-temperature superconductors, which appears to be a problem that can be solved but one that remains stubbornly beyond our reach.
As for what I call extraordinary physics, there we encounter problems involving fundamental uncertainties. This includes the question of what types of particles can exist. With regard to cosmology, I hold a perhaps unorthodox view that the complete theory should not have singularities, in particular no energy density singularities. This not fully understood area of cosmology is characterized by the Planck length [1.616 × 10^–35 m] and Planck time [5.4 × 10^–44 s] I once tried to work in this direction but obtained no very significant results – see, for example, my 1971 paper in the Soviet Journal of Experimental and Theoretical Physics (33 242). In general, it is a huge terra incognita.
Of course, I feel it would be logical to include the problems of creation of life and the emergence of logical thinking in "future physics", although it is perhaps "extraordinary biology" rather than physics. It is impossible to know when success will be achieved in "extraordinary physics" but it may happen in the coming years.

Finally: when and where were you happiest (if such a state of mind exists)?

You do feel happier when you produce results. However, I would not really say that one's previous achievements make one happy.

New LateX tool

2009-07-11T14:52:00.000-07:00

After using for some time this script in order to have LateX in my blogger posts I've come across this other script that also allows me to have equations inserted in my posts.

It works in a cleaner way. It is more powerful than the previous script and the displayed equations have a much prettier look.

In conclusion I'll be changing the script I use in order to have LateX in Blogger together with Luca's script.

Let's talk about numbers

2009-07-11T11:26:00.000-07:00

In this post I intend to tell a concise and perfected story of how and why numbers came to be. Please don't expect this to be a factual account of all the trial and tribulations that accompanied the process. So, for instance, I won't dwell in all the misconceptions that plagued negative numbers for a long time, nor will I tell in a detailed way how complex numbers came to be (even though I'm saving this bit for a later post).

Here the story will have no false starts, no dead-ends and everything will flow perfectly and rationally.

— 1. Operations —

Let us first introduce the concept of an operation. For us an operation will be a process that will allows to act on a set of numbers to obtain another number:

Addition, which is represented by the symbol $ {+} $, as an operation that picks up two numbers and obtains a third one. In symbols: $ {a+b=c} $.
Subtraction is defined as the inverse operation to addition and is represented by the symbol $ {-} $. So, we'll say that $ {c-b=a} $ if it is $ {a+b=c} $.
Multiplication is represented by the symbol $ {\times} $ or $ {\cdot} $ and also associates two numbers to a third one: $ {a \times b = c} $.
Division is the inverse operation to multiplication and is represented by the symbol $ {/} $. So we'll say that $ {c/b=a} $ if it is $ { c = a \times b} $. Another way of representing $ {c/b} $ is by the symbol $ {\dfrac{c}{b}} $.

Fortunately for us as human beings these are not the only operations available but for now they are all that we need.

Some readers may be taken aback with the level of abstraction that I used and with the fact that we still don't know what these numbers really are. To those readers I say that in Mathematics this is the normal way to present things. Yes, I'll do things in a more digestible way from now on (cringe worthy to mathematicians I'd assume) but nevertheless I wanted to show how things work out in the real world.

— 2. Numbers —

Well, on to numbers we go. The first numbers that we need are what called the natural numbers. The symbol to denote them is $ { \mathbb{N} } $ and they appeared and evolved from the need to count how many objects of a given class we possess:

$ \displaystyle 0,1,2,3,4,5,6,\ldots $

Where the symbols $ { \ldots } $ mean that that list doesn't have an end. Note that I included the number $ {0} $ in this list. The history of the number $ {0} $ is a fascinating one and I urge the reader to find out a little bit about it.

— 2.1. Operations with Natural numbers —

In order to break down the level of abstraction that we had while introducing mathematical operations we'll now see what they look like for natural numbers.

Addition. The first thing I want to say about addition in the special case of natural numbers is that we can see it like this:
$ \displaystyle m+n=p $

$ {m} $ is the number from where we start and $ {n} $ is the number of steps we'll walk till we stop in point $ {p} $
$ {0} $ is giving no steps at all, $ {1} $ is what we define as giving one step, and the natural numbers that follow are defined in relationship to $ {1} $. For instance: we define $ {2=1+1} $ and so we interpret $ {n+2} $ as giving two steps starting from point $ {n} $. All the natural numbers that follow have the definitions and interpretations that you expect them to have.
Subtraction is the inverse operation to addition. We can see addition, $ {m+n} $ as starting from $ {m} $ and giving $ {n} $ steps to the right, thus $ {m-n} $ can be seen as starting from $ {m} $ and giving $ {n} $ steps to the left.
Multiplication is defined to be consecutive additions. By that we mean that we interpret $ { 2 \times 3} $ as being $ {2+2+2} $ and $ {3 \times 2} $ as being $ {3+3} $. As we can see it is $ {2 \times 3 = 3 \times 2 = 6} $. In general we can say that for every natural number it is $ {m \times n = n \times m = p} $
Division is the inverse operation to multiplication and thus can be seen as consecutive subtractions.
For example let's compute $ {6/2} $.
It is $ {6-2=4} $. Now we take the result and go subtracting $ {2} $ until we can.
$ {4-2=2} $, and $ {2-2=0} $. Since we were allowed to make three subtractions and the end result was $ {0} $ we say that $ {6/2=3} $ with remainder $ {0} $.
Another example is $ {7/3} $. This time it is $ {7-3=4} $, $ {4-3=1} $ and here we have to stop. This time we made two subtractions and ended up with $ {1} $. So what we say is seven divided by three is two with remainder one.

The beautiful thing about numbers and these four operations is that now a whole new realm of nifty and useful things pop out almost for free. All that we need to do is to think and to be ambitious.

Let us analyze with a more watchful eye to the inverse operations and let's see what we can do with them.

— 3. Augmenting the available numbers —

— 3.1. Negative numbers —

First off let s take a look at subtraction and the numbers we already have. The numbers are $ { \mathbb{N} = \left\lbrace 0,1,2,3,4,\ldots \right\rbrace } $. For instance $ {7-2=5} $ and $ {10000-1000=9000} $. This notion of inverse operation to addition seems to work really well. But we don't need to go very far to encounter some possible problems $ {3-7} $. In our current number system we can't go seven steps to the left starting in three. But naturally we'd like for subtraction to work all the time and so we'll take matter into our own hands.

We'll just increase the numbers available to us. We define $ {-1=0-1} $ and $ {-2=-1-1} $, $ {-3=-1-1-1} $, and so on... Now subtraction is always possible. For instance the once problematic $ {3-7} $ just equals $ {-4} $ and the new number system that we end up with is $ { \mathbb{Z}=\ldots,-3,-2,-1,0,1,2,3,\ldots } $.

Besides providing for a way to subtraction being always possible negative numbers can also represent debts, heights below a reference level, etc.

— 3.2. Rational numbers —

After being provided with a whole new set of numbers via subtraction we will now construct a new subset of numbers in order to division always have a meaningful result.

Now the problematic cases are $ { \dfrac{m}{n} } $ with $ {n>m} $. Imagine that you had seven breads to divide for 10 people. How would you do that? If you limited yourself to numbers in $ {\mathbb{Z}} $ you couldn't share the bread. Hence to make division always possible (and remember that this problem certainly doesn't lack practical motivation) we have to further increase the scope of the numbers that are available to us. These new numbers will be called fractional numbers. To the set that is formed by the reunion of the set of fractional numbers and the set of integer numbers ($ { \mathbb{Z} } $) we'll give the name of rational numbers $ { \mathbb{Q} } $.

So for instance imagine that you have one bread and you want to share it between you and two friends. You just break the bread in three equal parts and there you go. The number that represents this process is $ {\frac{1}{3}} $. One of the disconcerting things about fractional numbers is that they allow an infinite number of representations:

$ \displaystyle \frac{1}{3} = \frac{2}{6}= \frac{3}{9} = \ldots $

This problem is circumvented with the notion of equivalent classes, but in a more pedestrian way just think about sharing breads and you'll see why those fractions do have the same value.

— 4. Catching up —

At this point we can be very happy about ourselves. We started off with just the ordinary natural numbers, the addition operation, the notion of inverse operation, some brain power and a lot of ambition.

After deciding what addition meant in the context of the natural numbers we introduced the notion of subtraction as being the inverse operation to addition. Multiplication was defined as being a form of shorthand to repeated additions. Imagine that you had to add fifty times the number twenty. Instead of writing $ {20+20+20+20+\ldots+20} $ one just writes $ {20 \times 50} $ and everything is said.

After defining multiplication in this way once again we wanted to know what the inverse operation would look like. We defined division as being consecutive subtractions and so once again an operation appeared as a short hand for an already known operation.

After having arrived to these four operations (starting from just one) we allowed ourselves to apply them to the natural numbers without any reservations. In that way we came to the conclusion that either the numbers we were working with weren't enough or that we had to put limits in to the applications of the inverse operations. The second option wasn't compatible with our principle of ambition so we had to extend the amount of numbers in our disposal:

$ \displaystyle \mathbb{N} \rightarrow \mathbb{Z} \rightarrow \mathbb{Q} $

— Glimpses of a near future —

In the next post we'll continue in our search of coherence and wider scope of application of inverse operations. Powers and radicals will be introduced and we'll see that the operation of extracting radicals forces us to once again widen the scope of our numbers, and this time not once, but twice!

Aristotelian physics

2009-07-06T11:33:00.000-07:00

As a way to start things up in this blog I decided to recycle one of my old University essays. This essay was done for a discipline that versed on the history of the ideas in Physics. This a subject that is very fond to me because I always like to see how concepts evolve and how people react to that natural evolution.

The essay was written in Portuguese and had four pages. I won't feign definiteness, completeness, nor authority in the topic of Aristotelian (meta)physics but nevertheless it was an essay that gave me a lot of pleasure writing and that can be a very helpful first introduction to some people.

In the course of this blog I intend to write a lot more about this topic (and give references to my sources) and hopefully things will get as far as the Newtonian revolution.

Without further ado:

— A very short introduction to Aristotelian physics —

The sun dwell in the afternoon sky. A man of distinguished appearance was followed by a group of young people. He Gestured and spoke in a tone of completion and gives the lesson as over. All except one young student disperse. The student advances and questions the man:

- It's the first lesson of yours that I watch and I would like to better understand your ideas about the world that surrounds us.

- Well, young man, speak and let us see what we can do. If I can help you I will, if not forgive me... to me what surrounds us is a Cosmos. Something that can be rationally understood and explained in a rational way. It is a hierarchical cosmos in which each being seeks to realize its nature. It is finite and focused on our planet...

- But why our planet? Why not another heavenly body?

- Because it is logical. Everything in the Cosmos seeks to realize its nature. Be sure of this: it will be needed in the following discussion. What is the farthest thing you can see?

- The stars. Every night they are there to illuminate the sky ...

- You said it! "every night"! And how you know that?

- From the old records which are compiled... Nothing ever changes

- Nothing! Are you sure? Think about it.

- ... You're right. Nothing except their positions. They move as a whole in a cyclical way.

- To sum up: since the dawn of times we see the same stars cyclically running the same patterns in the heavens? - the student agreed - Then what will be the more reasonable inference about where they are? What is so perfect that they can always be there to move so regularly?

- As many have said before, they must be embedded in a spherical surface... but why is that the most perfect geometric configuration?

- Firstly, because all its points are the same distance from the center. It has no beginning nor end, the order of the cosmos is the same: it is eternal.

- But why is the Earth the center of the Cosmos ...

- Patience, young one, we'll get there ... the cosmos is hierarchical. But what is its hierarchy? Do you think that we can have mixtures of perfect beings with imperfect beings?

- Of course not!

- You already said that the stars are eternal and unchanging. Are these the attributes of what is perfect or not? In our world there is only corruption, generation, change... Is this perfect? No! What can we conclude about these two worlds?

- That they are radically different ...

- In that case do you think that two worlds so different are made by the same elements?

- No. In no way can our world be be made with the same substance of which the stars are made...

- Until now we only spoke of the stars, but in the heavens there is also the planets. They are also always there with cyclical movements. From what has been said they have to be perfect. Like the stars, the planets are also organized in their celestial spheres ...

- So if I understand, all these spheres are concentric, and the Moon should be the first one. And it marks the difference between our world and the celestial world ..

- Exactly! From the Moon upwards the bodies are composed of a special element called the aether. But what will be the elements of our world? As Empedocle, I say that is composed of four elements: earth, air, fire and water. All these elements are grouped in their places. The first is the sphere of the earth, above it is the water sphere, above it is the air sphere, and the last sphere of the sub-lunar world is the fire one.

- But by the way you describe it the sub-lunar world would be static and not have the attributes we know it has...

- Yes indeed, if it wasn't disturbed. But the movement of celestial bodies causes the fire to fall into the regions of the other spheres and this mixes the elements. And from the mixtures and perturbations change arises in our world. Now let us conclude that the Earth is spherical: when lunar eclipses occur the shadow that the Earth projects on the moon is always curved. From the top of a mountain you can see further than down below. And finally: the fact that the stars we see changes with our location. All this tells us that the Earth is spherical.

- Yes. A little knowledge of geometry suffices to this conclusion being necessary.

- Tell me, how will something that is mostly composed of earth behave when left alone.

- It will move naturally to the center of the sphere of the earth.

- Depending on your location, what would change if you dropped a stone?

- Nothing different would happen. The stone would always fall vertically.

- It always would fall to the center of the earth. But didn't you say earlier that something composed mostly by earth wold seek, when left alone, to meet the center of the sphere of the earth? - The student agreed - So what is your conclusion?

- ...That the two centers coincide. Or in other words: the center of the Earth is the center of the Cosmos!

- Bravo! But we also necessarily conclude that the Earth is at rest. If the Earth were to move the clouds, the stones, would be resting in their spheres. As we see, this doesn't happen so the Earth does not move. We will call this vertical motion that you talked about as the natural movement in the sub-lunar world. I the spheres above the moon one the natural movement is circular or is the result of the composition of circular movements. The movements that are not natural we'll call violent movements. These occur only in the sub-lunar world and need a motor.

- You are right! It just takes look at our world to see that the movement of all things when left alone is vertical: falling rocks, rain falling from the clouds, the fire moving up a bonfire ... But is there some kind of law that governs this movement?

- All we have to do is to think. Imagine two bodies suspended at a same height but with different weights. The heavier one has a greater desire to head towards the center of the earth. As it hits the ground, it will do so with greater speed. Consider now this same body being dropped in the water from the same height. Since water is more dense it opposes greater resistance to movement and thus the body will reach the end of the route with a lower speed.

- That is: the speed of a body, when it moves naturally, is directly proportional to body weight and inversely proportional to the resistance of the medium.

- Bravo! That's right! Or said in an equivalent way: a heavier body dropped from a given height reaches the ground first than a lighter one. And the more resistant the medium, the more time the body needs to fall and when it reaches the ground it does so with a lesser speed.

- So, master, this is where Mathematics enter your explanations?

- No! Mathematics does not enter into my explanations. Only logic does. Mathematics deals with pure forms and in our world form is always associated with matter. The later, as we have seen, is impure. If we want to know about what surrounds us we shouldn't follow the road of Mathematics, but by observing and using or reason. What do we have in the world of pure forms to prevent the existence of vacuum? Nothing! And yet in our world the concept of vacuum is absurd. What would be the speed of a body in a vacuum?

- The vacuum, if it existed, wouldn't offer any resistance to movement ... So the speed of a body would be infinite... This concept is disgusting...

- Thus vacuum does not exist. Space is always associated with matter. When one ends the other also ends.

- Ah! This is why the Cosmos is finite! Since matter ends in the fixed sphere, space has to have an end too! It makes no sense in even talking about beyond the fixed sphere!

- That's right! Unfortunately we have to stop our conversation. It is late and I have much to do. Goodbye! Until a next time!

- Goodbye and thank you master!

The Moon drifted in the starry sky. With a sense of awe the student contemplated the sky. This feeling was not for the show itself, but for the new found understanding.

Luca's script in blogger

2009-05-10T09:16:00.000-07:00

Look at the document source to see how to ~~strike out~~ text, how to use different colors, and how to link to URLs with snapshot preview and how to link to URLs without snapshot preview.

There is a command which is ignored by pdflatex and which defines where to cut the post in the version displayed on the main page

Anything between the conditional declarations ifblog . . . fi is ignored by LaTeX and processed by latex2wp. Anything between iftex . . . fi is processed by LaTex and ignored by latex2wp.

This green sentence appears only in WordPress

This is useful if one, in desperation, wants to put pure HTML commands in the {\emph ifblog . . . fi} scope.

Lemma 1 (Main) Let $$ {\cal F}$$ be a total ramification of a compactifier, then

$$ \displaystyle \forall g \in {\cal F}. g^2 = \eta \ \ \ \ \ (1)$$

The (modifiable) numbering scheme is that lemmas, theorems, propositions, remarks and corollaries share the same counters, while exercises and examples have each their own counter.

Theorem 2 The adèle of a number field is never hyperbolically transfinite.

Proof: Left as an exercise. QED

Exercise 1 Find a counterexample to Theorem 2.

Exercise 2 (Advanced) Prove Lemma 1.

Note that accented characters are allowed. Unfortunately, Erdös's name cannot be properly typeset in HTML. (Note that to get the above approximation, you need to type backslash-H-space-o, rather than backslash-H-{o}. Both are good in LaTeX, but only the second is recognized by LaTeX2WP.)

It is possible to have numbered equations

$$ \displaystyle \frac 1 {x^2} \ge 0 \ \ \ \ \ (2)$$

and unnumbered equations

$$ \displaystyle t(x) - \frac 12 \geq x^{\frac 13} $$

Unnumbered equations can be created with the double-dollar sign command or with the backslash-square bracket command.

$$ \displaystyle f(x) = \int_{-\infty}^{x} \frac 1 {t^2} dt $$

It is possible to refer to equations and theorems via the ref, eqref and label LaTeX commands, for example to Equation (2), to Equation (1), and to Lemma 1 above.

eqnarray* is supported, but not eqnarray:

$$ \displaystyle \begin{array}{rcl} f(x) & < & x^2 - y^2\\ & = & (x+y) \cdot (x-y) \end{array} $$

You can nest a bold text inside an emphasized text or viceversa.

The theorem-like environments theorem, lemma, proposition, remark, corollary, example and exercise are defined, as is the proof environment.

The LaTex commands to type $, %, and & are supported, both in text mode and in math mode:

$$ \displaystyle \$ 30 + 10 \% $$

(Right now WordPress seems to have a problem with & in math mode: $$ {\&}$$.)

It it is possible to have tabular environments, both with borders (the border will not be displayed in the LaTeX preview), as in

blog	quality
what's new	excellent
in theory	poor

and without borders as in

$$ {a}$$	$$ {\rightarrow}$$	$$ {b}$$
$$ {\downarrow}$$		$$ {\uparrow}$$
$$ {c}$$	$$ {\rightarrow}$$	$$ {d}$$

(The tabular environments will be centered in WordPress, but not in the LaTeX preview.)

And it is possible to include a picture so that the pdf file produced with pdflatex imports it from a local image file (which has to be pdf, gif, jpeg, or png) and the WordPress post imports it from a URL.

The image command used to generate the above image has three parameter: a size parameter for either the width or the height, expressed in pixels (if different from the original resolution, the picture will be scaled), a URL for the location of the image (this will be used by WordPress) and a local file name (which will used by pdflatex).

It is possible to have numbered and unnumbered sections and subsections. References to label commands which are not in the scope of a numbered equation or a numbered theorem-like environment will refer to the section number, such as a reference to Section 1 below.

— Examples of Sections —

— And Subsections —

— 1. A section —

— 1.1. And a subsection —

— 2. Changing the style —

The file latex2wpstyle.py contains several definitions that determine the appearance of the WordPress translation. It should be self-explanatory to change the way sections, subsections, proofs and theorem-like environments are typeset, and to change the numbering scheme for theorem-like environments.

The variable $$ {M}$$ in latex2wpstyle.py contains a list of pairs of strings. For every pair, every occurrence of the first string in the document is replaced by an occurrence of the second before proceeding to the conversion from LaTeX to WordPress. If you want to use simple macros (which do not involve parameter-passing) then edit $$ {M}$$ to add support for your own LaTeX macros. (You will have to define the macros in macrosblog.tex as well, otherwise you will not be able to compile your LaTeX file and preview it.)

Some macros are already defined. For example, backslash-E produces an expectation symbol:

$$ \displaystyle \mathop{\mathbb E}_{x \in X} f(x) := \sum_{x\in X} \mathop{\mathbb P} [x] \cdot f(x) $$

Some more macros (see the LaTeX source)

$$ \displaystyle \B, {\mathbb R} , {\mathbb C}, {\mathbb Z}, {\mathbb N} , \epsilon $$

LateX is in the house!!!

2009-05-04T06:52:00.000-07:00

Since I plan to have a lot of Mathematics here I tried to find a solution on blogger that allowed me to post mathematical expressions in a convenient way. I found this page that allowed me to do just that. So my great thanks to him.

Anyway let us try to post a few expressions and see how they look like:

And things seem to be working just fine!

Tricki

2009-04-15T12:56:00.000-07:00

It's Tricky to rock a rhyme, to rock a rhyme that's right on time
It's Tricky...it's Tricky (Tricky) Tricky (Tricky)
It's Tricky to rock a rhyme, to rock a rhyme that's right on time
It's Tricky...Tr-tr-tr-tricky (Tricky) Trrrrrrrrrrricky

Run DMC aside I think that Tricki is a very good idea. But what is Tricki exactly some of you may ask? Well let me copy and paste from the wiki because I think that the person that had the idea for the site can explain a lot better than I could:

" The main body of the Tricki will be a (large, if all goes according to plan) collection of articles about methods for solving mathematical problems. These will be everything from very general problem-solving tips such as, “If you can’t solve the problem, then try to invent an easier problem that sheds light on it,” to much more specific advice such as, “If you want to solve a linear differential equation, you can convert it into a polynomial equation by taking the Fourier transform.” Some articles will be written at a very elementary level and some will be quite advanced, though, obviously enough, the usefulness of an article will increase dramatically if it can be made widely accessible. Some will concern particular areas of mathematics, such as algebraic geometry or probability, whereas others will concern techniques that are relevant to many different areas. "

I think that it is pretty obvious why a site like this can be very helpful to people all around the world that study math. Either as a Mathematics student or as a student of a field that needs Mathematics. Problem solving skills are very important and just to cite a few effort that have been made in shedding some light in this issue I'll link to two relevant sources: the first one is of course the classic of Polya: How to solve it (Wikipedia's page) and you also have this site that besides providing educational material to people also has some very nice help on getting started with LateX.

On a future post I'll address the notion of problem solving skills in the optic of Physicist and so I'll be more interested in the more practical aspects of the question. I won't discuss common proof techniques, but will, instead, talk about some very general ideas I try to follow when solving problems or trying to understand any bit of theoretical knowledge that might be handy to know.

New member

2009-02-20T15:20:00.000-08:00

it is with great pleasure that I announce the fact that I have a collaborator on this blog. He's a good friend and someone who has ideas and the will to discuss them in an open way, so he's more than welcome here.

I think he'll bring a lot of good new perspective to here and certainly the topics will be more diversified, which is something that will benefit this blog.

The gentleman in question goes by the name of Ruben and I'm sure that he'll be a great addition to this place.

What is this all about?

2009-02-12T10:05:00.000-08:00

On my other blog on Wordpress I have a pretty clear and definite plan. On that blog I'll write up my class notes, and other resources, with the goal of re-studying and having a stronger and more united grasp of physics.

Here I'll just post meta-questions. By meta-questions I mean such questions has: "Why is it necessary to build on an axiomatic basis the real numbers?", "Why are we also presenting physics in an axiomatic way?", "What is Quantum-Mechanics and what is my interpretation of it?". And of course a whole lot of other things similar to these ones.

These sort of questions and other issues that I feel that don't go along with the purpose of Climbing the Mountain will end up in here. I may review some book, I may discuss some particular point (or points) of some book I've read, I may post here some wacky reasoning/calculation/neat trick/thought that for some reason or another I find to be interesting. I'll post some interesting questions (to me at least) pertaining some field of knowledge that has some sort of relation with Physics and/or Mathematics. Please expect a true pot pourri of ideas and themes whose only identifying theme will be the fact that they come from the same guy.

Thanks for reading and enjoy the ride.