Category Archives: Autobiographical

Writing a paper in E-prime

Many top scientists communicate clearly, sometimes seemingly effortlessly. The papers by Einstein flow elegantly in clear and logical steps, almost as if choreographed, from one idea to the next. Some articles even have qualities more commonly seen in great works of art, for example, Dirac’s seminal book on quantum mechanics or Shannon’s paper introducing his celebrated entropy. What a pleasure to read! Most physicists similarly recognize Feynman as a master of clear communication.

Before I became a grad student, I had underestimated the importance of good and effective communication. My former PhD advisor, an excellent communicator, taught me the crucial role played by communication in scientific discourse and debate.

Let me explain this point in greater detail. As an illustrative example, imagine if Einstein had not written clearly. Then it may very well have taken much longer for his ideas to percolate and gain acceptance throughout the scientific community. Indeed, Boltzmann, in contrast to Einstein, wrote lengthy and admittedly difficult-to-read texts. Some of his critics perhaps  failed to grasp his seminal ideas. Disappointed and possibly depressed, he eventually committed suicide while still in his prime. Today, the top prize in the field  of statistical physics honors his name— the Boltzmann Medal. Nevertheless, it took many years and the efforts of other scientists (e.g. Gibbs) for the physics community to recognize the full extent of Boltzmann’s contributions.    Clear exposition can make a big difference.

In this blog post, I do not give tips or advice about how to write clearly. Good tips on how to write clearly abound.  Instead, I want to draw your attention to how this article does not contain a single instance of the verb “to be” or any of its conjugations or derived words, such as “being,” “was,” “is,” and so forth — excepting this sentence, obviously. The subset of the English language that remains after the removal of these words goes by the name E-prime, often written E’. In other words, E’ equals English minus all words derived from the above-mentioned verb.

Writing in E’ usually forces a person to think more carefully. Scientists need to communicate not only clearly, but with a slightly higher degree of precision than your typical non-scientist. I have found that fluency in E’ helps me to spot certain kinds of errors of reasoning. The key error of reasoning attenuated by the use of E’ relates to identification.   Too often, the referents of the grammatical subject and object become identified in standard English, where in fact no such identification exists in the real world.  E’ helps to reduce this improper identification, or at least to call attention to it.  The topic of E’, and of related subjects, such as  its ultimate historical origins in general semantics, the study of errors of reasoning, the nature of beliefs, cognitive biases, etc., would require too broad a digression for me to discuss here, so I recommend that interested readers research such topics on their own.

In my early 30s, soon after I obtained tenure in my first faculty position, I decided to write a full article entirely in E’.  What a wonderful and interesting exercise!  Of course, I did not find it easy to write in E’, but with few exceptions, the finished paper contained only E’ sentences.  Forcing myself to think and write in E’ helped me to give a better description of what we, as scientists, really did.  I would cautiously claim that writing in E’ benefited our paper, at least as far as concerns clarity and precision.  No longer do I publish papers in E’, but I learned a lot about how to write (and think) a little bit more clearly.

That paper, about an empirical approach to music, appeared in print in 2004 in the statistical physics journal  Physica A. It eventually ended up cited very well: 33 citations according to  Thomson Reuters’  Web of Science and 60 citations on Google Scholar, as of May 2016.  Most incredibly, it even briefly shot up to the top headline at Nature.com (click here to see)!  We had never expected this.

In that paper, my co-authors and I proposed a method for studying rhythmic complexity. The collaboration team included as first author Heather Jennings, a professor of music (and also my spouse). We took an empirical approach for comparing the rhythmic structures of Javanese Gamelan, Jazz, Hindustani music, Western European classical music, Brazilian popular music (MPB), techno (dance), New Age music, the northeastern Brazilian folk music known as Forró and last but not least: Rock’n Roll. Excepting a few sentences, the paper consists entirely of E’ sentences.

You can read the paper by clicking here for the PDF. A fun exercise: as you read the paper, (1) try to imagine how you would normally rephrase the E’ sentences in ordinary English; (2) try to spot the subtle difference in meaning between the English and E’ sentences.

 

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Are science and religion compatible?

This blog post explores whether or not science and religion are compatible. I use the term religion in the usual sense, to mean a system of faith, worship and sacred rituals or duties. Religions typically consist of an organized code or collection of beliefs related to the origins and purpose of humanity (or subgroups thereof), together with a set of practices  based on those beliefs. Can such belief systems be compatible with science?

Since this topic is controversial, I only reluctantly decided to write about it.  Being a physics professor and a research scientist, I decided not to flee debate on this issue (which is like the third rail of science). Instead, here I detail my thoughts in writing.

Actually, I spent decades trying to reconcile science and (organized) religion, however I made little or no significant progress. Eventually, after much hesitation and discomfort, I was forced to conclude that full reconciliation between science and organized religion may not be possible, even in principle.   Although this realization was initially surprising (and unpleasant) to me, I soon discovered new and more fulfilling ways of approaching issues such as ethics, morals and the purpose or meaning of life, which religion has traditionally monopolized.

1. Short answer: science and religion are incompatible

`Religion is a culture of faith and science is a culture of doubt.’  This statement is usually attributed to Richard Feynman.   Faith and doubt are indeed antagonistic, like water and fire. How can it be possible to fully reconcile religious views, which are based on faith, with the systematic doubt and the skeptical questioning that are intrinsic to the scientific method? Like many scientists, I too have concluded that full reconciliation of science and religion is not possible.

One caveat: obviously, if one removes the element of dogma and faith from religion, then reconciliation might be possible. But religion without dogma is more like a social club than a traditional religion. What would become of Christianity without faith in Jesus Christ? Can you imagine Islam without faith in the Koran?  So, by religion I always mean organized religion, with a set of teachings or dogmas.

Nevertheless, this caveat actually points to a possible way forward  at reconciliation of science and religion: if religions do become more like social clubs and less dogmatic, then disagreement with science can be minimized or even avoided. I see some movement in this direction. There is growing realization, even among people of faith,  that the arbitrary divisions of race, ethnicity and religion, for example, do not have a clear and well-established scientific foundation. In this context, it is admirable that the  new pope of the Catholic Church,  Pope Francis, has defended interfaith dialogue.  He has said, for example, that even atheists can be redeemed. This concession is a major advance, compared to the old threats about burning in hell in eternal damnation!  Moreover, by claiming that he would baptize even Martians, he has (perhaps inadvertently) signaled an openness to the possibility of extraterrestrial intelligence (i.e., aliens). Similarly, his emphasis on raising awareness of climate change is also most encouraging. These are all welcome developments. Other religions have also responded positively to the challenges brought on by science. The Dalai Lama is good example: a Buddhist religious leader who has shown interest in and kept an open mind about science. He has stated that “…if scientific analysis were conclusively to demonstrate certain claims in Buddhism to be false, then we must accept the findings of science and abandon those claims.” Pope Francis, the Dalai Lama and many others like them have contributed positively towards the reconciliation of science and religion. They are forward-thinking and broad-minded religious leaders. Maybe, in some sense, they have more in common with liberal social and political leaders than with the dogmatic defenders of religious orthodoxy and closed-minded conservatism. So I do see a ray of hope. While I welcome the positive change in attitudes brought by religious leaders like Pope Francis and the Dalai Lama, on the other hand the fact remains that their religions are still based on dogma. Religions still have too much dogma, too much superstition and too much bigotry. So, even considering the above caveat, overall I still feel  generally pessimistic about science and religion being compatible.

Below I explore these issues in some detail.

2. Dirac and Feynman on religion

In the list of the all-time greatest physicists, Newton and Einstein invariably take the top positions. Paul A. M. Dirac, of Dirac equation fame, is considered to be an intellectual giant, ranking just a few notches below Einstein or Newton. And Feynman, who usually ranks just below or comparable to Dirac, has rock star status in the physics community.

Feynman had the following to say about religion:

It doesn’t seem to me that this fantastically marvelous universe, this tremendous range of time and space and different kinds of animals, and all the different planets, and all these atoms with all their motions, and so on, all this complicated thing can merely be a stage so that God can watch human beings struggle for good and evil — which is the view that religion has. The stage is too big for the drama.

Dirac had the following to say about religion:

If we are honest — and scientists have to be — we must admit that religion is a jumble of false assertions, with no basis in reality. The very idea of God is a product of the human imagination. It is quite understandable why primitive people, who were so much more exposed to the overpowering forces of nature than we are today, should have personified these forces in fear and trembling. But nowadays, when we understand so many natural processes, we have no need for such solutions. I can’t for the life of me see how the postulate of an Almighty God helps us in any way. What I do see is that this assumption leads to such unproductive questions as why God allows so much misery and injustice, the exploitation of the poor by the rich and all the other horrors He might have prevented. If religion is still being taught, it is by no means because its ideas still convince us, but simply because some of us want to keep the lower classes quiet. Quiet people are much easier to govern than clamorous and dissatisfied ones. They are also much easier to exploit. Religion is a kind of opium that allows a nation to lull itself into wishful dreams and so forget the injustices that are being perpetrated against the people. Hence the close alliance between those two great political forces, the State and the Church. Both need the illusion that a kindly God rewards — in heaven if not on earth — all those who have not risen up against injustice, who have done their duty quietly and uncomplainingly. That is precisely why the honest assertion that God is a mere product of the human imagination is branded as the worst of all mortal sins.

I do not accept arguments from authority. But it is nevertheless interesting to read about what these eminent physicists had to say.

3. Scientists abandon God and religion

Most scientists are non-religious. Many are atheist. A Pew survey from 2009 found that while over 80% of Americans believed in God, less than 50% of scientists believed in God. The percentage was actually 33% in that particular survey. These numbers are typical. For instance, among the members of the US National Academy of Sciences, more than 60% of biological scientists had disbelief in God (i.e., were what most people call `atheists’) according to a study from 1998. In the physical sciences, 79% had disbelief in God.

This issue is relevant in society because most politicians and people in leadership positions are, at least outwardly, sympathetic to religion if not actively religious. So there is at least this one important difference between the majority of scientists and the rest of society. Whereas most people are religious, most scientists are non-believers.

More worrying is that many politicians actively campaign against science and science education. We have all heard about attempts by the religious to eliminate (or water down) the teaching of Darwinian evolution in schools. At least in the West, these attempts have largely failed.

Fortunately, the voting population does not particularly crave a return to the dark ages. It  is easy to understand why. The experience of the last few centuries has shown that social and economic development is only possible when there is political support and commitment to science research and education. Science is responsible for the invention of the Internet, cell phones, radio, TV, cars, trains, airplanes, X-rays, MRIs, the eradication of smallpox, etc.   Rich and socially developed countries are precisely those in which science education and research are well funded. Economic pressures have thus led to investment in science and in science education.

At the same time, science has led to unintended consequences. The more a person is exposed to science, the less religious they are likely to become. (Possibly as a consequence, wealth is also negatively correlated with religiosity. In other words, on average the richer you are, the less religious you are likely to be.)

Especially among those with less science education, there is a fear that exposure to science and to “subversive” ideas such as Darwinian evolution will infect the minds of young people and turn them into “Godless infidels.”  In fact, fear is a constant theme in religion: fear of God, fear of divine punishment, fear of hell, fear of forbidden knowledge, etc.  Science education dispels such fears, and replaces it with the cultivation of curiosity, wonder, questioning, doubt and awe. Since fear is often used as an instrument of control and power, the loss of fear can be a setback for the power structures of organized religion. In this sense, science and science education sometimes directly threaten some religious movements.

Consider, as an example, suicide bombing as a form of jihad by Islamic militant organizations. It is perfectly fair for us to ask: is it even remotely plausible that these hapless suicide bombers correctly understood the scientific method? This is a rhetorical question, of course. A genuinely curious and scientifically literate potential candidate for suicide bombing would immediately ask questions, especially when faced with death by suicide. Is life after death a sure thing? Will Allah really reward a suicide bomber? How it is possible for the big-breasted and hot Houri girls and women to recover their sexual virginity every morning? The young man may then go on to ask: is there a remote possibility, perhaps, that such ridiculous claims are not a sign of pulling the wool over the eyes of the naïve young men in their sexual prime who crave sex and intimacy with women, but who are forbidden by religion to engage in casual sex? And why is recreational and free sex allowed in Paradise but not on earth? A scientifically literate young man would probably say `Thanks but no thanks, I’ll let you go first to set the example!’  Hence the fear and loathing of doubt, curiosity and questioning. Indeed, scientific illiteracy makes people gullible and easier to manipulate.

There is no denying the statistics: exposure to science is correlated with loss of religious faith. This raises two questions: (i) why does this happen and (ii) is this good or bad? I am mostly concerned here with question (i) and only briefly touch upon point (ii).

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Onsager’s solution of the 2-D Ising model: The combinatorial method

1. Introduction

In this blog post, I derive, step by step, the exact partition function for the ferromagnetic Ising model on the square lattice. The result is celebrated as “Onsager’s solution” of the 2-D Ising model. It was originally derived by Lars Onsager in 1942 and published in 1944 in Physical Review [1]. That paper revolutionized the study of phase transitions and what we now call critical phenomena [2].

Somewhat ironically, I first heard about the Ising model when I was working in industry. I was 20 and held a summer job at what was then known as British Telecom Research Labs (BTRL), near Ipswich in the UK. This was before I had ever seen a cell phone or heard of the Internet (although I knew about BITNET and JANET). I worked there in the summer of 1990 and again for a month or so around April 1991. My job at BT involved writing C implementations of multilayer perceptrons and Hopfield neural nets. In those days, BT was interested in implementing hardware neural networks and my boss mentioned casually to me that certain kinds of neural nets are basically just special cases of the Ising model. (Indeed, the Hopfield network is closely related to the Ising spin glass.) Thus began my fascination with the Ising model. Later, in 1994 in Boston, I took a course given by Bill Klein at BU on statistical mechanics, where we went through the solution of the 1-D ferromagnetic Ising model. Still, I never had the chance to study properly the 2-D Ising model. As a PhD student, I would almost daily pass by a poster with a background photo of Lars Onsager (with a cigarette in his hand), hung near the office door of my advisor Gene Stanley, so I was regularly reminded of the 2-D Ising model. I kept telling myself that one day I would eventually learn how Onsager managed to do what seemed to me, at the time, an “impossible calculation.” That was 1994 and I am writing this in 2015!

In what follows, I solve the Ising model on the infinite square lattice, but I do not actually follow Onsager’s original argument. There are in fact several different ways of arriving at Onsager’s expression [3–9]. The method I use below is known as the combinatorial method and was developed by van der Waerden, Kac and Ward among others and relies essentially on counting certain kinds of closed graphs (see refs. [3,10–13]). I more or less follow Feynman [3] and I have also relied on the initial portions of ref. [13].

2. The 2-D Ising model

Consider a two dimensional lattice {\Bbb Z^2} where at each point of the lattice is located a (somewhat idealized) spin-{\tfrac{1}{2}} particle. Consider a finite subset of this lattice, of size {L\times L} and let {i,j=1,2,\ldots L^2}. Let {\cal N} denote the set of pairs of integers {(i,j)} such that spins {\sigma_i} and {\sigma_j} are nearest neighbors. In the ferromagnetic 2-D Ising model with nearest neighbor interactions, spins {i} and {j} interact if and only if {(i,j)\in \cal N}. Each spin {\sigma_i} can assume only 2 values: {\sigma=\pm 1}.

Consider a system of {N=L^2} spins. The Hamiltonian for a spin configuration {\sigma=(\sigma_1,\sigma_2,\ldots \sigma_N)} is given by

\displaystyle H(\sigma)= -J \displaystyle\sum_{i,j\in \cal N} \sigma_i \sigma_j ~. \ \ \ \ \ (1)

The sum over the nearest neighbors should avoid double counting, so that {(i,j)} and {(j,i)} are not counted separately. Without loss of generality, we will assume {J=1} for simplicity.

3. The canonical partition function

In the theory of equilibrium statistical mechanics, the canonical partition function contains all the information needed to recover the thermodynamic properties of a system with fixed number of particles, immersed in a heat bath, details of which can be found in any textbook on statistical mechanics [3-6,14].

I prefer to define the partition function as the two-sided Laplace transform of the degeneracy {\Omega(E)} of the energy level {E}. But traditionally, the partition function is defined as a sum or integral over all possible states of the system:

\displaystyle Z(\beta) = \displaystyle\sum _{\sigma} e^{-\beta H(\sigma)} ~. \ \ \ \ \ (2)

The two ways of thinking are equivalent. The Laplace transform variable {\beta} is related to the thermodynamic temperature {T} via {\beta =1/k_B T}, where {k_B} is the Boltzmann constant. What follows is the exact calculation of {Z(\beta)}.

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Lévy flights of wandering albatrosses: fact or fiction?

A very recent and remarkable study appears to overturn the conventional wisdom about how wandering albatrosses move when they search for food. A new paper published in the Proceedings of the National Academy of Sciences USA (PNAS) by Nicolas E. Humphries et al. looks at whether or not the idea of Lévy flight foraging is supported by empirical data for two species of albatrosses: the wandering albatross and the black-browed albatross. A Lévy flight is a type of random walk in which the zig-zag patterns contain rare but extra long jumps.  In mathematical terms, Lévy flights contain a power law tailed probability density function of jump lengths. The latest results add a new twist to the controversy surrounding wandering albatrosses.

A little background: More than a decade ago in 1994, as a graduate student of Professor Gene Stanley at Boston University, I analyzed flight data of wandering albatrosses.  Our motivation was to try to gain insight into nebulous concepts such as the “free will” of animals.  If  free will really exists and confers real advantages, then it ought to operate in complex and sophisticated behaviors such as foraging.  At least, that was our thinking.  The research progressed rapidly and, in 1996, we published in Nature a paper titled Lévy flight search patterns of wandering albatrosses in which we reported what we thought was evidence of Lévy flights.  A year later, in 1997, I returned to Brazil.  In Natal and later in Maceió, I continued to investigate this phenomenon.  In 1998, my collaborators and I made some unexpected discoveries about previously unknown properties of Lévy flights.  In 1999, Sergey Buldyrev, Shlomo Havlin, Marcos da Luz, Ernesto Raposo, Gene Stanley and I published these results in another paper in Nature.  We showed analytically  that Lévy flights can optimize random searches under conditions of scarcity when the targets are revisitable and randomly located.  (Actually we also assumed some other simplifying conditions, such as negligible learning and memory, but it turns out that these issues do not significantly alter the main findings of the 1999 paper.)

The main reason why Lévy flights increase efficiency is that they can
reduce the expensive habit (known as “oversampling”) of Brownian random walkers to revisit previously visited sites.  They also increase, relative to ballistic motion, the chance of reaching nearby targets at the cost of reaching far away targets, thereby reducing overall traveled distances.  These and many other issues are discussed in our new book, The Physics of Foraging.

Over the next 5 years or so, a growing number of studies added weight to the Lévy flight foraging hypothesis.  Naturally, so did attempts to overturn or falsify the hypothesis.  As Karl Popper explained, scientists do not just verify hypotheses, rather they work to falsify them. This is in large part how science advances.   Supplanting old theories with new and better theories is how our understanding grows and improves.

The Lévy flight foraging hypothesis exploded into controversy in 2007, when another  paper published in Nature by A. M. Edwards et al.  questioned the validity of the hypothesis.  The hypothesis holds that animals should have evolved to move in a superdiffusive Lévy flight type of behavior under conditions of scarcity, because such behavior improves search efficiencies and encounter rates according to mathematical predictions.  Early papers from the 1980s and early 1990s looked at Lévy walks and similar behavior of microorganisms. But it was only in 1996 that the idea of Lévy flight foraging became a hot topic, when we reported evidence  (apparently) showing that Diomedea exulans, the wandering albatross, performs Lévy flight search patterns.   But there was a problem with the data. In 2007, in a paper authored by A. M. Edwards and many others,  we reported evidence which corrected the 1996 results.

Journalists had a field day.  For example, it was claimed by a journalist (Alexandra Witze of Science News) that the 1996 albatross results had been ‘debunked.’   After 2007  few people thought that albatrosses used Lévy flight patterns to forage.

It is in this context that the results reported in this new PNAS paper are simply extraordinary.  The crucial point is that this study directly contradicts the 2007 paper of Edwards et al.  How could this be?  The authors provide an explanation for the inconsistency: Whereas the 2007 study only looked at pooled data sets, the present paper looks at individual birds.  If I understood the paper correctly, the authors have re-tested the 2007 data and found that the behavior of individual birds can be strikingly different from the average behavior.  What I understood was that individual birds do not always use the Lévy behavior, so when you average the data for many birds, the signal-to-noise ratio degrades and the power law tail is less clear.  The authors also reconfirm what was suspected all along, viz., that the 2007 paper did not explicitly take into account power law truncation. But the bigger effect seems to be the pooling.

Maybe albatrosses do in fact use Lévy flight foraging patterns after all.