Field of Science

Bridging the gaps: Einstein on education

This is my latest column for 3 Quarks Daily.

The crossing of disciplinary boundaries in science has brought with it a peculiar and ironic contradiction. On one hand, fields like computational biology, medical informatics and nuclear astrophysics have encouraged cross-pollination between disciplines and required the biologist to learn programming, the computer scientist to learn biology and the doctor to know statistics. On the other hand, increasing specialization has actually shored up the silos between these territories because each territory has become so dense with its own facts and ideas.

We are now supposed to be generalists, but we are generalists only in a collective sense. In an organization like a biotechnology company for instance, while the organization itself chugs along on the track of interdisciplinary understanding across departments like chemistry, biophysics and clinical investigations, the effort required for understanding all the nuts and bolts of each discipline has meant that individual scientists now have neither the time nor the inclination to actually drill down into whatever their colleagues are doing. They appreciate the importance of various fields of inquiry, but only as reservoirs into which they pipe their results, which then get piped into other reservoirs. In a metaphor evoked in a different context - the collective alienation that technology has brought upon us - by the philosopher Sherry Turkle, we are ‘alone together’.

The need to bridge disciplinary boundaries without getting tangled in the web of your own specialization has raised new challenges for education. How do we train the men and women who will stake out new frontiers tomorrow in the study of the brain, the early universe, gender studies or artificial intelligence? As old-fashioned as it sounds, to me the solution seems to go back to the age-old tradition of a classical liberal education which lays emphasis more on general thinking and skills rather than merely the acquisition of diverse specialized knowledge and techniques. In my ideal scenario, this education would emphasize a good grounding in mathematics, philosophy (including philosophy of science), basic computational thinking and statistics and literature as primary goals, with an appreciation of the rudiments of evolution and psychology or neuroscience as preferred secondary goals.

This kind of thinking was on my mind as I happened to read a piece on education and training written by a man who was generally known to have thought-provoking ideas on a variety of subjects. If there was one distinguishing characteristic in Albert Einstein, it was the quality of rebellion. In his early days Einstein rebelled against the rigid education and rules of the German Gymnasium system. In his young and middle years he rebelled against the traditional scientific wisdom of the day, leading to his revolutionary contributions to relativity and quantum theory. In his old age he rebelled against both an increasingly jingoistic world as well as against the mainstream scientific establishment.

Not surprisingly, then, Einstein had some original and bold thoughts on what an education should be like. He held forth on some of these in an address on October 15, 1931 delivered at the State University of New York at Albany. 1931 was a good year to discuss these issues. The US stock market had crashed two years before, leading to the Great Depression and mass unemployment. And while Hitler had not become chancellor and dictator yet, he would do so only two years later; the rise of fascism in Europe was already evident.

Some of these issues must have been on Einstein’s mind as he first emphasized what he had already learnt from his own bitter Gymnasium experience, the erosion of individuality in the face of a system of mass education, similar to what was happening to the erosion of individuality in the face of authoritarian ideas.

“Sometimes one sees in the school simply the instrument for transferring a certain maximum quantity of knowledge to the growing generation. But that’s not right. Knowledge is dead; the school, however, serves the living. It should develop in the young individuals those equalities and capabilities which are of value for the welfare of the commonwealth. But that does not mean that individuality should be destroyed and the individual becomes a mere tool of the community, like a bee or an ant. For a community of standardized individuals without personal originality and personal aims would be a poor community without possibilities for development. On the contrary, the aim must be the training of independently thinking and acting individuals, who, however, see in the service of the community their highest life problem…To me the worst thing seems to be for a school principally to work with methods of fear, force, and artificial authority. Such treatment destroys the sound sentiments, the sincerity, and the self-confidence of the pupil. It produces the submissive subject. It is not so hard to keep the school free from the worst of all evils. Give into the power of the teacher the fewest possible coercive measures, so that the only source of the pupil’s respect for the teacher is the human and intellectual qualities of the latter.”

Einstein also talks about what we can learn from Darwin’s theory. In 1931 eugenics was still quite popular, and Darwin’s ideas were seen even by many social progressives as essentially advocating the ruthless culling of ‘inferior’ individuals and the perpetuation of superior ones. Where Einstein came from, this kind of thinking was on flagrant display right on the doorstep, even if it hadn’t already morphed into the unspeakable horror that it did a decade later. Einstein clearly rejects this warlike philosophy and encourages cooperation over competition. Both cooperation and competition are important for human progress, but the times clearly demanded that one not forget the former.

“Darwin’s theory of the struggle for existence and the selectivity connected with it has by many people been cited as authorization of the encouragement of the spirit of competition. Some people also in such a way have tried to prove pseudo-scientifically the necessity of the destructive economic struggle of competition between individuals. But this is wrong, because man owes his strength in the struggle for existence to the fact that he is a socially living animal. As little as a battle between single ants of an ant hill is essential for survival, just so little is this the case with the individual members of a human community…Therefore, one should guard against preaching to the young man success in the customary sense as the aim of life. For a successful man is he who receives a great deal from his fellow men, usually incomparably more than corresponds to his service to them. The value of a man, however, should be seen in what he gives and not what he is able to receive.”

In other words, with malice toward none, with charity toward all.

And what about the teachers themselves? What kinds of characters need to populate the kind of school which imparts a liberal and charitable education? Certainly not the benevolent dictators that filled up German schools in Einstein’s time or which still hold court in many schools across the world which emphasize personal authority over actual teaching.

“What can be done that this spirit be gained in the school? For this there is just as little a universal remedy as there is for an individual to remain well. But there are certain necessary conditions which can be met. First, teachers should grow up in such schools. Second, the teacher should be given extensive liberty in the selection of the material to be taught and the methods of teaching employed by him. For it is true also of him that pleasure in the shaping of his work is killed by force and exterior pressure.”

If Einstein’s words have indeed been accurately transcribed, it is interesting to hear him use the words “grow up” rather than just “grow” applied to teachers. I have myself come across stentorian autocrats who inadvertently reminded students that their charges were in fact the adults in the room. They definitely need to grow up. Flexibility in the selection of the teaching material is a different matter. To do this it’s not just important to offer as many electives as possible, but it’s more important to give teachers a wide berth within their own classes rather than constantly being required to subscribe to a strictly defined curriculum. Some of the best teachers I had were ones who spent most of their time on material other than what was required. They might wax philosophical about the bigger picture, they might tell us stories from the history of science, and one of them even took us out for walks where the topics of discussion consisted of everything except what he was ‘supposed’ to teach. It is this kind of flexibility in teaching that imparts the most enriching experience, but it’s important for the institution to support it.
What about the distinction between natural science and the humanities? Germany already had a fine tradition in imparting a classical education steeped in Latin and Greek, mathematics and natural science, so not surprisingly Einstein was on the right side of the debate when it came to acquiring a balanced education.

“If a young man has trained his muscles and physical endurance by gymnastics and walking, then he will later be fitted for every physical work. This is also analogous to the training of the mental and the exercising of the mental and manual skill. Thus the wit was not wrong who defined education in this way: “Education is that which remains, if one has forgotten everything he has learned in school.” For this reason I am not at all anxious to take sides in the struggle between the followers of the classical philologic-historical education and the education more devoted to natural science.”

The icing on this cake really is Einstein’s views on the emphasis on general ability rather than specialized knowledge, a distinction which is more important than ever in our age of narrow specialization.

“I want to oppose the idea that the school has to teach directly that special knowledge and those accomplishments which one has to use later directly in life. The demands of life are much too manifold to let such a specialized training in school appear possible. Apart from that, it seems to me, moreover, objectionable to treat the individual like a dead tool. The school should always have as its aim that the young man leave it as a harmonious personality, not as a specialist. This in my opinion is true in a certain sense even for technical schools, whose students will devote themselves to a quite definite profession. The development of general ability for independent thinking and judgement should always be placed foremost, not the acquisition of special knowledge. If a person masters the fundamentals of his subject and has learned to think and work independently, he will surely find his way and besides will better be able to adapt himself to progress and changes than the person whose training principally consists in the acquiring the detailed knowledge.”

One might argue that it’s the failure to let young people leave college as ‘harmonious personalities’ rather than problem-solvers that leads to a nation of technocrats and operational specialists of the kind that got the United States in the morass of Vietnam, for instance. A purely problem-solving outlook might enable a young person to get a job sooner and solve narrowly defined problems, but it will not lead them to look at the big picture and truly contribute to a productive and progressive society.

I find Einstein’s words relevant today because the world of 2018 in some sense resembles the world of 1931. Just like it did because of the Great Depression then, mass unemployment because of artificial intelligence and automation is a problem looming on the short horizon. Just like it had in 1931, authoritarian thinking seems to have taken root in many of the world’s governments. The specialization of disciplines has led colleges and universities to increasingly specialize their own curricula, so that it is now possible for many students to get through college without acquiring even the rudiments of a liberal arts education. C. P. Snow’s ‘Two Cultures’ paradoxically have become more entrenched, even as the Internet presumably promised to break down barriers between them. Meanwhile, political dialogue and people's very world-views across the political spectrum have gotten so polarized on college campuses that certain ideas are now being rejected as biased, not based on their own merits but on some of their human associations.

These problems are all challenging and require serious thinking and intervention. There are no easy solutions to them, but based on Einstein’s words, our best bet would be to inculcate a generation of men and women and institutional structures that promote flexible thinking, dialogue and cooperation, and an open mind. We owe at least that much to ourselves as a supposedly enlightened species.

Who's the greatest physicist in American history?

A photo of an impish Richard Feynman playing the bongos appears in Ray Monk's sweeping biography of Robert Oppenheimer. It is accompanied by the caption "Richard Feynman, Julian Schwinger's main rival for the title of greatest American physicist in history". That got me thinking; who is the greatest American physicist in history? What would your choice be?

The question is interesting because it's not as simple as asking who's the "greatest physicist in history". The answer to that question tends to usually settle on Isaac Newton or Albert Einstein; in fact few American physicists if any would show up on the top ten list of greatest physicists ever. But limit the question to American physicists and the matter becomes more complicated. Contrast this to asking who's the greatest American chemist in history; there the answer - Linus Pauling - appears much more unambiguous and widely agreed upon.

Any discussion of "greatest scientist" is always harder than it sounds. By what measure do you judge greatness?: A single, monumental discovery? Contributions to diverse fields? Theory or experiment? Creation of an influential school of physics? Or by looking at lifetime achievement which, rather than focusing on one fundamental discovery, involves many important ones? There are contenders for "greatest American physicist" who encompass all these metrics of achievement.

Here's what's concerning: Even a generous, expansive list of contenders for "greatest American physicist" in history is embarrassingly thin compared to a comparable list of European physicists. For instance, let's consider the last three hundred years or so and think up a selection which includes both Nobel Laureates and non-Nobel Laureates. The condition is to only include American-born physicists.

Here's my personal list for the title of greatest American physicist in history, in no particular order: Joseph Henry, Josiah Willard Gibbs, Albert Michelson, Robert Millikan, Robert Oppenheimer, Richard Feynman, Murray Gell-Mann, Julian Schwinger, Ernest Lawrence, Edward Witten, John Bardeen, John Slater, John Wheeler and Steven Weinberg. I am sure I am leaving someone out but I suspect other lists would be similar in length. It's pretty obvious that this list pales in comparison with an equivalent list of European physicists which would include names like Einstein, Dirac, Rutherford, Bohr, Pauli and Heisenberg; and this is just if we include twentieth-century physicists. Not only are the European physicists greater in number but their ideas are also more foundational; as brilliant as the American physicists are, almost none of them made a contribution comparable in importance to the exclusion principle or general relativity.

Note that I said "almost none". If you ask who's my personal favorite for "greatest American physicist in history", it would not be Feynman or Schwinger or Witten; instead it would be Josiah Willard Gibbs, a man who seems destined to remain one of the most underappreciated scientists of all time but who Einstein called "the greatest mind in American history". Feynman and Schwinger may have invented quantum electrodynamics, but Gibbs invented the foundations of thermodynamics and statistical mechanics, a truly seminal contribution that was key to the development of both physics and chemistry. 

It's hard to overestimate the importance of concepts like free energy, chemical potential, enthalpy and the phase rule for physics, chemistry, biology, engineering and everything in between. In fact, so influential was Gibbs's work that it inspired that of Paul Samuelson - who unlike physicists, is actually agreed upon as the greatest American economist in history. If you really want to discuss lists of great American physicists (or scientists in general), you simply cannot exclude Gibbs. In my dictionary Gibbs's contributions are comparable to that of any famous relativist or atomic physicist. Unfortunately Gibbs also remains one of the most little known scientists in America, largely because of his introverted nature and tendency to publish groundbreaking papers in journals like the Proceedings of the Connecticut Academy of Sciences.

More importantly though, the sparse list of great homegrown American physicists makes two things clear. Firstly, that America is truly a land of immigrants; it's only by including foreign-born physicists like Fermi, Bethe, Einstein, Chandrasekhar, Wigner, Yang and Ulam can the list of American physicists start to compete with the European list. Secondly and even more importantly, the selection demonstrates that even in 2018, physics in America is a very young science compared to European physics. Consider that even into the 1920s or so, the Physical Review which is now regarded as the top physics journal in the world was considered a backwater publication, if not a joke in Europe (Rhodes, 1987). Until the 1930s American physicists had to go to Cambridge, Gottingen and Copenhagen to study at the frontiers of physics. It was only in the 30s that, partly due to heavy investment in science by both private foundations and the government and partly due to the immigration of European physicists from totalitarian countries, American physics started on the road to the preeminence that it enjoys today. Thus as far as cutting-edge physics goes, America is not even a hundred years old. The Europeans had a head start of three hundred years; no wonder their physicists feature in top ten lists. And considering the very short time that this country has enjoyed at the forefront of science, we have to admit that America has done pretty well.

The embarrassingly thin list of famous American physicists is good news. It means that the greatest American physicist is yet to be born. Now that's an event we can all look forward to.

Why the world needs more Leo Szilards

The body of men and women who built the atomic bomb was vast, diverse, talented and multitudinous. Every conceivable kind of professional - from theoretical physics to plumber - worked on the Manhattan Project for three years over an enterprise that spread across the country and equaled the US automobile industry in its marshaling of resources like metals and electricity.

The project may have been the product of this sprawling hive mind, but one man saw both the essence and the implications of the bomb, in both science and politics, long before anyone else. Stepping off the curb at a traffic light across from the British Museum in London in 1933, Leo Szilard saw the true nature and the consequences of the chain reaction six years before reality breathed heft and energy into its abstract soul. In one sense though, this remarkable propensity for seeing into the future was business as usual for the Hungarian scientist. Born into a Europe that was rapidly crumbling in the face of onslaughts of fascism even as it was being elevated by revolutionary discoveries in science, Szilard grasped early in his youth both a world split apart by totalitarian regimes and the necessity of international cooperation engendered by the rapidly developing abilities of humankind to destroy itself with science. During his later years Szilard once told an audience, "Physics and politics were my two great interests". Throughout his life he would try to forge the essential partnership between the two which he thought was necessary to save the human species from annihilation.

A few years ago Bill Lanouette brought out a new, revised edition of his authoritative, sensitive and outstanding biography of Szilard. It is essential reading for those who want to understand the nature of science, both as an abstract flight into the deep secrets of nature and a practical tool that can be wielded for humanity's salvation and destruction. As I read the book and pondered Szilard's life I realized that the twentieth century Hungarian would have been right at home in the twenty-first. More than anything else, what makes Szilard remarkable is how prophetically his visions have played out since his death in 1962, all the way to the year 2014. But Szilard was also the quintessential example of a multifaceted individual. If you look at the essential events of the man's life you can see several Szilards, each of whom holds great relevance for the modern world.
There's of course Leo Szilard the brilliant physicist. 

Where he came from precocious ability was commonplace. Szilard belonged to the crop of men known as the "Martians" - scientists whose intellectual powers were off scale - who played key roles in European and American science during the mid-twentieth century. On a strict scientific basis Szilard was not as accomplished as his fellow Martians John von Neumann and Eugene Wigner but that is probably because he found a higher calling in his life. However he certainly did not lack originality. As a graduate student in Berlin - where he hobnobbed with the likes of Einstein and von Laue - Szilard came up with a novel way to consolidate the two microscopic and macroscopic aspects of the science of heat, now called statistical mechanics and thermodynamics. He also wrote a paper connecting entropy and energy to information, predating Claude Shannon's seminal creation of information theory by three decades. In another prescient paper he set forth the principle of the cyclotron, a device which was to secure a Nobel Prize for its recognized inventor - physicist Ernest Lawrence - more than a decade later.

Later during the 1930s, after he was done campaigning on behalf of expelled Jewish scientists and saw visions of neutrons branching out and releasing prodigious amounts of energy, Szilard helped perform some of the earliest experiments in the United States investigating fission, publishing key papers with Enrico Fermi and Walter Zinn in 1939. And while he famously disdained getting his hands dirty, he played a key role in helping Fermi set up the world's first nuclear reactor. As the scientists celebrated the historic moment with a bottle of Chianti, Szilard seems to have stood on the balcony and said, "This will go down as a dark chapter in the history of humanity". Once again he saw the Faustian bargain that the scientists were making with fate.

Szilard as scientist also drives home the importance of interdisciplinary research, a fact which hardly deserves explication in today's scientific world where researchers from one discipline routinely team up with those from others and cross interdisciplinary boundaries with impunity. After the war Szilard became truly interdisciplinary when he left physics for biology and inspired some of the earliest founders of molecular biology, including Jacques Monod, James Watson and Max Delbruck. His reason for leaving physics for biology should be taken to heart by young researchers - he said that while physics was a relatively mature science, biology was a young science where even low hanging fruits were ripe for the picking.

Szilard was not only a notable theoretical scientist but he also had another strong streak, one which has helped so many scientists put their supposedly rarefied knowledge to practical use - that of scientific entrepreneur. His early training had been in chemical engineering, and during his days in Berlin he famously patented an electromagnetic refrigerator with his friend and colleague Albert Einstein; by alerting Einstein to the tragic accidents caused by leakage in mechanical refrigerators, he helped the former technically savvy patent clerk put his knowledge of engineering to good use (as another indication of how underappreciated Szilard remains, the Wikipedia entry on the device is called the "Einstein refrigerator"). Szilard was also finely attuned to the patent system, filing a patent for the nuclear chain reaction with the British Admiralty in 1934 before anyone had an inkling what element would make it work, as well as a later patent for a nuclear reactor with Fermi.

He also excelled at what we today called networking; his networking skills were on full display for instance when he secured rare, impurity-free graphite from a commercial supplier as a moderator in Fermi's nuclear reactor; in fact the failure of German scientists to secure such pure graphite and the subsequent inability of the contaminated graphite to sustain fission damaged their belief in the viability of a chain reaction and held them back. Szilard's networking abilities were also evident in his connections with prominent financiers and bankers who he constantly tried to conscript in supporting his scientific and political adventures; in attaining his goals he would not hesitate to write any letter, ring any doorbell, ask for any amount of money, travel to any land and generally try to use all means at his disposal to secure support from the right authorities. In his case the "right authorities" ranged, at various times in his life, from top scientists to bankers to a Secretary of State (James Byrnes), a President of the United States (FDR) and a Premier of the Soviet Union (Nikita Khrushchev).

I am convinced that had Szilard been alive today, his abilities to jump across disciplinary boundaries, his taste for exploiting the practical benefits of his knowledge and his savvy public relations skills would have made him feel as much at home in the world of Boston or San Francisco venture capitalism as in the ivory tower.

If Szilard had accomplished his scientific milestones and nothing more he would already have been a notable name in twentieth century science. But more than almost any other scientist of his time Szilard was also imbued with an intense desire to engage himself politically - "save the world" as he put it - from an early age. Among other scientists of his time, only Niels Bohr probably came closest to exhibiting the same kind of genuine and passionate concern for the social consequences of science that Szilard did. This was Leo Szilard the political activist. Even in his teens, when the Great War had not even broken out, he could see how the geopolitical landscape of Europe would change, how Russia would "lose" even if it won the war. When Hitler came to power in 1933 and others were not yet taking him seriously Szilard was one of the few scientists who foresaw the horrific legacy that this madman would bequeath Europe. This realization was what prompted him to help Jewish scientists find jobs in the UK, at about the same time that he also had his prophetic vision at the traffic light.

It was during the war that Szilard's striking role as conscientious political advocate became clear. He famously alerted Einstein to the implications of fission - at this point in time (July 1939) Szilard and his fellow Hungarian expatriates were probably the only scientists who clearly saw the danger - and helped Einstein draft the now iconic letter to President Roosevelt. Einstein's name remains attached to the letter, Szilard's is often sidelined; a recent article about the letter from the Institute for Advanced study on my Facebook mentioned the former but not the latter. Without Szilard the bomb would have certainly been built, but the letter may never have been written and the beginnings of fission research in the US may have been delayed. 

When he was invited to join the Manhattan Project Szilard snubbed the invitation, declaring that anyone who went to Los Alamos would go crazy. He did remain connected to the project through the Met Lab in Chicago, however. In the process he drove Manhattan Project security up the wall through his rejection of compartmentalization; throughout his life Szilard had been - in the words of the biologist Jacques Monod - "as generous with his ideas as a Maori chief with his wives" and he favored open and honest scientific inquiry. At one point General Groves who was the head of the project even wrote a letter to Secretary of War Henry Stimson asking the secretary to consider incarcerating Szilard; Stimson who was a wise and humane man - he later took ancient and sacred Kyoto off Groves's atomic bomb target list - refused.

Szilard's day in the sun came when he circulated a petition directed toward the president and signed by 70 scientists advocating a demonstration of the bomb to the Japanese and an attempt at cooperation in the field of atomic energy with the Soviets. This was activist Leo Szilard at his best. Groves was livid, Oppenheimer - who by now had tasted power and was an establishment man - was deeply hesitant and the petition was stashed away in a safe until after the war. Szilard's disappointment that his advice was not heeded turned to even bigger concern after the war when he witnessed the arms race between the two superpowers. In 1949 he wrote a remarkable fictitious story titled 'My Trial As A War Criminal' in which he imagined what would have happened had the United States lost the war to the Soviets; Szilard's point was that in participating in the creation of nuclear weapons, American scientists were no less or more complicit than their Russian counterparts. Szilard's take on the matter raised valuable questions about the moral responsibility of scientists, an issue that we are grappling with even today. 

The story played a small part in inspiring Soviet physicist Andrei Sakharov in his campaign for nuclear disarmament. Szilard also helped organize the Pugwash Conferences for disarmament, gave talks around the world on nuclear weapons, and met with Nikita Khrushchev in Manhattan in 1960; the result of this amiable meeting was both the gift of a Schick razor to Khrushchev and, more importantly, Khrushchev agreeing with Szilard's suggestion that a telephone hot-line be installed between Moscow and Washington for emergencies. The significance of this hot-line was acutely highlighted by the 1962 Cuban missile crisis. Sadly Szilard's later two attempts at meeting with Khrushchev failed.

After playing a key role in the founding of the Salk Institute in California, Szilard died peacefully in his sleep in 1964, hoping that the genie whose face he had seen at the traffic light in 1933 would treat human beings with kindness.

Since Szilard the common and deep roots that underlie the tree of science and politics have become far clearer. Today we need scientists like Szilard to stand up for science every time a scientific issue such as climate change or evolution collides with politics. When Szilard pushed scientists to get involved in politics it may have looked like an anomaly, but today we are struggling with very similar issues. As in many of his other actions, Szilard's motto for the interaction of science with politics was one of accommodation. He was always an ardent believer in the common goals that human beings seek, irrespective of the divergent beliefs that they may hold. He was also an exemplar of combining thought with action, projecting an ideal meld of the idealist and the realist. Whether he was balancing thermodynamic thoughts with refrigeration concerns or following up political idealism with letters to prominent politicians, he taught us all how to both think and do. As interdisciplinary scientist, as astute technological inventor, as conscientious political activist, as a troublemaker of the best kind, Leo Szilard leaves us with an outstanding role model and an enduring legacy. It is up to us to fill his shoes.

The Ten Commandments of Molecular (and other) Modeling

Thou shalt not extrapolate too much beyond the training data.

Thou shalt prioritize simple experiments over complex models.

Thou shalt never forget the difference between accuracy and precision.

Thou shalt never try to woo thy audience with pretty pictures or tales of fast GPUs.

Thou shalt not worship “physics-based” models that are not actually physics-based.

Thou shalt not be biased toward favorite models and should use whatever gets the job done.

Thou shalt use good statistics as much as possible.

Thou shalt always remember that modeling is a means and a tool, not an end unto itself.

Thou shalt always understand and explain the limitations of thy models.

Thou shalt never forget: Only good experiments can uncover facts. The rest is crude poetry and imagination. 

The only two equations that you should know

“Chemistry”, declared the Nobel laureate Roger Kornberg in an interview, “is the queen of all sciences. Our best hope of applying physical principles to the world around us is at the level of chemistry. In fact if there is one subject which an educated person should know in the world it is chemistry.” Kornberg won the 2006 Nobel Prize in chemistry for his work on transcription which involved unraveling the more than dozen complicated proteins involved in the copying of DNA into RNA. He would know how important chemistry is in uncovering the details of a ubiquitous life process.
I must therefore inevitably take my cue from Kornberg and ask the following question: What equation would you regard as the most important one in science? For most people the answer to this question would be easy: Einstein’s famous mass-energy formula, E=mc2. Some people may cite Newton’s inverse square law of gravitation. And yet it should be noted that both of these equations are virtually irrelevant for the vast majority of practicing physicists, chemists and biologists. They are familiar to the public mainly because they have been widely publicized and are associated with two very famous scientists. There is no doubt that both Einstein and Newton are supremely important for understanding the universe, but they both suffer from the limitations of reductionist science that preclude the direct application of the principles of physics to the everyday workings of life and matter.
Take Einstein’s formula for instance. About the only importance it has for most physical scientists is the fact that it is responsible for the nuclear processes that have forged the elements in stars and supernova. Chemists deal with reactions that involve not nuclear processes but the redistribution of electrons. Except in certain special cases, Einstein therefore does not figure in chemical or biological processes. Newton’s gravitational formula is equally distant for most chemists' everyday concerns. Chemistry hinges on the attraction and repulsion of charges, processes overwhelmingly governed by the electromagnetic force. This force is stronger than the gravitational force by a factor of 1036, an unimaginably large number. Gravity is thus too weak for chemists and biologists to bother with in their work. The same goes for many physicists who deal with atomic and molecular interactions.
Instead here are two equations which have a far greater and more direct relevance to the work done by most physical and biological scientists. The equations lie at the boundary of physics and chemistry, and both of them are derived from a science whose basic truths are so permanently carved in stone that Einstein thought they would never, ever need to be modified. The man who contributed the most to their conception, Josiah Willard Gibbs, was called "the greatest mind in American science" by Einstein. The science that Gibbs, Helmholtz, Clausius, Boltzmann and others created is thermodynamics, and the equations we are talking about involve its most basic quantities. They apply without exception to every important physical and chemical process you can think of, from the capture of solar energy by plants and solar cells to the combustion of fuel inside trucks and human bodies to the union between sperm and egg.
Two thermodynamic quantities govern molecular behavior, and indeed the behavior of all matter in the universe. One is the enthalpy, usually denoted by the symbol H, and roughly representing the quantity of energy and the strength of interactions and bonds between different atoms and molecules. The other is the entropy, usually denoted by the symbol S, and roughly representing the quality of energy and the disorder in any system. Together the enthalpy and entropy make up the free energy G, which roughly denotes the amount of useful work that can be extracted from any living or non-living system. In practical calculations, what we are concerned with are changes in these quantities rather than their absolute values, so each one of them is prefaced by the symbol ∆, indicating change. The celebrated second law of thermodynamics states that the entropy of a spontaneous process always increases, and it is indeed one of the universal facts of life, but that is not what we are concerned with here.
Think about what happens when two molecules – of any kind – interact with each other. The interaction need not even be an actual reaction, it can simply be the binding of two molecules to one another by strong or weak forces. The process is symbolized by an equilibrium constant Ke, which is simply the ratio of the concentrations of the products of the reaction to the starting materials (reactants). The bigger the equilibrium constant, the more the amount of the products. Ke thus tells us how much of a reaction has been completed and how much reactant has been converted to product. Our first great equation relates this equilibrium constant to the free energy of the interaction through the following formula:
∆G0 = -RT ln Ke
or, in other words
Ke = e-∆G0/RT
Here ln is the natural logarithm to base e, R is a fundamental constant called the gas constant, T is the ambient temperature and ∆Gis the free energy change under so-called 'standard conditions' (a detail which can be ignored by the reader for the sake of this discussion). This equation tells us two major things and one minor thing. The minor thing is that reactions can be driven in particular directions by temperature increases, and exponentially so. But the major things are what's critical here. Firstly, the equation says that the free energy in a spontaneous process with a favorable positive equilibrium constant is always going to be negative; the more negative it is the better. And that is what you find. The free energy change for many of biology's existential reactions like the coupling of biological molecules with ATP (the “energy currency” of the cell), the process of electron transfer mediated by chlorophyll and the oxidation of glucose to provide energy is indeed negative. Life has also worked out ingenious little tricks to couple reactions with positive (unfavorable) ∆G changes to those with negative ∆G0 values to give an overall favorable free energy profile.
The second feature of the equation is a testament to the wonder that is life, and it never ceases to amaze me. It attests to what scientists and philosophers have called “fine-tuning”, the fact that evolution has somehow succeeded in minimizing the error inherent in life’s processes, in carefully reining in the operations of life to within a narrow window. Look again at that expression. It says that ∆G0 is related to Ke not linearly but exponentially. That is a dangerous proposition because it means that even a tiny change in ∆G0 will correspond to a large change in Ke. How tiny? It should be no bigger than 3 kcal/mol.
A brief digression to appreciate how small this value is. Energies in chemistry are usually expressed as kilocalories per mole. A bond between two carbon atoms is about 80 kcal/mol. A bond between two nitrogen atoms is 226 kcal/mol: this is why nitrogen can be converted to ammonia by breaking this bond only at very high temperatures and pressures and in the presence of a catalyst. A hydrogen bond - the "glue" that holds biological molecules like DNA and proteins together - is anywhere between 2 and 10 kcal/mol.
3 kcal/mol is thus a fraction of the typical energy of a bond. It takes just a little jiggling around to overcome this energy barrier. The exponential, highly sensitive dependence of Ke on ∆G0 means that changing ∆G from close to zero to 3 kcal/mol will translate to changing Ke from 1:99.98 in favor of products to 99.98:1 in favor of reactants (remember that Ke is a ratio). This is a simple mathematical truth. Thus, a tiny change in ∆G0 can all but completely shift a chemical reaction from favoring products to favoring reactants. Naturally this will be very bad if the goal of a reaction is to create products that are funneled into the next chemical reaction. Little changes in the free energy can therefore radically alter the flux of matter and energy in life’s workings. But this does not happen. Evolution has fine-tuned life so well that it has remained a game played within a 3 kcal/mol energy window for more than 2.5 billion years. It's so easy for this game to quickly spiral out of hand, but it doesn’t. It doesn’t for the trillions of chemical transactions which trillions of cells execute everyday in every single organism on this planet.
And it doesn’t happen for a reason; because cells would have a very hard time modulating their key chemical reactions if the free energies involved in those reactions had been too large. Life would be quickly put into a death trap if every time it had to react, fight, move or procreate it had to suddenly change free energies for each of its processes by tens of kilocalories per mole. There are lots of bonds broken and formed in biochemical events, of course, and as we saw before, these bond energies can easily amount to dozens of kcals/mol. But the tendency of the reactants or products containing those bonds to accumulate is governed by these tiny changes in free energy which nudge a reaction one way or another. In one sense then, life is optimizing small changes (in free energy of reactions) between two large numbers (bond energies). This is always a balancing act on the edge of a cliff, and life has managed to be successful in it for billions of years. It's one of the great miracles of the universe.
The second equation is also a relationship between free energy, enthalpy and entropy. It's simpler than the first, but no less important:
∆G = ∆H - T∆S
The reason this equation is also crucial to the operation of the universe is because it depicts a fine dance between entropy and enthalpy that dictates whether physical processes will happen. Note that entropy is multiplied by the temperature here and the sign is negative. So if it decreases in a process then ∆S becomes negative and the overall product (T∆S) becomes positive. In that case the change in enthalpy needs to be negative enough to compensate, otherwise the free energy will not be negative and the process won't take place. 
For instance, consider the schoolboy experiment of oil and water not mixing. When oil is put into water, the water molecules have to order themselves around oil molecules, leading their entropy to decrease and become negative. The attraction between water and oil on the other hand is weak, so the change in enthalpy does not compensate for the change in entropy, and oil does not mix. This is called the hydrophobic effect. It's a fundamental effect governing a myriad of critical phenomena; drugs interacting with signaling proteins, detergents interacting with grease, food particles attracting or repelling each other inside saucepans and human bodies. On the other hand, salt and water mix easily; in this case, while the entropy is still unfavorable because of the ordering of water molecules around salt molecules, the enthalpy is overwhelmingly favorable (negative) because the positive and negatively charged sodium and chloride ions strongly attract water.
Because temperature is part of the equation it too plays an important role. For instance consider a phenomenon like a chemical reaction in which the change in entropy is favorable but quite small. We can then imagine that this reaction will be greatly accelerated if T is high, making the product of it and the entropy large. This explains why the free energy of chemical reactions can be made much more favorable at high temperatures (there is a subtlety here, however: making the free energy more favorable is not the same as accelerating the reactions, it's simply making the products more stable. The difference is between thermodynamics and kinetics).
Even the origin of life during which the exact nature of molecular interactions was crucial in deciding which ones would survive, replicate and thrive was critically dependent on enthalpy and entropy. When little oily molecules called micelles repelled water molecules because of the unfavorable entropy and enthalpy described above, they sequestered themselves into tiny bags inside which fragile molecules like DNA and RNA could safely isolate themselves from the surrounding water. These DNA and RNA molecules could then experiment with copying themselves at leisure, not having to worry about being hydrolyzed by water. The ones with higher fitness survived, kickstarting the process which, billions of years later, finally led to this biped typing these words on his computer.
That's really all there is to life. We all thus hum along smoothly, beneficiaries of a 3 kilocalorie energy window and of the intricate dance of entropy and enthalpy, going about our lives even as we are held hostage to the quirks of thermodynamic optimization, walking along an exponential energy precipice.
And all because Ke = e-∆G0/RT

Book review: "John von Neumann" by Norman Macrae

What could one possibly say about good time Johnny von Neumann that isn’t known? His was widely considered to be the fastest, most wide-ranging, most original mind of the twentieth century. That the same man who laid the mathematical foundations of quantum mechanics also became the father of both game theory and (with Alan Turing) the modern computer is simply astonishing. In addition he helped build the atomic bomb, axiomatized set theory as Hilbert’s assistant, did early work in weather prediction and helped the United States get started on ICBMs. And he was just getting warmed up. It seemed that even the greatest scientists of the 20th century could talk about von Neumann only in superlatives. For instance, Enrico Fermi said that von Neumann often left him feeling that he knew no mathematics at all. And Hans Bethe said something truly interesting: if there were ever a race of superhumans, he wondered if the members of this more advanced species would not resemble Johnny von Neumann.

Norman Macrae – highly accomplished editor of The Economist for thirty years - brings a light touch, a genuine appreciation and an amusing style to the story of von Neumann, a bit like von Neumann’s mind itself. Macrae is not a scientist so this book is more about von Neumann the human being than von Neumann the mathematician. Johnny was a certified child prodigy who at age eight could multiply two eight digit numbers in his head, already knew six languages and had a photographic memory because of which he could recite entire books by heart. Combined with these qualities was an amazing ability to concentrate, and with great speed. Infused with a lifelong passion for history, he finished all 40 volumes of a multi-volume series of world history in a few days, and rumor has it that he would take two books with him to the bathroom, just in case he finished one and still had time left. When asked a question he would instantly jump ten steps ahead and solve a whole family of related questions. He would invent entire fields on train and plane trips. It helped tremendously that he grew up as the son of a wealthy banker in Budapest, at a time when the empire was crumbling. Along with his fellow émigrés Edward Teller, Eugene Wigner and Leo Szilard, he became part of the group of Martians, men with rarefied intelligence who had apparently disguised themselves to fit well in human society. Many years later, Wigner who won the Nobel Prize and was one of the foremost theoretical physicists of his time was asked why Hungary in that short time produced so many geniuses like himself. You are wrong, said Wigner. Hungary produced only one genius, Johnny von Neumann. But anti-Semitism and political turmoil certainly helped, as did Hungary’s high schools which were among the best in the world.

Macrae’s book does an excellent job tracing von Neumann’s background and the political developments during and between the wars in Europe. Only Szilard was more prescient in seeing the cloud of fascism blow over the continent. Along with Einstein, von Neumann thus became one of the first scientists to leave Europe in the early 30s. He settled down at the Institute for Advanced Study, helping other émigrés like Kurt Gödel (whose incompleteness theorem he was the only one to truly appreciate when Gödel first presented it in 1931) settle down. It was at the institute that he built his famous computer until the pure mathematicians who wanted to preserve the sanctimony of the place howled and shut it down, passing a motion not to build anything smacking of engineering in the place ever again.

Equally comfortable in the highest reaches of both pure and applied science, von Neumann quickly adapted to his country’s war needs, first working on ballistics and then making himself the country’s foremost expert on shock waves. It was this work that led to an invitation from Oppenheimer at Los Alamos, and he contributed a key idea to the plutonium implosion bomb regarding the squeezing of plutonium using explosive lenses. Everyone could rely on him to transform their thinking in a few minutes or hours. After the war he continued to be the most highly valued defense consultant in the country, advising every government organization except the Coast Guard on ballistics, nuclear weapons, submarines and missiles. And all this time he continued to lay the foundations of computing and its applications to fields as diverse as hydrogen bomb design and meteorology, first with J. Presper Eckert and John Mauchly at the University of Pennsylvania where he worked on the ENIAC, and then with Julian Bigelow and Herman Goldstine at the Institute for Advanced Study (a story nicely told in George Dyson’s book “Turing’s Cathedral”). Von Neumann came up with the idea of the stored program, realizing along with a select few others that numbers could be used to encode both instructions and their results. The von Neumann architecture remains a part of computers to this day. In the last days of his life he also worked on self-replicating automata and drew prescient parallels between human and machine thinking. The latter were published as “The Computer and the Brain” which still makes for highly original reading. There is little doubt that many of von Neumann's prophesies about the weather, the brain and computing are still waiting to be tested.

But enough about his amazing mind and scientific feats which are well known. Two things about von Neumann from the book are perhaps not as widely known or appreciated. First was his unusual personality. Von Neumann defied the image of the mathematician with his head in the clouds. He exuded bon homie and loved fast cars, rich food, dirty jokes, expensive suits (he wore one even on a mule during a hike in the mountains) and raucous parties (in the middle of which he could intensely concentrate on and solve important mathematical problems). Unlike other mathematicians he felt more at home with generals and admirals rather than with fellow academics. Unfortunately, similar to other great scientists, his family life was messy. His first wife Mariette left him for another man, partly because of his self-obsession with mathematics. His second wife Klari was very intelligent but also neurotic, and she committed suicide by walking into the ocean a few years after his death. Fortunately his daughter Marina became one of the most prominent women of her time, serving as the first woman on Nixon’s council of economic advisors and then on the board of directors of General Motors. She has written a first-rate memoir of her own life titled “The Martian’s Daughter”.

What really stands out, however, is Johnny's perpetually jovial nature and his friendships. During those troubled times he called himself a cheerful pessimist. Of all the important people of the time, he is the only one I know who managed to be friends with people who were each other’s sworn enemies. For instance, he testified on Oppenheimer’s behalf at the latter’s infamous security trial. The man who orchestrated the trial was Oppenheimer’s sworn enemy Lewis Strauss, the vindictive politician who had initially offered Oppenheimer the position of director of the IAS in Princeton. And yet Strauss remained one of von Neumann’s most steadfast supporters and genuine admirers, nominating him to the Atomic Energy Commission and making sure he got special treatment during his last days when he was struck down too early by cancer. Von Neumann similarly stayed good friends with both Edward Teller and Stanislaw Ulam who clashed severely over a priority dispute about the hydrogen bomb. Everyone seemed to like Johnny, and the main reason was that he would stay away from public insults or political arguments, often defusing political disagreements with a ribald joke. His philosophy was simple: keep on doing good work and be a decent person and then don’t worry about anything else. You are not responsible for what others think about you; a philosophy that Richard Feynman said he imbibed from Johnny.

The other interesting thing about von Neumann was his politics. Unlike the vast majority of his friends, he was a conservative, right wing Republican. His opinion that the US should be ready not just for a preemptive strike on the Soviet Union but a preventive one was admittedly an extreme one. He did not seem to realize that even talk of such things would raise the risk of nuclear annihilation. And yet there was a certain logic to his thinking. If a hydrogen bomb or a whole fleet of ICBMs was possible, he wanted the United States to be the country that got them first. Many others thought similarly, and while one can disagree with their thinking, one can also understand their internal logic. In our own times, von Neumann provides a role model of how people can vehemently disagree on politics and still be good friends.

Von Neumann’s death was heartbreaking, and there is no sugarcoating it. Here was a man whose sole purpose for existence seemed to be to think. When he realized at the early age of fifty-three that he had terminal brain cancer, he also realized that his mind would soon stop thinking. This was unimaginable for him and he had nightmares. He had so many military secrets in his head that he was moved to a special suite at Walter Reed Hospital in Washington and surrounded day and night by armed security guards to make sure he did not accidentally give out secret information. His friends constantly visited him and came away distraught. Lewis Strauss once recounted an extraordinary spectacle; this recent immigrant lying on a hospital bed, surrounded by the secretaries of the army, navy, air force and defense and the chairman of the joint chiefs of staff, everyone hanging on to his every word and squeezing every bit of information from his extraordinary mind. His brother Michael would read Goethe’s Faust to him, and until the very end, with his prodigious memory he would start reading the next page automatically. His daughter Marina recalls one particularly heartbreaking incident when he called her into his hospital room and asked her to ask him to add and divide simple numbers to test whether his mind was still intact. She could not stand it after a few minutes and left the room.

As the end drew near, von Neumann had a Catholic priest summoned and converted to Catholicism, hedging his bets, hoping that the small chance of an afterlife promised by religion might yet save him. Everyone was horrified and confused by this action. They could not believe that the most rational mind of the century would cling to this seemingly irrational thinking. On the other hand, his act could point at truths that only Johnny von Neumann’s mind could access that were entirely rational. I would like to believe the latter.

My favorite popular math books

When I was growing up I loved reading popular science, and some of my favorite popular science books were about mathematics and mathematicians.

Although there were scores of interesting volumes, the following ten especially inspired and entertained greatly.

1. Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter: Mesmerizing. The kind of book which seems otherworldly when you first encounter it. It introduces you to so many profound human and mathematical creations: impossible drawings, undecidable propositions, Bach's fugues. Gödel, Turing, Achilles and the Tortoise…great figures abound in this narrative. Even if you don’t understand everything you know you are part of something special. Gödel’s theorem is one of the greatest intellectual achievements in human history, and Hofstadter tells it well (the best account of it for laymen, however, is Nagel and Newman’s “Gödel’s Proof”). A feast all around.

2. Logicomix: An epic search for Truth by Apostolos Doxiadis and Christos Papadimitriou: A delightful animated exposition of Bertrand Russell and his search for mathematical truth. Does a great job bringing out Russell's passion, his madness, his personal indiscretions. The exaggerated drawings of mathematicians and philosophers like Wittgenstein and Frege as well as Russell’s fears and dreams add to the delight.
3. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth by Paul Hoffman: There was nobody as obsessed with mathematics as the Hungarian mathematician Paul Erdos. Unmarried, traveling around the world, this math fanatic would randomly knock on mathematicians' doors, asking them if "their mind was open." The book is filled with choice anecdotes of Erdos installing himself unannounced in people's houses and almost forcing them to do math 24/7. These collaborations led him to be the most prolific mathematician since Euler, publishing almost a thousand papers. A singular personality, and Hoffman tells his story so well.

4. The Man Who Knew Infinity by Robert Kanigel: Another singular personality about whom much is now known, partly thanks to the recent movie. Ramanujan belongs to the category of people like Einstein, Newton and von Neumann in being a true genius. His collaboration with Hardy is one of the great human relationships of all time.

5. Fermat's Enigma by Simon Singh: Another example of a mathematical obsession. Fermat's Last Theorem unsuccessfully challenged some of the great minds in science since its cryptic conception: Gauss, Riemann, John Nash, Heisenberg. All failed to solve it until Andrew Weil worked on it for fifteen years without telling anyone about it and announced the proof at a conference that left the world reeling. Singh is really good at explaining the admittedly very abstruse mathematics behind the theorem, and does a great job describing the follies and triumphs of the trails of mathematicians trying out their hand at Fermat.

6. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire: Derbyshire does for the Riemann hypothesis what Singh does for Fermat. Another great journey through both history and math, culminating in a great puzzle which challenged the finest minds of the field and which still awaits resolution.

7. A Beautiful Mind by Sylvia Nasar: John Nash's mind was a maelstrom of tragedy and triumph. Before the movie came out few people had heard of him, but Sylvia Nasar is brilliant at telling us the story of both his great achievements in economics and pure mathematics (which ironically are considered even more important than his work in game theory) as well as his frightening and sad slide into mental illness followed by an almost miraculous - albeit partial - recovery. Nasar also does an outstanding job describing the post-war mathematical milieu which was in part driven by the exigencies of the Cold War as well as the colorful characters around Nash and their intense competition with each other. A truly beautiful book.

8. One Two Three . . . Infinity: Facts and Speculations of Science by George Gamow: Just like it did many teenagers, this book delighted and completely gripped my attention when I read it. Gamow was a highly creative scientist and a joker, and both these qualities shine in this wonderful volume. Cheeky, irreverent Gamow takes us through a slew of amazing ideas in science including mathematics, atomic physics and genetics. But it is the first chapter saying that one can compare two different infinities that blew my mind and keeps it blown wide open to this day. I haven’t come across a more mentally stimulating and simply astounding idea than the one about countable and uncountable infinities.

9. What is the Name of This Book? by Raymond Smullyan: As much a logic puzzle book as a math book, the riddles in this volume are utterly entrancing. Similar to Martin Gardner, the bearded, wizardlike Smullyan lays before us a smattering of treats in math and logic. The puzzles set up simple premises and then get diabolical and delightful: for instance one extended piece sets up an island of knights and knaves, one of whom always lies and the other always tells the truth. From this premise follow tons of interesting questions, some of which are quite thorny (and I certainly cannot claim I have solved most of them). The coup de grace is a meeting with Dracula whose existence or lack thereof you are supposed to ascertain through a series of logical questions. Brain fun beyond measure.

10. What is Mathematics? by Richard Courant and Herbert Robbins: A classic book with a preface by Einstein, leading the readers through scores of important mathematical ideas, theorems, proofs and analyses. What I love about the book is the diversity of topics: topology, analysis, geometry, number theory, combinatorics and calculus all receive their dues, and the appendices are filled with summaries of interesting and unsolved problems like the Goldbach Conjecture. A solid, serious journey through key concepts of mathematics.

11. (Bonus) Men of Mathematics by Eric Temple Bell: This book, first published in 1937, is known to have started many famous mathematicians like John Nash and Freeman Dyson on their chosen path. While it did not lead me to any such paradise, I thoroughly enjoyed reading the extremely vivid descriptions of the lives, times and work of famous mathematicians like Fermat, Galois, Euler, Gauss, Riemann and Abel. Bell presents the lives of these people as heroic and romantic tales. The book is not the most accurate description of their work, but by presenting leading mathematicians as rogues, unscrupulous rascals and romantic idealists, it makes mathematics seem more of a human endeavor than any other book I know.