On the Beauty Of Physical Theories

Finding a Theory of Everything is one of the most sought-after objectives for scientific research in today’s era. And needless to say, that theory will be mathematical. One’s understanding of the universe is doubtlessly incomplete without realizing it from the mathematical standpoint.

There is no limit of the extent to which numbers can amaze and intrigue us. As a matter of fact, did you know that presumably there are not more than 10200 atoms in the entire visible universe; did you know that the farthest we have peaked in the universe is only about 1.3 x 1023 km away (a number no bigger than the Avogadro’s number)? Do not be fooled by how small these numbers seem, if you can write five zeros on a sheet of paper in one sec, it would take you about 10 million million lifetimes to write just enough zeroes equal to the second number mentioned above.

“Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius – and a lot of courage – to move in the opposite direction.” In this phenomenal statement, Einstein explains the trend that was being introduced and strongly followed by his contemporaries during the early 1920s and 1930s. People had begun to realize that the Universe is already prodigiously complicated, the quantum world of the constituent particles of the universe vastly contributed to that. This complexity is not just because of the light-years of calculations that the testing experiments necessitate, we are also faced by a barrier of incapability to observe many quantum events to a desired degree of accuracy. We have nothing better than light that would effectively interact with those tiny particles but simultaneously not disturb the system in any way, and also cause in us a sensation for that phenomenon. Light with sufficiently low wavelength to interact with subatomic particles just happens to be too energetic to be catalytic.

Inspite of these difficulties, we have no shortage of potential candidates for the so-called Grand Unified Theories or GUTs. We are looking for theories that would explain each interaction in the universe concisely and beautifully, possibly which uses a single but comprehensibly elaborate equation, accounting for each and everything around us. You must be wondering what I mean by a candidate theory being beautiful. By ‘beautiful’, I mean that the theory should use mathematics that we already have some idea about, and that every subsequent step of explanation must have some resemblance with the earlier explanations. There is high probability that a theory which satisfies these criteria must be right in most possible scenarios, even if the experiments suggest otherwise. In one such instance, Einstein was once told about an experiment done by some experimentalist David Miller which showed that the observational data did not match the predictions that the former’s General Theory of Relativity dictated. Einstein is known to have reacted indifferently, stating that the claim was preposterous, nonsensical and will probably turn out to be wrong, which it did. There are some other sophisticated instances that support the notion of a ‘beautiful’ theory being true.

Science, in its entirety, survives upon two of its inseparable resources – theory-building and experimentation. It is not necessary for a theory to be hypothesised before an experiment is carried out to test it. Newton’s laws appeared in the reverse order of scientific process, while Einstein’s theories came forward in order to explain what was already out there. A theory stays a hypothesis unless and until any strong experimental observation is not suggested that supports it, a reason why the latter had to wait for a decade to publish his General Theory of Relativity in 1916. On similar grounds, the hypothesis that a unique field called the Higgs field is responsible for the mass of every substance in the universe is soon to become a theory based on the recently concluded experiments at the Large Hadron Collider.

Every physicist is a mathematician, but every mathematician needn’t be a physicist. The orthodox view among physicists is that mathematics was invented as a language and a tool to deal with the physical world. Eugene Wigner[1], however, has pointed out the “unreasonable effectiveness of mathematics in the natural sciences”— that is, if mathematics is merely invented, how is it that mathematics that was invented to explain things at the “human scale” also applies at very small scales (nuclear) and very large scales (cosmology)? Michael Atiyah[2]’s own view is that mathematics originates from the physical world but is organized and developed by the human brain. Moreover, this relationship is complicated by the fact that the brain is itself a part of the physical world and is thereby affected by it and cannot be completely segregated from it. Indeed, an evolutionary point of view is that humans evolved by natural selection so that the human mind is adapted to and reflects physical reality. Mathematical thinking is thus an incidental consequence of evolution.

Physicists and mathematicians strongly believe that the Creator, if there was one, would have made it simple, and beautiful to the interested eye. Now how does one go about making a theory “beautiful”? It can be done by reinventing methods to represent the same thing in ever shorter spaces, different treatments to picture the same matter. We don’t need any new calculations to prove anything or in order to make it beautiful. Whatever mathematics we would ever need, we already have it.

I would hereby like to quote Sir Michael Atiyah from the article “Two Cultures”, written by Tim Gowers because it says a lot about how I feel about mathematics.

Some people may sit back and say, “I want to solve this problem” and they sit down and say, “How do I solve this problem?” I don’t. I just move around in the mathematical waters, thinking about things, being curious, interested, talking to people, stirring up ideas; things emerge and I follow them up. Or I see something which connects up with something else I know about, and I try to put them together and things develop. I have practically never started off with any idea of what I’m going to be doing or where it’s going to go. I’m interested in mathematics; I talk, I learn, I discuss and then interesting questions simply emerge. I have never started off with a particular goal, except the goal of understanding mathematics.’

If it is true that pure mathematics can be divided into two broad cultures – theoretical and experimental aspect – with not a great deal of communication between them, one can still ask whether this matters. In my opinion it does. One reason is that this situation has many undesirable practical consequences. For example, mathematicians from one culture may well find themselves making decisions that affect the careers of mathematicians from the other. If there is little mutual understanding between the two cultures, then making such decisions fairly, which is difficult under the best of circumstances, becomes even harder. A second effect is that potential research students who are naturally suited to one culture can find themselves under pressure to work in an area from the other, and end up wasting their talent.


‘Beauty is truth, truth beauty,—that is all      

Ye know on earth, and all ye need to know.’                                ~ John Keats [Ode on a Grecian Urn]

For many years of my past, I had come to believe that studying mathematics is tedious and the subject didn’t come to make much sense at all. In some cases, this is true. The human mind strives to apprehend that the precious Time it spent on solving problems had some significance, nothing that it did was a waste, risking loss of such time when it occupy itself with something more sensible. For a person, nothing is more important than the realization that his/her life has some meaning, that they are not living in vain.

As John Keats mentions in the verse, Beauty corresponds to Truth and if something is true, it surely must be beautiful (although he might not have meant it in the context of math). As a matter of truth, mathematics does not look anything but ugly with respect to all the arrays of equations we write for it, and this is irrevocably true. The beauty of mathematics doesn’t lie in the symbols; beauty doesn’t work that way. As an analogy, imagine climbing the stairs of the Empire State Building to its pinnacle, it’s a hard and strenuous job indeed. We had to find a solution to save time and strength; this solution must lie within the volume of the building, and it did. Elisha Otis came to our rescue with his invention of the elevator. In a similar way, James Clerk Maxwell formulated his famous equations in quite a lengthy fashion, if only all that could be said in a short and concise space (a criterion for simplicity). Gibbs saved the day with his invention of vector analysis, with which people were able to bring all the information into four, powerfully simple equations. Wouldn’t that have been lovely? As I said, beauty doesn’t lie in the operators and operands; it lies in the ability to simplify. Compared to the long array of Maxwell’s equations, Gibbs’ treatment truly was beautiful.

As is often said, don’t judge a person by his/her looks but by his/her inner character. Similarly, study something disregarding what it requires; study it to be marvelled at what it can do for you.

Notes:

[1] Eugene Wigner (1902 –1995) was a Hungarian American theoretical physicist and mathematician who is known for having laid the foundation for the theory of symmetries in quantum mechanics.

[2] Michael Atiyah’s work was inspired by theoretical physics are responsible for some subtle corrections in quantum field theory. He was awarded the Fields Medal in 1966 and the Abel Prize in 2004.

References

Murray Gel Mann’s TED Talk on the character of Beauty in Physical Theories

http://terrytao.wordpress.com/2007/12/28/einsteins-derivation-of-emc2/

http://golem.ph.utexas.edu/category/2007/04/the_two_cultures_of_mathematic.html