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Physics and mathematics: boundaries and interactions

In the fast comments under the previous posting, we have had some discussions with Q2 about the boundaries between physics and mathematics. Does Yau's theorem belong to theoretical physics? Are the differential equations governing the vector field flows tools of physics? My answers were essentially Yes, while the answers of Q2 were No.

The separation of wisdom and research to physics and mathematics is largely a social phenomenon - one that is affected by some objective features of reality (including the Universe around us as well as the Platonic Universe of mathematical ideas) but one that can also be influenced by personal and political decisions, by social conventions, and by fashionable trends.

Commercial: See also Feynman's lecture about the relation of physics and mathematics
In ancient Greece, people did not distinguish physics and mathematics. In fact, all of us were philosophers, the lovers of wisdom. The crowning achievement of that era, the Euclidean geometry, later became a part of pure mathematics. As Einstein emphasized, it can also be interpreted as the oldest branch of physics: statics of perfectly solid bodies.

Let's jump to the era between Galileo and Gauss. We find many true heroes of thinking squeezed into several centuries. The separation of the quantitative thinkers into mathematicians and physicists is slightly arbitrary, and in fact it is mostly a result of the historical perspective that only appeared in the new era. Those thinkers who liked to do (or look at) some experiments or observations are counted as physicists - Newton, Maxwell, and others - while others are usually classified as mathematicians - Leibniz, Euler, Gauss. The actual theoretical work of these two groups did not differ much. If your physics department could hire Euler in the Fall, I think you would hire him. Even if we call them mathematicians, they could still count as extraordinary theoretical physicists.

One century ago or so, people decided to separate mathematics and physics. The main substantial, non-sociological reason behind these developments was the discovery that our intuition can often fail. Newton used to be convinced that he was directly "reading" the laws of physics from the real world: this is the only way how we can interpret his quote "hypotheses non fingo" (I am not inventing any hypotheses). In the 19th century, people realized that this can't be possible.

Non-Euclidean geometries were discovered. Suddenly, the opinion that the Euclidean geometry is the only mathematically possible geometry, which would also imply that it must be true in the real world, collapsed. The mathematicians had to build on firmer fundaments instead of the fundaments that collapse whenever the physicists find something surprising about this dirty world - and whenever the mathematicians find that some insights about the real world aren't as logically inevitable as they seemed to be previously.

So the mathematicians realized that they could (and should) build their structures without any links to the observable physics and the intuition from everyday life whatsoever. One starts with a set of axioms, and using well-defined and "obviously meaningful" rules of logic, he (and today also she) can prove the validity of some other statements. Modern mathematics was born.

The birth of pure mathematics was an important moment in the history of thinking. Nevertheless, it did not change the fact that a majority of the most interesting questions and results was directly or indirectly linked to the real world as understood at the given moment of the past, or at least to the real world as understood from a more complete future perspective. Also, the intuition from the everyday life was no longer necessary for mathematics. Again, it still helped many mathematicians even though many of them decided (and are still deciding) to obscure this fact. ;-)

The key reason for the separation of mathematics and physics was described by Einstein as follows: whatever is rigorous cannot be directly applied to the real world, and whatever can be directly and accurately applied to the real world is not rigorous. Of course, such a rule could break down as soon as we find the complete theory of everything that could be formulated rigorously and that would be physically accurate at the same moment. But we're not there yet, which is why we can still separate the fields.

Mathematicians themselves had to discover some purely mathematical and surprising facts, especially about the solution of paradoxes in set theory and Gödel's theorems
  1. about the incompleteness - the existence of an unprovable and undisprovable assertion - and
  2. about the unprovability of the internal consistency of a system of axioms

which are valid for all consistent systems of axioms that can mimic the set of integers and its usual properties.

But these interesting insights occured inside the world of mathematics after its velvet divorce with physics - and most physicists are more or less certain that these logical games have no physical (or even measurable) consequences whatsoever. For example, we can't design an experiment that would decide whether the axiom of choice is true, false, or undecidable or whether the Zermelo-Frenkel set theory is better than the Gödel-Bernays framework.

The velvet divorce - inspired by the split of Czechoslovakia - was not good enough for a certain extremist group of mathematicians who preferred a divorce according to the Yugoslav example. The group called Bourbaki started to publish boring, mechanical books that were based on the ideology that physical intuition must always be assassinated whenever it appears near the iron curtain separating mathematics from the rest of the world of ideas. Because I think that the mathematicians should mostly be ashamed of this chapter of their history ;-), let me say nothing else about that movement.

During the last decades, the iron curtain started to disappear again, especially in the context of geometry and related disciplines where the gap or wall between the cultures of mathematicians and the culture of physicists is finite, shrinking, and penetrable.

I am convinced that there exists some general organization of deep mathematical ideas - something that God or Nature had to know when He or She was designing the world(s). In this organization, the main ideas have certain mutual relationships and a hierarchy. Even if you think about deep questions in mathematics only, I am convinced that the identity of the most interesting generalization(s) of a mathematical structure has an objectively well-defined answer that can in principle be found, plus minus the error margin proportional to the social conventions.

Moreover, all these very deep ideas eventually turn out to be important for theoretical physics. I can't prove this assumption but I believe that it is consistent with the whole history of mathematics and physics, as I understand it, combined with my personal appraisal of the values of different ideas in mathematics. This appraisal, of course, values general insights about robust and continuous structures (those that are useful for predictive natural science) much more than special insights about particular discrete structures (that are useful for creating many new games in recreational mathematics).

When Newton was solving the differential equations relevant for the Kepler system, he was solving not only an abstract mathematical problem but also an extremely important physical system. Some of the modifications, deformations, and generalizations of these equations and other equations turned out to be more important for physics, some of them were less important for physics, but I think that no one would question that the insights about the solutions to differential equations are important for natural sciences, and in this sense they belong to the natural sciences.

They can only be isolated as "mathematics" if someone decides that some subproblems should only be solved by the people from one group, and other problems should be solved by the people from another group, and that these groups should not be encouraged to look behind the boundaries of their fields of expertise. This arrangement is nothing more than a social policy that does not say much about the true internal relationships between different ideas and insights. You may decide that you're not interested in anything outside your narrow field; but such a decision can't change what is actually there behind these walls.

When Jacobi studied the theta functions, he did not know much about string theory. But it was his fault, so to say - and the fault of other scholars before him who were not able to do what Lenny Susskind et al. could do in the late 1960s. ;-) Today, we know that when Jacobi proved his obscure identity, he also proved a necessary condition for spacetime supersymmetry in superstring theory formulated in the RNS variables.

In fact, we know much more. The theta functions and similar functions are related to the partition sums and correlators of a physical system called the worldsheet. This insight provides us with natural generalizations and new important unanswered problems along the same lines. I am convinced that the answer of string theory to the question

  • "What should we do with the theta functions in the following century?"

is more or less unique. It is not a coincidence that most of the 21st century papers that talk about theta functions use them as the partition sums of a string or a dual, equivalent physical system unified with the strings in string theory. I would bet that the extraterrestrial aliens would find the same application of the theta functions.

The same comment applies to many other insights that have become important parts of physics in general and string theory in particular. Yau's proof of Calabi's conjecture was presented as pure mathematics. In the decades that followed, it was realized that it is primarily an extremely important result in theoretical physics. That does not mean that Yau is suddenly a pure physicist; he is still a mathematician although he is now co-authoring a large number of physics papers. But it does mean that the natural and interesting generalization of his insights has a very powerful physical interpretation.

Similar observations hold in the case of mirror symmetry. If you only define mirror symmetry as the fact that for every Calabi-Yau manifold "M", you can find a Calabi-Yau manifold "W" whose Hodge diamond is rotated by 45 degrees, it can look like an abstract mathematical problem. Or perhaps even a sophisticated exercise from recreational mathematics.

However, if you actually try to solve some more general problems of this kind and to extend the result into a stronger statement, you will inevitably be led to string theory. Also, string theory will allow you to solve some problems from "recreational mathematics" much more efficiently than what the mathematicians who are ignorant about string theory can do.

In string theory, the "elementary" description of mirror symmetry involving the Hodge diamonds of manifolds "M" and "W" above is just a very tiny portion of a much more general conclusion that reveals the equivalence between "two" physical systems that look very different a priori but that can be shown to be isomorphic, including the infinite number of new observables that both of them admit and that were ignored in the paragraph about the Hodge diamond.

When people study important mathematical results carefully enough, they will inevitably be led to their natural generalizations. In other words, they will be forced to discover the role that these mathematical insights play within physics or within string theory. I could continue with many examples such as those in knot theory, Chern-Simons theory, and their extension via topological string theory and perhaps the full string theory, but the main point of this essay is of philosophical nature, so let me avoid too many examples.

Many proofs in mathematics use various ad hoc inequalities or they assign mathematical structures different roles than those that would be viewed as natural ones from a physicist's viewpoint, but I believe that none of these physically unexpected features can be quite unique, fundamental, or universally important. All of them are technicalities that could be replaced by different technicalities and the true important result is the proof modulo the choices of these technicalities. Only the properties of the mathematical objects that are natural from the physics viewpoint can be truly important and deep.

Of course, you might think that this statement simply means that the physicists should be defined as those who are thinking in a deep way, but I still feel that what I want to say is more than just a definition of "physics".

The paragraphs above make it clear that I believe that the distance between string theory as theoretical physics on one side and mathematics of string theory studied using the tools of pure mathematics on the other side will be diminishing throughout the remainder of the 21st century. The term "mathematics of string theory" mostly describes properties of various "continuous" mathematical structures. But it is not hard to imagine that very discrete subfields of mathematics such as number theory will be incorporated into this powerful system of ideas, too.

Of course, there will always be differences between people who study pure sciences and applied sciences, and between people who use their hands vs. heads, but these differences will be viewed as sociological barriers while the actual, intellectual barriers between the ideas of different fields - and between physics and mathematics in particular - will continue to melt down.

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