Mathematics and the Real World (28 page)

In the nineteenth century a Norwegian mathematician, Sophus Lie (1842–1899), found groups that described symmetries of differential equations. These are today referred to as Lie groups, and one of them is the SU(3) group, which Gell-Mann and Ne’eman suggested constituted the basis of the arrangements of the hadrons.

Yet not all the elements in the group were represented by particles that were known when Gell-Mann and Ne’eman put forward their hypothesis. In particular, in a lecture delivered by Gell-Mann at the European Organization for Nuclear Research (CERN) in Geneva, which currently has the largest particle accelerator, he presented the SU(3) group and its properties and noted that a particle was missing; he called it omega minus. That particle would have completed the group model. Moreover, its existence would have absolutely contradicted another model for classifying hadrons proposed by Japanese researchers. Gell-Mann, however, was apparently not familiar with the literature on experiments then being performed. One of the people who attended Gell-Mann's talk was Luis Alvarez, of the
University of California, Berkeley. He was a famous scientist who later received the Nobel Prize in Physics in 1968, and among his many scientific achievements he put forward the hypothesis that dinosaurs became extinct due to a meteorite impacting on the Earth. Alvarez stated that the missing particle, omega minus, had been identified seven years previously by Yehuda Eisenberg, a physicist at the Weizmann Institute of Science in Rehovot, Israel. Eisenberg had made his discovery using the technique of placing photographic plates at the height of the atmosphere, but that was an isolated occurrence not backed by a mathematical explanation, so the experimental discovery of the particle entered the catalogue of particles without attracting any great attention. Following Gell-Mann's lecture, several groups of experimenters started again to look for that same omega minus using the bubble-cell technique until in 1964 a team of physicists led by Nicholas Samios at the Brookhaven Laboratory in New York eventually again succeeded in identifying the missing particle several times and thus corroborated Eisenberg's finding.

Therefore, it can be claimed that in a certain sense, ex post facto, the existence of the particle was predicted by group theory. Enlisting group theory to characterize and classify elementary particles scored its first success. The order in which the elementary particles are recorded moved up from the level of simply a table to a mathematical theory, which enables sophisticated predictions to be suggested and examined.

It should also be noted, however, that there is no fundamental or logical explanation for the fact that group theory and the structure of the elementary particles match each other. From a conceptual viewpoint, that agreement might remind us of the match that Plato found between the world and the four elements of nature that it comprises, and the five perfect solids, or the matching, with the detailed calculations, that Kepler found between the paths of the six celestial bodies and the perfect solids (see sections 10 and 17). Will we in the future assess the role of group theory in describing the elementary particles in the same way as we assess Kepler's model of the perfect solids?

The combination of the existing mathematics, physical principles, and technology of experiments in physics resulted in the discovery, mapping,
and understanding of the structure of a larger number of elementary particles and the interaction between them. In addition to the two forces already known, that is, gravity and electromagnetism, two new forces were discovered: a strong nuclear force and a weak nuclear force. Gell-Mann used mathematical principles and put forward the hypothesis that particles exist that are parts of protons and that have a fractional charge. These are the quarks, combinations of which make up the various protons. Although quarks cannot be isolated, their existence was proven beyond all doubt, and this earned Gell-Mann the Nobel Prize in Physics in 1969.

Since then the picture has broadened, and other experimental findings have been added, as well as many mathematical items. Yet the picture is still not complete or final. Right now an experiment is underway, at enormous expense, at the Geneva particle accelerator, intended to reveal the particle known as the Higgs particle (or Higgs boson), named after the British physicist Peter Higgs, who predicted the existence of the particle already in 1964. Initial reports indicate that a new particle has been found in the range of mass and energy in which the Higgs particle is predicted to be. Higgs and his colleague François Englert, who independently made the same prediction, won the Nobel Prize in Physics for 2013. If the initial findings are confirmed, this will corroborate the model known as the standard model. If it transpires that the particle discovered is not the Higgs particle, the physicists will have to rethink the picture of the subatomic world and perhaps will have to adopt a new mathematics to do so.

32. THE STRINGS RETURN

Physicists who deal with elementary particles are engaged in completing the picture of the subatomic world of these particles, but the model on which they are working is not consistent with the order that prevails between the collection of these particles and the theory of gravity. Moreover, to describe different particles, different versions of Schrödinger's equation must be used. In light of past successes in finding mathematical models that brought together different theories, physicists today feel obliged to
find one theory, one equation, that will enable them to explain the whole of the subatomic world. The attempt to incorporate all subatomic phenomena within one equation is the development of a mathematical system known as string theory.

From the aspect of the relation between mathematics and nature, string theory presents another stage. Maxwell's revolution presented mathematics that described physics whose components could not be perceived directly but whose effect on other physical quantities could be measured and whose predictions, such as the existence of electromagnetic waves, could be confirmed or disproved. String theory is mathematics that describes physics whose components cannot be perceived and whose effect on other physical elements also cannot be measured at this stage. Moreover, at this time, the theory does not provide predictions that can be confirmed or denied, and it does not seem that it will be able to provide such predictions in the foreseeable future. Is this the picture of the world? Is this physics?

Some of my physicist colleagues deny that this is physics and state that those involved in the theory are just mathematicians. Others are prepared to include that community under the umbrella of physics (also because their work is theoretical and does not compete for expensive research resources). There are also physicists who believe that despite the fact that currently it is difficult to even imagine such a situation, the day may come when the way will be found to examine string theory with experimental methods and to obtain benefit from it.

So what is string theory? The mathematical system of string theory is essentially similar to the systems that define the world of elementary particles, combined with geometric elements. The solutions to these equations are the basic elements that the theory presents. Since the brain cannot analyze mathematics without recognizable metaphors, string theory is described and examined via interpretations of those solutions. We will also relate only to the interpretation of the theory.

The strings are firstly particles of miniature size. They are hundreds of thousands times smaller than the elementary particles (this explains why they cannot be perceived, as the means for perceiving the tiniest particles are based on the elementary particles). These strings are wave solutions,
but unlike the electron, for example, which is described as a point particle that revolves around the nucleus of the atom, a string is described as a body that has a length, all of which vibrates and moves like a wave. Hence its name “string.” Once a particle that has a length is permitted, the question immediately arises as to if its ends are joined, like a ring, or are they attached to a plane, or are they free? It turns out that all of these are solutions to the equation, solutions that give strings of various types. The different strings create structures from which, so it is hoped, the subatomic structure can be derived. However, it transpires that for those structures to exist, the physical space must have certain surprising properties. For example, the space must have more than the four dimensions in which we can perceive, that is, three spatial dimensions and time. This means actual physical directions, but the distance along each direction is too small for us to perceive or measure in any way. The number of additional dimensions depends on the specific theory. The latest versions refer to ten or eleven dimensions. Moreover, those equations that describe the strings also have solutions of another type, sorts of membranes that are likely to be huge. Do these solutions have a physical interpretation or perhaps even implementation? If so, those membranes could contain worlds in addition to our own, worlds that we cannot perceive or communicate with, although the distance between us and them might be the smallest. Furthermore, collisions between those worlds are possible and could perhaps cause huge explosions, like the big bang that led to the transformation of energy into mass and, according to the accepted theory, created the world we know today.

To the reader whose reaction at this stage is “I don't understand,” I would say you are not alone. The writer of these lines does not understand much more, if to understand means to be able to translate the metaphors into mathematical language with real implications. This “understanding” means attempting to build a bridge between intuition about the world around us, intuition based on evolutionary development over millions of years and therefore limited by our senses, and the mathematical product that comes to describe situations that are so alien to what our senses teach us. Will this mathematics rise to the challenge of describing a real world? Time, apparently a very long time, will tell.

33. ANOTHER LOOK AT PLATONISM

We return now to the discussion of the connection between mathematics and nature and first remind the reader of the main difference between Plato's and Aristotle's (and their successors’) approaches to the essence of mathematics as a description of nature. Plato claimed that mathematics has an independent existence in the world of ideas in which mathematical truth is absolute. Humans can reveal this truth via logic. The starting point of research that will reveal the truth is axioms, which must be derived from nature, but it must be borne in mind that nature itself only mimics the ideal mathematical truth. Aristotle, however, claimed that mathematics itself has no independent significance or even existence. If axioms are found that reflect the truth of nature, the conclusions that can be drawn by means of mathematical formalism will apply to the description of nature. Specifically, the closer the axioms are to the truth of nature, the better the mathematical conclusions to describe nature will be. Plato and Aristotle agreed that there would be differences between mathematical conclusions and actual observations of nature. For Plato these differences were disturbances. For Aristotle the differences derived from inaccurate axioms, or in more modern terms, the inaccurate construction of the mathematical model. Neither of them discusses the essence of the difference between mathematics and nature.

The Aristotelian approach to the applications of mathematics can be summarized by the statement that
mathematics is a very good approximation of nature
. Furthermore, when a mathematical model does not describe nature as it really is, the model must be corrected. As stated, this is Aristotle's approach according to which mathematics does not have an independent existence. To reach the correct description of nature, we start with an approximate model and correct it and make it consistent with reality by comparing the results derived from the model with empirical data.

New research into mathematics and its applications suggest another way of looking at the relation between mathematics and nature, one that is closer to Platonism and may be called Applied Platonism. This approach can be summarized by the statement that
nature is a very good approximation of mathematics
.
Moreover, in some cases Platonic mathematics in the world of ideas inherently contains contradictions to basic laws of nature. Nevertheless, nature tries to copy it and has no choice but to approximate mathematics. The following is an example of this.

We have referred to the least action principle as the purpose underlying motion in nature. In the same way the minimization-of-energy principle also serves a purpose. Objects in nature strive to reach a situation of minimum energy, at least a local minimum; in other words, they will stay in the local minimum state unless they are subject to an external force. John Ball of the University of Oxford, England, and Richard James of the University of Minnesota in the United States examined the structure of an elastic object under stress. Their approach was a mathematical one. They wrote the expression for the energy of a body under stress and looked for the structure that would minimize the energy. They succeeded in solving the mathematical problem, and the result was that the mathematical solution is not applicable in nature. Mathematics required that the molecules in the elastic body simultaneously arrange themselves in two different ways. Clearly that is impossible in nature. Does it mean that the minimization-of-energy principle is not correct in this case? Laboratory experiments provided the surprising answer: The structure of the object in nature is an
approximation
of the mathematical result. The volume that the object occupies divides into microscopic parts so that in each of the tiny parts the molecules arrange themselves in one of the forms that constitutes the mathematical solution. Specifically, in each relatively large microscopic volume, both arrangements that together constitute the minimum energy appear and in the right proportions so that the average over the macroscopic surface is very close to the mathematical minimum. Nature tries to converge to the ideal solution that mathematics found, but it is not achievable.