The primitive connection between mathematics and the physical world make use of everyday experience at the macroscale. With quantum phenomena, one encounters a regime far removed from ordinary experience. One relies on laboratory experiments to define the quantum world; the experimental process refines ordinary everyday experiments such as for counting and logic. One cannot expect to find results that appeal to ‘‘common sense.’’ Here, one must make a choice as to whether the new situation can be described by existing mathematical infrastructure or it requires the invention of new infrastructure. In either case, one must set up a mathematical system, which exists only in the human mind, in order to describe the system of nature. That is, ‘‘mathematics belongs to and exists in the human mind whereas nature belongs to God.’’ Mathematics and nature have separate but intertwined existences. As will become evident later, one must allow for the possibility that human observation (a physical process, not a mathematical one) can affect the state of the natural system.
The primitive connection between mathematics and the physical world make use of everyday experience at the macroscale. With quantum phenomena, one encounters a regime far removed from ordinary experience. One relies on laboratory experiments to define the quantum world; the experimental process refines ordinary everyday experiments such as for counting and logic. One cannot expect to find results that appeal to ‘‘common sense.’’ Here, one must make a choice as to whether the new situation can be described by existing mathematical infrastructure or it requires the invention of new infrastructure. In either case, one must set up a mathematical system, which exists only in the human mind, in order to describe the system of nature. That is, ‘‘mathematics belongs to and exists in the human mind whereas nature belongs to God.’’ Mathematics and nature have separate but intertwined existences. As will become evident later, one must allow for the possibility that human observation (a physical process, not a mathematical one) can affect the state of the natural system.
Quantum theory is the soul of contemporary physics. It was discovered in an adventurous way, under the urge to solve the puzzles posed by atomic spectra and blackbody radiation. But after its invention, it immediately became clear that it was not just a theory of specific physical systems: it was rather a new language of universal applicability. Already in 1928, the theory had received solid mathematical foundations by Hilbert, von Neumann, and Nordheim, and this work was brought to completion in the monumental work of von Neumann, in the form that we still study nowadays. The theory is extraordinarily successful, and its predictions have been confirmed to an astonishing level of precision in a large spectrum of experiments. However, almost 90 years after von Neumann's book, quantum theory remains mysterious. Its mathematical formulation – based on Hilbert spaces and self-adjoint operators – is far from having an intuitive interpretation. The association of physical systems to Hilbert spaces whose unit vectors represent pure states, the representation of transformations by unitary operators and of observables by self-adjoint operators – all such postulates look artificial and ad hoc . A slightly more operational approach is provided by the C * -algebraic formulation of quantum theory – still, this formulation relies on the assumption that observables form an algebra , where the physical meanings of the multiplication and the sum are far from clear. In short, the postulates of quantum theory impose mathematical structures without providing any simple reason for this choice: the mathematics of Hilbert spaces is adopted as a magic blackbox that “works well” at producing experimental predictions. However, in a satisfactory axiomatization of a physical theory the mathematical structures should emerge as a consequence of postulates that have a direct physical interpretation. By this we mean postulates referring, e.g., to primitive notions like physical system, measurement, or process, rather than notions like, e.g., Hilbert space, C * -algebra, unit vector, or selfadjoint operator. The crucial question thus remains unanswered: why quantum theory? Which are the principles at the basis of the theory? A case that is often invoked in contrast is that of Special Relativity theory, which directly follows from the simple understandable principle of relativity.
In the last two decades, the scientific community has witnessed a surge in activity, interesting results, and notable progress in our conceptual understanding of computing and information based on the laws of quantum theory. One of the significant aspects of these developments has been an integration of several fields of inquiry that not long ago appeared to be evolving, more or less, along narrow disciplinary paths without any major overlap with each other. In the resulting body of work, investigators have revealed a deeper connection among the ideas and techniques of (apparently) disparate fields. As is evident from the title of this volume, logic, mathematics, physics, computer science and information theory are intricately involved in this fascinating story. The inquisitive reader might focus, perhaps, on the marriage of the most unlikely and intriguing fields of quantum theory and logic and ask: Why quantum logic? By many, “logic” is deemed to be panacea for faulty intuition. It is often associated with the rules of correct thinking and decision-making, but not necessarily in its most sublime role as a deep intellectual subject underlying the validity of mathematical structures and worthy of investigation and discovery in its own right. Indeed, within the realm of the classical theories of nature, one may encounter situations that defy comprehension, should one hold to the intuition developed through experiencing familiar macroscopic scenarios in our routine impressions of natural phenomena. One such example is a statement within the special theory of relativity that the speed of light is the same in all inertial frames. It certainly defies the common intuition regarding the observation of velocities of familiar objects in relative motion. One might be tempted to dismiss it as contrary to observation. However, while analyzing natural phenomena for objects moving close to the speed of light and, therefore, unfamiliar in the range of velocities we are normally accustomed to, logical deductions based on the postulates of the special relativity theory lead to the correct predictions of experimental observations. There exists an undeniable interconnection between the deepest theories of nature and mathematical reasoning, famously stated by Eugene Wigner as the unreasonable efficacy of mathematics in physical theories.
Humans live in many different worlds, each of which has its own representation in mind space. The world of the scientist will be called the physical realms. The laws of physics are an integral part of the real world, and humans acquire a natural understanding of these laws without regard to any mathematical formulation. While the experimental physicist studies the behavior of the real world, the theoretical physicist develops and studies its representation in the mathematical world (i.e., mind space). It is not necessary to know the mathematical equations to be able to carry out a physical investigation. This can be performed through observation and recording of the data numerically, and using the processes of interpolation and extrapolation. Practicing physicists usually find that they must adopt the roles of both the theoretical as well as the experimental physicist to achieve the aims of an experiment. Observers play a vital role in physics. They become important when their state of motion changes, or when their presence affects the results of the experiment. The observer can be either animate or inanimate. Unlike inanimate observers, animate observers have access to mind space, which allows them to make decisions from collected information.
“I refute it thus!” Samuel Johnson famously dismissed Bishop George Berkeley's argument for the unreality of matter by kicking a large stone (Boswell, 1823). In the light of modern physics, however, Johnson's simple reasoning evaporates. Apparently solid matter is revealed, on closer inspection, to be almost all empty space, and the particles of which matter is composed are themselves ghostly patterns of quantum energy, mere excitations of invisible quantum fields, or possibly vibrating loops of string living in a ten-dimensional space–time (Greene, 1999). The history of physics is one of successive abstractions from daily experience and common sense, into a counterintuitive realm of mathematical forms and relationships, with a link to the stark sense data of human observation that is long and often tortuous. Yet at the end of the day, science is empirical, and our finest theories must be grounded, somehow, “in reality.” But where is reality? Is it in acts of observation of the world made by human and possibly non-human observers? In records stored in computer or laboratory notebooks? In some objective world “out there”? Or in a more abstract location? The ground of reality When a physicist performs an experiment, he or she interrogates nature and receives a response that, ultimately, is in the form of discrete bits of information (think of “yes” or “no” binary answers to specific questions), the discreteness implied by the underlying quantum nature of the universe (Zeilinger, 2004). Does reality then lie in the string of bits that come back from the set of all observations and experiments – a dry sequence of ones and zeros? Do these observations merely transfer really-existing bits of information from an external world reality to the minds of observers, or are the bits of information created by the very act of observation/experiment? And – the question to which this entire discussion is directed – are bits of “classical” information the only sort of information that count in the reality game, or does an altogether different form of information underpin reality? In short, where is the ontological ground on which our impression of a really-existing universe rests?