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Reasonable Effectiveness of Mathematics

Basudeba Mishra

Published Oct 1, 2015• 0 citations
OpenAlex

Abstract

Is mathematics the only language of science? Is there an equation for the observer? If not, can science ignore observer as unscientific. Was Wigner right when he talked about the unreasonable effectiveness of mathematics? Was Godel's incompleteness theorem based on natural mathematics? What is the basis of number theory? If we re-envision classical and quantum observations as macroscopic overlap of quantum effects, we may solve most problems. The physics community blindly accepts rigid, linear ideas about the nature of space, time, dimension, etc. These theories provide conceptual convenience and attractive simplicity for pattern analysis, but at the cost of ignoring equally-plausible alternative interpretations of observed phenomena that could possibly have explained the universe better.

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Prespacetime journal•Jan 1, 2011•0 citations

Introduction

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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.

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•Jan 1, 2017•0 citations
Is mathematics the only language of science? Is there an equation for the observer? If not, can science ignore observer as unscientific. Was Wigner right when he talked about the unreasonable effectiveness of mathematics? Was Godel's incompleteness theorem based on natural mathematics? What is the basis of number theory? If we re-envision classical and quantum observations as macroscopic overlap of quantum effects, we may solve most problems. The physics community blindly accepts rigid, linear ideas about the nature of space, time, dimension, etc. These theories provide conceptual convenience and attractive simplicity for pattern analysis, but at the cost of ignoring equally-plausible alternative interpretations of observed phenomena that could possibly have explained the universe better.
The view of the interdependent genesis of the two realms of dualistic manifestation; the realm of ‘physicality’, which is the objective aspect of the dualistic manifestation from the deeper, unitary, implicate (to use a term for levels of non-duality used by physicist David Bohm) dimension of Mindnature, and the subjective realm of individuated ‘mentality’ solves a crucial puzzle that has bothered many physicists and mathematicians. Eugene Wigner, for instance, referred to what he considered to be the ‘unreasonable effectiveness’ of mathematics in describing and explaining the physical world of ‘nature’; he called mathematics a ‘miracle’ and ‘a wonderful gift that we neither understand nor deserve.’ However, if both the realm of mentality and physicality emerge from a deeper level of universal Mindnature then it is surely not such a great mystery that mathematics, itself a product of mind, produces the conceptual patterns generated and followed by the ‘physical’ functioning of reality.
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.
When we consider and analyze physical events with the purpose of creating corresponding mathematical models we often assume that the mathematical apparatus used in modeling, at least the simplest mathematical apparatus, is infallible. In particular, this relates to the use of infinitely small and infinitely large numbers in arithmetic and the use of Newton's definition of a limit in analysis. We believe that is where the main problem lies in contemporary study of nature (physics, genetics). We have introduced a new concept of “Observer's Mathematics—Mathematics of Relativity” (see references). Observer's mathematics creates new arithmetic, algebra, geometry, topology, analysis and logic which do not contain the concept of continuum, but locally coincide with the standard fields. The concept of a quantum arises automatically from the human thought process, rather than from physics. From this point of view classical mechanics is brought closer to quantum mechanics.
It is argued that the main reason of crisis in quantum theory is that na-ture, which is fundamentally discrete, is described by continuous mathematics. Moreover, no ultimate physical theory can be based on continuous mathematics because, as follows from Gödel’s incompleteness theorems, any mathematics in-volving the set of all natural numbers has its own foundational problems which cannot be resolved. In the first part of the paper inconsistencies in standard approach to quantum theory are discussed and the theory is reformulated such that it can be naturally generalized to a formulation based on discrete and finite mathematics. Then the cosmological acceleration and gravity can be treated simply as kinematical manifestations of de Sitter symmetry on quantum level (i.e. for describing those phenomena the notions of dark energy, space-time background and gravitational interaction are not needed). In the second part of the paper motivation, ideas and main results of a quantum theory over a Galois field (GFQT) are described. In contrast to standard quantum theory, GFQT is based on a solid mathematics and therefore can be treated as a candidate for ultimate quantum theory. The presentation is non-technical and should be understandable by a wide audience of physicists and mathematicians.
In this thesis I argue that many problems in the philosophy of science and mathematics (in particular, the unreasonable effectiveness of mathematics in physics) can only be addressed within a broader metaphysical framework which provides a coherent world view. I attempt to develop such a framework and draw out its consequences. The attempt is in two parts: first, I develop a speculative framework based on an analogy to set theory; then I combine elements of the framework with ideas from Leibnizian monadology and consistent histories quantum theory to introduce (what I call) quantum monadology. The two parts focus on different aspects of the problem and should be viewed as stages on the way to a final formulation. The inspiration for the thesis came from Plato’s Timaeus and Wigner’s comments on quantum mechanics. As it turned out, Leibniz’s Monadology became a third key source.