Zenith Grant Awardee
Donald Spector
Hobart and William Smith Colleges
Project Title
Set Theoretic Forcing and Information Theory
Project Summary
Physics uses numbers that can in principle be specified to infinite precision. There is a well developed theory of information that incorporates key ideas from physics, and there are arguments suggesting that, at a fundamental level, the theories of physics should be theories of information, yet the existing mathematical correspondence of physics and information breaks down when we consider numbers specified to infinite precision. Mathematicians, in dealing with the surprising finding that there are different sizes of infinities, have developed a technique known as forcing. In my research, I will demonstrate that the mathematical technique of forcing provides the tools necessary to characterize how much information is encoded in arbitrarily well specified numbers. The techniques of forcing will provide a way to understand how experiments can squeeze out more information by going to every higher precision, how theoretical physics can track the information flow from input to output, and how a theory of information can be developed that works whether we have a finite or an infinite number of possible outcomes. The resulting, more powerful theory of the relationship between physics and information would also represent the first real-world application of the abstract mathematical technique of forcing.
Technical Abstract
Theories of physics, because they are defined over the real numbers, involve the specification of infinite – even uncountably infinite – amounts of information. Yet existing characterizations of information are ill suited to systems with an infinity of possible values. I propose to address the question of how to measure the information in cases where the number of possibilities is infinite by using the set theoretic technique of forcing. Note that a chaotic system, despite its sensitive dependence on initial conditions, behaves physically in some particular way, indicating its ability somehow to track its initial state to infinite precision. Similarly, scientific measurements can never obtain the precision of a real number, but can, in principle, be refined arbitrarily well. I will use forcing to mathematize the notion of information in such contexts, and in a way that offers a useful characterization of information when an infinite number of outcomes is possible. Different forcing methods will provide insights into different ways in which information is embedded in continuous systems, with the associated notions of generic real numbers that arise in forcing constructions extending notions inherent in maximum entropy methods and in the Asymptotic Equipartition Theorem of conventional information theory.
QSpace Latest
PressRelease: Shining a light on the roots of plant “intelligence”
All living organisms emit a low level of light radiation, but the origin and function of these ‘biophotons’ are not yet fully understood. An international team of physicists, funded by the Foundational Questions Institute, FQxI, has proposed a new approach for investigating this phenomenon based on statistical analyses of this emission. Their aim is to test whether biophotons can play a role in the transport of information within and between living organisms, and whether monitoring biophotons could contribute to the development of medical techniques for the early diagnosis of various diseases. Their analyses of the measurements of the faint glow emitted by lentil seeds support models for the emergence of a kind of plant ‘intelligence,’ in which the biophotonic emission carries information and may thus be used by plants as a means to communicate. The team reported this and reviewed the history of biophotons in an article in the journal Applied Sciences in June 2024.