Academic journal article Cosmos and History: The Journal of Natural and Social Philosophy

An Introduction to Mathematical Metaphysics

Academic journal article Cosmos and History: The Journal of Natural and Social Philosophy

An Introduction to Mathematical Metaphysics

Article excerpt


In these early years of the new millennium, there has been considerable academic concern over the stubbornness of Cartesian dualism and the conceptual difficulty of uniting the mental and physical sides of reality. Meanwhile, it has gone all but unnoticed in academic circles that the formal aspect of this challenge was met decades ago, and in the most effective possible way. This paper contains a brief introductory account of the work in question, including a highly simplified description of the logicomathematical reasoning needed for the proper high-level theoretical description of reality. Everything follows from the requirements of this objective.

Aside from the author's own writings, there is as yet no well-defined field of mathematical metaphysics. Being an explanation of the author's previous work, this paper is largely an exercise in self-reference. The same applies to its bibliography. Conveniently enough, this is well in accord with a major theme herein emphasized, namely, the self-referential nature of reality.


Whether or not any given scientist or philosopher of science chooses to admit it, science is heavily invested in the idea that there exists a valid, comprehensive theory of reality or "theory of everything" (ToE), and indeed there must. For reality as a whole has structure and history, and in principle, that structure and history are isomorphic to a valid comprehensive theory of reality that can be written in a language of sufficient expressive power, the existence of which would seem to be implied by the fact that reality is accessible to the mind and senses. In a pinch, one need merely wave expansively at the universe and declare that the theory shares its structure.

Unfortunately, getting any farther than this entails a few difficulties. For example, we must decide how reality should be studied in order to discover its structure, and by whom. A closely related issue, which we encounter before even getting around to establishing a valid correspondence between theory and content, is how to identify a language of sufficient expressive power to contain a valid, comprehensive description of reality.

If there were a scientific consensus on these issues, it might well go something like this:

"Reality should be studied by physical scientists using the tools and methods of the physical sciences, and there can be no better language for expressing their findings than the language of physics. For after all, physics is the most fundamental science, that to which all other sciences can supposedly be reduced."

But this gives rise to yet another problem: physics is not self-explanatory. If physics is regarded as an expression of the structure of reality, then clearly it is real, and a comprehensive theory of reality must explain its every part and aspect. But then in order to qualify as a comprehensive theory of reality, physics must explain itself, its correspondence to reality, and arguably the biological origins and mental activity of physicists in whose minds it exists. In fact, physics as we know it can explain none of these things, or even the natural law of which it supposedly consists. Such explanations are prohibited by its methodology as encoded in the scientific method, according to which it must constantly be tested by observation and experimentation.

Technically, it is not permitted to depart from the world of the senses.

As a matter of logic, the task of explaining such things as physics, the possibility of physics, and the relationship of physics to the physical universe requires a metalanguage of physics, a higher-order language in which the "object language" of physics can be an object of reference. Let a metalanguage capable of these functions be called "metaphysical". In order to properly refer to its content, the required metaphysical metalanguage must include physics, understood as the theoretical aggregate of physical insight, as a sublanguage, along with distinctions and classifiers suitable for distinguishing between reality and its complement or negation. …

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