Quantum Entanglement: “Spooky action at a distance”, or an inflection towards the extension of classical set-theoretic description? Part II

The most fundamental mathematical objects used in quantum mechanics are the same objects that makes up the structure of set theory – our entire number system. So, it is only natural that the success of the canonical Zermelo-Fraenkel Axiom of Choice (ZFC) in its description of classical mechanics and its subsequent application to classical computation should be employed in dissecting the structure of quantum mechanical phenomena; but the failure of this exercise, if evident, is very much so in our attempt at applying our understanding and experiences of quantum mechanics as a foundation for quantum computation – quantum entanglement takes quantum computation to a level that no classical system can simulate.

If you missed the previous part of this newsletter, please read for continuity: https://www.linkedin.com/pulse/quantum-entanglement-spooky-action-distance-towards-felix-wejeyan-nkt0c

Although, even before the phenomena of quantum superposition and quantum entanglement was encountered there had been lots of mathematical ideas about extending the ZFC set-theoretic description to accommodate diverse mathematical structures. Neumann, Barnay and Godel (NGB) extended ZFC into an axiomatic system which is used to give a description of a collection of sets called Classes. It is considered as a conservative theory as it doesn’t prove any new theory that the language of ZFC hasn’t already proven, but creates a sort of convenient approach. A proper extension of ZFC was provided by internal set theory, Morse-Kelly (MK) set theory and Tarski-Grothendieck (TG) set theory whose extension laid an axiomatic foundation for category theory. Given these beautiful extensions of ZFC why isn’t there a complete structural description of quantum entanglement as applicable to quantum computation?  

“Pure mathematics is, in its way, the poetry of logical ideas” – Albert Einstein

A Physicist’s Apology: It has always been difficult to describe mathematical formalism and structures using the set-theoretic language of membership relation, equivalence and identities, as various mathematicians develop their different styles and ways of expressions, and the conventional standard still hasn’t been able to make the matter more succinct or any less difficult. So, being that I would be trying to describe the mathematical structure of quantum entanglement (a phenomenon which contemporary physicist and mathematician are still struggling to comprehend, talk less understand) using the worst ever mathematical and logical tool – English language – which has been replaced with more sophisticated contemporary mathematical notations, I would like to apologize beforehand for any inaccuracies and clumsiness in my presentation, and would implore the reader to make necessary comments, corrections, amendments and addition after reading this text for the uncomprehending reader to further digest.

The major primitive notions that bind all set-theoretic descriptions into its structure are the binary and membership relations. Set theory begins with a fundamental binary relation between a mathematical object  and a set . While the superposition and entanglement relation has proven to be of a fundamentally higher form of relation between mathematical entities, say A as stipulated by the wave-function, and other mathematical entities B, C, D, etc., who are themselves related to each other to an extent determined by the quantum superposition properties of the wave function of each entity, and the nature of the entanglement involved. Can one say that in the quantum universe, sets are defined by probabilistic boundaries? Then, how can one describe an empty set, and other properties of set in this universe?

"In the vastness of the universal set, even the smallest element holds significance, a testament to the interconnectedness of all things."

The universe of consideration for all set-theoretic formulations is of paramount importance as shown by the “Tarski Universe”, which forms the major distinguishing factor for the extension of ZFC by TG set theory. The Tarski universe and the Grothendieck universe are described more like a universal set: they provide a set in which all mathematics can be performed. Does the Tarski universe subsume the quantum universe? Or is it the other way round, and the quantum universe subsume the Tarski universe? Or are they both distinct and different part of a bigger universe that interacts to define our experienced reality? In order to avoid set-theoretic paradoxes, the universe of consideration has to be properly tailored and described such that it is not a member of itself, and at the same time it is not a set of all sets according to the comprehension principle.

According to M. Makkai, 1998, set theory is governed by equality predicates, and one of such predicates is the membership predicate. This implies that in set theory, if x belongs to a set A, then one can assume that it is an absolute member of the said set with all certainty. If we assume the A to be a set of characteristics that describes the entire attributes of a wave function, then quantum mechanics tells us that sometimes it is probabilistic as to whether x belongs to A or not.

"The universal set is the bedrock upon which set theory is built, the foundation of all mathematical structures."

When we allow mathematical objects that are not in a particular universe into a different universe of computation, we are bound to encounter paradoxes and contradictions, unless we revise and enlarge our sense of universality and equivalent relation by involving other primitive notions and relations that we encounter/experience. This enlargement should also be considered for the logical foundation upon which the mathematical structure is to be built. Given that the theoretic description given by quantum superposition cannot be clearly delineated by a membership relation, neither can it be described by fuzzy sets, and the equivalent relation proposed by quantum entanglement cannot be described by equations or any form of mathematical equivalence, how then can the partition and the subsequent combinatorics of this “quantum universe” be described as axiomatic?

In other words, in order to describe a universe (quantum universe) that would subsume the Tarski universe, the phenomena of quantum superposition and entanglement has to be imbibed as primitive notions in the structural description of this new universe. Following this train of thought implicitly implies that we would have to revise/redesign/extend the mathematical foundations of this enlarged universe beyond that which has already been established by set theory. Of a truth, as Henri Poincare feared, set theory should now be regarded as a disease from which our generation has to be recovered from. If this is true, from where do we begin to initiate this recovery process?

It has been observed that most of the fundamental axioms of ZFC seems particularly tailored for proof. So, at this point it is time to go back to the sewing machine and begin to design axioms for a quantum universe for which phenomena such as quantum superposition and entanglement would play a fundamental and independent role; but still in such a way as to be consistent enough to fit into, or rather subsume our overall standard mathematical models – a theory of everything, if you like.

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without the gorgeous trappings of painting or music." —Bertrand Russell

There are numerous mathematical works and ideas that aims at designing a new and encompassing foundations for mathematics, and one major characteristics of each proposition/theory is its description of morphism and identity. The transcendental set-theoretic description used isomorphism as the criterion of identity, and this is at the bottom of the universe of all things that constitute a set (Jean-Pierre Marquis, 2006). In an extended version of a universe of sets, isomorphism between various sets, class, groups or categories doesn’t lead to a singular identity – rather, they lead to various identities.

Of further importance is the logical foundation upon which propositions and proofs can be made in this quantum universe. The comprehension axiom is designed with a Boolean logical foundation, and this limits any sense in thinking of a universal set, imposes an empty set as an element of every other set, and corners the outcome of all predicate into a binary outcome. In other words, every formal system, like the set-theoretic description, uses sentences that can be described as closed under logical implication. From the many propositions of quantum logic, it is evident that the way in which we describe an axiomatic system as “complete” has to go beyond the conventional true or false Boolean structure to accommodate more possibilities with relative consistency within a formal proof system. (https://www.linkedin.com/pulse/quantum-logic-gates-beauty-behind-mind-thinking-process-felix-wejeyan-xhd7c).

"There is a great danger in the present day lest science- teaching should degenerate into the accumulation of disconnected facts and unexplained formulae, which burden the memory without cultivating the understanding." ~J. D. Everett

Whenever there is a lack of equality between mathematical objects or entities, the law of the excluded middle breaks down and propositions and arguments can no longer be stated in a meaningful way – at least in a Boolean meaningful way. There are plethora of mathematical propositions, ideas and theories about how to redesign the foundations of mathematics to cater for many phenomena which our canonical set-theoretic description seems to be deficient at describing. There are many more pieces of thoughts, logic and ideas about a comprehensive mathematical structure of everything, that are scrambled and scattered throughout numerous research papers and thesis that would bamboozle the most analytic mind, and still, we have no complete mathematical structure.

Would a complete and tall mathematical structure that is able to hold all of our experienced reality in a concise axiomatic framework using formal proof and without “acceptable” contradictions continue to elude our intellectual capabilities like the Tower of Babel, just because we cannot seem to be able to speak one mathematical language? What then are the potentials (scientific, technological, social, cultural, etc.) available to the human race when they are finally able to unite all mathematical language into this tall and complete mathematical structure made of formal systems, and that stretches into the heavens?

This article is for instructional and didactic purpose. If you found it informative and inspiring please like, share and re-post.