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

It has been observed that most of the time classical theories tries to corner physical phenomena in nature and its processes into a sometimes “unreasonable” binary relation. An epistemological example would be a statement such as “Have you subscribed to this newsletter? If yes, breathe; if no, do a backflip”. From a universal point of view, everyone can breathe, and everyone needs to breathe, but that doesn’t invariably mean that everyone who breathes has subscribed to the newsletter. On the other hand, not everyone can do a backflip – I tried to learn how to as a child, but gave up because I couldn’t trust gravity enough after I have flipped myself off the ground into the air. For those of us who cannot backflip, we are left with only one option, and that is to breathe/subscribe. But is this really true? Is this an ideal/realistic ontological commitment, or have we been cornered into two seemingly “unreasonable” options? ?

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

Quantum mechanics has come to question our classical ontological commitment on a fundamental level, and in such a way that so many surprising and counterintuitive theories with experimental confirmation has been proposed to push the boundary of our understanding and experience about what to believe truly exist, and what entities are fundamental to our theoretical description of nature. Quantum entanglement is one of these quantum phenomena that has stirred the minds and thoughts of the science community to reconsider the fundamental entities of our physical theories.

A theory is committed to those and only those entities to which the bound variables of the theory must be capable of referring in order that the affirmations made in the theory be true – W. Quine, 1948.

In 1897, J.J. Thompson through experiments with cathode ray proved the existence of electrons. The result from his experiments shows invariably that electrons exist, and this can be translated in the language of first-order logic to mean that there exists x, such that the predicate E(x) describes the state of x. We are therefore ontologically committed to the acceptance of the existence of an electron (x) as a physical entity in our world, for which the predicate E(x) is said to be true or false as guided by Thompson’s experiment. This is one amongst many of the ontological commitments we have made about the existence of some physical and abstract entities as part of our universe, whose interactions can be used to describe phenomena that makes up our reality: therefore, we design theories, and select from among the designed theories which to endorse as true among some other rival theories. ?

Analyzing the structure of classical theories provides a clear ontology, specifying the fundamental entities describing matter in space-time, and that physical objects can be described individually by the maximal set of properties they instantiate, whose values are independent of the performance of observations and always well defined (A. Oldofredi et al, 2018). On the other hand, the physical objects of quantum mechanics cannot be specified in terms of a maximal set of properties as their values are not determinate until a measurement is performed. Regardless of this obstacle, experiment has shown that quantum phenomena are govern by mathematical structures, and it is our responsibility to investigate these structures in order to build a proper and useful quantum ontological commitment.

Mathematicians do not study objects, but relations among objects; they are indifferent to the replacement of objects by others as long as relations do not change - Henri Poincare.

The structural trend in contemporary mathematics has always been attributed to the name Nicolas Bourbaki, a man who never existed. Nicolas Bourbaki was the nom de plume of some of the greatest French mathematicians in the 1930s who believed that using a strict method of axiomatization, one could build up the whole of mathematics on the sole basis of a few fundamental “mother structures” (i.e. algebraic structures, order structures and topological structures) and their combinations (D. Aubin, 1997). According to L. Corry 1992, they began their monumental treatise with a set-theoretical foundation upon which to design the rest of mathematics from; but later down the road, it became clear that set theory wasn’t going to be an adequate foundation that would hold such an ambitious structure – the whole of mathematics – together. ?

The set theoretical language that was used in the beginning of the treatise Bourbaki formulated was further tuned down to a more natural, and somewhat informal, language. Although the path taken by these valiant mathematicians changed over the course of time, the structural character of mathematics that was founded by Bourbaki has been described as an informal approach, and in an attempt to formalize this approach, the term “mathematical structure” has led to many more mathematical conundrums than the originally sort after clarity that could have been provided by a single structure (L. Corry, 1992).

Later generations will regard set theory as a disease from which one has to be recovered – Henri Poincare.

Therefore, the structural analysis used here, although comprises of contemporary set-theoretic language, isn’t designed in the informal approach of Bourbaki, neither is it in a formal approach. It is rather a tailored description of the fundamental nature of quantum entanglement as an extension of the characteristics, properties and operation of standard set-theoretic formalisms using a structural description that blurs the line between a formal and informal approach.

We saw from the previous part of this publication that the classical set-theoretic formulation has been designed on a structure (with predicate logic of binary relations – equality and membership relations) having number systems that fits perfectly into the structure (just like branches and leaves on a tree) as to make it seem complete – axiomatic (and beautiful). And yes, when describing classical phenomena and processes, it excels so well that we have come to believe in its predictive power to a high degree of certainty; as long as all the variables are in the right place, then we know exactly what to expect from a classical process. Just like the human body is held together by our skeletal system made up of networks of bones, ligaments and muscles, the mathematical structure of our set-theoretical description of classical processes is held together by a network and combination of the existential relation, membership relation, equivalence relation (equality) and identity relations (M. Makkai, 1998).

As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality – Albert Einstein.

Mathematical structures can be stretched, bent and twisted to withstand all transformations of the relations between the mathematical objects that are encountered in classical mechanics, even up to the counterintuitive descriptions of the theory of relativity, until the first surprise – quantum superposition – was encountered within quantum mechanics. This was a surprise because the membership relation dictates that the nature and properties of a mathematical object can be succinctly described by its equivalent relation to every other mathematical object within the set of which it is a member; quantum superposition came to blur this neat boundary with an extensive layer of probabilities. This probability isn’t deterministic as those encountered in our usual classical description, which we believe is due to hidden variables and randomness: No! This probability was certain, persistent and somewhat intrinsic to all quantum processes, and there was no way to escape it or go around it.

This notion deeply troubled Albert Einstein when thinking of quantum mechanics, as he had found that the usual set-theoretic structure was sufficient to describe special and general theory of relativity, even though these theories made propositions that were highly counterintuitive and subsumed the already established theories of classical mechanics. He, Einstein, believed something has to be missing (hidden) somewhere within the theory (variables), so later on, himself, Boris Podolsky and Nathan Rosen came up with a thought experiment and a subsequent theory (EPR paradox) to show that the then contemporary quantum mechanics (the wave function particularly) is incomplete, and there might be some hidden variables which should account for the intrinsic probabilities which seems to plague quantum theory. To make their point, they described a quantum system which after interacting, when separated still behaves in a way that the state of one cannot be described without the other. They thought they were solving a problem, but rather they highlighted a bigger problem within quantum mechanics – quantum entanglement!

In order to give a proper structural analysis of quantum entanglement in light of all that is already known, at this point, it is imperative to first analyze the properties, nature and differences between mathematical objects as used in classical mechanics and those used in quantum mechanics. https://www.dhirubhai.net/pulse/quantum-mathematical-objects-used-describing-invented-felix-wejeyan-fducc

To be continued…

Please share your thoughts on the differences between the mathematical objects used in classical mechanics and those used in quantum mechanics, and the set-theoretic implications of this difference.?

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