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The Problem With Equality

By Roy D. Follendore III

Copyright (c) 2005 by RDFollendoreIII

Everyone in the United States of America should be familiar with the phrase: "All men are created equal." It represents an fundamental democratic ideal upon which our form democracy was founded. As our nation grew our people came to realize that the ideal of equality is a a complex and difficult concept to manifest within organized reality. The Civil War and the historical battles over civil rights were fought because of organized differences in the notions of equality. But as much as we may have been socially impacted, notions of equality are more deeply engrained within science. Within this essay I bring up the point that there are both physical as well as logical reasons for us to misunderstand the notions of equality. We often don't accurately communicate ideas about equality because we tend to misplace our meaning and this is because equality exists at the junction somewhere between our logical and physical rationalizations. These fundamental problems that we have with respect to the various core ideas of equality both have and will continue to be seen within the critical systems that we engineer.    

November 19, 2005

In different ways I have written the essence of this article several times before, or at least I have tried to. It seems like such a simple thing to write about but it isn't. I want to say that even the most simple of things are not as simple as we would have them to be. I want to say that the notion of Occam's razor is as complex as any other philosophy and I want to state the reason why. I think that I can do that because of a 'rational' flaw in the ointment that exists between the universe of logic and the physical universe in which our corporeal bodies  exist. This flaw is something that we too easily overlook because it does not 'fit' with our logical notions of perfection.

There are two completely overlooked truths to the science of logic.

Our existence of everything within the physical universe is complex and therefore the concept of "equality" is complex.

The symbolic simplification of things within our universe is not equivalent to the "existence" of its reality.

Notions of equality presupposes that the things that are equal preexists. One must also presuppose that any statement of equality must exist in accordance with the fundamental notion that logic itself exists. According to an article in Encyclopedia Britannica, ( http://www.britannica.com/eb/article-36304) "The interest in the notion of existence is connected with the question of what entities a theory commits its holder to or what its “ontology” is. The “predicate of existence” just mentioned recalls Quine's criterion of ontological commitment: “To be is to be a value of a bound variable”—i.e., of the x in ($x) or in ("x). According to Quine, a theory is committed to those and only those entities that in the last analysis serve as the values of its bound variables." This ontological commitment has everything to do with the inherent problems of applying concepts of equality.

Obviously the term "equal" defines itself as the root word with respect to the notions of equality and equivalence. To be equal it is not necessary to be similar. On the other hand, to be equal is to be equal in all ways that are meaningful. To be equal is an unequivocal statement that supposedly specifies a truth. Within a one or zero universe of true and false, the term equal represents the probability of one rather than the probability of zero. When the concept of the word equality involves an absolute probability that implies something less than the absolute existence of truth, the specification of equality no longer exists. The rue of our thinking about equality is the way in which we have been indoctrinated by our learning experience.  

When you were a small child your Mother, Father or teacher probably taught you the formal notion of equivalency.  ( 1+1=2 )  At the time you probably thought that it was foolish.  Children ask themselves the question, How could the marks "1+1" be the mark "2" and they take the word of their parents and teachers that they know the correct answer. As children we all learn the concept that to be equal does not mean that two things are literally the same things. Even if we were to say that "1" = "1" the rational problem that we face is that the first "1" is a different physical symbol from the second "1". The manifestations of the logical and physical concepts of symbolisms have quite different implications. This is a neat trick, the almost transparent seam that flows throughout mathematics. With respect to the physical manifestation that represent calculated results we must show one concept, while logically we must intend something else. There is a corollary to this logical seam that manifests itself in the physical world.    

At the macro view, most things may appear to remain constant. For instance the sun and the moon and their relationship to the earth does not appear to change.  From night to night, week to week and year to year the planets continue their relentless relationships that never seem to change. The occasional burst of light in the heavens that may be witnessed with the naked eye can be explained through a complete sense of continuity. It is within this sense that the large view of our physical existence is manifested within the essence of our logic. At the grand scale of mathematics, the relationships of logical perfection has been defined by the relationship of whole numbers. The logical patterns of numbers have infinite patterns that describe consistency through the notion of cardinality. This means that Zero is always less than One, One is always less than Two... and so on all the way to the impossible notion of infinity. This is where another transparent seam that mathematicians don't like to talk about exists.  In order to convey the notion of infinity, we must take a shortcut.

There is no way to express the term infinity in "tangible" ways. It is a word for something that does not physically exist and never will. This is because no one has ever, or will ever be able to actually count the value of infinity.  Logic is a second order way of thinking that can never be fully expressed by the physical mechanism that endows it. There is not enough time to physically manifest the potential of mathematical cardinality and the life expectancy of the atoms that would make up the presentation is not long enough if were to attempt to do so. The beauty of this transparent seam is that the notion of logical mathematics as we have been taught is limited by both time and space and we choose to simply overlook it.  Who knows how many seams exist? In order to bear witness to these seams of logic we must constantly be willing to reflect between the logical mathematical and the physical boundaries. This turns out to be something that can be pretty difficult to do.

Physics has definitively shown us that at the micro level, the actual makeup of things can never completely be equal because things are constantly changing and with that in mind let's return to the logical notion of equality. The concept of equality is based on the idea of differentiation. If two things can be differentiated then they are not identical.  The degree to which they are identical determines the truth of their equality. Equality is ultimately a statement of absolute relevance. Either something is equal or it is not and therefore equality exists or it does not exist. If equality between two specified things do not exist then those two things may still be similar. Within the physical universe, because time is a dimension of all things, because of their passage through the dimensions of time and space, two separately identifiable things can always be differentiated. But at the quantum level of physical existence we also know that every single individual thing is changing at different rates. The "average" rate of decay of a particular type of atom is one thing, the "specific" rate is something else. If we are to also consider the dimensional placement of particles that make up things then the differences in things are more instantaneous.   

Because their atoms are constantly in motion, it becomes both obvious and inevitable that even the most featureless rock that exists in empty space is not the same rock a moment later. Atomic particles decay and the elementary nature of physical things evolve. Theoretically the gravitational fields of every atom is constantly interacting with every other individual and aggregate atom.  Effect affects everything else.  Physical science has shown that even the observation of the smallest atomic particles affect them and when physical objects are juxtaposed differently they become individually different.  

The physical notion of equality is relative to scope. Things that are physically equivalent are never really equal in the sense that they are the same.  Physics demonstrates that two separate things that can not possibly be expressed as being different can exist and therefore while "A" can never equal "B", they may be equivalent. In logic the mathematical symbols that are being used to represent absolute notions of equality are physically equivalent but are not absolutely equal. This means that depending on the perspective, while 1 "equals" 1, 1 is also "unequal" to 1. Associative and commutative rules of mathematics depend on both specific physical and logical notions of context.

At this point most mathematicians might argue that as far as they are concerned this is a moot point to make and perhaps within their field of inquiry that may be so. Why get all wrapped around the axel of semantics? But there are practicalities that are important about these concept that are important because opportunities for the useful application of mathematics is largely being overlooked within the design of computer applications. The primitive psychological point of equivalent logic is this; When we first learn mathematics we begin to buy into the seamless deception that mathematics accurately represents our physical universe.  As children we once noticed the transparent seams within mathematics that as adults we have forgotten.

Functionally we too often imply notions of the relationships of physical and logical equalities but very often we don't generally apply them.  There are lessons to be learned and there are certain advantages to be had from those lessons which will allow us to do things that can't be accomplished by lesser means. There are also dangers of isolating the notions of equivalence which all systems analysts should be constantly aware. Within computer science we should make a concerted effort to be wary of engineers who construct systems based on complex mathematical alchemy and call it physical manifestations of reality. After all, the models that can be created with logic are never "equal" to physical reality, at best they are only approximate equivalents.

From an educational perspective there is a dire need to change the way that we approach the teaching of mathematics. Ideas such as logical equivalence in set theory need to be expressed in such a way that is sensitive to the alternative manifestations of physics. Students need to be better able to better integrate ideas between our physical universe and expressions of logical thinking. The invisible seams of logic become apparent when as a society we attempt to reconcile the differences between fables of perfection that are presented through logical models and facts of reality. To close the functional gap of "logical and physical equalities" we will inevitably need to develop a better and more robust language for communicating. The language that we will require will need to encompass the differences in physical manifestations while maintaining relevance to logical structure. This has been successfully attempted in object oriented computer languages through within most aspects of human to human interaction its formal complexity has made it largely incomprehensible. Most educated adults, including those who are may use object oriented programming are simply not capable of communicating in terms of both the realistic and logical complexities of objects.

The subject of dealing with logical and physical problems with equality is not simply a matter of existential philosophy.  There are tangible realities in the immediate future that can be achieved if we are able to successfully handle its challenges.  How man chooses to reinvent his perceptions of the relationships between physical reality and logic will change the rate at which man will be able to deal with increasingly more relevant and more complex problems. 





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