On Mathematics As Language
By Roy D. Follendore III
Copyright (c) by RDFollendoreIII
May 15, 2005
As a University Professor, a major problem that I have found is the way that a great many of my students unrealistically think about communicating ideas through mathematics. Too often they are not able to explain what they may easily be able to calculate. They also can't explain why they have been taught to calculate and because of this they have lost touch with their creative intuition about the subject matter of their analysis. To them the practice of mathematical logic is a rubric. Empirical notions presented to them out of context with that rubric are either ignored or completely misunderstood. Even experienced engineers who come back to University as students and that have the best skills at memorizing and computing mathematical formulas have few other ways for creatively expressing what they know through written language. This problem must somehow be dealt with within universities because a rigid mathematical mind set becomes the limiting factor for teaching new opportunities related to the design of creative solutions for humanity's most difficult problems. Mathematics is obviously important and in fact I believe that mankind's need to be able to create new and useful forms of mathematical logic is far too important to allow the mathematicians creativity to be lost to the limitations of an absolute academic tautology.
The problem is that for those who are completely intellectually bound by the rigidity of mathematical logic, there can be no proof without numbers. In an age were catastrophic problems seem to arise faster than solutions, some intellectuals are waking up to the fact that this is a practical and fundamental roadblock for the future of our society. As children we are not initially taught by our mother and father to primarily communicate through numbers. There is a reason for this. A child's learning first takes place through empathy, not logic. As children our belief system is based upon our parental dependence. Things are so because those whom we depend upon and trust have told us so. But as we mature we find that we have no recourse than to test our independence through logic. We find that many things that were taught to us as children by our parents are simplistic, or just plane wrong. We eventually learn that parents teach their children what they need to know in order to get along and grow into maturity, not necessarily what is or is not absolutely true. The point is that young adults tend to be willing to accept the relationship of mathematical terms than the ambiguous nature of the physical universe which exists around us. Mathematics seems to magically make the ambiguous properties of reality disappear and become functions that then somehow might magically be expected to be understandable. All things seem to become possible if they can be expressed within a formula. Of course this is not true either. Perhaps the most important axiom of mathematics is that it is a wonderful illusion.
As a communicator, I tend to perceive mathematics as a special kind of human language, one that sacrifices flexibility for the advantages of an absolute consistency that may not necessarily exist. Within human language there are practical and useful contexts and creative meanings that come with the use of ambiguities. This is not the case for mathematics. Ambiguities within mathematics are specified as thought they are countable. A probability is defined as a countable definite. By this I mean that mathematics is not easily able to effectively communicate ideas when the true nature of a subject is unknown or when a subject’s absolute relationship to other known or when unknown things are implied or unexpressed as direct statements. From the perspective of language, one might easily argue that the assignment of a mathematical probability to an absolute statement have every right to be logically ambiguous and incorrect; just as might an author of a bad murder mystery might write, “The murderer killed him and therefore the murdered man lives on in court.” Within mathematical logic if for instance a murdered man is represented by a 1 and the same man is represented by a -1, then mathematics has little value if the man is represented by both 1 and -1. The point is that human language is not really about absolute expressions of clarity arising from consistency but about creative intellectual opportunities that arise from ambiguity.
It is within this context that the ultimate battle within mathematics exists and I suspect that it begins in the form of the ‘=’ sign. This is the mathematical balancing point, and as such the intellectual ‘equivalent’ of the physical fulcrum. Physical and mathematical ambiguities begin with the difference between the concepts of being equal and equivalent. But math is intellectual, not truly physical. Physical and intellectual concepts are not necessarily the same things, nor should they always be considered to be, and yet the way we are educated as children, it becomes too easy to get the philosophy of mathematics confused with physical reality.
Because of its regimented specification of consistency with respect to its core concepts of logical truth, mathematics requires that we be willing to have faith in an omnipotent perspective of a universe where that physical fulcrum ‘=’ is the "appropriate" intellectual balancing point. Mathematics is a language of precision and accuracy that speaks to a fantasy universe where consistency is the law. On the other hand, 'native' human language is largely about the philosophical ambiguities of equivalencies that fly in the face of consistency. Within our natural universe balance is not necessary because there is no requirement for a left and right side of a universal formula to exhibit equality in terms of perspective or phenomena. The law of thermodynamics states that for every action there is an 'equal' and opposite reaction. It is the concept of opposition which is the fly in the ointment of mathematics. Opposition within physical phenomena radiates to us from the quantum level where according to Heisenberg, our very observation changes the expression of particles which surround us. There simply can be no mathematical measurement for that and there is also no logical reason not to believe that similar physical phenomena which may exist all around us is simply ignored by our minds. Our brain evolved to discriminate noise and in doing so make sense of our physical surroundings, not disassociate and turn our senses into noise.
Man is a witness, and only a theoretical concept like God is actually able to calculates the true nature of physical equality. The perspective of a godlike omnipotence that equality implies that relative outcomes are known or can be expected to be known through predictions by the symmetry of mathematical form. But this does not mean that within an asymmetric physical universe the best thinkers have a deeply desperate need for creating symmetry where there is none. We still need to fight for peace, kill for love, and force people to vote for democracy, and each of these seemingly asymmetrical things do imply a kind of symmetry between intellectual ideas and actions. We mathematical mortals have the need to use the consistencies and symmetry of mathematics to express the inconsistencies of asymmetric chaos.
Words are often deliberately ambiguous for the reasons that inconsistency is not allowable or useful within mathematics. Words have allowed us the ability to consider the bifurcation of forms of logic in ways where the ‘why’ may not be rationally justifiable through the ‘how’. Simultaneously in a nonsymmetrical way of thinking the ‘how’ may not be rationally justifiable through the ‘why.’ From an anthropological perspective, I suspect that human language has a greater rational tendency of importance toward the ‘why’ over the ‘how’ of a subject matter for practical physical reasons. The reason for this is that individuals have historically needed to be able to communicate intent more than method. If a group of hunter gatherers for instance are relatively equivalent in hunting for food, then intent becomes far more important than their explicit methods. Diversity is very much a good thing when both methods are successful. But then, the intentions of the individuals to share the bounty resulting becomes important. This suggests that the probability of such a group of individuals being successful in a complex environment is greater if they take coordinated individual action as opposed to a single unified action. We can often see this kind of independent behavior within nature. The such things as natural communication patterns of animals can be predicated by expressing these kinds of asymmetrical relationships. I think that there is a great deal we modern mathematically oriented men and women may learn from considering such things.
Mathematically we humans are and should remain mortals because we must choose to coexist with the beauty of our imperfections. That is who we are. We are perfectly imperfect and we desperately require our native language to create unbalanced equations that entertain us as well as to allow us to understand ourselves. Society is not inherently mathematical, nor can any fundamental science which is based upon fixed a system of mathematical logic also be completely rational. Mathematics as we know it is therefore not a complete language through which we can operate. If we are to continue to survive as a species we must be willing to evolve the evolution of our logic with our ability to reason. The moment that we choose to move past the physiological boundaries of natural language and become completely consistent with logic, (if and when that becomes a possibility), we may no longer be human.
Copyright (c) 2001-2007 RDFollendoreIII All Rights Reserved