Mathematics

Division by zero the Quest for a Mathematical Proof of God



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“Dr James Anderson, a computer science professor at the University of Reading (UK) earned a bit of fame and notoriety in 2006 by claiming he had solved, “a 1200 year old problem,” most notably, he offered a proof that division by zero is possible. Dr Anderson defended the criticism of his claims on BBC Berkshire on 12 December 2006, saying, ‘If anyone doubts me I can hit them over the head with a computer that does it.’” - http://en.wikipedia.org/wiki/James_Anderson_(computer_scientist)

The purpose of pointing this out has little to do with Dr. Anderson’s claims, or any other rhetoric regarding the “1200 year old” problem. Certain elements of mathematics have exercised the right to divide by zero for years. Mathematicians and physicists like Riemann and Einstein have not only been doing this but also making progress in areas of science that make significant differences in our understanding of nature. However, the persistence of a problem causing it to continue to resurface over a span of time that great makes it worthy of attention.

Instead, the focus of this paper will generally follow a philosophical inquiry identified by Immanuel Kant. 

“The vastness of Kant's influence on Western thought is immeasurable. Over and above his influence on specific thinkers, Kant changed the framework within which philosophical inquiry has been carried out from his day through the present in ways that have been irreversible. In other words, he accomplished a paradigm shift.” - http://en.wikipedia.org/wiki/Immanuel_Kant

Best characterized by the notion, “we already have all the answers we just need to ask the right questions.” Kant felt that human beings exercise some fundamental misapplication of knowledge, that is, we have lots of intellectual ability, but for some reason we are unable to apply it appropriately. This results in misconceptions based upon preconceived notions we are unable to shed without a significant and rigorous amount of reflection.

For example, prior to 1905 a great deal of physicists spent thousands of dollars and hours trying to reveal evidence of a, “luminiferous” ether. It took the mental faculties of Albert Einstein to dispel this fundamental obstacle. Noticeably, the matter was reconciled through re-interpretation. That is, the mathematics was telling humanity something and once that “something” was understood, the mathematics did not change. Einstein’s interpretation simply clarified that we were making use of the wrong constants and variables. Once time was identified as a variable and the speed of light became a constant, everything (but the mathematics) changed. This fundamental crisis in modern science instituted a paradigm-shift and ushered in a complete scientific revolution. However, this ability to develop new perceptions regarding the natural world can be characterized as similar to “sleepwalking.” Philosophers of science such as Karl Popper (The Logic of Scientific Discovery) and Thomas Kuhn (The Structure of Scientific Revolutions) have shown that human minds do not consciously develop these ideas, but rather, stumble upon them as the result of exposure to elements of the concepts and ideas over time.

Similar observations in history offer scenarios where the matter of human interpretation was the basic obstacle between progress and understanding. Another example would be Karl Friedrich Gauss’s belief that Euclid’s fifth axiom fails (the parallel postulate). The resulting change in mathematics ushered in the era of Georg Friedrich Bernhard Riemann’s, non-Euclidean geometry, which allows division by zero and correctly superimposes 182-degree triangles on the surface of the Earth.

It has long been stated that division cannot be allowed when using real numbers because it introduces nagging consistencies in logic. Examples such as 1 = 2 are (correctly) shown, as reasons for the justification that division by zero is not allowed. Even more mystifying is the notion that division by zero remains undefined. Perhaps, reasons behind the mystery become evident when understanding that all difficult problems in philosophy are typically set aside for later while things more easily understood are contemplated for furthering progress in human understanding. Therefore, it is not unreasonable to assume the properties of dividing by zero, are simply set aside, to be investigated later. This allows mathematics to move forward with useful applications while the nagging and poorly understood relevance of division by zero is set aside with an arbitrary rule that suspends its effects on the real number system.

Kurt Godel made progress in furthering mathematical understanding with his “incompleteness theorem,” which asserts that mathematical theories, which are complete, will remain inconsistent. Alternatively, theories, which are consistent, will remain incomplete. This would seem to shed some light on the mystifying parameters surrounding zero. Notably, multiplication and division are inverse operations, yet, with respect to zero, this is not allowed. That is, it is ok to multiply by zero but division by zero is not allowed. This momentarily suggests these are not inverse operations while at all other times multiplication and division are inverse operations.

Not altogether differing from the example above is the changing definition of what a quantity such as zero represents at times asserting it is simply a, “place-holder” on the number line. This implies zero holds a place equal in value and quantity as any other number in the line. This “place” effectively becomes every bit as representative of a quantity as any other place on the line. As a result, this makes zero at a minimum equal in value to at least one, or negative one. Other systems of math suggest that division by zero will yield infinity, which is indicative of anything other than nothing (similarly, its quantity is not the fixed equivalent of any other number on the line). Seemingly, for such a rigorously defined science as mathematics, these perplexing notions require additional investigation in order to develop a consistent definition for the properties of zero.

The purpose of pointing out these nagging little problems is not to develop some nonsensical dilemma designed upon simple semantics. However, semantics has everything to do with the problem. Seemingly, what zero represents and how it is to be handled is entirely based upon a system of semantics related to each specific situation. Zero seems to be at the root of a fundamental philosophical conundrum. Simply put, “even nothing is something, the something it is – is nothing,” at other times, its value is as significant a variable as time is in relativity.

Word games outlined in the form above can easily be interpreted as a simplistic desire for the author to be some sort of smart-alec pointlessly complicating a very simple problem for purposes not altogether clear. This is not the intent, furthermore, the development of semantics offered in the answer above are simply the result of evolving definitions for the meaning of zero in differing situations.

In addition, the evolving definitions that reflect a system of semantics surrounding zero clearly are not as conclusive as results shown by Dr. Anderson, the IEEE arithmetic model, Riemann’s non-Euclidean geometry, and Einstein’s use of that geometry to build a model representative of the observable world. That is, there are very real machines and applications that perform just fine using division by zero as an order of operations. Therefore, there is something unique about the concept of zero that is not entirely understood by human minds. The result of this problem is most likely the consequence of interpretation.

Two of the most significantly perplexing abstractions developed by the human mind are concepts of the existence of God and the concept of a number. Both remain elusive for concrete definition and proof beyond immaterial abstraction. They are two of the most fantastic inventions of human reason and consistently elude formal proof. Are numbers real? Is God real? Trying to provide a definitive answer to these questions has plagued the greatest of minds well beyond a period of 1200-years.

Furthermore, God and zero both entertain some unique similarities. The fact that notions such as “division by zero,” and the “presence of a personal God,” continue to spark vigorous debate is testimony to the absence of the necessary and sufficient explanations required for definition. This should not be interpreted to mean that this author believes in the presence or absence of either entity (numbers or God). Instead, the purpose of this article is designed to inspire the imaginations of the reader in the hope that someday clarity and understanding for both of these perplexing dilemmas will receive resolution.

While it is certain that the absence of clear definition regarding division by zero introduces properties that to a large degree remain incomprehensible, the fact that we can contemplate such possibilities suggests that someday a relevant theoretical formula will offer enough structure for this operation to offer plausible explanations about nature.

Similarly, there may be some effective method in which a proof of God (or the absence of) may become a reality. In fact, earlier philosophical attempts such as the ontological, cosmological, and teleological arguments (for the existence of God) offer more promise in this area than any ideas regarding division by zero in the real number system. However, there are some interesting coincidences observed when using zero in mathematical operations. For example,

Let one be equal to the existence of God

Let zero be equal to the absence of God

Then the statement 1 = 1 is an identity (true statement)

Hence, adding or subtracting from both sides:

1 – 1 = 0                and        1 + 1 = 2

Conversely, multiplying or dividing by both sides:

1 x 1 = 1                and        1/1 = 1

So far, no problem arises and the system remains logically congruent.

However, looking further, we find:

Given 0 = 0 (starting out with the absence of God as an assumption)

Adding or subtracting from both sides:

0 – 0 = 0                and        0 + 0 = 0

Conversely, multiplying or dividing by both sides:

0 x 0 = 0                and        0/0 = 1 or, is undefined? (ending up with the presence of God as a result).

Dividing both sides by zero equals either, one, or remains undefined. Hence, even nothing is something, the something it is, is nothing. The ability to create something from nothing seems to resemble the fundamental scientific premise for the origin of the Universe. The suggestion that there is a God or the answer is undefined is complicated by the fact that the symbol for infinity (a sideways figure eight) is also the symbol for undefined. This seems to allude to the proposition that if God is undefined, God is also infinite.

The purpose here is not to advance notions that mathematics can prove the existence of God. Instead, by examining the behavior of zero in division coincidentally reflects a property that seems to match what modern science suggests heralds as the origin of our Universe. Curiously, this could be the result of people like Einstein and Riemann developing systems that use this property as a fundamental aspect of their theories. Alternatively, this could be an accurate reflection of the origins of the Universe.

In addition, if relevant aspects of any mathematical theory designed to make accurate predictions regarding real properties of nature must generate a geometry consistent with observation, it would seem to follow that the mathematics of such a theory would also offer evidence of the presence or absence of an intelligent designer. That is, if such an entity exists, an accurate theory of cosmology should make predictions regarding such an entity. In this case, the cosmology of physics and the cosmology of hermeneutics (not necessarily verbatim with biblical content) should become congruent if such a concept of God is valid. Alternatively, such a theory should also be able to prove irrefutably no such being exists.

At present, we can make the assertion that the very same biological forces within the human mind that conceived notions of God, also conceived explanations (Einstein’s General theory of Relativity) of the Universe consistent with those notions. This would mean empirical science is little more than the organized witchcraft of some disciplined empirical mediums (scientists). On the other hand, human minds have developed a system (science) that empirically derived the proper answers to the function of the Universe and an inherent result of those empirical discoveries is a persistent notion of the plausibility of a God.

Finally, whatever the reader wishes to believe, the point of the paper is to suggest that perhaps there is another possibility with respect to division by zero. That possibility is that we do not yet understand completely what allowing division by zero means (the results of such equations and definitions are incomprehensible or nonsensical). However, that is the meaning of undefined and this is not the equivalent of impossible. Furthermore, the result of division by zero does not affect the operations of numbers, but rather, requires the correct interpretation of those operations. If there is anything relevant to division by zero, it likely involves interpretation by human beings – not anything to do with mathematical functions.

Someday, the notion of dividing by zero may become elevated to the equivalent of a crisis. At that time, a paradigm-shift will result in a new method of thinking. This may usher in an era of unprecedented advances and consequently a new scientific revolution. Certainly, at the moment Dr. Anderson, the IEEE arithmetic model, Riemann’s non-Euclidean geometry, and Einstein’s use of that geometry are all examples of division by zero being used in real-world applications. Furthermore, Einstein’s equations (using Riemann’s geometry where division by zero is allowed) reflects what is really observed in nature. This makes the 0/0 = (1, infinite, or undefined) example uncanny in terms of a proof for the presence of a God.


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