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A number cannot be divided by zero. This is what mathematics today claim. But think again. Some years ago, it was not possible to have a number whose square results in a negative number. But then… we thought… why not? And now we have imaginary numbers with tons of applications. Zero is a useful tool and a curse. It represents “nothing”, which philosophically does not exist. But yet, we see it every day in our calculations. Everything is a matter of definition and the sentence “division by zero does not have any meaning” is such a case. If we decide to give meaning to it, then it will instantly have. In a sense we have already allowed division by zero. Consider cases when we want to find the limit (lim) of a fraction whose denominator tends to zero. Mathematics have a great deal of weird theories – why not a theory where division by zero is allowed? The limit is only set by us. And even though many think that allowing division by zero we introduce many inconsistencies in mathematics, consider this: According to Gödel no theory based on specific set of axioms can prove its consistency. Could a theory full of antiphasis be actually the most consistent theory of them all? Thinking logically never served the progress of Science. Zero entails antinomies. And life is full of them. Perhaps the best way to describe the world would be to simply accept what our radical heart want but our cold stubborn logic rejects… Discover the world! Divide by zero!

Below you can find some articles (from other sites) and/or ideas regarding the question “Is division be zero possible?”

Collection of relates papers

Paper 1

To Continue with Continuity Metaphysica 6, pp. 91–109, a philosophy paper from 2005, reintroduced the (ancient Indian) idea of an applicable whole number equal to 1/0, in a more modern (Cantorian) style. [http://www.metaphysica.de/texte/mp2005_2-Cooke.pdf, http://en.wikipedia.org/wiki/Division_by_zero]

Paper 2

[http://www.helium.com/items/582924-division-by-zero-is-it-really-impossible]

A beautiful question and the most sought after answer no doubt. Dividing by zero is mathematically impossible, for now. Not too long ago we thought we couldn’t land on the moon and we did that.

The simple question is: how is it possible for us to divide something by nothing. This would be the answer for time travel and from what I understand is what a black hole is supposed to represent. Having such a huge amount of gravity that it will break it down beyond a singularity into nothingness.

I find this question more of a philosophical one. If I were to put something into a black hole and create a zero value for something that used to exist where does it go? If we could divide by zero we could prove Einstein wrong. We could travel through time.

I had an awesome math professor in college who was all about dividing by zero and talking about the endless amounts of universes and dimensions within them. He would say that if we were to be able to travel in time it wouldn’t do us much good. Since on the return trip we would not be in the same dimension as the one we left. It comes down to the story of traveling back into time to prehistoric ages and stepping on a butterfly.

I guess until we solve the problem the answer really has to be nunreasonable or reasonable to belief in god or the big bang theory. Although the big bang theory does seems like it has makes more sense. Personally I would like to belief in a combination of the big bang and a spiritual connection. There could have been a singularity, call it god if you will, from which the universe was created. In either case you would have to accept that you are in fact godly since we were all created from this singularity. Maybe we’ll have this under another discussion in the philosophy section.

Paper 3

[http://www.helium.com/items/1242048-dividing-by-zero-bhaskari-dr-james-anderson]

In 2006, Dr James Anderson, a computer science professor at the University of Reading (UK) boldly announced that he had solved a very important problem. It was a problem that has perplexed academics and anyone of a vaguely scientific or mathematical ilk since about 800AD. The big announcement, carried by the BBC, was that Dr Anderson had devised a proof that it was possible to divide by zero.

His proof is not possible to write here because of the mathematical symbols involved, but it was based on the idea of using 0/0 and what he called a “transreal” number. Rather than being a proof, Anderson’s workings assume that it is possible from the start, so it is a logical fallacy. What it did do though was create an uproar in academic circles. Given that there hasn’t been a major discovery in the field of mathematics for a very long time, perhaps they felt a little aggrieved at all Dr Anderson’s attention. As a number of critics pointed out, you can use Dr Anderson’s logic to prove anything:

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0 x 1 = 0 and 0 x 2 = 0; therefore

0 x 1 = 0 x 2; so dividing both sides by zero gives:

1 = 2 (which is clearly rubbish.  And you can substitute 1 and 2 with anything).

It assumes that the very thing to be proven is valid at the start and is clearly an abuse of logic. Monty Python employed a similar logic in the audio version of their classic ‘Monty Python and the Holy Grail’.  It goes something like this:

Statement 1 = I like to eat kippers for breakfast,

Statement 2 = Kippers live in water, and

Statement 3 = Water comes from rain.

And the brilliant conclusion:

If I don’t eat kippers for breakfast, it will not rain.

Anyway, to move away from Dr Anderson and to some discussion that supports the proposition that it is possible to divide by zero.

In the 12th century, an interesting Indian mathematician and astronomer by the name of Bhaskara came up with a brilliant treatise called the Lilavati.  This dealt with all manner of arithmetic concepts, one of which was the properties of zero and things that you could do with it. I say interesting because he is also known as Bhaskara II and Bhaskara Acharya (which means Bhaskara the Teacher) and he was one of a long line of mathematicians and astronomers. One can only imagine what their dinner table conversation would have been like.

Bhaskari was into astrology, fair enough given his interest in mathematics and astronomy, and one of the enduring legends concerning him has to do with his daughter, Lilavati (his work on arithemtic is named after her and said to be a gift to her). Bhaskari read her horoscope as she was planning to wed and somehow worked out that her husband to be would die soon after their marriage unless the wedding took place at a particular time. To help work out when that time would be, he took a cup and made a tiny hole in it before placing it in a water filled vessel. He aligned the volume of the cup and the hole in such a way that the cup would sink during the hour when she needed to be wed. Like Eve and the Forbidden Fruit, she was told to go nowhere near the thing and couldn’t help herself. Sure enough, her curiousity led to disaster. Her nose ring popped out and fell into the cup, disturbing it so that she ended up marrying at the wrong time. Her husband died shortly after the wedding.

Bhaskari’s big pronouncement was an important one – that any finite number divided by zero yielded infinity. He stated it in slightly more grandiose terms, namely “In this quantity which has zero as its divisor there is no change even when many [quantities] have entered into it or come out [of it], just as at the time of destruction or creation when throngs of creatures enter into and come out of the infinite and unchanging [Vishnu]”. Regardless of the flowery language, it was a brilliant deduction that prevails to the current day.

To think of this in more understandable terms, if you take any number, for argument’s sake we’ll use 148. If you deduct 2 from it, it becomes 146 and you can do this 74 times until it becomes zero. If you deduct 1, you can do twice as many times, ie 148 times. What this means is that as you reduce the number you are deducting (in this case going from 2 to 1), you can do the exercise more often (74 to 148). Now, subtract zero off your 148. It stays as 148. You can keep doing this forever and it will always be 148.

As Bhaskari suggested, you can do this an infinite number of times, and on this basis, you can clearly divide a number by zero. Anything divided by zero yields an answer of infinity.  Some critics may argue that this is not a real number but that is iirelevant for the purposes of this debate.  It is an answer.  Division by zero is possible.

Paper 4

[http://www.helium.com/items/1360161-division-by-zero-is-it-really-impossible]

Division by zero is done on the Riemann sphere in the complex plane and its result tends to infinity whereas zero divided by any other number tends to one. It is also important to note that in the real number system (and on better graphing calculators), division by zero is not impossible but remains undefined. http://en.wikipedia.org/wiki/Riemann_sphere

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Georg Friedrich Bernhard Riemann, German mathematician, student of Carl Friedrich Gauss, and mathematical genius developed numerous theories in mathematics such as Riemann partitions, The Generalized Riemann Hypothesis, Non-Euclidean Geometry, and extensive work in the realm of imaginary (complex) numbers, and was the mathematician Einstein selected for use in the General Theory of Relativity. Riemann’s work in Non-Euclidean geometry just happens to fit exactly with what is actually experienced when taking measurements on the surface of the Earth.

That is, triangles laid upon the surface of the Earth add to 182 degrees instead of the Euclidean 180 degrees. This is due to the failure of the Euclidean fifth axiom, the Parallel Postulate. Its failure is immediately evident to anyone looking at a spherical model of planet Earth where longitude lines at the equator cross at the North Pole proving failure of the parallel postulate (parallel at the equator and crossing at the North Pole). Carl Friedrich Gauss suspected this as well as the existence of non-Euclidean geometry, and the number of primes less than a given magnitude yet did not disclose choosing to let his students discover and receive credit for the items. His most significantly accomplished student was Georg Friedrich Bernhard Riemann.

It is fascinating to note that real numbers generally tend to give imaginary results and imaginary numbers tend to give real results. Meaning that Euclidean results tend to give ideal answers verses real answers and non-Euclidean geometry gives answers consistent with what is really observed in our natural world. This is not to detract from Euclidean mathematics merely that a perfect triangle is merely an idealization and not typically encountered in the real world. Certainly, man can create one but trying to find one in the real world would be as challenging as trying to find a straight line. A straight line laid out on the surface of the Earth would tend to describe a geodesic. Nevertheless, this remains for another topic.

Since the consequences of Riemann’s interpretation of non-Euclidean geometry tend to reflect what is actually observed here on Earth, it is tantalizingly imminent that Einstein would indeed use this mathematical model for the description of gravitation. How else could one conceivably proceed? This also provides compelling evidence of Riemann’s stature as a founding father of modern physics, a circumstance that will inevitably become more prevalent upon a conclusive solution to the Riemann hypothesis.

In order to provide some degree of logical reasoning for the necessity of division by zero in some (but not all) circumstances, examine the incompleteness theorem presented by Kurt Godel.

“Godel showed that truth cannot be contained within the limits of strict logic. Only if we allow paradox can truth completely reveal itself in form. These two sides of Godel’s proof represent the apophatic (via negativa) and cataphatic (via positiva) approaches to truth, respectively. In the apophatic approach, one adheres to strict logic to show that any attempt to represent or speak truth necessarily failsthe truth is beyond all rational comprehension. In the cataphatic approach, on the other hand, one embraces paradox and the coincidence of opposites to demonstrate the tangible presence of truth in all its limitless expressions. Like the ancient mathematics of Pythagoras, Godel’s mathematical proof can be seen as a symbol of profound truths about the relationship between the limited and the unlimited, form and formlessness, transcendence and immanence, Godel’s postmodern mathematics undermines any attempt to fixate on any totalizing axiomatic system for mathematical discourse, and reveals the essential ambiguity, openness, and emptiness of mathematical activity.” Thomas J. McFarlane Spring 2000 Revised and Edited for the Web March 2004.

Inherent to Godel’s proof is the conclusion that if a system is consistent, then it is incomplete. Conversely, if a system is complete, then it must be inconsistent. Considering this fact when examining such a precept as the impossibility of division by zero, we would be relegated to an inconsistent mathematics. Alternatively, entertaining the possibility of division by zero generates consistency, yet leaves us with an incomplete understanding of mathematics, which is precisely the circumstance with which we are dealing. Therefore, in some circumstances, one can in fact divide by zero.

Related sources

  • http://en.wikipedia.org/wiki/Bernhard_Riemannnn
  • http://en.wikipedia.org/wiki/Riemann_sphere
  • Mathemati cal Poetics of Enlightenment Thomas J. McFarlane Spring 2000, Revised and edited for the web March 2004. http://www.integralscience.org

Paper 5

[http://www.roangelo.net/logwitt/logwitt8.html]

At school we were taught that mathematics is about the properties of numbers and that numbers are abstract objects. How we were to verify this (how to read off from the object its properties), we were not taught. Perhaps we thought that if only we knew more mathematics…. What we were taught at school were grammatical myths — fanciful inventions about the meanings of mathematical signs that have no relation whatever to how those signs are used. And knowing more mathematics will not enlighten us here, because this is a philosophical not a mathematical problem.

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The sign ‘2/x’ is an instruction telling us to divide 2 by whatever number ‘x’ is replaced with. But replace ‘x’ with ‘0’ and we do not know how to go on. We ask our teacher who answers that “division by zero is impossible”. But what kind of impossibility is this? The impossibility is grammatical — i.e. the sign ‘2/0’ is an undefined combination of signs (in just the way that ‘2/+’ is). In a word, division by zero is only impossible because ‘division by zero’ is undefined (language); that is the only meaning ‘impossible’ has here. [Note 1]

But how do I know? Because I only talk about what I know — about the use of mathematical signs that I learned at school. And what I do not know does not concern me here, because: what a sign is used to do in a particular context is the sign’s meaning in that context. (Wittgenstein’s Lectures on the Foundations of Mathematics p. 13-4). What I can describe are some ways that mathematical signs are used, namely, the ways I and many other people were taught in our early school years. I won’t try to talk about anything beyond addition, subtraction, multiplication, division and elementary algebra — because that is the mathematical language I still remember and these are still the techniques that I use in my daily life.

The Philosophy of Mathematics is not concerned with this or that particular mathematical calculus (metaphorically: “game governed by strict rules”), but with the nature of mathematical calculi as such. But to investigate these calculi, it must look at particular calculi, and it may discover that all the things we call ‘mathematical calculi’ do not have a common nature. But I know nothing about that; I myself am acquainted with very few calculi and not at all with what is called “higher mathematics”. All that concerns me here is trying to understand the nature of the mathematical language I learned at school — e.g. the ways it is like or unlike natural language. School left me and many other people with a lot of confused ideas about the nature of mathematics, and one task of philosophy is to clear up that confusion.

When we don’t know how to go on, what we need is a rule.

If A x C = B x C then A = B. But then if C = 0, any number can be proved to be equal to any other number. But we do not want our mathematics to be that way. So we make the rule ‘but C may not equal zero’. (WLFM p. 221-2) But why? Is it because something is inherently wrong with a contradiction, e.g. 4 = 5? Or is it because we want our mathematics to have applications outside itself? We do not want, for example, any amount of money to be equal to any and all other amounts. But if we allowed C = 0, would that be the end of mathematics? Or would it just be a different mathematics? ‘C = 0 is a rule mathematicians resort to when they get tired.’ Why not? Because can’t we imagine a people who only ever used mathematics the way we use chess — i.e. as a game with no subject matter outside itself?

Confronted with a contradiction, or any other undefined sign, the student does not know how to go on. But what do we do when we are confronted with a contradiction? We proscribe it, and move on. We make a rule: ‘but C may not equal zero’, or ‘but 2/0 is undefined’.

When we teach a child chess, we give it the rule: the bishop only moves along diagonals. Suppose we play a game and the child moves its bishop along a diagonal and clear off the board (the child says it wants to protect the bishop from attack). We now state the rule: but the bishop must remain on the board. But would it be the end of chess if we allowed the child’s move?

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