James A. D. W. Anderson

Research and background

Anderson is a member of the British Computer Society, the British Machine Vision Association, Eurographics, and the British Society for the Philosophy of Science. He is also a teacher in the Computer Science department (School of Systems Engineering) at the University of Reading. He was a psychology graduate who worked in the Electrical and Electronic Engineering departments at the University of Sussex and Plymouth Polytechnic (now the University of Plymouth). His doctorate is from the University of Reading for (in Anderson's words) "developing a canonical description of the perspective transformations in whole numbered dimensions".

He has written two papers on division by zero and has invented what he calls the "Perspex machine".

Anderson claims that "mathematical arithmetic is sociologically invalid" and that IEEE floating-point arithmetic, with NaN, is also faulty.

Transreal arithmetic

Anderson's transreal numbers were first mentioned in a 1997 publication, and made well-known on the Internet in 2006, but not accepted as useful by the mathematics community. These numbers are used in his concept of transreal arithmetic and the Perspex machine. According to Anderson, transreal numbers include all of the real numbers, plus three others: infinity ( ∞ ), negative infinity ( − ∞ ) and "nullity" ( Φ ), a numerical representation of a non-number that lies outside of the affinely extended real number line. (Nullity, confusingly, has an existing mathematical meaning.)

Anderson intends the axioms of transreal arithmetic to complement the axioms of standard arithmetic; they are supposed to produce the same result as standard arithmetic for all calculations where standard arithmetic defines a result. In addition, they are intended to define a consistent numeric result for the calculations which are undefined in standard arithmetic, such as division by zero.

Transreal arithmetic and other arithmetics

"Transreal arithmetic" closely resembles IEEE floating point arithmetic, a floating point arithmetic commonly used on computers. IEEE floating point arithmetic, like transreal arithmetic, uses affine infinity (two separate infinities, one positive and one negative) rather than projective infinity (a single unsigned infinity, turning the number line into a loop).

The main difference is that IEEE arithmetic replaces the real (and transreal) number zero with positive and negative zero. (This is so that it can preserve the sign of a nonzero real number whose absolute value has been rounded down to zero. See also infinitesimal.) Division of any non-zero finite number by zero results in either positive or negative infinity.

However, in IEEE arithmetic, division of zero by zero is still considered indeterminate. The reason for this is simple: A statement about the quotient of two numbers is understood in mathematics as another statement about multiplication. Specifically, if

a ÷ b = c

this is understood as simply another way of saying that

a = b × c

Thus, if for some number c

0 ÷ 0 = c

then this is just another way of saying that

0 = 0 × c

But in fact this is true for all real numbers c . And that is precisely the reason that mathematicians do not assign a single value to 0 ÷ 0 but rather label it "indeterminate". Assigning a value to 0 ÷ 0 , even a newly fabricated "number", misses the point entirely.

In IEEE arithmetic, the value of 0 ÷ 0 is therefore represented by the symbol Not a Number (NaN) (Not a Number). NaN is not meant to be a number, but rather an error message conveying the fact that the arithmetical operation the computer just attempted cannot be assigned a single number as an answer – even if + ∞ and − ∞ are considered numbers. Because N a N is an error message and not a number, it is not considered equal to anything, even itself. That is, the comparison N a N = N a N evaluates to false.

Here are some identities in transreal arithmetic with the IEEE equivalents:

The main difference between transreal arithmetic and IEEE floating-point arithmetic is thus that nullity compares equal to nullity, whereas NaN does not compare equal to NaN.

Anderson's analysis of the properties of transreal algebra is given in his paper on "perspex machines".

Due to the more expansive definition of numbers in transreal arithmetic, several identities and theorems which apply to all numbers in standard arithmetic are not universal in transreal arithmetic. For instance, in transreal arithmetic, a − a = 0 is not true for all a , since Φ − Φ = Φ . That problem is addressed in ref. pg. 7. Similarly, it is not always the case in transreal arithmetic that a number can be cancelled with its reciprocal to yield 1 . Cancelling zero with its reciprocal in fact yields nullity.

Examining the axioms provided by Anderson, it is easy to see that any term which contains an occurrence of the constant Φ is provably equivalent to Φ . Formally, let t be any term with a sub-term Φ , then t = Φ is a theorem of the theory proposed by Anderson.

Media coverage

Anderson's transreal arithmetic, and concept of "nullity" in particular, were introduced to the public by the BBC with its report in December 2006 where Anderson was featured on a BBC television segment teaching schoolchildren about his concept of "nullity". The report implied that Anderson had discovered the solution to division by zero, rather than simply attempting to formalize it. The report also suggested that Anderson was the first to solve this problem, when in fact the result of zero divided by zero has been expressed formally in a number of different ways (for example, NaN).

The BBC was criticized for irresponsible journalism, but the producers of the segment defended the BBC, stating that the report was a light-hearted look at a mathematical problem aimed at a mainstream, regional audience for BBC South Today rather than at a global audience of mathematicians. The BBC later posted a follow-up giving Anderson's response to many claims that the theory is flawed.

Applications

Anderson has been trying to market his ideas for transreal arithmetic and "Perspex machines" to investors. He claims that his work can produce computers which run "orders of magnitude faster than today's computers". He has also claimed that it can help solve such problems as quantum gravity, the mind-body connection, consciousness and free will.