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Oh, the nullity

Posted on Fri, 8 Dec 2006 12:24:23 +0100 - Keywords: debian

Erich Schubert writes:

[A rather erroneous rebuttal of Anderson’s “new” tool]

Please. If you are going to use maths to refute a paper, at least read the paper, not the BBC news article.

• You cannot directly write 2×(0/0) = (2×0)/0, no axiom allows that; you cannot “prove” that 2×Φ = Φ, it is a result given by axiom A15.
• You cannot directly write (0/0)+(1/1) = (0×1+1×0)/(0×1), again, no axiom expresses this kind of distributivity. Axiom A4 however states that Φ+1 = Φ.
• Same for (0/0) / (0/0) = (0/0) × (0/0) and (0/0) × (0/0) = (0×0)/(0×0). To prove that Φ/Φ = Φ you should use axioms A22 and A15.
• An equation cannot equal nullity, you probably meant “expression”.
• In the last step of your “simple calculation why you must not be allowed to divide by zero”, you divide both expressions by Φ, which is allowed by axioms A22 and A15, but the final statement is wrong, you wrote 2 = 1 instead of Φ = Φ.

In fact, dividing by zero can be useful in calculus. What the guy did is design a consistent system (I trust him at least on this) that resembles the IEEE floating point algebra, except that in his system Φ = Φ while NaN ≠ NaN. Such a distinction makes it possible to write and solve equations where Φ may be involved. Of course the final result may yield Φ but the important part is that it is consistent, proven, and remains valid whatever the input values.

What should be really questioned is:

• The usefulness of it all, of course. As a mathematic tool, since many simple theorems and tools (which often go far beyond what we learn in school, such as simple recursion) do not directly apply and need to be proven again, which is a tedious task. As a computer science tool, the benefits of having Φ = Φ versus NaN ≠ NaN remain yet to be explained.
• Teaching that to kids.
• The lack of a serious peer-reviewed publication. At the moment it’s in the proceedings of the Vision Geometry conference.

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