On terminology for approaching the multiplicative inverse of zero

Jan Bergstra, University of Amsterdam,

March 2016

Posted by James Anderson

What can be said about the value of 1/0? This question has led to may speculations and increasingly the issue constitutes an incentive for the development of mathematical and logical theory.

Different groups work on different options while promoting their view on the matter. In several cases a combination of design choices is captured in a single name for a kind of structure. Here are some examples of that approach:

(i) Wheels are structures with numbers in which 1/0 is an unsigned infinite value, while 0 x 1/0 is an error value for which I will write "a” as an abbreviation of additional value) that propagates though all operations. In a wheel 1/0 = -1/0, and 1/0+1/0 = 1/0. The error values enters the picture via multiplication: 0 x 1/0 = a.

(ii) transreals, transrationals are number systems in which 1/0 is positive signed infinite. Now 1/0 and -1/0 are different and 1/0 - 1/0 (= 1/0 + -1/0) = a.

(iii) meadows are number structures for in which 1/0 = 0.

(iv) common meadows are structures in which 1/0 = a.

Each of these approaches has a long history. For instance wheels stand in the tradition of the work of Riemann. Wheels formalise the rules of calculation in the Riemann sphere. Working with 1/0 = 0 has shorter history. The technical work on mathematical logic based on 1/0 = 0 seems to have started in papers by Komori and Ono some decades ago, the work on meadows proceeds along that line of research. More recently Saitoh and his colleagues promote 1/0 = 0 as a step forward in the design of classical mathematics while providing may examples for its use. Systematically working with transreals has been taken up by Anderson. The above list of approaches is incomplete, some authors insist that 1/0 = 1, and fixing a value of 1/0 still leaves room for differences in setting a value for 0/0.

In this contribution I wish to propose an alternative approach to naming the various design decisions on which the different approaches are based. The idea is that instead of (or in addition to) naming structures one may also provide more detailed naming for the function x -> 1/x.

PARTIAL INVERSE: this is the inverse function for which 1/0 is considered to be nonexistent. I expect that for most mathematicians the partial inverse constitute the most plausible option.

NATURAL INVERSE: the inverse function with 1/0 equal to unsigned infinite. This choice dates back to Riemann and may be the first significant proposal for dealing with 1/0 as a mathematical entity.

SIGNED NATURAL INVERSE: the inverse function with 1/0 denoting a positive infinite value.

SYMMETRIC INVERSE: the inverse function extended that takes value 0 on 1/0. The symmetry refers to the point symmetry in the graph of x->1/0 as defined in this manner.

COMMON INVERSE: the inverse function that takes value a (the euro value mentioned above) on 1/0. The justification of this name is that in computing it is quite common to consider 1/0 an error.

The usefulness of this naming scheme lies in the improved possibility to compare different approaches. Rather than speaking of wheels versus transreals one may speak of number algebras (or structures) with (unsigned) natural inverse versus number algebras with signed natural inverse. The approach of Saitoh reveals a preference for doing mathematical work with symmetric inverse over working with partial inverse.

My own current preference is to work with symmetric inverse and I have done som work on structures with common inverse. A survey of that work can be found on the following website which is maintained by Kees Middelburg:

https://meadowsite.wordpress.com

Jan Bergstra, University of Amsterdam,

March 2016

Posted by James Anderson

What can be said about the value of 1/0? This question has led to may speculations and increasingly the issue constitutes an incentive for the development of mathematical and logical theory.

Different groups work on different options while promoting their view on the matter. In several cases a combination of design choices is captured in a single name for a kind of structure. Here are some examples of that approach:

(i) Wheels are structures with numbers in which 1/0 is an unsigned infinite value, while 0 x 1/0 is an error value for which I will write "a” as an abbreviation of additional value) that propagates though all operations. In a wheel 1/0 = -1/0, and 1/0+1/0 = 1/0. The error values enters the picture via multiplication: 0 x 1/0 = a.

(ii) transreals, transrationals are number systems in which 1/0 is positive signed infinite. Now 1/0 and -1/0 are different and 1/0 - 1/0 (= 1/0 + -1/0) = a.

(iii) meadows are number structures for in which 1/0 = 0.

(iv) common meadows are structures in which 1/0 = a.

Each of these approaches has a long history. For instance wheels stand in the tradition of the work of Riemann. Wheels formalise the rules of calculation in the Riemann sphere. Working with 1/0 = 0 has shorter history. The technical work on mathematical logic based on 1/0 = 0 seems to have started in papers by Komori and Ono some decades ago, the work on meadows proceeds along that line of research. More recently Saitoh and his colleagues promote 1/0 = 0 as a step forward in the design of classical mathematics while providing may examples for its use. Systematically working with transreals has been taken up by Anderson. The above list of approaches is incomplete, some authors insist that 1/0 = 1, and fixing a value of 1/0 still leaves room for differences in setting a value for 0/0.

In this contribution I wish to propose an alternative approach to naming the various design decisions on which the different approaches are based. The idea is that instead of (or in addition to) naming structures one may also provide more detailed naming for the function x -> 1/x.

PARTIAL INVERSE: this is the inverse function for which 1/0 is considered to be nonexistent. I expect that for most mathematicians the partial inverse constitute the most plausible option.

NATURAL INVERSE: the inverse function with 1/0 equal to unsigned infinite. This choice dates back to Riemann and may be the first significant proposal for dealing with 1/0 as a mathematical entity.

SIGNED NATURAL INVERSE: the inverse function with 1/0 denoting a positive infinite value.

SYMMETRIC INVERSE: the inverse function extended that takes value 0 on 1/0. The symmetry refers to the point symmetry in the graph of x->1/0 as defined in this manner.

COMMON INVERSE: the inverse function that takes value a (the euro value mentioned above) on 1/0. The justification of this name is that in computing it is quite common to consider 1/0 an error.

The usefulness of this naming scheme lies in the improved possibility to compare different approaches. Rather than speaking of wheels versus transreals one may speak of number algebras (or structures) with (unsigned) natural inverse versus number algebras with signed natural inverse. The approach of Saitoh reveals a preference for doing mathematical work with symmetric inverse over working with partial inverse.

My own current preference is to work with symmetric inverse and I have done som work on structures with common inverse. A survey of that work can be found on the following website which is maintained by Kees Middelburg:

https://meadowsite.wordpress.com

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- James AndersonOwnerAll of Jan's five types of inverse are defined by the properties of inverting zero as 1/0 -> x. But we also need to characterise 1/x -> y. We need to add to Jan's scheme.Mar 12, 2016
- James AndersonOwnerIf 0 = 0/x then zero has, potentially, as many reciprocals, x/0, as there are admissible choices of x. This number could be greater than the cardinality of the natural numbers so we cannot hope to name all possible inverses using an ordinary language. Any nomenclature for "inverses" will be partial. It will be predicted on the algebraic structures that anyone wants to describe that way. As I said before, the most we can hope for is a nomenclature that suits Jan and all the field folks but which does not block our work on transfields.Mar 13, 2016
- James AndersonOwnerLet’s concentrate on “inverses” or, as I prefer, “reciprocals” of zero.

The reciprocal of zero may be empty, i.e. non-existent. This is the “e-reciprocal” of zero. It corresponds to Jan’s “partial inverse.”

The reciprocal of zero may be an absorptive element. This is the “a-reciprocal” of zero. It corresponds to Jan’s “common inverse.”

The reciprocal of zero may be a negative infinity. This is the “n-reciprocal” of zero. Jan did not consider this case.

The reciprocal of zero may be zero. This is the “z-reciprocal” of zero. It corresponds to Jan’s “symmetric inverse.”

The reciprocal of zero may be a positive infinity. This is the “p-reciprocal” of zero. This is Jan’s “signed natural inverse.”

The reciprocal of zero may by an unsigned infinity. This is the “u-reciprocal” of zero. This corresponds to Jan’s “natural inverse.”

This characterises six inverses of zero. In order to characterize wheels and transreals we would have to add nomenclature for the inverses of negative, positive and unsigned infinities and for the inverse of the absorptive element (our nullity.)Mar 13, 2016 - James AndersonOwner"Wheels," a note, Anton Setzer, 1997:

"The idea to extend the quotient field by allowing fractions with denominator 0 is due to P. Martin-Lof."

This earlier attribution agrees with Carlstom's attribution and contradicts the assertion that Wheels are inspired by Riemann.

The attribution to Riemann can be further falsified by a geometrical argument.

Lay off a line in the ground plane. Consider Riemann projections onto this line. Exclude the projection that passes only through the north pole of the Riemann Sphere. Now the projections onto the line correspond to real numbers. In particular all of the points on the line are ordered.

All elements of a Wheel are unordered so they do not correspond to points on the line obtained by projection of the Riemann Sphere.

Hence Wheels are not related to the Riemann Sphere.Mar 13, 2016 - James AndersonOwnerIn all of the number systems we have been considering:

The reciprocal of every infinity is zero. This is the "z-reciprocal" of infinity.

The reciprocal of every absorptive element is the absorptive element. This is the "a-reciprocal" of the absorptive element.

The reciprocal of every empty element is the empty element. This is the "e-reciprocal" of the empty element.

Thus we do not require any further nomenclature.

Jan do you have any comments on this nomenclature?Mar 13, 2016 - James AndersonOwnerThere is, of course, a seventh reciprocal. The blindingly obvious "reciprocal" of a finite number that is the multiplicative identity. Let's call this the "i-reciprocal."

Now the term "reciprocal" is analysed into seven case.Mar 13, 2016