http://www.boundaryinstitute.org/articles/Relation-arithmetic_Revived_v3.pdf
Despite its great promise, relation-arithmetic didn’t get very far. The > problem is that the most important combining operators for relations, such > as cross product and join, are not invariant under similarity. That is, if > A is similar to A’ and B is similar to B’, it does not follow that the > cross product or join of A and B is similar to the cross product or join of > A’ and B’. This can be seen from a very simple example. Consider a unary > relation R with only one tuple (its table has one row and one column). Let > the value in that row and column be v. Replacing v by any other value w > produces a relation R’ that is similar to R. Now consider the cross product > (Cartesian join) RR; it consists of the ordered pair vv. But the cross > product RR’ consists of the ordered pair vw, so RR and RR’ are not similar, > despite the similarity of their components. > Again there is an analogy to geometry. You can’t combine geometric shapes > into a single shape if all you know is their similarity classes, since that > doesn’t tell you their relative sizes. For that you need to know their > congruence classes, which requires that they be parts of a single > encompassing “context shape” called space. Is there an analogous solution > to the problem of combining relations? > In fact there is. Fortunately, congruence has a relational analogue. To > make use of it, however, we can no longer work with relations given > separately but must work with partial relations within a single > encompassing context relation. By a partial relation will be meant any > subset of cells in a relation table (see Section 2). Given a context > relation C, part A of C is called congruent to part A’ of C if we can map > the rows and columns of C that define the members of A onto those that > define the members of A’ in such a way that corresponding cells in A and A’ > have the same values. If we substitute congruence for similarity in the > definition of relation-number, then operators like product and join can in > fact be defined in an invariant way, and Russell’s conception of > relation-arithmetic makes sense. Since Russell’s definition of these words > is not in general usage, this substitution should not produce confusion, so > let us hereby make it: > *A relation-number is defined as an equivalence class of partial relations > under congruence. * https://web.archive.org/web/20060927064015/http://www.boundaryinstitute.org/articles/Structure_Theory_v7.pdf "I think relation-arithmetic important, not only as an > interesting generalization, but because it supplies a symbolic technique > required for dealing with structure. It has seemed to me that those who are > not familiar with mathematical logic find great difficulty in understanding > what is meant by 'structure', and, owing to this difficulty, are apt to go > astray in attempting to understand the empirical world. For this reason, if > for no other, I am sorry that the theory of relation-arithmetic has been > largely unnoticed." Bertrand Russell in "My Philosophical Development" There is a poisonous scholarly article out there titled "Whatever Happened to Relation Arithmetic" that fails on a number of levels, not the least of which is the pedantry of a guy named Newman whose critique amounts to claiming that it is futile to impute structure to the empirical world because any correlation we might find may be spurious -- among other nonsense such as that Tarski's model theory provided arithmetic operators adequate to the task set forth by Russell. ------------------------------------------ Artificial General Intelligence List: AGI Permalink: https://agi.topicbox.com/groups/agi/T52749d62f73acb31-Mfdad06f8f9cce486c7cad97e Delivery options: https://agi.topicbox.com/groups/agi/subscription
