Where are we on relations and negative numbers?
P.A.R.P. = Pivoted Arbor Rings Puzzle
When the engineer has solved puzzle, he may pause to think: "ah! a mathematical relation!". Even before the solution has occurred to him, his thought may alight upon mathematical relations: in system and world.
So is Stanley on the money, when she accuses Livostein (see the link to right of this page) of a theft in the heyday of New Mathematics?
The good thing is the rallying and revival now available.
Engineers get some leeway with design. Motorcars, for example, can built to many different designs. Ontologies are a bit different. There is room for the opinion that people do not have much leeway in the design of a fundamental ontology.
It is time, surely, to regroup and to think of set only when there is a relation graph needed. For elementary level mathematics students, that used to be the way; maybe not everywhere, not at every school, but logical naming is cool. When all of a referent has been taken away by drawing exceptions, zero is the referent. Here is the pure, real zero. It is not a set of any kind. Nor is it a mathematical subtraction that initially brings it on. And although subtraction sometimes arrives at zero, subtraction may have the propensity to take us beyond zero into a territory of negatives that exception cannot reach.
In the old way of thinking, 1 comes from expression 0 + 1. A student discovers 0 and then the ideas of algebraic operation and expression shortening. Subsequently it is the properties of numerals that dominate the first elementary lessons. One does not need to reach into the territory of philosophy at this stage. Nobody need care whether a set-theoretic concept annoints the lessons.
That the objective of analysis broadly conceived is intimate with the theory of relations is amply demonstrated in one of the affidavits discussed under the Doll's Writ section of this website (see link to right of page). Therein a frame of reference is worked out suitable for the definition of the joule as a unit of energy. Eh, but it is the teaching of this theory of relations that is compromised by the constructivists' set-theoretical paradigm in mathematics. Hence to the discussion imagined between Stanley and Livostein, in 1969.
Stanley & Livostein
(a discussion about relations and sets - for teachers)
Proof of Euclid's Fifth
(proceeding from the investiture of straightness with a relational linearity)