Dog Mathematics

by J. Marshal Futrelle and W. C. Wendling

Mathematics has been traditionally conceived of as a purely theoretical endeavor, founded in such esoteric fields of investigation as set theory and topology. That mathematicians eschew field work and data-gathering to test their theories can be readily ascertained by the style of dress and attention to personal appearance exhibited by most theoretical mathematicians. But in recent years a new and much more worldly branch of mathematics has made impressive strides. The authors present here a rigorous and empirically testable dog mathematics.

Axioms

The axioms of dog mathematics were initially observed in a mixed canine-feline rural setting in West Virginia, and elaborated with an intensive (one could almost say immersive) ethnography of several highly feline urban apartment settings. Briefly stated, the axioms are as follows:
1. dogs + dogs = dogs
2. dogs - dogs = cats
3. cats + cats = -dogs
It is immediately apparent to the more mathematically astute reader that dog mathematics bears little relationship to garden-variety number theory. The oft-repeated claim that dog mathematics is inherently meaningless because of one of several variations on the following proof,
1. dogs + dogs = dogs [axiom 1]
2. dogs = dogs - dogs [subtracting dogs from both sides]
3. dogs = cats [as per axiom 2. patently absurd!]
is a mere theoreticist ploy. No one has ever observed anyone subtracting dogs from both sides of anything.

It's Raining Cats and Dogs

Addition and subtraction are some of the most easily observable aspects of dog or dog/cat environments. A less frequent but as readily observable phenomenon is multiplication. There are three primary axioms regarding multiplication which have been borne out by our observations:
4a. dogs * dogs = dogs
4b. cats * cats = cats
4c. dogs * cats = undefined
Division has rarely, if ever, been observed. Numerous animal species have been observed being divided by dogs and cats, in particular by wolves and lions. We hypothesize that the following relationships would obtain among dogs and their feline counterparts, if we could ever observe such events:
dogs / dogs = dogs
cats / dogs = dogs
cats / cats = cats
dogs / cats = cats
Obviously dogs and cats do not have roots[1]. They cannot be raised to powers; though dogs, especially retrievers, have been observed apparently in possession of small natural logs, which they carry in their mouths.

Higher Mathematics

The further elaboration of dog mathematics is beyond the scope of our current work. Although dog topology is presumably relatively straightforward, dog geometry may be intractable[2] due to the volume of data involved and a puacity of models for fur dynamics. The most promising branch of higher dog mathematics is pack theory. But in order to make advances in this crucial area, it's clear that much more empirical observation needs to be undertaken. We are actively seeking grants to cover the cost of dog food and squeaky toys.

[1] One wonders, for instance, if theoretical mathematicians have ever seen a truly square root; the roots we have observed have all been gnarled.

[2] Although preliminary observations indicate that dogs + pie = dogs.