Saturday, June 30, 2012

A Possibly New Primary Arithmetic Theorem:

The primary arithmetic theorem "Variance" (T9) of Laws of Form (hereafter "LoF") can be written in square-bracket notation as

[[pr][qr]] = [[p][q]]r

(square-bracket is preferable to paren notation in text, since it doesn't conflict with the literary use of parentheses).

(The theorem interpreted as logic—see Appendix 2 of LoF—might read " '( p OR r ) AND ( q OR r )' is equivalent to '( p AND q ) OR r'.)

The theorem itself is proven in LoF by application of Theorem 16, "The bridge" (see the final remark below), showing it holds in either case or value of r, reducing to

[] = []

when r is dominant ( r = [] ) and

[[p][q]] = [[p][q]]

when r is recessive ( r = ).

Let p be dominant instead, and the theorem reduces to

qr = qr

while letting q be dominant yields

pr = pr

But let p be recessive, and the theorem reduces to

[[r][qr]] = r

while letting q be recessive yields

[[pr][r]] = r

The left-hand expressions of these equations are of course equiformal, since

[[r][qr]] = [[qr][r]]

and the equations therefore represent a single form of equation.

This equation can be rewritten as

[[a][ab]] = a

(since ab = ba).

It holds for all cases; dominant a yields

[] = []

while recessive a yields


and both cases of b yield

a = a

This equation is not presented in LoF as a primary arithmetic theorem, nor does it seem to be a consequence of the algebra.

This suggests that it is a new primary arithmetic theorem and might, therefore, be a new, third initial of the algebra as well, opening a whole new world of consequences.

(The theorem interpreted as logic might read " 'a AND ( a OR b )' is equivalent to 'a' ".)

Should the theorem prove indeed to be new, it is suggested that it be called "Volution", its forward step

[[a][ab]] -> a

"devolution", and its reverse

[[a][ab]] <- a

"evolution": This nomenclature is itself suggested by the reflection (LoF primary algebraic consequence C1) and internal iteration (C5) implicit as first steps in the latter, which recall the genetic duplication so critical to biological evolution, allowing mutation of a duplicate gene without losing the function of the protein encoded by the original.

The theorem might be written

If successive spaces sn, sn+1, sn+2 are arranged so that sn, sn+1 are distinguished by one cross, and sn+1, sn+2 are distinguished by two crosses, sn+2 being thus in two divisions, both divisions sharing an expression, one containing an unshared expression, then the value of the resultant expression in sn is the shared expression.

Note that

[[ab][ac]] = a

does not hold for recessive a or dominant b or c.

Finally, the above arose from a continuing skeptical consideration of LoF Theorem 16, "The bridge".

Keywords: binarics, binaries, enarics, enaries, George Spencer Brown, Laws of Form, LoF