Laws of Form
From the Simple English Wikipedia, the free encyclopedia that anyone can change
Laws of Form ( LoF) is a book by G. Spencer-Brown, published in 1969. It is about mathematics, philosophy, and logic. Spencer-Brown called the mathematical systems of LoF the "primary algebra" and the "calculus of indications."
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[change] The book
LoF grew out of the author's work in electronic engineering, and out of an evening course on mathematical logic he taught at the University of London. LoF has been published in several editions and translations, and it has never gone out of print. LoF is a short book, and its mathematical part is only 55 pages long.
Spencer-Brown's philosophy was influenced by Ludwig Wittgenstein, R. D. Laing, Charles Peirce, Bertrand Russell, and Alfred North Whitehead. Much of LoF touches on what is now called cognitive science. When LoF was written, cognitive science did not exist.
[change] Reception
LoF was praised in the Whole Earth Catalog and has become a cult classic. Those who admire LoF see it as a pioneering work about the "mathematics of consciousness." They see the calculus of indications and the primary algebra as a way to think about a fundamental mental action: the ability to distinguish. LoF argues that this ability is the foundation of human cognition and consciousness. According to LoF, the primary algebra reveals new connections among mathematics, cognition, and the philosophy of language and mind.
[change] The mathematics
Let 0 and 1 be the two basic primitive values of Boolean algebra. Let AB denote a binary operation of Boolean algebra. Let (X) stand for the Boolean complement of X. Then the calculus of indications is simply Boolean arithmetic reduced to the two equations 11=1 and (1)=0. These are the only "axioms" in LoF.
The primary algebra is mainly a simpler notation for Boolean algebra, except for one thing. In Boolean algebra, () is not defined. () is "empty" complementation (the complementation of "nothing"). On the other hand, in the primary algebra () is defined, and stands for one of 0 or 1. (()) stands for the other primitive value, and is the same thing as the blank page.
Let A and B be any two expressions of the primary algebra. The primary algebra is made up of equations of the form A=B, and these equations are treated in the same way as the equations of the number algebra taught in all schools. Standard methods of logic seldom use equations. LoF argues that doing elementary logic with the primary algebra is easier. In particular, if A is a tautology in logic, then one of A=() or A=(()) holds in the primary algebra.
LoF proves that the primary algebra:
- Cannot prove both A=B and A/=B. Hence the primary algebra is free of contradiction (is consistent);
- Can always prove whichever of A=B and A/=B happens to be true. (The primary algebra is complete.)
Hence the primary algebra is a well-behaved piece of mathematics. It can be useful even if the philosophy and cognitive science of LoF are wrong or uninteresting.
[change] Reference
- Spencer-Brown, George, 1997 (1969). Laws of Form. E. P. Dutton.
