I was recently made aware of Spencer-Brown's work (see his book) on what is known as Laws of Form (LoF) or Calculus of Indications. I did find some very sparse literature on the subject, including a well written draft by Kauffman. I am naturally skeptical of relatively unknown theories that make grandiose claims, so mathematical rigor is always something I am very happy to see.
On the other hand, I have to admit that - besides some subjective mathematical "elegance" - I was not able to identify any new results, i.e. statements we did not know before. Moreover, it seems there's evidence in the literature, that the theory of LoF is simply a reformulation. For example, wikipedia informs us that:
- Banaschewski (1977) argues that the primary algebra is nothing but new notation for Boolean algebra.
- Schwartz (1981) proved that the primary algebra is equivalent — syntactically, semantically, and proof theoretically — with the classical propositional calculus.
Of course, what counts as novel, groundbreaking or simply useful is in the eye of the beholder, as it is very hard to quantify these concepts. (Incidentally, there is a wonderful talk by Timothy Gowers on the topic of the Importance of Mathematics that touches on the idea of 'usefulness').
Regardless, I have a very practical question:
Is there at least one new theorem, shorter proof of an existing theorem, or any completely new statement derived from the theory of LoF that no other existing theory was able to reproduce?
If not, can you help me get a better understanding as to why the theory of LoF still has supporters to this day - even within the mathematical community? For example, would it be fair to say that "elegance" (for those who deem the theory worthy of the tag) has a non-negligible impact on research which should not be underestimated?