Talk:Laws of Form/Archive 3

"simplifies proofs in Boolean algebra"

The complexity of proofs in Boolean algebra is a highly technical problem, known as the Boolean satisfiability problem (SAT). To prove that "A entails B" (i.e., to prove a theorem in Boolean algebra, where A and B are arbitrary formulas) is equivalent to showing that "not (not A or B)" is not satisfiable (i.e., there is no possible assignment of truth values to A and B such that "not A or B" is false; that is, "A implies B" must be true). There are benchmarks for evaluating algorithms for SAT. See the external links section of Boolean satisfiability problem. Has pa been evaluated on these benchmarks? I doubt it. I think the claims that pa "... dramatically simplifies proofs in Boolean algebra, and in sentential and syllogistic logic" are not adequately supported and should be deleted. --Pdturney 23:42, 30 October 2006 (UTC)

Section 5.4 of Meguire (2007) includes a dozen or so propositional logic problems whose worked solutions in standard undergrad texts struck me as unusually involved. Meguire shows how each of these problems can be cracked using the primary algebra in a mere 5-8 lines. J0, J1, C1, and C2 suffice for a large majority of the steps. I take this as suggestive evidence in favor of the primary algebra "dramatically simplifying proofs in Boolean algebra and sentential logic."

More generally, I would love it if a working computer scientist, e.g., P D Turney, Ph.D., were to run a computer "horse race" among competing axioms sets for Boolean algebra, with the primary algebra being one of the contenders. While I am critical of aspects of Stephen Wolfram's work, p. 1175 of his New Kind of Science reports the outcome of just such a horse race. I dare not claim that the primary algebra points to a proof or refutation of P=NP. But I do conjecture that algorithms building on the primary algebra will run faster. Bricken tells me he has raised venture capital to explore this conjecture.132.181.160.42 06:42, 4 September 2007 (UTC)

introductory text

"The primary algebra (chapter 6), an algebraic structure that is a provocative and economical notation for the two-element Boolean algebra..."

1. "an algebraic structure ... is a ... notation" - How can a structure be a notation? This doesn't make sense.
2. "provocative" - this adjective does not fit with Wikipedia:Neutral point of view
3. "economical" - this requires a citation to a paper that carefully defines what it means for a notation to be economical and then proves that primary algebra is more economical than some other standard system, either by a theoretical proof or an empirical demonstration with a large set of benchmark formulas; one or two examples is not sufficient Wikipedia:No_original_research

--Pdturney 20:50, 6 November 2006 (UTC)

Change it to "The primary algebra (chapter 6) is an economical notation for the two-element Boolean algebra..." if you prefer. "Provocative" simply refers to the reaction the primary algebra has elicited among fans and detractors over the past 40 years. For my part, I like a little verbal tabasco sauce in my daily Wiki cocktail!

The primary algebra is the most economical syntax known for Boolean algebra and sentential logic, simply because it requires only 1 (LoF) or 2 (boundary algebra) symbols in addition to sentential letters. Meguire (2003) and its revision give many examples of how the primary algebra drastically simplifies and shorten proofs. This is a more important economy. As for the Praeclarum Theorema, compare PM, MetaMath (a website that rederives a lot of extant math from ZFC embedded in the late Tarski's first order logic), John Sowa's existential graph proof, and the critique thereof in Meguire (2003). I trust the primary algebra demonstration given in the entry will beat all rivals: it is short, invokes few prior results, and easy to teach to nonspecialists. Turney wants a more formal and algorithmic approach to rating the economy of a formal language such as the primary algebra. I will gladly assist him or a student in devising a horse race of that nature.132.181.160.42 06:53, 27 August 2007 (UTC)

Intensely harsh criticism of LoF

I am considering adding a section which conveys the sentiment of some mathematicians including Conway : Laws of Form is an elaborate exercise in delusions of grandeur. Some mathematicians argue that LoF is both good and original; but the part that is good is not original, and the part that is original is not good. LoF is seen as a mountain of flowery language piled upon boolean logic, which is a completely understood system (there is no incompleteness theorem for it.) In the few pages where he goes off the rail into imaginary truth values, he retreats into hand waving.

They consider LoF a beautifully written intellectual hoax.

Thoughts? CeilingCrash 19:18, 6 June 2007 (UTC)

Much of this article chews over the issues you have just raised. I conclude that when LoF was written and published, the 1960s, the primary algebra was quite original except for those who had assimilated Peirce's alpha existential graphs. To my knowledge, only two people had done so at that time: Don Roberts and Jay Zeman, authors of PhD theses on Peirce's graphical logic, completed in 1963 and 1964, respectively. LoF 's notation was scooped by Nicod in 1917, and by papers Peirce wrote in 1886 but that were not published before 1976. But I do not conclude that LoF was so unoriginal as to be worthless.

I agree that a lot of LoF deserves stern criticism and even dismissal. Significant parts of LoF are bombastic, and many of its conjectures are falsified by the subsequent history of mathematics. (E.g., the 1976 proof of the Four Color Theorem owes nothing whatsoever to the "imaginary truth values" of LoF 's chpt. 11.) LoF is not much of a contribution to the philosophy of mathematics, because its author did not know much philosophy or mathematics. When Spencer-Brown thinks he is being philosophical, he is mostly just fawning over Russell, Wittgenstein, and R D Laing.

Nevertheless, I do not conclude that LoF is worthless or a hoax; rather, it is an independent rediscovery of Peirce's point that the conventional notations and proof methods for Boolean algebra and the sentential connectives are unnecessarily complicated. If the primary algebra were taken seriously, I conjecture that it would be a good deal easier to teach logic to nonspecialists. I never forget that LoF emerged out of lecture notes for adult evening classes. GSB did indeed mostly just "rederive Boolean algebra and logic." But his version of that algebra and logic are more elegant in nontrivial ways, even if Peirce did scoop the Cross notation.132.181.160.42 06:58, 27 August 2007 (UTC)

It is not the job of Wikipedia editors to determine the veracity of claims and counter claims (cross or no cross). If you want to add a referenced section that conveys these sentiments you should. If they can be countered by any cited counter claims, that should be added as well. -- ☑ SamuelWantman 22:35, 6 June 2007 (UTC)

Probably good to do so, since they represent the conventional mathematical view of what Spencer-Brown attempted. However, I would like to present some opposing thoughts.

One very harsh review of LoF that I read, amounted to a dismissal on the grounds that all he did was re-derive Boolean logic. That reviewer entirely missed the point of the book. One might, by the same logic, dismiss Whitehead and Russell because all they did was re-derive set theory. LoF is beautifully written, from the point of view of mathematical elegance. Now you must understand that to a mathematician, "elegance" is defined as doing the mostest with the leastest. This means that something very elegant as mathematics, can be very difficult and clumsy as presentation. Chapter 2 contains a good example of this, where Spencer-Brown goes through some gyrations which would be unnecessary if he had used the principle of substitution; but it did not yet exist in LoF (that substitution is a principle generally accepted in mathematics, is irrelevant here, since LoF explictly starts out with as near to nothing as can be managed).

I believe the "delusions of grandeur" accusation derives from the various prefaces to the book. The man is laboring under some delusion about who he is, in my opinion. But he was sane when he wrote the book.

The criticism you quote ("the part that is good is not original, and the part that is original is not good") is correct but misleading. I do not believe I misstate the case, to say that the first ten chapters of LoF are to logic as Euclid is to geometry. The fact that others described logic before him does not detract from the elegance of Spencer-Brown's presentation. It is sad that the original part of his work, in chapters 11 and 12 (bringing self-reference into play explicitly, rather than indirectly and implicitly as all others have done), is not of the same quality as the first ten chapters.

On the other hand, the fault is not entirely in Spencer-Brown's writing. Self-reference represents a very different view from what mathematics usually deals with. In mathematics, however the form of an expression may change, the content of that expression is unchanged. This is embedded so deeply that no mathematician I have talked to can even recognize the possibility of another view. Yet a mathematical treatment of (say) an SR flipflop requires the ability to express something whose form is constant but whose content differs at different times or in different instances. I have asserted to an instructor of mathematics that, in these two (identical) expressions:

${\big \{}Q=\lnot (R\lor Q'),\,\,\,Q'=\lnot (S\lor Q){\big \}}$

${\big \{}Q=\lnot (R\lor Q'),\,\,\,Q'=\lnot (S\lor Q){\big \}}$

one instance of Q was TRUE and the other instance of Q was FALSE. The instructor told me that I was speaking nonsense; Q could be TRUE in both expressions, or Q could be FALSE in both expressions, but it could not be different in different instances of the expression. The fact that this is the mathematical expression of an SR flipflop, and that two different flipflops can indeed hold two different values even though they both have the same form, was completely opaque to him.

Until one learns to shed the usual mathematical conception of form and content, one will not understand a treatment that is contrary to that conception.

— SWWright^Talk 23:15, 6 June 2007 (UTC)

Your experience does not surprise me. Most working mathematicians know no logic or Boolean algebra or foundations generally. I once asked a working mathematician with an interest in logic "How many of your colleagues could be trusted to manipulate the existential and universal quantifiers?" His reply "About none." Still "...the ability to express something whose form is constant but whose content differs at different times or in different instances." strikes me as nothing radical.132.181.160.42 (talk) 03:07, 6 April 2009 (UTC)

Leibniz

I've seen in one or two other places the claim that Leibniz "invented Boolean algebra". That strikes me as a non-standard claim. Unless I'm wrong about that, then the claim should be qualified with "who says so" and cited everywhere it appears, not presented as accepted fact. --Trovatore 21:50, 23 July 2007 (UTC)

Read the link to the historical research of Wolfgang Lenzen. Or Google his name. He is Professor at Osnabrueck in Germany. Nicholas Rescher, in a 1954 Journal of Symbolic Logic article, came close to appreciating Leibniz's precursor role.132.181.160.42

The primary algebra is not trivial

The primary algebra is, to a fair extent, just a streamlined notation for the two-element Boolean algebra 2. So what? Well, 2 is mathematically rich, despite being decidable and even though all finitary Boolean algebras have the same theorems (in this respect Boolean algebras are very different from groups). How is 2 rich? There's now an ample Wiki entry answering this question: click on Boolean algebras canonically defined. The technical level is high in parts, and some of it could be better written, but this entry bears close reading by anyone truly curious about the mathematical (as opposed to the logical) significance of the primary algebra. Charles Peirce wrote 120 years ago that mathematics (especially abstract algebra) precedes logic, and not vice versa, as Frege and Russell would have it. Logics are interpretations of bounded lattices. Logicism runs afoul of the limitative theorems, which Spencer-Brown very regrettably belittled in LoF.132.181.160.42 (talk) 04:22, 14 December 2007 (UTC)