Talk:Laws of Form

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Philip Meguire, 22.10.05: I am now the author of most of this entry, while being none of mathemacian, logician, or philosopher. I respect the Laws of Form, seeing them as a major simplification of the 2 element Boolean algebra (2) and of the truth functors of elementary logic. The formalism should be extendable to arbitrary finite Boolean algebras and to first order logic. In any event, the primary algebra could greatly simplify the teaching of logic to nonspecialists, such as philosophy majors, electrical engineers, and computer scientists.

About Spencer-Brown's bolder claims, I reject some (e.g., his belief that LoF eliminates any need for type or set theory) and am thoroughly agnostic about others (e.g., imaginary truth values could revolutionize mathematics and electrical engineering).

Philip, I offer my very sincere congratulations on your mathematically rigorous treatment of Spencer-Brown's work. You've done a great piece of work yourself -- similarly respectful of LoF, the calibre of this article had previously been something that I could not even reference. Please stay flexible and with us as this continues to evolve. The article may need a separate section to discuss the philosophical and even theological implications of LoF, but there are critical points in your very well done discussion that provide segue and foundations for this future section. Again...very nicely done. --AustinKnight 19:59, 28 October 2005 (UTC)

Philip Meguire, October 31. I have taken the liberty of editing your additions. I teach university and find it very natural to edit other people's writing.

Well done, again. I didn't like the interruption of flow myself, and was clearly too focused on the segue opportunity presented by the "first distinction" dialogue. I've reedited to somewhat 'sandwich' the ineffable allusions via both a re-assertion of the linkage with the "first distinction" and the LoF Notes section for Chapter 12. --AustinKnight 14:27, 31 October 2005 (UTC)

Philip Meguire. I've expanded your discussion of Confucianism by including my favorite quote from the Analects.

The article seems complete to me except with regard to one important dimension: LoFs dialogues on the imaginary. I don't necessarily disagree with you, Philip, that "G." overreached, but at the same time he was dealing with the imaginary, and so one clear characteristic of that 'set' is that it is boundless. Given that imaginary numbers are quite "real," it'd be interesting and perhaps valuable to capture S-B's thoughts in our article re. the imaginary, while at the same time sticking to a rigorous mathematical treatment of it. Philip, do you think you could you take a shot at this? --AustinKnight 14:28, 7 November 2005 (UTC)

Philip Meguire, November 7. I have concluded that LoF strongly overstates the value of imaginary truth values for logic, mathematics, and engineering. Moreover, what chpt. 11 of LoF seems to groping after was anticipated by the work of the Russian logician Bochvar (Russian original 1939, English translation 1981). I am happy to let someone else add a section summarizing chpt. 11 and mentioning possible extensions (I have yet to encounter work building on LoF in a serious way; the nearest exceptions are the curious books by Nathan Hellerstein). Incidentally, the imaginary numbers are no more boundless than the reals. Granted, complex numbers are two dimensional, but Cantor showed that complex numbers have the same order of infinity as the reals.

Quite right...I should have stated "infinite" vs. "boundless." My computer science/math days are long ago and more than a bit rusty.
I am intrigued by the nature of imaginary numbers and sense (right or wrong) more than a bit of resonance in other matters of the rational mind and the demonstrated limitations thereof. I am hopeful that Spencer-Brown is at least conceptually onto something with respect to the coupling of: (1) the concept of the distinction as the root of cognition, and (2) the concept of an imaginary dimension to logic.
Clearly, imaginary numbers are already of value in the real world...I'm hopeful that the extension of white/black binary logic can benefit from their counterpart. I'll take a shot at this, perhaps. If so, the creaking hinges of mathematical thinking that you hear will be entirely my own fault. --AustinKnight 13:54, 9 November 2005 (UTC)


In 1963 I attended a lecture series given by Brown (as then known) at University College London. This article considerably extends the content of that series; I guess the lectures were a try-out for the book.

Brown referred to his work on the control of lifts (elevators) as a significant driver in the development of the Laws.

I kept my lecture notes. The purpose of this 'discussion' entry (Feb, 2005) is to offer a view of my course notes, as background material for anyone who may be examining the early history of the subject. This is the web site:

   http://www.tooke-picarel.co.uk/LoF/

Richard H. Pickard, Norwich, UK



Richard Shoup has published an interesting article elaborating upon the imaginary values. At the end of the article there is a nice overview of correspondences between classical circuit notation, Boolean Algebra and the Calculus of Indications aka Laws of Form. [1]


If anybody feels compelled to elaborate upon the self-referential forms - please do. The reference to Spencer-Browns talk in 1973 is, unfortunately, the best I'm up to at the moment. I think it necessary to at least give a hint to this element of the "Laws of Form" which is crucial to many a discussion about "paradox" and the still lingering theory of types.

Contents

[edit] Unmarked state

This should be merged in (it was on an orphan page). Charles Matthews 07:51, 7 October 2005 (UTC)

[edit] Criticism

What does the "Resonance in religion & philosophy" have to do with Laws of Form? LoF is a formal system for logic. The "Resonance..." section quotes from a collection of religious texts that appear to have no relationship with LoF. Maybe someone could make the connection clearer. Or conversely maybe they could tell me why there aren't similar discussions in articles for other branches of logic? sigfpe Nov 1, 2005

Concerned Cynic, Nov. 4 2005. Adding the section "Resonance" is Austin Knight's preference. Once I committed to humouring him in this respect, I added the quotes from Genesis, Confucius and the rectification of names, Royce, and Wheeler.

Syntactically, the Laws of Form are no more than a streamlining of the Boolean algebra 2 and propositional logic in equational form, and monadic predicate logic. The Laws are not isomorphic to first order logic, but I am confident they can (and will) be made so.

Relation to LoF? That book reveals that Spencer-Brown believes in some God, and that he was very much caught up in the bohemian Zen mysticism (Alan Watts, Suzuki, etc.) of a half century ago. Add to that a fascination with the enigmatic Wittgenstein (popular among persons educated at Oxbridge 40-60 years ago) and Ronald Laing (the radical psychiatrist), and you can see why LoF became a cult classic (a phrase that Austin Knight will not let me include in the article!).

That God, logic, and order in the natural and human world are interconnected was argued by, e.g., Plato, Aristotle, some Chinese classics, Aquinas, Leibniz, Kant, the late Whitehead, Charles Hartshorne, and the curious American analytic philosopher Richard Milton Martin (whose Wikipedia entry I wrote). -- User:132.181.160.61

In the U.S., at least, usage of the term "cult" of any form is highly pejorative...with good reason, as we have had some truly nasty ones. It also implies some sort of at least loosely-formed organization, of which there certainly is nothing substantial for LoF that I am aware of.

As my original notes indicated, there is little avoiding of such topics around the ostensibly mathematical writings of Spencer-Brown. As User:132.181.160.61 notes in the article, S-B was highly paradoxical in his writings...sometimes to the point of being virtually opaque, but clearly with the intent of such ties as noted above.

BTW, I predicted this section at least as much as contributed to it, but did not create it. Someone else did using the original title "Analogies," and I thought to replace that with the term "Resonance" as a sort of homage to Spencer-Brown...whose work I also very much respect.

As to ties with LoF: User:132.181.160.61's list of reknown philosophers who would tie these topics together also goes, of course, into all of those currently listed in "Resonance." Historically, these topics are wedded by quite a substantial collection of individual thinkers and belief systems. It'd be intellectually dishonest to assert otherwise. Specific, referenceable ties to LoF, as also indicated in the article, include the language surrounding the "first distinction" and the Notes to Chapter 12. --AustinKnight 23:38, 2 November 2005 (UTC)


User:132.181.160.61 = Philip Meguire! I added Royce and Wheeler.

The 'History' tab above provides a good bit of clarity re. editors. It really is best to sign all 'Talk' work via the 2nd button from the right at the top of an 'Edit' page. Cheers, --AustinKnight 04:18, 4 November 2005 (UTC)

[edit] Featured Article? Perhaps.

This was a very enjoyable article on an enjoyable book. Many of us have been influenced by it. Congratulations to those of you who have improved it. Ancheta Wis 16:35, 5 November 2005 (UTC)

[edit] Lead section

Would it be possible to conver this into prose? - Ta bu shi da yu 14:42, 22 December 2005 (UTC)

[edit] Funny You Should Mention It (FYSMI)

I am wikiworkinup to a rewrite, but will need to discuss much before I do. Maybe later today. Jon Awbrey 15:06, 22 December 2005 (UTC)

[edit] Incidental Incremental Improvement Issues (I^4)

I will list here the issues that arise as we discuss improving this article toward the point where the coverage of Laws of Form (LOF), the book by George Spencer Brown and the corresponding formal system or system of forms, plus the necessary formal and historical connections to C.S. Peirce's "Logic of Relatives: Qualitative and Quantitative", including his Logical Graphs, in their dual interpretation as entitative graphs and existential graphs, his "Qualitative Logic" manuscripts, his alpha graphs, beta graphs, gamma graphs, along with whatever else comes along, will be more generally understandable and useful to the reader.

NB. Please excuse all the acronyms -- they're just how I keep track of things in my own mind and notes. Jon Awbrey 18:00, 22 December 2005 (UTC)

[edit] Query 1

  • I guess the first issue that comes to mind is this: What's a good way to coordinate the content here with the closely associated and/or overlapping content in the Charles Peirce article?
I submit that HTML cross referencing works just fine here.202.36.179.65 11:31, 28 December 2005 (UTC)
  • This may be more acutely critical since I also see a message saying that this article is approaching or already passed through some kind of "singularity of size" (SOS). Is that still an issue, or is it now an obsolete concern?
Wikipedia politely suggest that entries over 32KB in size are perhaps bigger than optimal. I hear tell that that can be ignored with impunity. The 32KB is the largest segment of HTML code I can edit in one bite on the iBook I use at home. The generic Windows system in my office has no such limitation.202.36.179.65 11:31, 28 December 2005 (UTC)
  • Incidentally, how do I create a redirect from C.S. Peirce, which is the more usual name in logical and mathematical journals, not to mention large parts of the Peirce literature? Jon Awbrey 18:44, 22 December 2005 (UTC)
Wikipedia is committed to Charles Peirce. The entry Charles Sanders Peirce is deemed dead. If there is a way of creating and managing aliases for Wikipedia entries, I know nothing about it.202.36.179.65 11:31, 28 December 2005 (UTC)
I created a redirect for C.S. Peirce. Here's how it is done: Create a new page where you want to have a redirect. The easiest way to do this is follow a link like the one above (C.S. Peirce), or just use the search box to go to the page you want to create. type #REDIRCT and then a link to the page you want to be redirected to. In this case it was: #REDIRECT [[Charles Peirce]] -- Samuel Wantman 20:25, 6 March 2006 (UTC)

[edit] Query 2

Another issue that arose somewhere between my 2nd and 3rd reading of the articla has to do with the lower case acronym pa for primary algebra. The last thing I want to do here is institute any sort of paternity suit, so ... Philip, I guess I'll address this to you in particular, just in case you have any kind of "personal attachment" (PA) to this usage. But seriously, now, here're the problems that I'm having:

  • To the mathematical community of interpretation, PA almost reflexively suggests "Peano Arithmetic", so I think I can see why you may have wanted to de-escalate the potential hash-clash there. But I'm not so sure that the lower register pa really does the trick, as you can't really hear the piano, in print or see the piano in normal speech -- no, I mean the other piano.
In my published work, I use "PA" to denote the primary arithmetic and "pa" the primary algebra. I write both in Helvetica, to very clearly demarcate them from the rest of the ms, which is in Palatino. I am quite aware that to many well versed in formal systems, PA brings to mind "Peano arithmetic," but you are the first to complain about this homonymy. GSB, William Bricken, and Jeffrey James have argued that one can derive the natural numbers from boundary methods, but doing so requires jettisoning A1 and exiting the PA. Hence the primary arithmetic and the Peano arithmetic are incompatible, and "PA" can refer to both with little risk of ambiguity.202.36.179.65 11:42, 28 December 2005 (UTC)
Okay, I was in a bit of a jolly mood that day, but the practical point that I'm trying to make is a bit like this. If I want to explain this to somebody at a party -- yes, I confess, I have -- or over a bad cellphone connection, or in some other noise-filled environ, then I can't depend on such nuances of fontology and intonation to 'make a distinction', as it were. Jon Awbrey 14:48, 29 December 2005 (UTC)
It is hard enough to exposit math and logic in writing. To expect, further, that they be communicable in speech is a Big Ask. To expect, even further, that they be communicable viva voce in a "noise-filled" environment" is simply too much. Formal systems require and deserve our quiet contemplation. I find speech so ineffective that I have found university lectures on such matters of little use; a lecture is an Index Sequential data structure, strongly dominated by Random Access structures, such as the printed page.202.36.179.65 15:59, 14 March 2006 (UTC)
  • Be that as it may, the more serious problem that I'm having is this: My own experience expositing LOF and Peirce's various systems of logical graphs tells me that it's best to do it in very gradual stages. Jon Awbrey 19:56, 22 December 2005 (UTC)
I believe that the entry pretty much does as you prefer here.202.36.179.65 11:42, 28 December 2005 (UTC)
Well, I guess I'll start contributing a little more of what I have found preferable, and then we can talk about it with something less hypothetical in mind. Jon Awbrey 15:00, 29 December 2005 (UTC)
  • Among other things, this means sorting out the "primary arithmetic" (PA) and the "primary algebra" (PA) from each other and introducing them to the reader or student in two distinct stages. I think you see the problem. One of the ways that I've addressed this problem in the past has been to dub the primary arithmetic "Par" and to dub the primary algebra "Pal", or fully capitalized variants thereof. What do you think? Could you live with that? Many Regards, Jon Awbrey 19:56, 22 December 2005 (UTC)
I italicized "pa" here simply out of a desire not to be typographically aggressive. If we are to deviate from the status quo, I would prefer keeping "pa" but writing it in bold rather than italics.202.36.179.65 11:43, 28 December 2005 (UTC)

[edit] Resonances in religion, philosophy, and science

Not sure where to put this note... but someone should seriously take a look at the wiki on Parmeides. ~wblakesx

You almost certainly mean Parmenides; I've added same today. --24.153.209.20 13:06, 22 August 2006 (UTC)

Philip (?), For my part I find this sort of material personally fascinating, but I'm thinking that it might go better toward the end of the article, a rest from the formal rigors, as it were, especially given the problematic reception that this book has had over the years from a diversity of readers who do not appreciate diversity to the same extent. What do you think? Jon Awbrey 15:54, 4 January 2006 (UTC)

The shift you propose has been done (by someone other than me) and I like the result. Turning to LoF, I do not agree that its reception by readers has been problematic, or that it has had difficulty appealing to a diversity or readers. If anything, the serious problems with LoF are:
  • Those who know logic and math dismiss it as "mere" Boolean algebra. They point to its confused assertions about set and type theory, and about metamathematics, and dismiss it out of hand;
  • Its many readers who know little about formal systems are almost invariably overawed by its enigmatic and paradoxical assertions, concluding that there are rigorous grounds for abandoning rigor!
In the history of LoF, there are two near-tragedies.
  1. When Spencer-Brown wrote LoF, Peirce's 1886 papers using the very notation GSB was proposing were mss gathering dust at Harvard. LoF cites vol. 4 of Peirce's Collected Papers, but GSB completely missed the 100+pp that volume includes on the existential graphs. The alpha graphs are isomorphic to the primary algebra; beta and gamma go further.
  2. Shortly after LoF was completed, George Lakoff and others began building 2nd generation cognitive science. There are strong affinities between the "container image-schema" of Lakoff's Women Fire and Dangerous Things and LoF's "distinction." There are further affinities between LoF and Where Mathematics Comes From, a work which discusses possible cognitive origins of Boolean algebra, sentential logic, and elementary set theory in some detail. I am astonished that fans of GSB and fans of Lakoff are like ships passing in the night, except perhaps in that tiny part of the universe that lies between my ears ;<) On the other hand, a few Peircians do politely acknowledge LoF.202.36.179.65 17:00, 14 March 2006 (UTC)

[edit] Boundary of a boundary is zero

I deleted this remark:

A2 captures the essence of Wheeler's sentence "The boundary of a boundary is zero", quoted above.

Wheeler is referring to an axiom of algebraic topology that is very different in form from the law of crossing. Jon Awbrey 20:00, 10 January 2006 (UTC)

Wheeler almost surely had nothing Boolean in mind when he wrote "The boundary of a boundary is zero." And I do not doubt that he had topology in mind. But that does not necessarily invalidate my sentence to which you object. An important aspect of boundary mathematics is the way it points to all sorts of unwitting analogies and connections in the realm of abstraction. For the record, Lou Kauffman, a topologist and knot theorist, agrees with the sentence to which you object. Moreover, Boolean algebra can be grounded in elementary topology; for an exposition, see chpt. 2 of Rosser's 1969 monography on simplified independence proofs in set theory. A major unsolved problem is marrying boundary mathematics to the large corpus of topological mathematics, by either fleshing out what Rosser began in 1969, or by drawing on the laws of form to devise a non-set theoretic foundation for topology. The relation between the laws of form and topology must be clarified eventually, otherwise topologically literate mathematicians will sneer at our loose use of "boundary" and "distinction". The entry does not argue that the Laws of Form constitute an approach to 2 that is innocent of set theory, because to my thinking that is a working hypothesis, not a settled fact. Unlike many LoF fans, I have no real quarrel with the large body of academic work in math and logic. We can learn much from it, even it has missed some elementary insights.202.36.179.65 16:32, 14 March 2006 (UTC)
  • JA: I know exactly what Wheeler is referring to, and will get you a standard ref when I get time. There is no analogy here because the formal properties are totally different. The boundary operator in algebraic topology has the properties d0 = 0, dd = 0, so all higher powers of d are 0, whereas the operator () has the form, (()) = blank, ((())) = (), and so odd and even powers alternate beyween () and blank. Jon Awbrey 16:33, 14 March 2006 (UTC)
In my view, this controversy points out the need to flesh out an intriguing little 2x2 table in Shoup's website, one claiming that changes to I1 and I2 yield finite numbers, sets, and multisets. The Holy Grail for me is a unified boundary formalism for lattices, number systems up to real analysis, sets, point set topology, and discrete math. Perhaps even groups and categories.202.36.179.65 17:27, 14 March 2006 (UTC)

I have drawn Lou Kauffman's attention to the difference of opinion aired in this section. He agrees that there is a useful analogy between A2 and the boundary operator of algebraic topology. If all powers of d are 0, then d and 0 can be seen as interpreting (), in which case d0=0 and dd=0 are both instances of A1. So if linking A2 and algebraic topology is a mistake, I say replace A2 with A1. In any event, there is a crying need for a boundary reformulation of group and lattice theory. Doing so will afford much insight.202.36.179.65 10:00, 16 September 2006 (UTC)

[edit] Speculation on Spencer-Brown's sources

  • JA: Your statements about Spencer-Brown's sources remain for the time being in the realm of wholly unsourced speculation. I'm told that they have moderately adequate libraries at Cambridge and Oxford, and transatlantic aeronautic transport was commonly available to the average scholar even in those primitive times. There were microfilm editions of Peirce's Nachlass at many university libraries in the early 70's just from my personal acquaintance. But the essentials of Peirce's graphical systems are abundantly clear from what is found in CP, and Spencer-Brown evinced a clear insight into their character. Jon Awbrey 17:28, 14 March 2006 (UTC)


LoF was completed in 1967 and published in 1969. The Peirce Nachlass was microfilmed in 1964. Did any British university library have a copy of this microfilm before 1967? I rather doubt it, if only because to this day Peirce studies have been weak in the UK. Did Spencer-Brown fly to Boston and spend 1-2 weeks in the Harvard's Houghton Library and find there the Peirce mss that were completely unknown until Carolyn Eisele published an important excerpt in 1976? I rather doubt it, because the Robin catalogue of the Nachlass was published only in 1967. Without that catalogue, GSB would have had no idea where to look in the vast Nachlass. How many British libraries acquired a copy of the Univ. of Mass Press edition of the Robin catalogue before 1968? Meanwhile, Spencer-Brown's Cross notation was well established in lectures he gave beginning in 1963, if not earlier. If Spencer Brown had any awareness of Peirce's graphical logic, he's kept very very quiet about that fact for now 40 years. LoF reveals that GSB, when writing LoF, was aware of Peirce's Collected Papers because LoF cites vol. 4 of said papers. Ironically, that volume is the one containing 115pp devoted to the existential graphs. If the initials of the primary algebra included what Peirce called (De)Iteration, I would suspect that GSB could have appropriated aspects of the existential graphs without attribution. But the initials of LoF betray GSB's close reading of Huntington's postulates, not Peirce's graphical logic.202.36.179.65 10:14, 16 September 2006 (UTC)

[edit] Are abbreviations like 'LoF' & 'pa' necessary?

Most book articles don't require one-shot acronyms. It's not clear that they're useful here. It seems like a self-important bias to tax readers with new jargon. It's not obvious why this:

LoF argues that the pa reveals striking connections among logic, Boolean algebra and arithmetic, and the philosophy of language and mind.

...would be preferable to this:

"Laws of Form argues that the primary algebra reveals striking connections among logic, Boolean algebra and arithmetic, and the philosophy of language and mind."

--AC 06:57, 10 October 2006 (UTC)

pa seems to be an especially poor abbreviation, given that primary arithmetic and primary algebra both have the initials pa. Pdturney 13:10, 10 October 2006 (UTC)

I agree with Pdturney and AC; pa is a poor abbreviation. Make it primary algebra, the length is easily compensated for by gained clarity. Apart from that most sincere thanks and congratulations for the huge improvements this page has seen in the last 2 years or so. I think this will be a most valuable starting point for research into both the fundamentals and the history of both logic and mathematics. RBF, 13 October 2006

[edit] "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)

[edit] 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)

[edit] 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.
— SWWrightTalk 23:15, 6 June 2007 (UTC)

[edit] 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

[edit] 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)