## Reading "The Laws of Form", by George Spencer-Brown.

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Laws of Form responds to a long-standing mathematical paradox relating to infinite series of numbers. At the end of the nineteenth century, the founder of set theory, Georg Cantor, found that the infinite is itself differentiated, with different infinite series coming in different sizes. Moreover, if one counted all the infinite series, one would again find an infinite series, but one whose number, including itself when counted, must be larger than any countable (cardinal) number (Davis, 2000, p. 67). This problem led Russell to consider paradoxes in logic and the membership of sets: Extraordinary sets are self-including, such as ‘a set of all things not sparrows.’ This set itself also belongs to all things not being sparrows. Ordinary sets, on the other hand, have no such self-referentiality, for instance the set: ‘all things that are sparrow’ which, clearly, does not include the ‘set’ itself. But what about a set containing all ordinary sets? Would that not at once have to be larger than the number of all the sets it contains, as it, itself, is one such set (Davis, 2000, p. 67)? As Russell (1919, p. 136) states:
The comprehensive class we are considering, which is to embrace everything, must embrace itself as one of its members. In other words, if there is such a thing as “everything,” then, “everything” is something, and is a member of the class “everything.”
Whitehead and Russell’s (1910) Principia Mathematica proposes a stopgap intervention by excluding paradoxes from the domain of logic: sets cannot be members of themselves!.
Spencer Brown’s biography placed him directly into this debate, having worked with the two foremost logicians of the time, Russell and Wittgenstein. His solution to the problem was formed when he worked on practical electrical and engineering assignments.
https://livrepository.liverpool.ac.uk/3101665/1/Spencer%20Brown%20submission%20(1)%20(1).pdf

The above is not the book, but something of a biographical note.

Here is the Book:
http://www.siese.org/modulos/biblioteca/b/G-Spencer-Brown-Laws-of-Form.pdf

No special rules. Read, comment, ask questions, as thou wilt. We might get somewhere, and we might not. In your own time, then...
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Golly, as I suggested this as a candidate for a discussion group, I ought to make some kind of contribution. :yikes: At the moment, the best I can do is link to the Wikipedia entry, which at least provides an introduction, but having suggested it, I notice this sentence in particular:

LoF's mystical and declamatory prose and its love of paradox make it a challenging read for all.

I don't know if it's really up my alley, what with its background in mathematics, but I feel that the more mathematically-literate members of the community might have an interest in it. But I will endeavour to absorb some of it, hopefully some spark of inspiration might be communicated.
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Oh, and I also thought, wonder what's on YouTube about this, and lo, a free, online course on G Spencer Brown's Laws of Form. (Beautiful English diction, by the way.)

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Ok, I'm seeing a re-work of Boolean logic with a sort of pseudo-Hegelian dialectic thing going on.

A poor idealist's Tractatus?
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Most mathematicians brush aside the Russell Paradox and its circumvention by classes. For me, this book would be a challenging read. I went part way into it and bailed.
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Thanks guys. I'm going to be watching those videos with the nice clear soothing voice. A couple of practical problems I have:

1. The pdf I linked won't allow quotes.
2. My keyboard does not have the cross symbol.

Not fatal, but annoying.

For 2 I think we could use brackets, thus:

[ ], [[ ]], [[ ] [ ]], [a], [[[a] [[b] [c]] [ ]] Not very clear, and it might be better to alternate square and curly by depth, thus:

[{ } { }], [{[a] [{b} {c}] { }]

I still don't like it much, any better ideas? Does mathjax or whatever it is have the solution?

Ok, I'm seeing a re-work of Boolean logic with a sort of pseudo-Hegelian dialectic thing going on.

A poor idealist's Tractatus?

Boolean logic is developed from the very simplest foundations, and then extended with imaginary values. But I don't think anyone ever solved an actual problem using Tractatus...

Come along for the ride. Maybe you can help us get as far as you did, Maybe we can help each other get a bit further...
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The pdf I linked won't allow quotes.
2. My keyboard does not have the cross symbol.

I think if you download said pdf and open it in an acrobat application, you can do a lot more with it, like copy text from it etc. I happen to subscribe to Adobe Acrobat Pro for about 30 bucks a month as I’m a tech writer (and it’s hellishly expensive to actually buy) and it’s a tool of the trade, but there are other PDF apps that allow you to copy text, which you might not be able to do inside your browser. Also, if you download a PDF and save it, you will find that MS Word can open it thereby converting it to text (although I don’t know how it would cope with all the special characters and typesetting in this particular text).

In respect of the Cross symbol, there *might* be some combination of characters that stands for it, although that’s totally off the top of my head. Worth researching it though.

I’m still um-ing and ah-ing about whether to really try and get into it, as my back-list is perennially full of ‘things I should have read already’. But that video series is golden - apart from the splendid voice, he’s a discovery in his own right, seems an exceedingly erudite and learned gentleman in the literary arts.
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$\left. {\overline {\, a \,}}\! \right|$

From MathType. Place math and /math, enclosed by square brackets on either end and try: \left. {\overline {\, a \,}}\! \right|
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I’ve noticed ‘Laws of Form’ but when I tried reading it, found it quite daunting. Maybe we should start a discussion group on it.
I'm not a logician, mathematician, or electrical engineer, but I am somewhat informed on the philosophical concept of Form. Especially as it applies to essential or causal Information --- To Enform : the act of creating recognizable forms : designs ; patterns ; configurations ; structures ; categories. Generic Information begins in the physical world as mathematical ratios (data points ; proportions, 1:2 or 1/2) in a starry sky of uncountable multiplicity. Hence, we begin by clumping cosmic complexity into symbolic zodiac signs relating to local significance. In an observing mind, that raw numerical data can be processed into meaningful relationships (ideas ; words). Or, in a mechanical computer, those ratios are analyzed reductively into either/or (all or nothing) numerical codes of digital logic : 100%true vs 0%true. This is probably the most elemental form of categorization, ignoring all degrees of complexity or uncertainty.

The Wikipedia article on Brown's book, Laws of Form, notes a primary requirement for the human ability to know (grasp intellectually) any Form in the world, first "draw a distinction"*1. Rather than sketching an arbitrary encirclement, this precondition seems to assume that the categorizing mind is trying to "carve nature at its logical joints". First a particular "form" (thing) must be selected (differentiated) from the universal background (the incomprehensible multiplex) of manifold Forms (holons) adding-up to a complete system (universe ; all-encompassing category). A holon (e.g. steak) is a digestible bit or byte from a larger Whole Form (e.g. cow), a comprehensible fragment. Human Logic requires a rational (ratio-carving) knife & fork for its comestibles. But, is the world indeed inherently logical in its organization, or do we have to use the axe-murderer approach : whack, whack?

Semiotician Gregory Bateson defined Information as "the difference that makes a difference"; referring to personally significant meaning in the subjective mind. Plato's theory of Forms defined them, not as phenomenal objects, but as noumenal categories of thought : "timeless, absolute, unchangeable ideas". Aristotle went on to classify human thoughts into distinguishable categories*2. More recently, modern neuroscientists have attempted to discover how the human brain filters incoming sensations into recognizable "classes of things"*3 (e.g. dog vs cat ; apple vs orange). Each of those categorical Forms is a meaningful distinction for the purposes of a hungry human mind.

Brown's book is over my head, but the notion of logical categories seems to be necessary for understanding how the human mind works as it does. And that need for pre-classification may provide some hint as to why we tend to overlay the real world with an innate template, in order to begin to understand its complexity of organization. First, we draw a circle around a small part of the whole system. Then, with manageable pieces, we can add them up into broader categories, or divide them into smaller parts, right on down to the sub-atomic scale, where our inborn intuitive categories begin to fall apart, becoming counter-intuitive. Hence, the weird notion of Virtual Particles. Is there a natural limit on our ability to encapsulate? Or can we go on imagining novel Forms forever? :smile:

*1. Laws of Form :
"The first command : Draw a distinction"
https://en.wikipedia.org/wiki/Laws_of_Form
Note --- In mathematics the distinctly-defined categories, of things that logically go together, are called “Sets”. However, so-called “set theory paradoxes” are not necessarily logical contradictions, but merely counter-intuitive. Does that mean the human mind can imagine sets or categories that don't fit into the brain's own preformed pairings?

*2. Aristotle’s Categories :
Hence, he does not think that there is one single highest kind. Instead, he thinks that there are ten: (1) substance; (2) quantity; (3) quality; (4) relatives; (5) somewhere; (6) sometime; (7) being in a position; (8) having; (9) acting; and (10) being acted upon
https://plato.stanford.edu/entries/aristotle-categories/
Note --- Perhaps the "single highest kind" of category is the universe itself.

*3. Category Learning in the Brain :
The ability to group items and events into functional categories is a fundamental characteristic of sophisticated thought. . . . . Categories represent our knowledge of groupings and patterns that are not explicit in the bottom-up sensory inputs.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3709834/
Note --- Our incoming sensations are typically randomized by repeated interactions & reflections (e.g echoes). So the brain/mind must sift out the grain from the chaff. Hence, evolution seems to have winnowed the winning organisms down to a few with the "right stuff" for correctly categorizing the fruits & threats of the game of life. Those inputs may include novel Forms that our ancestors have never encountered before in eons of evolution. So how can we make implicit Logical patterns explicit enough for categorical assimilation?
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Ok, I'm just going to say a few things by way of my own 'introduction' and then start.

First, we have the text, and we have the videos from Wayfarer above, and I at least am not going to very much attempt to further expound or teach as such, nor to actually use the system - I am not competent to.

The meat of the book is a formal system. Spencer-Brown might have claimed,"The" formal system. The reason for saying that is that he starts from as near to nothing as possible, and from this almost nothing, manages to 'prove' many of the axioms of other formal systems in common use. And this is one of the difficulties of the book, that a laborious effort is undertaken to show the very simplest most obvious things that we have taken for granted since forever. One tends to read along thinking one has understood, and then one reaches a blank incomprehension at some point...

Formal systems always begin like the voice of God, commanding the world into existence: "Let there be light!" So the text intends you to create a universe in your mind of a particular form, and although it necessarily does so through an already shared language, it intends you to keep all the distinctions of your experience and language that you use to read the text separate and outside the new universe that you create according to the text. First prepare a blank space in the mind, with no thought in it: and begin.

Construction
Draw a distinction.
Content
Call it the first distinction.

But this is already Chapter 2! There is no way it can be the first distinction, because already in chapter 1 we have distinguished what a distinction is, and produced a pair of axioms.

Of course, it has to be confessed that we are not gods, and that our minds are not blank; so we have already made a distinction in the mind between the ordinary mind full of thoughts and distinctions and the blank space of the mind in which we are going to construct this formal system. We insist on the 'continence' of that distinction, that we will keep out all our everyday thoughts, and we maintain that continence by calling our new distinction 'the first'. And that distinction is mentioned again as the unwritten mark under which all this formality subsists.

What appears on the blankness of the paper, or in the emptiness of the mind is a mark, that is a name, a boundary, and an instruction all at once because those things cannot be distinguished. And all we have to help us is our axioms.

Axiom 1. Philosophy[of science] and philosophy[of religion] are philosophy.
Axiom 2. Philosophy of philosophy is not philosophy.

So in the philosophy of science one asks 'what is science' and tries to answer, and in the philosophy of religion, one asks , what is religion, and tries to answer, but in the philosophy of philosophy, if one asks what is philosophy, one has put into question the process of putting things into question, and silence is the best one can hope for. (you don't have to agree, I'm just giving shape to the way the axioms work with a familiar example.)

"And the evening and the morning was the first post."
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I find it interesting that this comes out of an electrical engineering background because I had always thought there were some neat similarities between Hegel's Logic (a similar sort of project) and Claude Shannon's Mathematical Theory of Information (coming from electrical engineering).

A code with just a single undifferentiated signal carries no information. We can conceive of this as a single 1 observed/received an infinite number of times, or an infinite series of 1s. They both convey the same information, which is no information. There is no variance, all observations are identical and thus contentless.

Thus, pure undifferentiated signal, like Hegel's pure immediacy, pure undifferentiated being, collapses into nothing. Being and non-being are opposites and yet pure being collapses into nothing.

But this nothing is not empty, we have all being contained here, just devoid of determinateness. So this is like a sound wave on an oscilloscope. It's not the absence of a wave, but rather a wave of infinite frequency and amplitude. As we approach the limit of frequency, the peaks and troughs get closer and closer together, until eventually they are in the same place, cancelling each other out. This is a silence, but one that is pregnant.

The move to difference is what gives us more. From reading 1, over and over, to the combination of 1 and 0. Or as Hegel has it, being sublates nothing and we get the world of becoming, where being is constantly passing away into nothingness.
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Interesting use of the first chapters.

Something I'm stuck on, from a first reading of the first two chapters, is the distinction between letting and calling. I think I have to read "Let" as "Call a function" or something like that. It's naming an instruction rather than naming a distinction.

EDIT: Actually, thinking that through -- calling a function, more generally, a relation, would just be a distinction with a map.
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(you don't have to agree, I'm just giving shape to the way the axioms work with a familiar example.)

I get a different shape? More like, axiom 1 is about saying a thing (e.g. "Romeo!")... which, if you do it again, is only to reinforce that first statement (or state-naming!).

While, axiom 2 is about changing sides on the issue ("rather, thou art some other, that smell as sweet")... which, if you do it again, is only to undo that first change. And probably end up where it started.

Assertion and negation, basically?
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Something I'm stuck on, from a first reading of the first two chapters, is the distinction between letting and calling. I think I have to read "Let" as "Call a function" or something like that. It's naming an instruction rather than naming a distinction.

'Let' is a command from on High. This how it shall be henceforth. 'Let x be the number of angels that can dance on the head of a pin. 'Let' happens outside the formal system to create it. 'Call' is an action that happens inside the the system. You can call the distinction into being by making the distinction, that is by writing the sign. and If you write it twice in a row you call and recall.

and the distinction is:
I get a different shape?
Assertion and negation, basically?

It's completely abstract. It is cross (the boundary) and cross back, assertion and negation, on and off, 0 and 1, but what is important is that there is but one sign, that is the crossing of the boundary, the boundary itself and the mark of distinction that names the difference between the two sides of the boundary.

The significance of my little example is that it is close to home. This is a philosophy forum, and so we ought to have a very clear idea of what philosophy is, and therefore what it is not. But that turns out to be intractable and interminably controversial. But applying the idea of 'nesting' as 'negation' allows me to say very simply what philosophy is not, and why it is so difficult to be clear about in normal discourse.

The usual problem, and the problem with your picture, is that we tend to give already an equal meaning to the negative and positive. Here, there is no symbol for zero, and the nearest we can get is the something of something, or a blank space.

The other sense that is important in all this is the distinction between the observer and the observed. This give a sense of the inequality of meaning and meaninglessness that is fundamental. This formal system is all about self-reference, and thus to make a distinction is to put oneself on the map. The mark in this sense is like graffiti on the toilet wall. "Kilroy was here." Significantly, Kilroy never indicates where he was not.

It’s worth thinking about the ‘dictums’ in terms of Laws of Form – for they put the ineffable at the heart of the operation. ‘You can’t have a blue universe,’ I can hear him say. Of course you can’t. If you’re proposing a blue universe, then you’re also, by definition, expressing the contradictory of a ‘non-blue universe’ at the same time. Since there can only be one universe, the whole idea is preposterous.
It’s a key difference that applies to logic – with Boole committing a logical fallacy by indicating ‘the universe’, as 1 and nothing as 0, as Boole does, when introducing his binary approach to algebraic logic. 1 and 0 are both marks on the page, but he does not acknowledge the space in which they stand. In Boole’s thought – and, I would argue, in Luhmann’s, 1 and 0 are both marks. They’re contradictory terms, but the underlying unity isn’t acknowledged. Both Luhmann and Boole mark the unmarked state. Thus, in Boole’s work, and in Luhmann’s, it appears as a mere sign: [*]

Spencer-Brown never makes this error. He symbolises it the unmarked state by making it equivalent to the piece of paper it is written on: [*]
ibid.

[* The original presents here the equations denoted the form of condensation and cancellation respectively that can be found on page 5 of the Laws of Form, that I cannot reproduce here.]
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'Let' is a command from on High. This how it shall be henceforth. 'Let x be the number of angels that can dance on the head of a pin. 'Let' happens outside the formal system to create it. 'Call' is an action that happens inside the the system. You can call the distinction into being by making the distinction, that is by writing the sign. and If you write it twice in a row you call and recall.

Alright, that helps. So we have our meta-language which we're speaking now, and that differs from the formal system being created with the use of the meta-language.

Reading Chapter 1-2 (for some reason I'm finding them linked as I read this the first time -- like I can't talk about chapter 1 without chapter 2, and vice versa) again I can see the opening of 2 as a re-expression of Chapter 1, like The Form needed to be explicated before talking about forms out of the form, and the form takes as given distinction and indication which it also folds together as complementary to one another.

But then I get stuck right after "Operation" is introduced. "Cross" is a name for an instruction. Instruction, from just a bit before, leads to the form of cancellation. But what is the connection between states and instructions? Reading "Operation" again I'm reminded of the First canon "what is no allowed is forbidden". The name operates already as an instruction.

And then I get stuck on "continence", even though that was part of the opening. "Continence" is the name of the only relation between crosses, and that relationship is such that the cross contains what's inside, and does not contain what is not inside of it.

But this is where I really got lost entirely: What is going on from "Depth" to "Pervasive space", or are these concepts that, like the first chapter, will become elucidated by reading chapter 3? Like a puzzle unfolding?
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Re-reading Depth I think I'm getting it this time. (I'm doing this in bits -- in the morning I like philosophy to wake up my mind, and in the afternoon I like philosophy to take a little mental break to something totally different)

$\left. {\overline {\, a \,}}\! \right|$

So s(sub"0") is the blank page surrounded by an unwritten cross. So in this example there are 2 crosses which pervade a which is then named c. In this case that would be the pervading space.
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Some more on chapter 2 --

For 2 I think we could use brackets, thus:

[ ], [[ ]], [[ ] [ ]], [a], [[[a] [[b] [c]] [ ]] Not very clear, and it might be better to alternate square and curly by depth, thus:

[{ } { }], [{[a] [{b} {c}] { }]

So, following along with the axioms in chapter 2...

[][] = []

and axiom 2

[{}] = .

?

I tried messing with 's bit of code and looking for tutorials but got lost in the information web. If there's a easy link to figuring out how to embed multiple crosses, @jgill, I'd be happy if you could pass it along because it does look prettier, and if I can figure out the syntax it's probably not that hard to embed multiple crosses.

Part of me is wondering if we can read Axiom 1 as the line above, and axiom 2 as the crossing of the line above. So when we, while using the form of the meta-language to parse order, draw a segment from left to right that is the law of calling. And when we draw a segment perpendicular to the calling that is a crossing. So if we cross again we negate, but it's easier to see that when we embed the original cross within a series of crosses rather than a series of lines coming off of the original calling.

This makes sense at a purely formal level because they complement one another -- the calling and the crossing are perpendicular but simultaneously need one another in order to be a calling or a crossing. In a sense the perpendicularity of the crossing removes some of the form of space of the meta-language, but not quite because the space of this formal system is defined by the cross rather than by a set of axioms describing space. Perpendicularity can be defined by reference to the cross, rather than the other way about, and from that we can name the space "Cartesian" if we take space to have an infinite series of crosses. (not Euclidean, that would be harder, or at least different, I think) (a bit speculative here.... just trying to think through the ideas towards something familiar) ((Also -- it'd probably have to be two orthogonal and infinite cross-spaces to define the Cartesian plane))

Then Chapter 2 is the use of the axioms to draw a distinction -- a form taken out of the original form of calling-crossing. Which, from chapter 1, is perfect continence.

Is it right to read "construction" as what's happening in the rest of the chapter? That's the impression I get -- if distinction is perfect continence then drawing a distinction will accord with what is given -- distinction and indication. (interestingly, comparing 1 and 2, we can interpret the cross as a kind of circle, but with the space-properties of this formal, rather than a geometric, system)

But what we get is the space cloven by the first distinction is* the form, and that all others are following this form. The space is cleaved by a cross indicating/distinguishing, but distinction is the form by which we can indicate an inside or an outside. In a way we could look at the cross as a mere mark rather than an intent. It would have content but it would not be a* used signal.

The notion of "value" is really interesting to me. The value is marked/unmarked, at the most simple. The name indicates the state, and the state is its value insofar that an expression indicates it. And then with equivalence we are able to compare states through the axioms. At this point I think we can only hold equivalence between the basic axioms, which turns out to have an inside and an outside, and gives a rule for "condensation" and "cancellation". So in a way the value is just what is named at this point, but there's still a distinction to be had between marked and unmarked due to the law of crossing canceling rather than reducing to the original name.

Then the end of the chapter is what follows from everything before. "The end" as I'm reading it starts at "Operation" -- this is where we can now draw a distinction, having constructed everything prior, and it entails some properties about the system being built such as depth, shallowness, and a need to define space in relation to the cross.

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Reading Chapter 1-2 (for some reason I'm finding them linked as I read this the first time -- like I can't talk about chapter 1 without chapter 2, and vice versa) again I can see the opening of 2 as a re-expression of Chapter 1, like The Form needed to be explicated before talking about forms out of the form, and the form takes as given distinction and indication which it also folds together as complementary to one another.

Yes, I deliberately started at chapter 2, because that's the point at which something happens. I could liken it to a new game we have to unpack - fun for all the family, and you're trying to understand the rules while I'm looking at the pieces. It's a construction set of nesting boxes, and the rules set out what you can do and what you can't do. Some of what is going on is making sure that you don't have a box half in and half out of another box, or a box that is inside a box it is also outside. That's continence - like brackets, you can have any number of them inside another brackets and any number of brackets within brackets, but they mustn't overlap { [ ] [ ] [ {} ] } is ok, but { [ } ] is incontinent - the inner square bracket is leaking out of the curly bracket.

Forget about the 'a' for the moment, that comes later when we do algebra. For now, its computation/arithmetic we have the mark that we are also reading as a boundary between a marked and unmarked state and also calling a cross (c) and interpreting as an instruction to cross the border. It's no more confusing than switching a switch to switch things around. :wink:

But this is where I really got lost entirely: What is going on from "Depth" to "Pervasive space", or are these concepts that, like the first chapter, will become elucidated by reading chapter 3? Like a puzzle unfolding?

Depth is easy, It's how many boxes within boxes we're at. count the lines you have to cross to get out. Each line is a c for cross, each space is an s. this is just housekeeping - labelling the shelves in the cupboard.

but perhaps the way to understand is to read through first, and then go back and worry at the terms when you have a grasp of the 'idea of the game'. and all this 's' and 'c' is just a way of talking about

The idea of the game, at first, anyway, is that the stop light is on when the train is in the tunnel and off when the train is not in the tunnel. Mark, or no mark. And that game is what comes next.
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Cool! I think I'm actually following so far then from what lookedlooks* like a very intimidating book.

*EDIT: I shouldn't get cocky, I just started.
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If there's a easy link to figuring out how to embed multiple crosses, jgill, I'd be happy if you could pass it along because it does look prettier, and if I can figure out the syntax it's probably not that hard to embed multiple crosses.

$\left. {\overline {\, \left. {\overline {\, \left. {\overline {\, a+b \,}}\! \right| \,}}\! \right| \,}}\! \right|$

\left. {\overline {\, \left. {\overline {\, \left. {\overline {\, a+b \,}}\! \right| \,}}\! \right| \,}}\! \right|
put in the math boxes on either end, as before.
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$\left. {\overline {\, \left. {\overline {\, \left. {\overline {\, a+b \,}}\! \right| \,}}\! \right| \,}}\! \right|$

$\left. {\overline {\, a \,}}\! \right|$

$\left. {\overline {\, \left. {\overline {\, * \,}}\! \right| \,}}\! \right|$

*Been messing with it to try and figure out how it works, but updating the quote to reflect the code I'm using -- right now I'm uncertain why there's a gap between the top line and the cross-line in the embedded cross I Think I got it now.I'm going to respond to this post to make it appear more user friendly though. See the post below for better instructions.

I notice that if I do not put anything but a space where the "*" presently is that I get a negative symbol popping up, and also I'm still uncertain where that gap is. My hope, in the long run, is to offer strings which people can simply copy-paste with clear delineations for plug-and-play. If I'm running across a limitation rather than just messing up then perhaps "*" could serve as a blank space? But that kind of ruins the effect too.
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but perhaps the way to understand is to read through first, and then go back and worry at the terms when you have a grasp of the 'idea of the game'. and all this 's' and 'c' is just a way of talking about

The idea of the game, at first, anyway, is that the stop light is on when the train is in the tunnel and off when the train is not in the tunnel. Mark, or no mark. And that game is what comes next.

Yeah that makes sense, given how chapter 1 didn't even begin to make sense without chapter 2. I'll keep along. I'm still figuring out the accounting, and how to make the crosses pretty.
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My hope, in the long run, is to offer strings which people can simply copy-paste with clear delineations for plug-and-play.

That would be really useful. I had a little go at getting an empty cross and failed miserably, but that sort of confirms the thesis that the world has fallen in love with symbolising the unmarked state and naming the nameless. 0 And thanks @Jgill for your assistance.

This is the difference between GS-B and Boole, there is no 1:— here, everything takes place in The Hole in the Zero a largely irrelevant but excellent science fiction story from the same era.

(It inflates my petty smugness a wee bit that the implementation of the crosses utilises a series of brackets in roughly the same way I suggested we might do, but thought better of because the result was unreadable, even if the structure was right.)
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$\left. {\overline {\, * \,}}\! \right|$

Put the following code in between a bracketed math, then the code, then a bracketed \math

\left. {\overline {\, * \,}}\! \right|

It took me a second to get the syntax but I read this as: Start at the left. Use the function "overline". Within the squiggly brackets the first "\," can be read as "start expression underneath the overline" and the "\," on the right hand side still inside the squiggly bracket can be read as "end expression underneath the overline", then we close what's underneath the overline with a closing squiggly, then we close the function we called "overline" with the second squiggly, and then "\!" can be read as "This is the end of the expression which started from the left", and then \right starts the ability to write on the right hand side, and we place "|", the alternate character on the "\" key, so that there's a long line written on the right hand side.

Reading it from the middle upward through the crosses:

1. \left. {\overline {\, * \,}}\! \right|
2. \left. {\overline {\, * \,}}\! \right|
3. \left. {\overline {\, * \,}}\! \right|
4. \left. {\overline {\, * \,}}\! \right|

And the others, while it's easy to get lost in the syntax as I did in my first attempt, are expansions upon this first function such that we put our single overline function with a right bracket into another version of itself, and on and on. I'll just post the code I used, though, because I think the above probably serves as a good enough users guide for copy-pasting the code.

$\left. {\overline {\, * \,}}\! \right|$

The code used within the math brackets:
\left. {\overline {\, * \,}}\! \right|

$\left. {\overline {\, \left. {\overline {\, * \,}}\! \right| \,}}\! \right|$

The code used within the math brackets:
\left. {\overline {\, \left. {\overline {\, * \,}}\! \right| \,}}\! \right|

$\left. {\overline {\, \left. {\overline {\, \left. {\overline {\, * \,}}\! \right| \,}}\! \right| \,}}\! \right|$

The code used within the math brackets:
\left. {\overline {\, \left. {\overline {\, \left. {\overline {\, * \,}}\! \right| \,}}\! \right| \,}}\! \right|

And to construct the crosses in the Fourth Canon in chapter 3:

$\left. {\overline {\, \left. {\overline {\, * \,}}\! \right| \left. {\overline {\, * \,}}\! \right| \,}}\! \right|$ $\left. {\overline {\, * \,}}\! \right|$

Which I did by copying the first code with a single cross, and then in place of the "*" I put the copy of the original code twice right where the original "*" was in the first code with a single cross. Then I just copied the code again in a separate Math bracket to have it sit alongside

EDIT: Or the code --

\left. {\overline {\, \left. {\overline {\, * \,}}\! \right| \left. {\overline {\, * \,}}\! \right| \,}}\! \right|[/math] $\left. {\overline {\, * \,}}\! \right|$
• 3.7k
Chapter 3 feels like a set up for chapter 4, which is what I said about 1 and 2 so I may just be in that habit. But I felt like it was all a set up for the final paragraph to make sense -- we have the initials of number and order for the calculus of indications, and Chapter 4 begins to actually write out some proofs from what has been written thus far.

There's something similar to this and using nested sets as representatives of numbers, I think. But then the value isn't numerical, but is rather the marked or unmarked state at its simplest. The first theorem of Chapter 4 points out that these initials are a starting point for building more complicated arrangements and the simple arithmetic of the crosses is what's needed to make sense of the calculus of the crosses.

I'm going to try and work out the proof here by arbitrarily using this arrangement as "a" --

$\left. {\overline {\, \left. {\overline {\, \left. {\overline {\, s(sub(d)) \,}}\! \right| \left. {\overline {\, * \,}}\! \right| \,}}\! \right| \,}}\! \right|$

s is contained in a cross.

All the crosses in which s(sub(d)) is within are empty other than the space in which s(sub(d)) is in. ("*" counting as the unmarked space)

The arrangement chosen uses both cases --

Case 1 -- there are two crosses that are empty underneath a cross next to one another such that s(sub(d)) could have been in either cross. They're equivalently deep.

Case 2 -- the crosses surrounding the two deepest crosses are alone within another cross

So using the steps of condensation and elimination:

$\left. {\overline {\, \left. {\overline {\, \left. {\overline {\, s(sub(d)) \,}}\! \right| \left. {\overline {\, * \,}}\! \right| \,}}\! \right| \,}}\! \right|$ --> $\left. {\overline {\, \left. {\overline {\, \left. {\overline {\, s(sub(d)) \,}}\! \right| \,}}\! \right| \,}}\! \right|$ Condensation

$\left. {\overline {\, \left. {\overline {\, \left. {\overline {\, s(sub(d)) \,}}\! \right| \,}}\! \right| \,}}\! \right|$ --> $\left. {\overline {\, s(sub(d)) \,}}\! \right|$ Elimination

And by the definition of Expression from chapter 1: "Call any arrangement intended as an indicator an expression" we can draw the conclusion that any arrangement of a finite number of crosses can be taken as the form of an expression. (since we're indicating the marked or the unmarked state)
• 3.7k
Originally I wanted to actually put the fourth cannon example underneath a bracket of its own, but I found it difficult to stack multiple bracketed maths within a single bracketed math so there's a bit of a limit there. The only difference, though, would have been that there would have been another step of elimination where the deepest space's value for a was the unmarked state rather than the marked state.
• 22.1k
I'm lost.

Wouldn't it be better to spend your time learning a more widely used version of predicate calculus?
• 3.7k
Heh, yes. Undoubtedly.

The obscure and the strange is one of those things that just nabs my attention. Also I had some notions back when learning baby logic that this book seems to run parallel to. Notions which after writing them down I threw out because they seemed nonsensical, but hey -- there was something interesting about how the calculus managed to deal with the notion of the philosophy of philosophy as an unmarked state rather than a marked state.
• 19.7k
Have a glance at the two pdf's that Unenlightened provided in the OP. LoF is not *just* presenting a new form of predicate calculus, it is intended to build from fundamental principles - the opening sentence states 'that the universe comes into being when a space is severed or taken apart'. Coincidentally, or serendipitously, I appended the following aphorism to my profile page several days before this one appeared: 'The fundamental condition of existence is alterity'. (I'm not as yet really going through it, I'm about 300 years behind on my own self-selected reading list.)
• 22.1k
'that the universe comes into being when a space is severed or taken apart'

Yeah, understood; hence my previous reference to Hegel.

But there are all sorts of issues. The parsing makes it look as if we only need negation, but that's not so. And there's an odd slide in the Appendix from propositions to individuals. And fuck knows what is happening in chapter eleven, where moving out of a plane is equated with bending time... or something.

I think there are good reasons that the book did not catch on.
• 19.7k
Fair enough. I kind of get the appeal but as said, I have other projects to pursue, although I'm very pleased that this thread has been created for the benefit of those interested in it.
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