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

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• 4.3k
Cool, that helps with what comes next in the reading too. I asked about the tunnel thing trying to understand Subversion (why doesn't Figure 2 already behave exactly like f in E1?) -- but if it's because 'f' cannot be substituted by mark/no-mark then what is allowed vs. what must be avoided has an understandable difference because in the first case we're burrying the function 'f" deeper, but in the second case we're bringing a variable deeper to the function. So there are times when we can use chapter 6 ConclusionsConsequences or even the initials of the calculus, but we have to be careful about when it's appropriate to do so.

EDIT: I say that but the next part is opaque to me this evening. Might have to poke around the conference website to see if they have already done work on this chapter. It's not explaining itself as well as the previous chapters have, or at least I feel stupider while reading it. ((Heh, OK, the journal they have set up doesn't have any issues in it. So maybe the conference will be the way to go: "Hey, uh, what's he talking about here and how do you check the oscillating functions?" -- seems they have some videos from the last conference that might be worth checking out if I can't guess through to a possible insight tomorrow morning: http://westdenhaag.nl/exhibitions/19_08_Alphabetum_3))
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OK it's not much, but I'm sharing something that did click this morning: Figure 1 is a graphical representation of E3, and E3 is kind of the same as E1 when a and b are equal to n. E2 always has a solution regardless of where you're at in the iteration, but E3 is indeterminate at every odd-depthed iteration. The imaginary state is with respect to time because:

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

is using the form from before but to represent E3 instead of giving us an expression with a determinate value. The oscillator function is the solution to E3: it's both m and n, and by adding a dimension of time we're able to give a solution to E3 which goes back and forth, as Figure 1 shows. So now I'm seeing the square waves not as switches but as marks in time as E3 goes back and forth between its two values.

Also the bit on steps from Chapter 6 clicked this morning -- we can count steps but they don't cross a boundary so in a sense you can have as many steps as you want, and GSB uses that to generate the infinitely expanding functions of re-entry. But also because we're not crossing boundaries we're sort of in this place where, due to the dimension of time, we can begin to use steps in the process.... something like that. And now it's time for work, but I thought this worth sharing because most of what I've posted has been more confusing than elucidating, and for once I thought I had an elucidating thought.
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Heh the morning routine has worked so far, but this morning I think I have an idea about E4, but GSB really is drawing on his extensive knowledge of electronics. I find myself going back to 's explanation of one-bit adders, and looking over electronics websites, but instead of bits E4 is changing the wave-form as it is "processed" through E4. E4 is also the first time GSB introduces re-entry from a deeper part of the expression to a more shallow part of the expression, where the prior example of demonstrating how markers work is the opposite. The task, for myself, is to see how the sentence immediately following E4 is true (and I'm not sure how to display a wave form here on the forum.): if I substitute the wave structure given for a then how is it that I can obtain either the marked-to-unmarked wave or the unmarked-to-marked wave at the end of the process? Somehow the two waves get "smoothed out", and I believe figure 4 goes some way to show how the crosses modulate pulses (as, I believe, we're meant to understand the statement following E4: a starts as marked or unmarked, and then the pulse is fed into E4 and somehow the pulse changes to a smoothed out marked-to-unmarked or unmarked-to-marked depending upon how it started -- which then, in turn, could be fed into a flip-flop. But here we're not dealing with memory, I don't believe; we're assuming memory and dealing with how expressions change waves)

But this is more or less me saying I think I need more homework to really work that out. My posts would just be guesses in the dark about rules I don't know, which would be even more confusing than the confusion I've already expressed :D

I'll pick through those videos from the previous conference to see if there are more worked out examples there. Else, LoF24 might be the best bet for understanding just how to operate on waves with the calculus.
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Heh the morning routine has worked so far, but this morning I think I have an idea about E4, but GSB really is drawing on his extensive knowledge of electronics. I find myself going back to ↪wonderer1 's explanation of one-bit adders, and looking over electronics websites, but instead of bits E4 is changing the wave-form as it is "processed" through E4.

It has been interesting to read along with this discussion. I get tantalizing hints at what the topic under discussion might be related to, but not enough to be able to say anything helpful for the most part.

I suppose it is a bit like trying to decrypt a foreign language.
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Heh, even reading the book I feel the same :D

There are gaps in my understanding of the book, still. I can say everything up to Chapter 11 mostly makes sense, now, but Chapter 11 is where the calculus suddenly changes -- and he spent 10 chapters making sense of the calculus before changing it all in one quick go.

For me it's mostly the logic that I find fascinating: it's a genuine logic that relies upon neither number or sentences (or truth!), and in the notes GSB even goes into connecting his logic to classical Boolean logic so there's even some sense in which we could say this is a "more primitive" logic. Or, at least, so the guess would go -- It'd be interesting if Boolean logic could, in turn, also derive GSB's LoF, giving a kind of "map" between both where you could simply choose which one you want to use.

So even if I don't quite grasp Chapter 11's operations, it's still pretty cool to be familiar with yet another example of a logic (as opposed to there being One True Logic, or some such).
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Electro-mechanically, the square wave is produced by a circuit that when it is on, turns itself off, AND when it is off, turns itself on. So i imagine a buzzer, a circuit that turns on an electromagnet that pulls a lever against a spring until the circuit is broken, and then the spring pulls the lever back and closes the circuit again - on and off forever.

Whereas, (I'm guessing here) for the memory circuit, there would be no spring, but instead 2 electromagnets that operate a dual switch that turns one on and the other off and contrariwise, vice versa.

And then we come to imaginary values, and my best guess is that it relates to the p expression you mentioned above.

Suppose we now arrange for all the relevant properties of the point p in Figure 1to appear in two successive spaces of expres- sion, thus.
P'p
We could do this by arranging similarly undermined distinctions
in each space, supposing the speed of transmission to be
constant throughout. In this case the superimposition of the
•two square waves in the outer space, one of them inverted by
the cross, would add up to a continuous representation of the marked state there.
— P.61

I'm too lazy to correct the expression - you know what it is...

Anyway, two undefined expressions one under a cross give 2 square waves exactly out of sync, give rise to what looks like a stable 'on' but isn't.

And that is where my understanding ends. How one gets from there to frequency doubling and halving is beyond me.
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Somehow the two waves get "smoothed out",

You mean added? When you mix two frequencies, you get a waveform that is a composite of four different ones: you get A, B, A+B, and A-B. This is superheterodyne theory. It's how AM radios work.

So if A is the radio carrier frequency, we want to piggyback audio frequency B on it by sending A and B through a mixer and then filtering for A+B.

Not sure if that's what you mean or not.
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That could very well be what's going on, actually. I said "smoothed out" because the example wave being fed into E4 is a series of marked-unmarked-marked-unmarked equally spaced out (where both the marked and unmarked square waves are equal in length), but then the output is a single wave which either starts unmarked-to-marked or marked-to-unmarked. Now that you mention adding it kind of does look like the output wave is equal in length to the input wave, it's just that the marked-unmarked-marked-unmarked wave became a long version of either marked-marked-unmarked-unmarked or unmarked-unmarked-marked-marked, depending upon E4's starting state.

So it could very well be addition! That appears addition-like. But to actually mean addition I'd have to be able to parse E4 better. I can see the input and the output, but I don't really understand how E4 operates on the input to obtain said output.

It wouldn't surprise me if you could relate this to radio wave-forms, though one thing that'd be different is that we're dealing with square waves, and my understanding of radiowaves is that they are not square waves. (but I do understand that electronic circuits use square waves sometimes -- but my understanding is not in a real, practical sense. Only I've seen square waves being used as examples while looking at websites while trying to make sense of the book)

And actually, now that you're here, I've started seeing how it might be possible to make counting more explicit -- which relates to the thread on Kripke's skepticism.
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Yeah, it's a bit of a stretch for me. I think at this point what would be best is for me to try and write out a synopsis of each chapter, summarizing the key ideas, while putting a little flag on Chapter 11 reading "needs further research"

Going back to the initial hook, I'd like to understand Chapter 8 a little better because of its relationship to your inference about the philosophy of philosophy being a reflection rather than a content.
• 14.9k
That could very well be what's going on, actually. I said "smoothed out" because the example wave being fed into E4 is a series of marked-unmarked-marked-unmarked equally spaced out (where both the marked and unmarked square waves are equal in length),

What do mean by "marked?"

A square wave would be used when you want something to blink on and off, like the hazard lights on a car, or the turn signal.
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The marked state. So the waveform, trying this out, looks like this on page 66

¯¯|__|¯¯|__

and it becomes either:

¯¯¯¯|____

or

____|¯¯¯¯

And E4 is... not easy to render here, but the link to the book is on page 1 of this thread, and E4 is on book page number 66
• 14.9k

I think this guy probably took a large amount of LSD before he started writing.
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I actually don't. I had the thought, but then looking at his notes and what he did that just doesn't follow. I think he's drawing on his own intuitions about how he used the logic he developed more than he's being very explicit at this point in the book -- in a way he could have ended on Chapter 10 and called it a day, but instead he's trying to deal with the problems of self-reference.

There's something there, but in a way that reminds me of my old calculus professor: he certainly knew what he was talking about, but he found it hard to dumb it down for the rest of us. We managed to make it through, but it wasn't because the professor was good at communicating what he obviously knew.

And here the topic is very abstruse -- we don't even have the familiar things like number to rely upon in thinking through the calculus. But that's exactly what makes it interesting to me.
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I think this guy probably took a large amount of LSD before he started writing.

So perhaps the answer to understanding chapter 11...
• 14.9k
Cool!
• 4.3k
Some random thoughts:

I can sort of see how the cross and variables could represent various electrical components. One of the thoughts I had about re-entry was how, since he's dealing with a very large electrical system he kind of can get away with treating a part of the electrical system as being dependent upon another part of the system in such a way that it's like it's infinite. Or he can summarize a large network of components which are the same in form, but however-many times over (I have no idea what even the ballpark estimation would be) through re-entry rather than having to write out every individual component which would make for a technically accurate but difficult to use map. With re-entry you can summarize a large chunk of components.

And in Appendix 2, page 117 GSB makes a note of how he believes the marked state summarizes a large chunk of the Principia Mathematica -- so I believe it's correct to read him as trying to compress details into something more user-friendly so he can think through the problems of the network (but then he's a mathematician, so he's also developing a math).

Though
perhaps the answer to understanding chapter 11..

That's not out of the question. And I'd go further and say it wouldn't undermine the text either. One of the stories from science I like to tell is about how the structure of Benzene was guessed at by Kukele, at least so he tells the story, when he had a very vivid day-dream of a snake eatings its own tail. The moral being for a science the inspiration isn't as important as whether the idea "works" (in Benzene's case, unified a number of observations into a single theory of its structure)

I don't see any reason to think that one is under an altered state of consciousness to then think that they are unable -- I'd prefer to say differently abled. There are people who see things without drugs, after all, though we also cannot substitute rigorous thinking with the possibly profound experiences people sometimes report hallucinogens having. On this topic I've always found Aldous Huxley's The Doors of Perception to be good. .

In the notes GSB notes his belief about the relationship between logic and mathematics is, on page 101-102:

What status, then, does logic bear in relation with mathematics? We may anticipate, for a moment, Appendix 2, from which we see that the arguments we used to justify the calculating forms (e.g. in the proofs of theorems) can themselves be justified by putting them in the form of the calculus. The process of justification can be thus seen to feed upon itself, an d this may comprise the strongest reason against believing that the codification of a proof procedure lends evidential support to the proofs in it. All it does is provide them with coherence. A theorem is no more proved by logic and computation than a sonnet is written by grammar and rhetoric, or than a sonata is composed by harmony and counterpoint, or a picture painted by balance and perspective. Logic and computation, grammar and rhetoric, harmony and counterpoint, balance and perspective, can be seen in the work after it is created, but these forms are, in the final analysis, parasitic on, they have no existence apart from, the creativity of the work itself. Thus the relation of logic to mathematics is seen to be that of an applied science to its pure ground, and all applied science is seen as drawing sustenance from a process of creation with which it can combine to give structure, but which it cannot appropriate

Which I find super interesting. It's kind of going into how math justifies itself, and in a way it seems GSB believes that logic is an applied mathematics, but that at bottom it all comes out of the void.
• 8.9k
. Logic and computation, grammar and rhetoric, harmony and counterpoint, balance and perspective, can be seen in the work after it is created, but these forms are, in the final analysis, parasitic on, they have no existence apart from, the creativity of the work itself. Thus the relation of logic to mathematics is seen to be that of an applied science to its pure ground, and all applied science is seen as drawing sustenance from a process of creation with which it can combine to give structure, but which it cannot appropriate

I am strongly reminded of Pirsig, here, when he talks about 'quality'. (Which is a fair candidate for the first distinction.) there's a bit in the Zen and the Art of Motorcycle Maintenance about how one judges the quality of an essay first, and forms the criteria for what 'makes a good essay' from the good essays rather than the other way round. And then, the English professor, it is, refuses to grade the students work on the basis that they know the quality of their own work already.

And yes, it was the age of LSD.
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When the sequences of cause and effect become circular (or more complex than circular), then the description or mapping of those sequences onto timeless logic becomes self-contradictory. Paradoxes are generated that pure logic cannot tolerate. An ordinary buzzer circuit will serve as an example, a single instance of the apparent paradoxes generated in a million cases of homeostasis throughout biology. The buzzer circuit (see Figure 3) is so rigged that current will pass around the cir­cuit when the armature makes contact with the electrode at A . But the passage of current activates the electromagnet that will draw the arma­ture away , breaking the contact at A . The current will then cease to pass around the circuit, the electromagnet will become inactive, and the
armature will return ro make contact at A and
If we spell out this cycle onto a causal sequence, we get the following:

If contact is made at A, then the magnet is activated.
If the magnet is activated, then contact at A is broken.
If contact at A is broken, then the magnet is inactivated.
If magnet is inactivated, than contact is made.

This sequence is perfectly satisfactory provided it is clearly understood that the if . . . then junctures are causal. But the bad pun that would move the ifs and thens over into the world of logic will create havoc:
If the contact is made, then the contact is broken. If P, then not P.
The if . . . then of causality contains time, but the if . . . then of logic is timeless. It follows that logic is an incomplete model of causality .
— Mind and Nature

This, in case anyone wonders, is why my next reading thread is

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