Original Transcript: http://www.lawsofform.org/aum/session2.html
SPENCER BROWN: I am aware of a number of different pulls as to which way we could go from here, and in a universe where there is a degree of exclusion, one way excluding the other in a finite amount of time, I'd like to get some consensus as to which way we might go from here. If we could just ask questions of people as to what we could profitably talk about next.
VON FOERSTER: I think it would be lovely if you would make again for us the very important distinction between algebra and arithmetic. Because the concept of arithmetic is usually--although every child knows about it, and it is plain everyone knows about arithmetic--and here arithmetic comes up in a more, much more, fundamental point. And I think, if this is made clear, I think a major gain will be made for everybody.
SPENCER BROWN: I'll do what I can to make the distinction clear. I was going to say, "make the distinction plain," which means to put it on a plane. I suppose that most people know that the meaning of the word "plain," if you look at its root, is just another word for plane, plane like blackboard [1 - The Indo-European root is pela:, flat, to spread. Related words in English are PLAIN, FLOOR, PALM, PLANET ("to wander," i. e. spread out) PLASMA, POETIC, POTTS]. To make plain is to put it on a plane. So that's what I will do. I will try to put this distinction between algebra and arithmetic on a plane.
The reason it should go on a plane is that in a three-space it is difficult to disentangle the connections. So we project it onto a plane. On a plane, we can take a plan, which is the same word. We can see the relationships of the points on inner space, which is not too difficult to comprehend.
Now to make a distinction between algebra and arithmetic, I should go to the common distinction which is made in the schoolbooks, where you have--there are two subjects in kindergarten, perhaps a little beyond kindergarten--a subject called arithmetic, which is taught to you first, and then we have algebra, which is what the big boys and girls get onto and look rather superior about.
First of all, let me explain--this word keeps coming up, "out on a plain"--that even arithmetic is not what is taught at school. Mathematics certainly isn't. What the child is first taught is the elements of computation--the computation of number, not of Boolean values. He is taught the elements of computation, which is wrongly called arithmetic. Whereas arithmetic is the….
Let's be clear, for the moment. I'll go back and start again. We should approach this slowly and deviously. I don't want to give the game away before we have got there. In arithmetic, so-called, which the child is--it is true that it doesn't begin with arithmetic, because the child is given an object, two object, three objects, four objects, and he is--I don't think that he is taught that there is something called a number, but he is then taught to write "one, two, three, four," etc., and he is not given that there is somewhere between that--that it one object, that is two objects and that and that, that, that—there is somewhere between these a non-physical existing thing called a number. As I point out in Only Two, a number is something that is not of this world . That doesn't mean to say that it does not exist--it surely does. In fact, there are many extant groups of numbers.
VON MEIER: Exstacy? Exstasis.
SPENCER BROWN: Well, yes, that's "outstanding." Existence, ex, out, stare, to stand, outstanding.
What outstands, exists. And numbers do not outstand, they do not exist in physical space, They exist in some much more primitive order of existence. But they, nevertheless, do exist. But not in the physical universe. This, by the way, is the first way to confound the material scientist who thinks that physical existence is all there is.
You ask him--"Well, you know that there are numbers?" He will perhaps have to say that there aren't any numbers; in that case, you can't beat him. If he admits that there is such a thing as a number, then you say, "Well, find it, where is it, show it to me, and he can't find it. It does not exist here.
Now, a child may come to learn very much later, here, there are objects arranged in groups, here are figures.
Number is to be found in another space. Not in this space.
However, these are the symbols, tokens of number, which can be in any form--Roman, etc. Playing around, saying "two plus three equals five," an elementary computation with numbers, is discovering relationships with numbers, and how they are constructed and what they do together. Sounds a bit rude, but that's what we do.
First Distinction, Observer. and Mark
LILLY: At the end of Chapter 12, you make a sort of covert statement. You do not develop it, and I'd like you to develop it a little further. You mention that it turns out that the mathematician is one of the spaces.
SPENCER BROWN: The mathematician?
LILLY: Yes, this one of the spaces.
SPENCER BROWN: The part of the observer?
"We now see that the first distinction, the observer, and the mark, are not only interchangeable, but, in the form, identical" . I don't see how you didn't get it already.
The convention is that we learn to grow up to play the game that we are taught - the play is that there is a person called "me" in a body called "my body," who trots about and makes noises and looks out through eyes upon an alien, objective thing we call "the world," or, if we want to be a bit grander, called "the universe," which the thing called "me" in "my body" can go out and explore and make notes about and find this, that, and the other thing, find a tortoise, and make notes about a tortoise, and drawings, etc.
The convention is that this tortoise is somehow not me, but is some object independent of me, which I in my body have found.
We also have a further convention--well, depending on what sort of people we are, if we are behaviorists, we may not think this--most of us think that the tortoise also sees life in much the same way--that it is a being that has "my shell," "my feet," "my tail," "my head," "my eyes," out of which I look through the hole in the front of my shell and I see objects, big things walking around on two feet, etc., which are different from me. And we think that the tortoise thinks that.
Now, supposing that this is-only a hypothesis. Supposing that--if there were a distinction--if there were that--only supposing that, if it could be, what would happen --well, if one imagined--supposing one imagined, well, this is me and that ain't me. Surprise, surprise, what ain't me is exactly the same Shape as what is me. Surprise, surprise.
Come to this another way. Take it philosophically. take it philosophically and scientifically. Scientifically, on the basis of, there is an objective existence which we can see with our eyes and feel with our fingers and hear with out ears, and taste with our tongued smell with our nose," etc., and then we take it to ants. Now ants can see ultraviolet light, which we can't see, and therefore the sky looks quite different tol take it to extremes.
If there are beings with senses, none of which compares with ours, how could they possibly see a world which compares with ours? In other words, even if one considered it scientifically, the universe as seen appears according to the form of the senses to which it appears. Change the senses, the appearance of the universe changes. Ask a philosophical question and you get a philosophical answer. What therefore is the objective universe that is independent of these senses? There can be no such universe, because it varies according to how it is seen, the sensory apparatus. Take this a little further, and we see that we have made a distinction which don't exist. We have distinguished the universe from the sensory apparatus.
But since the universe changes according to changes in the sensory apparatus, we have not distinguished the universe from the sensory apparatus. Therefore, the universe and the sensory apparatus are one. Wow, then, does it appear that it is so solid and objective looking?
Now, the answer to this profound question takes a lot of thought, but I will try to give it all to you in a very short time. Because it takes a whole series of remarkable, Re-markable, markable again ("a whole series).  tricks before it can be made to appear like this. But since, if there ain't no such thing, then any trick within the Laws of Form is possible, this happens to be one of the possible tricks. If there is no such universe, if there is only appearances then appearance can appear any way it can. You have only to imagine it, and it is so.
BATESON: Can you go into the proof?
SPENCER BROWN: The proof, my dear sir, has nothing to do with the objective world, the proof is mathematical. Nothing in science can be proved.
BATESON: I see.
SPENCER BROWN: It can only be seen. But where it is co-extensive with mathematics, in that, in fact, what is so in mathematics-- The basis of what is so in mathematics is what can be seen. Theorem and theatre have the same root . they are the same word. It is the spectacle  that we see, and the discipline of mathematics is to go to what is so simple and obvious that it can be seen by anyone. Without doubt, it can be seen. And from this--
BATESON: By turtles. Can it be seen by turtles?
SPENCER BROWN: I don't know whether turtles see it. If they do, they have a different discipline whereby they communicate it. We don't talk with turtles, and I can't answer that. I have never spoken to a turtle. But I am sure that turtles can see. Well, they can certainly contemplate reality. I don't know whether they need to see mathematical theorems. I don't know whether they play that game.
VON MEIER: Yes, they carry their numbers on their back--13 variations in the shells in a certain pattern. It's the second avatar of Vishnu, so that when you see the turtle, you're seeing it from the point at which the Ethologists named it God. They have named the serpent the first avatar of Vishnu. She's the cosmic turtle swimming in the sea. And things that run around, run around on the back of the turtle.
LILLY: This is called "maya-matics."
PRIBRAM: Why so solidly Why is objective, so-called reality, so solid?
SPENCER BROWN: Well, it has to be, after all. Oh, dear, what we need is a 20-year course to get to that point.
JEAN TAUPIN: Any reality is real, the moment you perceive it as real?
SPENCER BROWN: Well, "reality" means "royalty." The words have the same root . Whatever is real is royalty. And what is royalty but what is universal--the form of the families of England.
VON MEIER: The measure, the rex, the regulus.
SPENCER BROWN: Yes, that is true.
LU ANN KING: You said two things. The motive precedes the distinction. And then, later on, you said that one has to determine their relevance, that in searching for a clue, you also have to determine its relevance.
SPENCER BROWN: You have to make a relevant construction, yes.
KING: Well, just personally, what process do you--
SPENCER BROWN: How do I do it? Just contemplate. One also tries all sorts of ways to get familiar with the ground. Why? It may take you two gears to find a proof which could be exposed in five, fifty seconds, is that you get familiar with the ground. You try in many ways that won't work, and then you try and try and try and then you realize that trying-- One day you stop that. And then, almost certainly, you will find-- Or seeing it, seeing anything to be so
I had been working on the second-degree equations for five years at least. I was thoroughly familiar with how they worked, and so on--hadn't seen what they were, theoretically. I was wondering whether to put them in Chapter 11, or some other beautiful manifestation of the form, whereby you break up the distinction and it turns into a Fibonacci series. Well, I won't go into that now because it is another thing altogether. In Laws of Form there is only about one twentieth of the discoveries that were actually made during the research. There is enough for 20 books, mathematically, and I had to decide what I could put out, and what I could put into. But the actual research in London is 20 times of what is in the book. And I wasn't quite sure whether to put it in at this point--because the book had to be finished. I wasn't quite sure whether to put in, with this chapter, this beautiful breaking up of the truth where you get the rainbow, which turns into the Fibonacci series. You break up white light and you get the colors.
You break up truth and you get the Fibonacci.
VON MEIER: The logarithmic growth spiral? 
SPENCER BROWN: Yes. I decided in the end that it was more practical to put in the expressions which went into themselves, because we did have practical engineering uses for this. But I still didn't recognize the theory. We had been using it, my brother and I, in engineering, but we still didn't recognize what it was. So I sat down to write Chapter 11 and without thinking I wrote down the title. I wasn't sure what I was going to call it, but I wrote something without thinking. I looked at it and I found what I had written was "Equations of the second degree." Now, I was not aware of writing this down. The moment I had written it, that was--Eureka--that is what it was. The moment that I spoke of it to my brother and then to other mathematicians, it began to focus. Yes, of course. And then it was only a matter of an hour or so to go through and see the analogy, which I did on the blackboard this morning. - To see the paradoxes and everything, all the same, all existent in the ordinary common arithmetical equations of the second degree. And this is what we were doing in the thrown-out Theory of Types; the coming to the knowledge of what it is.
How actually does this happen? It happened something like that, after five years of scratching one's head but thinking, nevertheless, let's find out more about it. And then it comes, in a way, quite unexpectedly; in a way, really, for which one can take no personal credit.
RAM DASS: Is the five years the method to get to the space--from which all titles are, or there was an implication in what you said that your familiarity with the ground was the prerequisite for you. Then stopping trying and then out comes this thing, which is like a subliminal, or a latent, or something inherent in the analytical process--nothing more?
SPENCER BROWN: Well, I will distinguish the proceeding. It goes very much like the education of the child. The child is born knowing it all, and it immediately has this bashed out of it. It's very disturbing. So it learns the game then. It learns the game that is played all around it--and with variations, it is much the same game in any culture, whether it is the ghetto, or ten thousand years ago, or today in America, or today in England, or today in China, or wherever it might be. It's much the same thing, with variations, of course, in the particular cultural pattern. It has its original knowledge bashed out--it must be bashed. Those of us who -have gone back and remembered our births, remembered what we knew, and remembered the covenant we then made with those standing around our cradle, the realization that we now have to forget everything and live a life--
RAM DASS: Excuse me, is the word "know" the proper word to use? Doesn't that imply a knower and an object that is known? Couldn't you say that the infant was being it all?
SPENCER BROWN: If you like, yes. I am only using words--you see, the language is no good for talking this way. We have to use these imperfect terms, which are based on distinctions. And you are quite right, it is not knowing, it is only in its interpretation, knowing. It is like dreaming a dream. While the dream is going on, it isn't funny. But bring it out into the critical atmosphere of waking life--now it appears funny. The child is bringing out into this, and it remembers it and knowing it. That is the way it is taught the disciplines. Can't Have One Without the other
RAM DASS: But you never can get into knowing it. You can only get back into being it again. You can only know a segment or a--
SPENCER BROWN: Well, it's dual, of course, because getting back from the-- There is no enlightenment without un-enlightenment.
RAM DASS: There is no survival without un-enlightenment, actually.
SPENCER BROWN: Well, I'll come to that.
VON MEIER: The planted seed is always regarded by primitive cultures as having died.
SPENCER BROWN: Enlightenment is different for every form of culture, because every form of culture is a form on un-enlightenment. And the enlightenment matches it, as the form of enlightenment for our culture matches our culture. It matches the way in which we have been unenlightened.
Enlightenment by itself, there is no such thing, just as there is no black without white. But to be enlightened, having been un-enlightened, is not the same as having been un-enlightened before. Because one wasn't really unenlightened at all.
MAN: We need another word.
SPENCER BROWN: Ta. Before, you are neither enlightened nor unenlightened. Then you become unenlightened, from which you have to be enlightened. That is not the same. You remember your original unenlightenment.
PRIBRAM: Original lightenment.
SPENCER BROWN: Well, that would be, in a way, but it might hurt.
VON MEIER: When one sees the light for the first time, from the interior of logical models, or from the cosmic tortoise in the sea, or from the inside of the womb. You see light.
PRIBRAM: What happens ontologically is that somehow as you go on through those five gears you distribute the thing, get it split up into parts all over the place, and then, what seems to occur, is that some new constellation, new way of getting it put together again, occurs at that moment of enlightenment. It's some process of that sort.
MAN: After the five gears, what happens?
PRIBRAM: I don't know, I just got it to that stage.
SPENCER BROWN: Let's simply go through the procedure again. the covenant with the world that the child rapidly has to make is--"Right. I am not allowed to notice this. But the child perceives where the lines are drawn and not drawn, and then suddenly it realizes that is must put on the same blocks, otherwise it will not be accepted. There is a moment of sanity when this happens. However, it's "good-bye" for quite a long time, I don't know how long. If it is to survive, it's "good-bye," and "hello"- "Hello, world."
And now instead of it being able to deduct, because it sees that is-fully outlawed, now it goes through the game of those people who know best, and who are teaching it; and in order that they can have the game, and it can play it, it must pretend to know nothing, so that they can now pretend that they are bringing it up and educating it. And so it then has the-- It goes through the learner stage of playing the game, of looking at things and being surprised. "Oh, look at that "What's this for?" and so on.
And thus, the whole proceeding of playing the game that there is an objective world which you can run around and look at and pick flowers and bring them back and say, "Look."
It's when one gets very far in this game and begins to wonder what it's about and how it is that we do find something outside, and it does appear to have some structure, and so forth, and to come back and base it on what we are doing, that we begin to see--that we begin to ask the question, well,-what is there outside? We begin to realize that what is outside depends on what is inside.
One of the questions that we might ask is why we appear to see the same things. Why does it appear-- I can see the moon and you can see the moon. If you are a different shape from me, then you should see a different thing. Well, if you take it-back far enough, we have this, () (()) = ((())). In other words, from one mark, we have any number of identical marks. This is a process here.
Actually, mathematically, this arrangement is still only one distinction. It has essentially the same rules as this one. Just let's have a look and see where you are. Outside, outside, inside, inside, making one crossing. Outside, outside, inside, inside, outside, outside. There is no difference mathematically between that and that. You make the same number of crossings, you get the same thing. This is the inside and this is the outside.
In other words, we can form another illustration of, that is the same as this, in-the point where they condense; but even so, we can make it look like two.
Insofar as you and I see the same moon, we do so because it is an illusion that we are separate. We are the same being. We only appear separate for the convenience of filling space. Of course, we can't have empty space-we'll have to fill it up with something.
WATTS: Parkinson's Law.
SPENCER BROWN: Yes. And with only a limited material to fill it up with. So since space is only a pretense, the observer, in filling space, undergoes the pretense of multiplying himself, or stationing himself. But two people are only like two eyes in one of them. The scientific universe, the objective form which we examine with telescopes and microscopes, and talk about scientifically, is not the form which our individual difference distinguish, It's the form which our basic one-ness, our multiplicity condensed to one, ()()... = () -- It's the scientific, objective universe observed with the part of us that is identical for each of us. Hence its apparent objectivity.
What is called "objective" in science is where we actually use our individual differences, where we say, "Well, that's rather different from that," if, in fact, what we observe depends upon that, and so forth; therefore, that's not an objective distinction, that is something which is a personal view. And that is not what science is about.
WATTS: Well, how would you react to the remark that what you have been saying is a system that used to be called "subjective idealism," in which you have substituted the structure of the nervous system for the concept of mind?
SPENCER BROWN: Well, I can go along with the nervous system, because the nervous system is an objective thing in science as well as a thing we observe--as the constants of what is called a body, which is an extension into hypothetical space of a hypothetical object. I have never had this thing about brain at all. "Inside my something brain"; "my teeming brain." I have never felt that my brain is particularly important.
WATTS: Are we talking about the structure of the sense organs?
SPENCER BROWN: Yes, only to bring us back to the fact that we have made the distinction between the world and ourselves. I have played the science game to show that even in science, playing the science game, which is to say, "Right. The reality is thus: there is a distinct me with senses. There is an objective world with objects and lights and things flashing about, and when I see that window there it means because there is light coming through that window focused through the lens of my eye on my retina in a certain pattern, which goes through the nervous channels to the visual area of the brain, where it all project into a muddled, upside-down--" and so on, with the whole scientific story.
And the trouble with the whole scientific story is that it leaves us no farther, it leaves us no wiser than we were before. Because nowhere does it say, "And here, this is why that is how it appears."
But if you play that game, as I was doing for the purpose of illustration, one still finds that, operating philosophically, and saying "Suppose I change all my sensory forms, now the whole universe is changed, I am only doing this to show that even playing the science game, whatever game we play, must leave Us the same place. Even playing the science game, we see that there is no distinction between us and the objective world, except one which we are pleased to make.
KELLEY: Can you tell Us something about the Fibonacci development?
SPENCER BROWN: Since not everybody here has mathematical training, it is something which, if I have the breath, I will do later with you and perhaps a number of other mathematicians. To explain it to the people-beautiful ideas mathematically--it does involve some rather lengthy exposition.
KELLEY: Maybe another question that wouldn't be too far out: what's the definition of the accidental factorization?
SPENCER BROWN: I was hoping you weren't going to ask me that. It has been a long time since I've done this. Again, it is something that I would--
KELLEY: As I recall, I think that Godel's Theorem basically says that in an algebra, you don't have completeness and consistency.
SPENCER BROWN: Not in an algebra. In the common algebra of numbers you can't have both. This is where so many people go wrong over it.
KELLEY: But the result is no more general than from the common algebra of numbers?
SPENCER BROWN: No. For example, in this algebra you do have completeness. I have proved it. And consistence. I have proved both. In Laws of Form, you find proven consistency , and, in Theorem 17, proof of completeness.
KELLEY: O.K., now, does Godel's Theorem only apply to the algebra based on real numbers? That is, it's beyond the integral, it's a field, right? The field of real numbers, where you've got multiplication, addition, and associativity.
VON MEIER: A system at least as complex as arithmetic.
KELLEY: Well, I am trying to find out where the boundaries are.
SPENCER BROWN: It does, in fact, have interesting boundaries. This is a very common error among mathematically trained people, that it does, in fact, apply all over. It doesn't--it's not applicable to the primary algebra, which is both consistent and complete. It is not applicable in a modulus where algebraic factoralizations are the only factoralizations. Godel's Theorem doesn't apply. The modulus is both consistent and complete.
The ordinary algebra of number, not introducing the complete system--for example, the algebra of the positive and negative integers--now Godel's Theorem applies, provided you use both constants, multiplication and addition.
The difficulty is that you have got two voids--you have got a void of zero in addition and you have got a void of one in multiplication. The constant you put in makes no difference. Interestingly enough, it doesn't apply to the complete number system. 1 , , 1
KELLEY: It applies in an integral domain, but it doesn't apply in the field.
SPENCER BROWN: In the whole, in the complete field, using real and imaginary number, no. Complex numbers. It doesn't apply.
KELLEY: O. K., what if you have just the field of real numbers, not including complex numbers?
SPENCER BROWN: Then I think it applies.
KELLEY: Now I think I am beginning to get an idea of where the boundary is: when the field just includes real numbers, it applies; but when you have complex numbers, it doesn't.
SPENCER BROWN: It happens that way, yes. But it is beautifully illustrated in cases where you are working with a modulus where you have no accidental factoralizations. If you find the algebraic factoralizations, you have found them all. Whereas when you are not working with this modulus, when you are working with the integers, when you have found the algebraic factoralizations, you still haven't found them all. The others are called the accidental factoralizations.
(End of Session Two.)
1. The Indo-European root is pela:, flat, to spread. Related words in English are PLAIN, FLOOR, PALM, PLANET ("to wander," i. e. spread out) PLASMA, POETIC, POTTS. From related roots we get FLAEE, FLAG, PLEJA, PIANO, PLACENTA, FIAT, FICUNDER, PIANO', PEACE, OPIATE, bed.
2. See Note 4, pp. 134-5.
3. The illustration below that the algebraic system is in complete, since its rules do not generate all of its possible states.
4. In the United States, by the Chelsea Publishing Co., New York. A reprint Of Carnegie Destitute Publication #256 (28). By Leonard Eugene Dickson.
5. Multiplying the first three primes, 2 r 3 x 5, and adding 1.
6. P. 76, IOF. & e also Blake couplet quoted on p. 126, Only Two, "If you have made a circle to go into/Go into it yourself and see how you would do."
7. Re-markable, markable again ("a whole series).
8. Greek theasthai, to view.
9. L. species, a seeing, form (SPECIES), from Indo-European spek- to o serve. Related is i. speculum a mirror.
10. American Heritage Dictionary gives REAL, from IndoEuropean root rei-, "possession, thing" (Latin res, thing.); ROYAL from IE reg1- "to move in a straight line" (Latin rex, king, READ27 a Spanish coin; rectus, right, straight RF&YM RECTOR RECTUM; L. regular straight piece of wood, rule REGULATE, RULE; Middle Dutch rec. framework RACE; Sanskrit raiati, he rules RAJAH.
11. The Fibonacci series goes 0, 1, 1, 2, 5, 5, 8, 1D, 21, etc.
12. Theorems 3, 4.