AUM Conference Transcript - Session Two
Monday Afternoon, March 19, 1973
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.
Algebra and Arithmetic
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 thank, 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 . 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 wag 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, n 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.
When the child gets a bit older, he is taught what is called algebra. The first teaching of algebra that was given to me, it may be the same here, was that we were given things like "a plus b equals c; find c when a equals five and b equals twelve." And we all scratched our heads and learned to do this sort Or thing. Eventually we came to formulae that were algebraic, and were finally told things that were universally true. We were taught that an algebraic relationship is true irrespective of what numbers a and b stand for. In other words, as we learned algebra, we learned it as an extension of arithmetic.
As we got a little older still and went to the university, we learned different names; and-we were taught that, whereas, these were constants and these were called variables, you could learn the science of algebra without ever knowing what those words stood for at all, treating algebra as a possible system, and having derived, actually, your rules of what to do, in the case of an ordinary algebra of numbers, from experimenting with the arithmetic. Eventually you see what the rules are, and you operate and find things out without referring back to the constants.
I have given the game away now. This is the difference between an algebra and an arithmetic. The algebra is about the variables, or is the science of the relationships of variables. It is a science of the relationships of the variables when you don't know or don't care what constants they might stand for. Nevertheless, the constants aren't irrelevant, because whatever arithmetic this is an algebra of, if you were to substitute constants for this variables, a, b, etc., then these formulae still will hold.
A lot of people have said, you see, "How can you have an arithmetic without numbers?" as the primary arithmetic in Laws of Form is without numbers We will go back in a moment to that. But just at the moment will emphasize, or return to, for memory purposes, the fact that the definition-the difference between algebra and arithmetic is that arithmetic is about constants, the algebra is about variables. The arithmetic is a science of the relations of constants.
In a common arithmetic for university purposes, which for a less vulgar name is called the Theory of Numbers, is the same thing. The Theory of Numbers is arithmetic, it's common arithmetic. The Theory of Numbers, the most beautiful science of all in mathematics--I happen to like it myself, so I praise it--or one of the most beautiful, is the science of the individuality of numbers. A number theorist knows each number in its individuality. He knows about the relationships it forms, and so on, as an individual, as a constant. An algebraist is not interested in the individuality of numbers, he is interested in the generality of numbers. He is more interested in the sociology Of numbers That applies, whatever individual numbers come there; he has produced a rule where these people go there and there and there, and so on, and he's not interested in individuals at all.
A very interesting point here is the illustration of Godel's theorem in the difference between, in number theory, an algebraic factoralization of a number and an accidental factoralization. As you know, we know from Godel's Theorem that in the common arithmetic, that's the arithmetic of the integers, the algebraic representations, the rules of the algebraic manipulation of numbers, do not give you the whole story. It doesn't give you the complete story of what goes on in arithmetic. And so we have this factorial relationship--any number that is in that form, we know will factorize into that form. But there are what are called in Number Theory 'accidental factoralizations," which happen over and above and irrespective of any algebraic factoralizations that you can find. And this is a very beautiful illustration of Godel's Theorem, Nobody has ever used it. I think this is because, in general, mathematicians don't-understand Godel's Theorem or even know what it says. I have lectured to an audience of university mathematics teachers Of maybe rifts. "Can anybody tell me Godel's Theorem?" Not one. Not one knows what it is. It is one Of the extraordinary breaks which mathematics took about the turn Of the century. Where a logic broke off from mathematics, and the two, you know, despised one another; like in gliding and power flying, they weren't speaking the same count. Hence we have this tremendous break, this schizophrenia, in mathematics, where common illustrations Of one thing in another field Just aren't seen as such. Accidental factoralization is a most beautiful illustration of Godel's Theorem, if a somewhat technical one, in number theory .
Now having seen, therefore, the difference between algebra and arithmetic--simply that arithmetic is concerned with constants and algebra is concerned with variables-we have--well, as Whitehead points out in the Treatise on Universal Algebra, Vol. 1, he points out that Boolean algebra is the only for non-numerical algebra known. Shortly after that, there was a book written by Dickson, who is also a number theorist of some considerable fame, who wrote a very wonderful book called the History of the Theory Of Numbers, now published by Dover ; and anybody who is interested I think should get it because it contains all that would be Of interest, except a Very few later things. And he starts right at the beginning with amicable numbers, and shows that the early mathematicians, if they wanted to be friends with somebody, would find a pair Of amicable numbers, and they would then swap numbers and they would eat the number Of their friend, to keep the friendship. All this is in the History of the Theory of Numbers, by Dickson, which is a wonderful book. He also wrote a book called "Algebras and their Arithmetics," which--I don't think he actually said it, but it was obvious that every algebra has an arithmetic.
At the same time, mathematical popularized such as W. W. Sawyer were writing popular expositions Of various forms of mathematics, including Boolean algebra. And Sawyer heads his chapter on Boolean algebra, "The algebra without an arithmetic." This can't be. This can't be. If it is an algebra, it must have an arithmetic. And if any mathematician could write this--I am not blaming Sawyer; Sawyer was only standardizing what is common mathematics taught in universities today. He is standardizing the common confusion and block. The fact that mathematics teachers in universities today do not understand the difference between an algebra and an arithmetic, which is simple.
How to Find Laws of Form
So, to find the arithmetic of the algebra of logic, as it is called, is to find the constant of which the algebra is an exposition of the variables--no more, no less. Not, just to find the constants, because that would be, in terms of arithmetic of numbers, only to find the number. But to find out how they combine, and how they relate--and that is the arithmetic. So in finding--I think for the first time, I don't think it was found before, I haven't found it--the arithmetic to the algebra Of logic--or better, since logic is not necessary to the algebra, in finding the arithmetic to Boolean algebra, all I did was to seek and find a) the constants, and b) how they perform.
And the first four chapters of Laws of Form are just about this arithmetic, And the nine theorems, with which the two connective theorems later form what would be called in any other algebra postulates, are called here theorems, because they are proven. They are not postulates--you do not have to postulate. These are the basis upon which we can build the algebra. The initial equations... are the rules of the arithmetic--or rather, they are all the equations necessary for the arithmetic.
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VON MEIER: What geometries would follow from this?
SPENCER BROWN: None whatever.
VON MEIER: Would there be any relationship?
SPENCER BROWN: Well, geometries are sciences Of spaces Of this kind, of a three-space or something like that, where you already have measurement. In the initial space which we are concerned with, where you have just drawn the distinction, there is no size and therefore no measurement, and no geometry can follow from it. Or, if you like, this is the geometry of it. In other words, we are in a place where geometry and arithmetic condense. Later on we can see, of course, in the Euclidean geometry, for example, that we can express it algebraically and, therefore, arithmetically, without figures.
VON MEIER: This is a plan of geometric icons, then.
SPENCER BROWN: Well, the point is that you don't have any geometry as distinct from the arithmetic here, because if you go into the definition of mathematics in a textbook, you will see that it is the science of spatial relationships --it's about space. The simplest science, the simplest form of space, is of distinction.
Proof a Demonstration
KARL PRIBRAM: Is this related to the difference between demonstration and proof?
SPENCER BROWN: Ah, the difference between demonstration and proof is that a demonstration is always done by the rules. A computer can do a demonstration. We can demonstrate this, (((()()))) = : condense-we get--that goes out and we get that and that goes out, you see and it goes to nothing. And here, (((()))) = , we just condense twice and it is nothing-not condense, cancel twice and it is nothing. So you demonstrate that-there, that's demonstrated...that's 0. K., you see. And we can demonstrate algebraic forms. This is a demonstration of arithmetic and a demonstration of algebra as well.
Now proof is quite different. Proof can never be demonstrated. I will give an example of proof--one which is familiar, I am always giving it as an example; and those who have been given this example will forgive me if I give it again now, because it is a very beautiful theorem, a very beautiful proof by Euclid To show the difference between a proof and a demonstration, I go to the illustration--rather than to the Boolean form, I go to the common school form, the arithmetic of numbers, which is so familiar to us all, and therefore, I thinks better for illustration.
We can give a demonstration, a computer can demonstrate, we just follow the rules within the calculus. Where we have to prove something, we always have to find--we cannot find it with the rules within the calculus. In other words, no-computer will compute a proof.
And we take, for example--or for the purpose of illustration-Euclid's proof of his beautiful theorem;; the question asked, "Is the number of primes infinite?" As we see, the prime numbers, and it's obvious when you think of it, as they go on, they get sparser. It's very obvious that they will, if you consider it, because every time we have a new one, we have a new divisor which is likely to hit one of the numbers we're looking for to see if it's prime. If it hits it, if it divides into its then it won't be prime. So, the bigger the number, the less likely it is to be prime. A strange sort of statement. the science of certainty, taken in probability terms. Because the more primes there are that could divide into it. So for fairly obvious reasons, as we continue in the number series, the primes get, in general, further and further apart. there are fewer and fewer of them. And what Euclid asked was, do they get so thinly scattered that in the end they stop altogether? Or does this never happen?
This is an example, now, of a mathematical theorem. To make it into a theorem, you actually give the answer, you actually state the proposition, "The number of primes is endless." You may not be certain whether it's true or not; you may still be asking the question, do they come to an end or do they go on?
Well, to illustrate the difference between mathematical art, because it now needs an art to do the theorem, where it only used a technique, a mechanical application, to demonstrate something, and we don't need to do it ourselves, as computers can do it so much better, we will now do something that a computer can never do. Because what we are going to do is find the answer to this question-do the primes go on forever or not? We are going to find this answer quite definitely, and we are not going to find it by computation, because it cannot be found by computation; but it can be found like this. This is the way Euclid found it. Be said, supposing they come to a stop--all right, if they come to a stop, then we know they are going to go on for a long time until we come to big primes, but, if they do come to a stop, there will be some largest prime, call it big N. That's it. That is the last prime, the biggest of the lot. I* they come to a stop, there must be such a prime. Now, if there is such a prime, and there it is up there, let us construct a number which looks like this: all primes, every single one of them, up to and including Big N. Right. We have made this number by multiplying all the primes together Now, Big N being the largest, this is a number which;is made of all the primes there are, there isn't another prime. Because we have assumed that this is the largest.
On the hypothesis that this is the largest, this number is now all the primes multiplied together, and we'll do this multiplication and get the answers and we'll call the answer Big M. We'll take this number Big M, and we will add one. Now we will examine the properties of Big M Plus One. You see this is why arithmetic is so lovely: it is about individuals. Here is our number Big M, as an individuals here is Big M Plus One. It is a hypothetical number, actually, it is a nonexistent number--this is why we can't speak of numbers not existing, because some Of them do and some of them don't. Big M Plus One, let's examine its properties. Well, it is obviously hot divisible by any other prime, including N. because we know they all divide M; therefore every single prime leaves a remainder of one when we attempt to divide it into M Plus One. So M Plus One, therefore, must either be prime, because it is not divisible by any existing prime; or if it ain't prime, then it must be divisible by a prime which is larger than N. Therefore, by assuming that there is a biggest prime, call it N. we have ineluctably shown that this assumption leads, absolutely without any doubt, to the construction of a larger prime, which is either M Plus One or another dividing M Plus One. And that is how Euclid did it, and that is--there are many other proofs, of course, but it still is one of the simplest and most beautiful, and tb answer is absolutely certain that there is no largest prime, that they do go on forever. This cannot be done by a computer. Currently there is no computer that has done that.
BATESON: There must be intervening primes, you might say accidental primes, just like that accidental factorization business.
SPENCER-BROWN: Between when? Where
BATESON: Between primes that are made by multiplying sequences of primes and adding one.
SPENCER BROWN: It is not necessary to make primes, you see. This is not necessarily prime, you see.
BATESON: It is not necessarily prime?
SPENCER BROWN: No, it isn't.
MAN: Multiply three by five, add one, that's 16. Non-prime.
SPENCER BROWN: We have to add two. You get 33 and it's non-prime.
BATESON: Non-prime. Why in heaven's--
SPENCER BROWN: It doesn't have to be prime, you see.
MAN: Thirty-one Thirty-one! Not 33. Thirty-one is prime.
SPENCER BROWN: Right, we get 31  . But if you go out far enough, you will find that you get one that isn't prime. But they will be divisible by a prime bigger than the largest prime you have used. Bet's see if we can find one. Uhe-ya, here, wait a minute, 211 is prime, isn't it? I'm just thinking of the prime factorial plus one; at seven, it's two, one, one. That's prime factorial plus one. 211 is prime, as far as I know. We want a table of primes here. Not divisible by 13, is 4. . .we multiply the next one, 11, 2, 1, 1, 2, 1, 2, 3, 2, 1. Sorry, 11, 2, 1, O. 2, 1v O. 2, 1~~ 3, 1, O. and so it comes out 2311. Is that prime? Probably not. I am very bad at figures. Divisible by 13... 4...not divisible by 13....
Anyway, I do assure you that if you go on long enough, getting the final factorial, adding one, you will find one that is not prime; but that doesn't matter, because it will be divisible by a prime that is bigger than the biggest prime you have used to produce it. If it were always prime, you would have immediately a means--you would have a formula for producing primes, and this we haven't got. There is no formula for producing primes except going about it the hard way and seeing as they don't divide by anything.
Theorems and Consequences
Now, this is totally confused, the idea of the difference between demonstration And proof in mathematics. In fact, Russell, you see, in suggesting it, completely confused them, and people have done so ever since. What he called theorems are in fact consequences, they are algebraic consequences, which can be, in fact, demonstrated. And indeed, he says, "These theorems"--he calls them theorems, they are consequences--"can be proved." And then he does the demonstration and then he calls it "Dem." "Dem." is short for "demonstration."- The two words are used interchangeably and wrongly. There is a difference, and what can be demonstrated is done within the system and can be done by computer. And what cannot be demonstrated, but may be proved, cannot be done by computer. It must have a person to do it. No computer can prove it, because it is not proved by computation. The steps of this proof, Euclid's proof, were not computational steps. No one could do it on a computer, because we were not doing computation. We were divining the answer, we were divining what had to be done by making certain deductions and seeing what they led to. This was an artistic process, not a mechanical one.
The computer cannot do it because it is not computation. Computation is counting in either direction, no more, no less. There is nothing more to computation than that, nothing more.
MAN: I am trying to determine what it is that a human can do that a computer can't do.
SPENCER BROWN: Bet's go through the steps again. Where is the computing? We compute nowhere. There is no computation in this proof. Not a single computation can be made, not one. The whole process is a proof. In the whole process of a proof, there is not one single computation, nothing that a computer could do.
MAN: Well, there are two fake computations.
SPENCER BROWN: There are no computations. They were fakes because there were no such numbers. We were imagining doing a computation of a particular kind--we weren't actually doing it, because there were no numbers to put in the places. In fact, there only could have been a computation if our number Big N. being prime to the largest, happened to exist. res. If it happened to exist, and we knew what it was, then we could do this whole thing on a computer. But it doesn't happen to exist. But in order to find it doesn't happen to exist, we go through the imaginary steps of computing in this particular way, and then we find that if we did that we would find another number which is prime or contains a larger prime.
MAN: John Lilly, what does the biocomputer offer as the possibility of doing this, if the computer doesn't?
LILLY: Well, the biocomputer does the whole thing.
MAN: Say it again.
LILLY: The biocomputer invented the whole thing.
SPENCER BROWN: I know, as an engineer, the computer boys have vastly oversold their products by saying that they can do anything that the human mind can do, and this is not so. They cannot do the most elementary things that the human mind can do. And I blame, I blame Russell/ Whitehead for totally mixing up proof and demonstration. I! you go through Principia, there is not a single theorem, not one theorem.--I think I am right about that, Dr. Von Foerster--because what they call theorems are consequences.
Also, they had a precedent in that Euclid himself already rightly called this a theorem, calls it algebraic. His geometric consequences, he called theorems, they are not. So the confusion developed right at the beginning with Euclid, who called his geometric consequences, they can be computed, he called them theorems. Wrongly. So Euclid was the first offender. And from him, it just shows how we copied. We have copied his error through hundreds of years.
VON FOERSTER: I think the Q. E. D. thing makes it appear as though it were a demonstration--quod erat demonstrandum. It should not have been called demonstrandum.
SPENCER BROWN: I may be wrong, you see. My Latin--I have little of it--perhaps he was 0. K. He said quod erat demonstrandum, "this has been demonstrated. n It is 0. K. after a demonstration, it is misleading after a proof. And maybe he did not make this error, but we have. We have called them theorems when we should call them consequences. And this has been responsible for-- A vast system of error has grown up there. Because a computer has been found to be able to demonstrate consequences--because all you need is the calculating facility to do this. And consequently The demonstration of consequences, in other words, calculations, has been confused with the proof of theorems, which is another matter altogether. Because of this confusion, it has been thought that a computer therefore can do practically all that a man's mind can do. But it can't, because only the most minor function of a man's mind, done very badly, is to compute. And we have, in fact this tremendous emphasis, because of the confusion in mathematics--the difference between computation and actual mathematical thinking--which has led us to believe that computers have minds, can do what we can do.
For example, they put Russell's consequences on the Titan computer at Cambridge. It managed, with great hesitation and very slowly, to demonstrate a few of them, but the more complicated of them it couldn't demonstrate. However, it could have done it, eventually. It was very slow and expensive. Even here, what a computer can do, a man can do better if he gives himself to the problem, because he has the capacity of seeing in-a way the computer never can.
MAN: Can you say what that capacity is? That makes us different?
SPENCER BROWN: Can I say what it is? No, I can only represent it. I can only be it. Just as you are. How can one say what it is except to give examples Computing is 1, 2, 3, 4, 5, space, space, space, space, space, 6, 7, 8. That's five plus three is eight. that's how a computer does it.
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SPENCER BROWN: I have been asked earlier if I would go through with you the main mathematics of the book, and I think this is not possible because there is not enough time, and it is so varied an audience.
I did do this in London, but it was a series of 20 lectures. I gave it seasonally every gear, and even then it wasn't enough to do anything but in outline. Only in the last two or three lectures was it possible, having got most- of the audience to an understanding of what it was about, was it possible to show how it was related--how it could possibly relate to the disciplines of everyday life.
Unless there is a very strong expressed desire here, I don't feel that it would be terribly desirable, for the majority of people here, if I did actually go through technically and get as far as one could--because it wouldn't be far enough to draw on the conclusions that are possible after more detailed study. I don't think it could be done.
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It's Just not possible to do everything all at once. You can't make the rice grow by pulling on the stalks. If there is anything further that you would like to discuss now, we'll see what we can do. Or would you like me to suggest one?
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 blithe game that we are taught to 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 w 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 tolt @ ake 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. Row, 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  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?
The Turtle's Specialty
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.
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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.
Separating Figure and Ground
KING: Well, just personally, what process do you--
SPENCER BROWN: How do I do it? Just contemplate. One also tries all sorts of wags 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.
Educating the Child into the Material World;
The Covenant of the Cradle
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 your 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 f1mng 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 an knowing it. That is the wag 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 wag 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 wag, 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 wag 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. n 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 n "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.
* * *
Why We See the Same Things
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' 8 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 hypo thetical 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 Bay, "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 ides mathematically--it does involve some rather lengthy exposition.
Godel's Theorem: Completeness and Consistency
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 wag, 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.)
- She Indo-European root is pela:, flat, to spread. Related words in English are PLAIN, FIFrn, 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.
- See Note 4, pp. 134-5.
- The illustration below that the algebraic system is in complete, since its rules do not generate all of its possible states.
- In the United States, by the Chelsea Publishing Co., New York. A reprint Of Carnegie Destitute Publication #256 (28). By Leonard Eugene Dickson.
- Multiplying the first three primes, 2 r 3 x 5, and adding 1.
- 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."
- Re-markable, markable again ("a whole series).
- Greek theasthai, to view.
- L. species, a seeing, form (SPECIES), from Indo-European spek- to o serve. Related is i. speculum a mirror.
- American Heritage Dictionary gives REAL1 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.
- The Fibonacci series goes 0, 1, 1, 2, 5, 5, 8, 1D, 21, etc.
- Theorems 3, 4.