G. Spencer Brown AUM Conference
And with that I will leave the discussion to G. Spencer Brown: James.
G. SPENCER BROWN: Well, I don't know what to say. It is a great pleasure, a great honor, to be here. I don't feel that I deserve the honor in any way. It is also--I think I feel rather nervous, as this audience has so many and so different qualifications.
Mathematics and Logic
I don't hope to do anything but really answer any questions that anybody has to ask about the nature of what I was trying to do when I began to write Laws of Form and the very different answer, what I actually found, that had appeared when I had finished the book, which was not what I had set out to do.
I guess that is the only way that I can begin to talk about the work, which is as far as I am concerned entirely impersonal. It has as little to do with me personally as anything I can imagine. I have no particular attachment to it. I wouldn't do it again if anybody asked me to. I was conned into writing it by thinking that it would have an entirely different effect from what it did have; and, in completing it,
I unlearned what I learned, the kind of values that present-day civilization inculcates into us soon after we are born.
Logic, in other words, is itself not mathematics, it is an interpretation of a particular branch of mathematics, which is the most important non-numerical branch of mathematics. There are other non-numerical branches of mathematics.
Mathematics is not exclusively about number. Mathematics is, in fact, about space and relationships.
A number comes into mathematics only as a measure of space and/or relationships. And the earliest mathematics is not about number. The most fundamental relationships in mathematics, the most fundamental laws of mathematics, are not numerical. Boolean mathematics is prior to numerical mathematics.
LILLY: Have you formulated or recommended an order of unlearning?
SPENCER BROWN: I can't remember having done so. I think that, having considered the question, the order of unlearning is different for each person, because what we unlearn first is what we learned last. I guess that's the order of unlearning If you dig too deep too soon you will have a catastrophe; because if you unlearn something really important, in the sense of deeply imported in you, without first unlearning the more superficial importation, then you undermine the whole structure of your personality, which will collapse.
Therefore, you proceed by stages, the last learned is the first unlearned, and this way you could proceed safely. Related to what is in the books, we know they say that in order to proceed into the Kingdom, one must first purify oneself. This is the same advice, because the Kingdom is deep.
What we talk of in the way of purification is the superficial muck that has been thrown at us. First of all that must be taken off, and the superficial layers of the personality must be purified. If we go to the Kingdom too soon, without having taken off the superficial layers and reconstructed in a simpler way, then there is a collapse.
The advice is entirely practical. It is not a prohibition. There is no heavenly law to say that you may not enter the Kingdom of Heaven without first purifying yourself. However, if you do, the consequences may be disastrous for you as a person.
Now, the whole of the first degree equation in the Boolean form are in terms of these two states.
When you do this peculiar thing of making something self-referential that is making the answer go back into the expression out of which the answer comes, you now automatically produce this set of possibilities which are well-known in numerical mathematics and of which everyone's been terrified of looking at in Boolean mathematics.
And Russell/Whitehead were so frightened of these, that they just had a rule with no justification whatsoever that we Just don't allow it, we don't even allow people to think about this.
Now, why is it used so much? Because i is the state that flutters, is the oscillation. This has been totally overlooked in mathematics, that i is in an oscillatory state. Because in order to get over this paradox of x-squared equals minus one, we see that we can't use any ordinary form of unity so we invent in mathematics another form of unity and we-call it i, which is the root that satisfies that equation. And the root that satisfies that equation is that you have plus one, minus one, and here's a state between; and the root that satisfies that equation, whatever it is, it isn't. And this is why i is so useful in dealing with that kind of curve--because it is, by its very nature, that kind of curve. i is an oscillation.
Just as the space or the first distinction has no size, no shape, no quality other than being states. This is one of the things that tend to upset people. It is part of the mathematical discipline that what is not allowed is forbidden. That is to say, what you don't introduce, you can't use. And until you have introduced shape, size, duration, whatever, distance, you can't use it.
In the beginning of Laws of Form, we defined states without any concept of distance, size, shape--only of difference.
Therefore the states in Laws of Form have no size, shape, anything else. They are neither close together nor far apart, like the heavenly states. There is just no quality of that kind that has been introduced. It's not needed. -
The same with the first time. The first time is measured by an oscillation between states. The first state, or space, is measured by a distinction between states. There is no state for a distinction to be made in. If a distinction could be made, then it would create a space. That is why it appears in a distinct world that there is space.
Space is only an appearance. It is what would be if there could be a distinction.
Similarly, when we get eventually to the creation of time, time is what there would be if there could be an oscillation between states.
Even in the latest physics, a thing is no more than its measure. A space is how it is measured; similarly, time is how it is measured.
The measure of time is change. The only change we can produce-when we have only two states--the only change we can produce is the crossing from one to another.
LILLY: Is that frequency of oscillation either zero or infinity?
GREGORY BATESON: What about the "then' of logic? "If two triangles have three sides, etc., then" so-and-so. The "then" is devoid of time.
BATESON: So we add sequence without adding duration.
Make something self-referential, it either remembers or it oscillates. It's either what it was before or it's what it wasn't before, which is the difference between memory and oscillation.
WATTS: In introducing the word "before, haven't you introduced time? You have a sequence.
SPENCER BROWN: I have to apologize, because you realize that in order to make myself understood in a temporal and even a physical existences as by convention is what we are in, remember
Basically, to do what I am attempting to do is impossible. It is literally impossible, because one is trying to describe in an existence which has them--one is trying to describe in an existence which has certain qualities an existence which has no such quality. And in talking about the system, the qualities in the description do not belong to what we are describing. So when I say things like, "To oscillate, it is not what it was before; to remember, it is what it was before, I am describing in our terms, something that it don't have. But, by looking at them, you can see.
For example, Rolt, in his brilliant introduction to the Divine Names by Dionysius the Areopagite, begins describing the form at first, and then he actually describes what happens when you get the temporal existence. It is all the same thing, but he is describing it in terms of religious talk, theorems become angels, etc. When he comes to the place, which he says most beautifully, having described all the heavenly states and all the people therein, etc., and he says,
"All this went on in absolute harmony until the time came for time to begin."
This is quite senseless. But it is perfectly understandable to someone who has seen what happens, who has been there. One cannot describe it except like this. It is perfectly understandable. He had described the form and then he had done that, and this is the time for time to begin.
Mathematics and Its Interpretations:
There is just one question that I have been asked to answer, and I think it is something that you, Gregory, asked, wasn't it? to do with "not." Was the cross--the operator-was it "not." No it ain't.
If I can, I'll try to elucidate that. I am reminded of one of the last times I went to see Russell and he told me he had a dream in which at last he met "Not." He was very worried about this dream. He had a dream, and he met "Not, and he couldn't describe it. But by the time we are using logic, we have in logic "not."
We say: a implies b. I am assuming that we know the old logic functions. You can describe this, ~a, as "not a." Now that is not -- that is a shorthand for "not" in logic.
"Not" in logic means pretty well what it means when we are talking, because after all, logic is only mildly distinguished from grammar. Just as we learn after reading Shakespeare's sonnets that after all they are full of grammar. Some people seem to think that all we have to do is learn grammar to be able to write like that--not so. So, they're full of grammar--they're also full of logic.
Now in arguments, there are the variables, "if it hails, it freezes," and the forms; we can say in that case, it means the same thing as "either it doesn't hail, or it freezes," and find this is actually what "implies" means. We can break down "implies" into "not" and "or."
Now when we are interpreting whenever are using the mathematics...we write a for "it hails," and b for "it freezes." If it hails, then it freezes; either it doesn't hail or it freezes. And in the primary algebra we can write, "a cross b," (a) b.
But the formula is not about cars, and so on and so forth; nor is this formula about statements in logic. Just as here we have used a to represent the truth value of the sentence, "it hails," and b to represent the truth value of the statement "it freezes," we are in fact applying, because we recognize the structure is similar, the states of the first distinction to the truth values of these statements.
We recognize the form of the thing. And in fact, "not" ; is in this case, although it is represented by the cross, the cross itself is not the same as "not. Because if it were-well, we can see obviously that it isn't, because, in this form we have represented "true" by a cross and "false" by a space...if you represent "true" by a space and "false" by a cross, then wherewith our "not"?
We have swapped over and identified the marked state with untrue this time, and the unmarked state with true. And here we have identified it with untrue. Change over the identification, which we may do, and now here is the statement. And if this were "not," this would now have two "nots--but it is not "not." We have only made it representative of "not" for the purpose of interpretation, just as well can give a color a number and use that in altering an equation. But the number and the color are not the same thing. This is not "not" except when we want to make it so. But it has a wider meaning than "not" in the book.
WATTS: Well, it means that it is distinct from.
Marked State/Unmarked State
If you go back to the beginning of the book, you see—you remember this is not what really happens, because nothing happens. We represent what doesn't actually happen but might happen if it could.
We represent it in the following way: we may draw a closed curve to represent a distinction, say the first distinction. Now we have a form. And we will mark one state, so, in fact. The mark is, in fact, shorthand for something like that, because it is only a bracket we marked it with. If we don't mark it with a bracket, we find that we have to mark it with a bracket, as I show in the notes .
WATTS: Well, you have got it in the frame of the blackboard.
Let the name as the token indicate the state. I missed out a sentence. Let the token be taken as the name, and let the name indicate the state--right . Now, here, this indicates the state. We now derive our first equation from Axiom One--if you call a name twice or more, it simply means the state designated by the name by which you call it. So we have the first equation ()() = ().
Then, let a state not marked with a mark be called the unmarked state, and let any space in which there is no token of the mark designate the unmarked state. in other words; we did away with the second name. This is essential. It's the fear of doing away with the second name that has left logic so complicated. If you don't do away with the second name, you can't make the magic reduction.
BATESON: Are you saying that the name of the name is the same as the name?
What people have done is that they have given a name always to both states. There is no need to do that; you have got quite enough to recognize where you are, because you do a search, and if you find the mark, you know you are in the marked state. If you do a search and Y0U don't find it, you know you are in the unmarked state. So that, mathematically, is all that is necessary. So you don't do the second thing. There has been already fear, you see, to have a state unmarked.
MAN: How did the printer feel about this. It must have driven him crazy.
MAN: The American military documents, because of the number of pages that have to be printed, frequently have a blank page, And to be sure that nobody gets confused about it, there is always a statement on that page that says, "this page is deliberately left blank, which, of course, it is not.
SPENCER BROWN: You see, why it has taken so long for the Laws of Form to be written is that one has to break every law, every rule, that we are taught in our upbringing. And why it is so difficult to break them is that there is no overt rule that you may not do this--why it is so powerful is that the rule is covert.
There is no rule that is overt anywhere in mathematics which says this may not happen, it may not be done. And it is because I found no such rule that I gathered that it could be done, and that it must be done. If you don't do it, you are not doing the mathematics properly, and that is why it is all such a mess. This is only a social rule that you may not do it. And there is no mathematical rule that you may not do it; and in fact, you have to do it. Otherwise the mathematics is a mess and you can't get the answers because you are blocked.
Now to go on to what I was going to say, which is: next we want to use the mark, which could be a circle. We want to use it. We haven't, in fact, discovered its shape. In the second equation, we discover, really, what the shape is. And we'll see it is inevitable. Having marked one side-if there is no mark, then we know we are on the other sides
WALTER BARNEY: Those m's are outside the circle or inside the circle?
All mathematics in books is only an illustration of what cannot be said. This illustration is misleading because there is no outside or inside when you have drawn the first distinction. You have just drawn a distinction--we can illustrate it with a circle because it happens to be convenient. Then we mark one side, and we know, in this case that it is the outside.
First of all, we have taken the mark as a name. And if you call a name twice, you are simply indicating the same state twice, and indicating the same state twice is the same as indicating the same state once. Now, instead of just calling this m, let us give it certain properties. Let it be an instruction to cross the first distinction.
Now here is our illustration of the first distinction . Now this is why we've drawn this line on our blackboard, because here is an illustration of the first distinction. Here is a record of instructions referring to the first distinction--right.
Now let m, the mark, be taken as an instruction to cross the boundary of the first distinction. So, if one is here, m says go there. If one is here, m says go there. m is now not a name, so we can ring that for the moment, don't confuse yourself with that, m is now an instruction. And all m means is "cross." So whenever you hear or see it, you've got to step over the boundary. That is all it means. Now, we will produce more conventions.
We will say that we have got a number of crosses considered together, and these we will call "expressions." Now suppose you have this. We'll say--right--we'll represent m like that. And we'll say m means "cross" and we'll make a convention so that whatever is represented in here, you'll have crossed to get what is represented out there. So if there is nothing represented here, absence of the mark indicates the unmarked state. You cross when you are in the unmarked state and you find you are in the marked state. So out here by representation will be a value attributed to this mark, will be the marked state, and that is the value we attribute to that expression.
Now let us put the marked state in here, and we can do that simply by putting another cross in here. Now the convention is that wherever you see nothing you-are in the unmarked state. Wherever you see this, you must cross. So, here we are. We hear nothing, we see nothing, we are in the unmarked state.
Our instructions now say "cross," so we cross, and then our second instruction says "cross," so we cross. So here we are, we started here and we have crossed, and we have crossed here, and so we can derive our second equation, (()) = . So that all this says in mathematics is "cross." It does not say "not." It says "cross."
(End of first session.)
1. Transmission Of Spencer Brown's marks on the blackboard has been absorbed elsewhere in the system. We invite outside constructions. The general discussion concerns re-entry at an odd level and at an even level. If odd, as in (x), we get marked state in and unmarked out, an oscillation. If even, ((x)) we get marked in, marked out, a memory.