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It is recommended …. The landlord and the tenant are legally bound to abide by the terms of this agreement upon signing it. The landlord has agreed to give the property on …. We call them theorems of procedure. The last two theorems will serve as a gate of entry into a new calculus. We call them theorems of connexion. The new calculus will itself give rise to further theorems which, when they describe aspects of the new calculus without direct reference to the old, will be called pure algebraic theorems, or theorems of the second order.
In addition we shall find theorems about the two calcuJi considered together. The bridge theorem and the theorem of completeness are examples, and we may call them mixed theorems. Call any token of variable form after its form. Call any token of constant form.
Let indications used in the description of theorem 8 be taken out of context so that. Call this the form of transposition. Let the forms of position and transposition be taken as the initials of a calculus. Let the calculus be seen as a calculus for the primary arithmetic. Call it the primary algebra. Rule 1. This rule is a restatement of the arithmetical convention of substitution together with an inference from the theorems of representation. Rule 2.
This rule derives from the fact, proved with the theorems of connexion, that we can find equivalent expressions, not identical, which, considered arithmetically, are not wholly revealed. In an equation of such expressions each independent variable indicator stands for an expression which, being unknown except in as far as, by theorem 5, its value must be taken to be.
Hence its indicator may also be changed at will, provided only that the change is made to every appearance of the indicator. Indexing Numbered members of a class of findings will henceforth ,be indexed by a capital letter denoting the class followed by a figure denoting the number of the member.
The classes will be indexed thus. R Theorem T Certain equations, designated by E, will also be indexed, but the reference in each chapter will be confined to a separate set. Thus El, etc, in Chapter 9 will not intentionally be the same equations as El, etc, in Chapter 8. We shall proceed to distinguish particular patterns, called consequences, which can be found in sequential manipulations of these initials. We next use R1 to change an appearance of to an appearance of. We next use the licence allowed in th e definit ion p 6 of relation to change this to Pl.
We then use R2 to change every appearance of p in this equatio n to an appe arance of. I rq II. And lastly, we find. This completes a detailed account of each of six steps. We may now use T7 five times to find. We repeat this demonstration, and give subsequent demonstrations, with only the key indices to the procedure. The classification of consequences In classifying these consequences, there is no need to confine them rigidly to the forms above. The name of a consequence may indicate a part of the consequence as in.
Nor, as we already see in one case, are the classes of consequence properly distinct. What we are doing is to indicate larger and larger numbers of steps in a single indication. This is the dual form of the contraction of a reference, notably the expansion of its content.
We shed the labour of calculation by taking a number of steps as one step. But now if we allow steps in the indication of steps, we find that the resulting calculus is inconsistent. This agrees with our idea of the nature of a step which, as we have already determined, is not intended to cross a boundary.
A further classification of expressions The algebraic consideration of the calculus of indications leads to a further distinction between expressions. Expressions of the marked state may be called integral. The letter tn, unless otherwise employed, may be taken to indicate an integral expression.
Expressions of the unmarked state may be called disintegral. The letter n, unless otherwise employed, may be taken to indicate a disintegral expression.
Expressions of a state consequent on the states of their unknown indicators may be called consequential. In any case, -;]bl In case 0. Theorem 13 The generative process in C2 can be extended to any space not shallower than that in which the generated variable first appears. Proof We consider cases in which a variable is generated in spaces 0, 1, and more than 1 space deeper than the space of the variable of origin. Clearly no additional consideration is needed for further generation of g, and it is plain that any space not shallower than that in which g stands can be reached.
It is convenient to consider 12, C2, C8, and C9 as extended by their respective theorems, and to let the name of the initial or consequence denote also the theorem extending it. Theorem Canon with respect to the constant From any given expression, an equivalent expression not more than two crosses deep can be derived,.
The maximum number of steps is needed in case. Proof The proof is trivial for a variable not contained in the original expression e, and so we may confine our consideration to the case of a variable v contained in e, Now by Cl and T Suppose e is a cross. The content of e is the content of the space in which it stands, not the content of the cross which marks the space.
In general, a content is where we have marked it, and a mark is not inside the boundary shaping its form, but inside the boundary surrounding it and shaping another form. Thus in describing a form, we find a succession,. Indicative space If So is the pervasive space of e, the value of e is its value to So.
If e is the whole expression in So, So takes the value of e and we can call So the indicative space of e. In evaluating e we imagine ourselves in So with e and thus surrounded by the unwritten cross which is the boundary to S-I. Seventh canon. An unwritten cross is common to every expression in the calculus of indications and so need not be written.
Similarly, a recessive value is common to every expression in the calculus of indications and also, by this principle, has no necessary ,indicator there. In the form of any calculus, we find the consequences in its content and the theorems in its image. But demonstrations of any but the simplest consequences in the content of the primary arithmetic are repetitive and tedious, and we can contract the procedure by using theorems, which are about, or in the image of, the primary arithmetic.
For example, instead of demonstrating the consequence above, we can use T2. T2 is a statement that all expressions of a certain kind, which Its proof may be regarded as a simultaneous demonstration of all the simplifications of expressions of the kind it describes. But the theorem itself is not a consequence. Its proof does not proceed according to the rules of the arithmetic, but follows, instead, through ideas and rules of reasoning and counting which, at this stage, we have done nothing to justify.
Thus if any person will not accept a proof, we can do no better than try another. A theorem is acceptable because what it states is evident, but we do not as a rule consider it worth recording if its evidence does not need, in some way, to be made evident.
This rule is excepted in the case of an axiom, which may appear evident without further guidance. Both axioms and theorems are more or less simple statements about the ground on which we have chosen to reside. Since the initial steps in the algebra were taken to represent theorems about the arithmetic, it depends on our point of view whether we regard an equation with variables as expressing a consequence in the algebra or a theorem about the arithmetic.
Any demonstrable conseq uence is alternatively provable as a theorem, and this fact may be of use where the sequence of steps is difficult to find. Thus, instead of demonstrating in algebra the equation. By their origin, the consequences in the algebra are arithmetically valid, so we may use them as we please to shorten the proof.
In these proofs we evidently supposed the irrelevance of variables other than the one we fixed arithmetically. It may not at fLrst be obvious that we can ignore the possible values of the other variables, but the supposition is in fact justified in all instances and, indeed, in all algebras , as the following proo will show.
The bridge If expressions are equivalent in every case of one variable, they are equivalent. Let a variable v in a space Sq oscillate between the limits of its value m, n. Under this condition call Sq transparent. If the value of any other indicator in Sq is m, nothing will be transmitted through Sq. The transmission from v is the alternation between transparency and opacity in Sq and in any more distant space in which this alternation can be detected. It may at any point be absorbed in transmissions from other variables in the space through which it passes.
On condition that this absorption is total, call the band of space in which it occurs opaque. Under any other condition, call it transparent. From these definitions and considerations we can see the following principle.
Eighth canon. Principle of transmission With regard to an oscillation in the value of a variable, the space outside the variable is either transparent or opaque. Proof of theorem 16 Let s, t be the indicative spaces of e, f respectively. Let either of e, f contain a variable v, and let v oscillate between the limits of its value m, n.
Consider the condition under which both e andfare opaque to transmission from v. If e and f are equivalent after a change in the value of v, they were equivalent before. Consider either e or f transparent. Suppose the oscillation of v is transmitted to one indicative space and not to the other. By selecting an appropriate value of v, we could make e not equivalent to f, and this is contrary to hypothesis.
Thus if either of e or f is transparent, both are transparent. Thus any change in the value of v is transmitted to sand t. Therefore, if e and f are equivalent after a change in v, they were equivalent before.
We have seen that any demonstrable consequence in the algebra must indicate a provable theorem about the arithmetic. In this way consequences in the algebra may be said to represent properties of the arithmetic. In particular, they represent the properties of the arithmetic that can be expressed in forms of equation.
We can question whether the algebra is a complete or only a partial account of these properties. That is to say, we can ask whether or not every form of equation which can be proved as a theorem about the arithmetic can be demonstrated as a consequence in the algebra.
Completeness The primary algebra is complete. That is to say, if 1. We prove this theorem by induction.
We first show that if all cases of 1. We then show that the condition of complete demonstrability in cases of less than n variables does in fact hold for some positive value of n. Proof Suppose that the demonstrability of 1. Let a given equivalent rx, fJ contain between them n distinct variables.
Reduce the given d. By the proof o f TIS we may suppose the canonical form of rx to be. But by substituting constant values for v we find.
Now each of E3, E4, having at most n - 1 distinct variables, is demonstrable by hypothesis. Hence El-4are all demonstrable, and we can demonstrate. Thus we need to show that if J.
If rx, fJ contain no variable, they may be considered as expressions in the primary arithmetic. We see in the proofs of T that all arithmetical equations are demonstrable in the arithmetic. It remains to show that they are demonstrable in the algebra. This completes the proof.
We call the equations in a set independent if no one equation can be demonstrated from the others. Independence The initials of the primary algebra are independent. That is to say, given Jl as the only initial, we cannot find J2 as a consequence, and given J2 as the only initial, we cannot find Jl as a consequence.
Proof Suppose Jl determines the only transformation allowed in the algebra. It follows from the convention of intention that no expression other than of the form Pl p I can be put into or taken out of any space. But, in J2, r is taken out of one space and put into another, and r is not necessarily of the form "PI pl.
Therefore, J2 cannot be demonstrated as a consequence of n. Next suppose J2 determines the only transformation allowed in the algebra.
Inspection of J2 reveals no way of eliminating any distinct variable. But Jl eliminates a distinct variable. Therefore, Jl cannot be demonstrated as a consequence of J2, and this completes the proof. Hitherto we have obeyed a rule theorem 1 which requires that any given expression, in either the arithmetic or the algebra, shall be finite. Otherwise, by the canons so far called, we should have no means of finding its value.
It follows that any given expression can be reached from any other given equivalent expression in a finite number of steps. We shall find it convenient to extract this principle as a rule to characterize the process of demonstration. Ninth canon. Rule of demonstration A demonstration rests in a finite number of steps.
One way to see that this rule is obeyed is to count steps. We need not confine its application to any given level of consideration. In an algebraic expression each variable represents an unknown or immaterial number of crosses, and so it is not possible in this case to count arithmetical steps.
But we can still count algebraic steps. We may note that, according to the observation in Chapter 6 on the nature of a step, it does not matter if several counts disagree, as long as at least one count is finite. Consider the expression. There is no limit to the possibility of continuing the sequence, and thus no limit to the size of the echelon of alternating a's and b's with which tll b I can be equated. Now, since this form, being endless, cannot be reached in a finite number of steps from al b I , we do not expect it to express, necessarily, the same value as - ;j b I.
But we can, by means of an exhaustive examination of possibilities, ascertain what values it might take in the various cases of a, b, and compare them with those of the finite expression. Re-entry The key is to see that the crossed part of the expression at every even depth is identical with the whole expression, which can thus be regarded as re- entering its own inner space at any even depth.
We can now find, by the rule of dominance, the values whichf may take in each possible case of a, b. Thus the equation, in this case, has two solutions. It is evident, then, that, by an unlimited number of steps from a given expression e, we can reach an expression e' which is not equivalent to e.
We see, in such a case, that the theorems of representation no longer hold, since the arithmetical value of e' is not, in every possible case of a, b, uniquely determined. Indeterminacy We have thus introduced into e' a degree of indeterminacy in respect of its value which is not as it was in the case of indeterminacy introduced merely by cause of using independent variables necessarily resolved by fixing the value of each independent variable.
Degree We may take the evident degree of this indeterminacy to classify the equation in which such expressions are equated. Equations of expressions with no re-entry, and thus with no unresolvable indeterminacy, will be called equations of the first degree, those of expressions with one re-entry will be called of the second degree, and so on. It is evident that J1 and J2 hold for all equations, whatever their degree. It is thus possible to use the ordinary procedure.
But we are denied the procedure outlined in Chapter 8 of referring to the arithmetic to confirm a demonstration of any such equation, since the excursion to infinity undertaken to produce it has denied us our former access to a complete knowledge of where we are in the form. Hence it was necessary to extract, before departing, the rule of demonstration, for this now becomes, with the rule of dominance, a guiding principle by which we can still find our way.
Imaginary state Our loss of connexion with the arithmetic is illustrated by the following example. Plainly, each of E2, E3 can be represented, in arithmetic, by equating either 1 with the same infinite expression, thus. But equally plainly, whereas E2 is open to the arithmetical solutions I or , each of which satisfies it without contradiction, E3 is satisfied by neither of these solutions, and cannot, thereby, express the same value as E2.
Time Since we do not wish, if we can avoid it, to leave the form, the state we envisage is not in space but in time. It being possible. One way of imagining this is to suppose that the transmission of a change of value through the space in which it is represented takes time to cover distance. Consider a cross. An indication of the marked state is shown by the shading. Now suppose the distinction drawn by the cross to be destroyed by a tunnel under the surface in which it appears.
In Figure 1 we see the results of such destruction at intervals tl, t2,. Frequency If we consider the speed at which the representation of value travels through the space of the expression to be constant, then the frequency of its oscillation is determined by the length of the tunnel.
Alternatively, if we consider this length to be constant, then the frequency of the oscillation is determined by the speed of its transmission through space. Velocity We see that once we give the transmission of an indication of value a speed, we must also give it a direction, so that it becomes a velocity.
For if we did not, there would be nothing to stop the propagation proceeding as represented to t4 say and then continuing towards the representation shown in fa instead of that shown in f5. Function We shall call an expression containing a variable v alternatively a function of v. We thus see expressions of value or functions of variables, according to from which point of view we regard them.
Oscillator function In considering the indications of value at the point p in Figure 1, we have, in time, a succession of square waves of a given frequency. We could do this by arranging similarly undermined distinctions in each space, supposing the speed of transmission to be constant throughout. In this case the superimposition of the.
Real and imaginary value The value represented at or by the point or variable p, being indeterminate in space, may be called imaginary in relation with the form. Nevertheless, as we see above, it is real in relation with time and can, in relation with itself, become determinate in space, and thus real in the form.
We have considered thus far a graphical representation of E3. We will now consider El and its limiting case E2 on similar lines. Memory function The present value of the function f in El may depend on its past value, and thus on past values of a and b. In effect, when a, b both indicate the unmarked state, it remembers which of them last indicated the marked state. We shall call such a partial destruction of the distinctive properties of constants a subversion. We may note that, if we wish to avail ourselves of the memory property off, where I is an evenly subverted function, certain transformations, allowable in the case of an expression without this property, must be avoided.
We may, for example, allow. Time in finite expressions The introduction of time into our deliberations did not come as an arbitrary choice, but as a necessary measure to further the inquiry. The degree of necessity of a measure adopted is the extent of its application. The measure of time, as we have introduced it here, can be seen to cover, without inconsistency, all the representative forms hitherto considered. This can be illustrated by reconsidering EI.
Here we can test the use of the concept of time by finding whether it leads to the same answer i. For the purpose of illustration, we shall consider a finite expression first. It is seen from Figure 3 that such a finite expression is stable in one condition, and has a finite memory of the other, of.
It is plain that an endless extension of the echelon allows an endless memory of either condition, so that the concept of time is a key by which the contracted and expanded forms of fin E1 are made patent to one another. A condition of special interest emerges if the dominant pulse from a is of sufficiently short duration.
In this condition the expression emits a wave train of finite length and duration, as illustrated in Figure 4, The duration of the wave train, the frequency of its components, etc, depend on the nature and extent of the expression From an infinitely extended expression comes a potentially endless emission, and here again, the two ways contracted or expanded of expressing EI in relation with time give the same answer.
Without the key of time, only the contracted expression makes sense. Crosses and markers Consider the case where the expression in El represents a part of a larger expression. It now becomes necessary not. Since the whole is no longer the part re-inserted, it will be necessary in each case either t? In a simple subverted expression of this kind neither of the non-literal parts are, strictly speaking, crosses, since they represent, in a sense, the same boundary.
It is convenient, nevertheless, to refer to them separately, and for this purpose we call each separate non-literal part of any expression a marker. Thus a cross is a marker, but a marker need not be a cross. Modulator function We have seen that functions of the second degree can either oscillate or remember. A memory function remembers the same response to the same signal: a counting function counts it different each time.
Another way to picture counting is as a modulation of a wave structure. This is the way we shall picture it here. The simplest modulation is to a wave structure of half the frequency of the original. To achieve this with a function using only real values, we need eight markers, thus. We are now in difficulties through attempting to write in two dimensio! We ought to be writing in three dimensions. We can at least devise a better system of drawing three-dimensional representations in two.
Let a marker be represented by a vertical stroke, thus. Let the value indicated by the marker be led from the marker by a lead, which may, in the expression, divide to be entered under other markers, Now, for example, the expression. By using imaginary components of some wave structures, it is possible to obtain the wave structure at p with only six markers.
This is illustrated in the following equation. Here, although the real wave structure at i is identical with that at r, the imaginary component at i ensures that the memory in markers c and d is properly set. Similar considerations apply to other memories in the expression.
Coda At this point, before we have gone so far as to forget it, we may return to consider what it is we are deliberating. The whole account of our deliberations is an aCC0unt of how it may appear, in the light of various states of mind which we put upon ourselves. By the canon of expanding reference p 10 , we see that the account may be continued endlessly.
This book is not endless, so we have to break it off somewhere. We now do so here with the words. Before departing, we return for a last look at the agreement with which the account was opened. The conception of the form lies in the desire to distinguish. Granted this desire, we cannot escape the form, although we can see it any way we please. The calculus of indications is a way of regarding the form. We can see the calculus by the form and the form in the calculus unaided and unhindered by the intervention of laws, initials, theorems, or consequences.
The experiments below illustrate one of the indefinite number of possible ways of doing this. We may note that in these experiments the sign. We may also note that the sides of each distinction experi-mentally drawn have two kinds of reference. The first, or explicit, reference is to the value of a side, according to how it is marked. The second, or implicit, reference is to an outside observer. That is to say, the outside is the side from which a distinction is supposed to be seen.
Now the circle and the mark cannot in respect of their relevant properties be distinguished, and so. Let the value of a mark be its value to the space in which it stands. That is to say, let the value of a mark be to the space outside the mark. Now the space outside the circumference is unmarked. Now, since a mark indicates both sides of the circumference, they cannot, in respect of value, be distinguished.
Again let the mark m be a circle. Therefore, it is not, in this respect, a distinction. An observer, since he distinguishes the space he occupies, is also a mark. In the experiments above, imagine the circles to be forms and their circumferences to be the distinctions shaping the spaces of these forms. In this conception a distinction drawn in any space is a mark distinguishing the space. Equally and conversely, any mark in a space draws a distinction. We see now that the first distinction, the mark, and the observer are not only interchangeable, but, in the form, identical.
Chapter 1 Although it says somewhat more, all that the reader needs to take with him from Chapter 1 are the definition of distinction as a form of closure, and the two axioms which rest with this definition.
Chapter 2 It may be helpful at this stage to realize that the primary form of mathematical communication is not description, but injunction.
In this respect it is comparable with practical art forms like cookery, in which the taste of a cake, although literally indescribable, can be conveyed to a reader in the form of a set of injunctions called a recipe. Music is a similar art form, the composer does not even attempt to describe the set of sounds he has in mind, much less the set of feelings occasioned through them, but writes down a set of commands which, if they are obeyed by the reader, can result in a reproduction, to the reader, of the composer's original experience.
Where Wittgenstein says [4, proposition 7] whereof one cannot speak, thereof one must be silent he seems to be considering descriptive speech only. He notes elsewhere that the mathematician, descriptively speaking, says nothing.
The same may be said of the composer, who, if he were to attempt a description i. But neither the composer nor the mathematician must, for this reason, be silent. NOTES In his introduction to the Tractatus, Russell expresses what thus seems to be a justifiable doubt in respect of the rightness of Wittgenstein's last proposition when he says [p 22] , what causes hesitation is the fact that, after all, Mr Wittgenstein manages to say a good deal about what cannot be said, thus suggesting to the sceptical reader that possibly there may be some loophole through a hierarchy of languages, or by some other exit.
The exit, as we have seen it here, is evident in the injunctive facuIty of language. Even natural science appears to be more dependent upon injunction than we are usually prepared to admit.
The pro-fessional initiation of the man of science consists not so much in reading the proper textbooks, as in obeying injunctions such as 'look down that microscope'. But it is not out of order for men of science, having looked down the microscope, now to describe to each other, and to discuss amongst themselves, what they have seen, and to write papers and textbooks describing it. Similarly, it is not out of order for mathematicians, each having obeyed a given set of injunctions, to describe to each other, and to discuss amongst themselves, what they have seen, and to write papers and textbooks describing it.
But in each case, the description is dependent upon, and secondary to, the set of injunctions having been obeyed first. When we attempt to realize a piece of music composed by another person, we do so by illustrating, to ourselves, with a musical instrument of some kind, the composer's commands. Similarly, if we arc to realize a piece of mathematics, we must find a way of illustrating, to ourselves, the commands of the mathematician. The normal way to do this is with some kind of scorer and a flat scorable surface, for example a finger and a tide- flattened stretch of sand, or a pencil and a piece of paper.
Taking such an aid to illustration, we may now begin to carry out the commands in Chapter 2. First we may illustrate a form, such as a circle or near-circle. A flat piece of paper, being itself illustrative of a plane surface, is a useful mathematical instrument for this purpose, since we.
If, for example, we had chosen to write upon the surface of a torus, the circle might not have drawn a distinction. When we come to the injunction. Now, in this separate form, we may illustrate the command.
It is not necessary for the reader to confine his illustrations to the commands in the text. He may wander at will, inventing his own illustrations, either consistent or inconsistent with the textual commands. Only thus, by his own explorations, will he come to see distinctly the bounds or laws of the world from which the mathematician is speaking.
Similarly, if the reader does not follow the argument at any point, it is never necessary for him to remain stuck at that point until he sees how to proceed.
We cannot fully understand the beginning of anything until we see the end. What the mathematician aims to do is to give a complete picture, the order of what he presents being essential, the order in which he presents it being to some degree arbitrary. The reader may quite legitimately change the arbitrary order as he pleases. We may distinguish, in the essential order, commands, which call something into being, conjure up some order of being, call to order, and which are usually carried in permissive forms such as.
NOTES names, given to be used as reference points or tokens; in relation with the operation of instructions, which are designed to take effect within whatever universe has already been commanded or called to order.
Laws of Form was first published in and brings forth a new articulation of the foundations of thought. In Laws of Form we have a mathematical formalism based on one symbol and an approach to the question how the world would appear if a distinction could be drawn. Laws of Form does not answer the question how, given nothing as a beginning, a distinction can, indeed must, inevitably take place. This second question must, in its own structure, be left to each individual thinker. Nevertheless, Laws of Form, beautifully written and content free form is emptiness, emptiness is form is the most powerful mathematical text on the edge of nothing that has been produced since Euclid's Elements.
These papers are a tribute to Spencer-Brown and his singular achievement.
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