In this paper, a new 'anti-semantic' symbol is introduced; the inverted exclamation mark, '¡', warns the reader to confine his attention exclusively to the visual aspect of the row of marks on paper that it 'governs'. Single quotes, ' ', are not used as speech-marks, but merely function as abbreviations for 'so-called', while double quotes, " ", instruct the reader to carry what is, in effect, a normal tendency of the symbol to its 'empirical limit',1 by confining his attention as completely as possible to the word-sound, or sound-thought, which is his tonal equivalent for the symbol-sequence that they include.
1. This term seems preferable to the more customary 'logical conclusion'.
I am reasonably confident that the empirical premises employed in the following reasoning will strike my readers as sufficiently 'self-evident/exaudient'. The key-word in the argument is the adjective 'short'; this is because it is readily accepted, both by logical philosophers and the general public, as a member of the 'autological' class of words, defined by the logician Grelling as the class of those words with an extra-linguistic meaning that can be predicated of themselves. For reasons which will soon become apparent, it is probably the only one which could sensibly be considered as a candidate for membership of that 'class', and this uniqueness of status makes it an invaluable link between language and 'spatio-temporality'. The argument follows.
1. ¡(Short) is only accepted as a 'word' if and when its viewer assumes that there is a word-sound or sound-thought with which it is 'synonymous' - that is to say, that the extra-linguistic ranges of meaning of the written marks and the sound are 'identical'.2 The sound in question will be designated as "short".
2. 'Nonsense words' like Carroll's 'vorpal', where the range is 'zero', are not relevant to the present discussion.
2. It follows that ¡('short') could not possibly be accepted as 'autological' unless its extra-linguistic meaning were a property that is 'predicable of' - that is to say, 'readily attributable to' - both ¡(short) and "short".
3. Now the shortness of "short" is not 'absolute'; the attribute expresses a judgment concerning the length of the interval between the beginning and end of its utterance, relative to that which obtains in the case of other spoken words with which it is habitually associated.
4. Such judgments depend on the assumption that the 'tempo' of the utterer's speech remains approximately constant. "Short" would remain recognisable, up to a point, even if the "sh" and "or" sounds were to be considerably prolonged, but a judgment of its 'shortness' would no longer be in order.
5. Similar considerations are applicable to the shortness of ¡(short). We can slow down the 'tempo' at which we read it by using magnification, when it will appear as:- ¡(short) which is one of the longest ¡(word)s, i.e., word-like rows of symbols, in this paper. It would remain recognisable unless the degree of magnification were so great that it actually overflowed the margins of the visual field.3
3. Our quasi-permanent criterion for judgments of 'visual length' is the interval between our awareness of the opposite margins of the visual field, which varies very slowly, if at all, under 'normal' conditions.
Corollary to paragraphs 4 and 5:
Grelling used 'long' as his example of a 'heterological' word, because both "long" and ¡(long), under 'normal' conditions, are relatively 'short'. If, however, they were submitted to the treatment suggested above in the case of 'short', they would become 'autological'.
6. The phonetic structure of "short", and the interval between the beginning and end of the sound or sound-thought, embody no 'spatial' characteristics whatsoever; they occur 'within the mind' of each individual speaker, and are entirely 'temporal' in character.
7. The possibility of effecting quasi-instantaneous 'translations' from our awareness of a written 'word' to the corresponding spoken "word" or sound-thought, which constitutes 'literacy', is dependent to a substantial extent on the measure of structural correspondence between them. The 'synonymity' of a written and spoken word is certainly dependent, among other things, on the identity of their 'syntactical' structure, i.e., of the temporal sequence of the 'sames' and 'differents' in the parallel series of phonemes and symbols. And since speech, or speech thought, always antecedes writing, it is clear that the structural elements which provide the basis for this correspondence must be exclusively 'temporal'. Were a written word to be perceived 'simultaneously', it could not possibly have any syntax, and there would be no way of testing the correspondence.
The sequence of events involved in the visual perception of a written word is admittedly so rapid that it can readily be regarded as 'synchronous' by the standards of any 'tempo' in which we are 'normally' interested, but the whole cannot possibly be regarded as 'simultaneous'. Even at the ultra-rapid 'tempo' of 'thought', or 'visual memory', some 'time-lapse' must be considered to occur between the beginning and the end of the row of symbols.
Conclusion:
Television already provides strong support for Leibniz' view that visual spatiality is never 'instantaneous', but the outcome of 'confused perception'.4
4. The term 'confusion' should on no account be regarded as pejorative. Played in melodic sequence, two notes of music are 'clear and distinct'; played synchronously, in a chord, they are 'confused'. Sometimes we prefer the one, sometimes the other. For example, words which are ambiguous, i.e., have a 'confused' reference, are undesirable for use in logical argument, but invaluable for humorous and poetic purposes. Similarly, a concrete noun, which denotes a 'bundle' of sensible qualities, is much more 'confused' than an adjective which denotes only one. But the 'synchronicity' of its multiple denotations effects an abbreviation which is valuable for 'signalling' purposes.
The 'atoms of luminescence', which each viewer 'confuses' into a television 'picture' within his internally generated 'field of illumination', are the activations of individual 'phosphors' behind the screen, and none of these ever occur 'synchronously' relative to the tempi employed by the television engineer.
The televisual 'field of luminescence' replicates a 'précis', obtained by rhythmic scanning, of a structurally similar field which is presumed, on the basis of experience, to have stimulated the generation of an interesting 'field of illumination' within the body of the camera-man. The 'scanning' is needed in order to abstract rhythmic series of 'events' from an original 'confusion'; 'events' are all that can be reasonably thought of as travelling along the 'causal lines' presupposed by all physical theory, since nothing that requires a minimum of duration for its 'existence' can ever be thought of as 'starting'.
An 'atomic event' would be the acme of 'clarity and distinctness'; a rhythmic series of such events can still be fairly 'clear'. But if there is anything at all in physical theory, it is obvious that our internal 'fields of illumination' are the outcome of an almost fantastic 'confusion' of a prodigious multitude of such series, to wit, the termini of the 'causal lines' which stimulate us to generate them.
The argument from language strongly supports the view that no act of awareness of any extended portion of the visual field can ever rightly be regarded as 'simultaneous'. And since the visual field is assuredly the perceptual context in which 'synchronicity' approximates most closely to 'simultaneity', the latter concept would seem to be inapplicable to any phenomenal material whatsoever, except as an approximation which can serve as a useful fiction for descriptive purposes.
It follows that no 'geometry' which assumes it can be strictly applicable to any phenomenal material which involves measurements of 'distance' - and this includes all 'pointer-readings'. The verbal or symbolic expression of 'Timeless Truth' is also out of the question, in view of the temporal activity involved in the 'scanning' of written sentences; the 'syntax' of the scanning always plays a vital part in their use as 'vehicles of meaning'.
Editor's note: Those two final paragraphs are absent from the typescript. Their presence on a separate page in the manuscript is consistent with J.D.'s practice of using a new sheet whenever one side of its predecessor has been completely filled. It is evident, however, that the manual pressure placed on his pen while writing these paragraphs differed from that employed in committing the earlier part of this essay to paper. Perhaps, therefore, their contents express 'after-thoughts' rather than being simply a continuation of the immediately preceding part of his 'conclusion'…
For some reason or other it is generally considered that the enormous improvement in mathematical notation effected by the Arabs was in great measure due to their introduction of the zero.
This belief is nonsense. The improvements which they effected could have been achieved perfectly well without this symbol, although it does represent a conventional procedure implicit in this system.
The Arab improvements were threefold. Firstly, they introduced separate symbols for all the numbers from 1 to 9, thereby eliminating the clumsiness of the Roman system, which had to write VIII for 8. Secondly, they arranged the numbers in regular ascending powers of 10, eliminating intermediate signs such as V, L and D.1 Thirdly, they realised that if the successive digits were understood to signify successive powers of 10, there was no longer any need to use letters to denote them.
1. Incidentally, it would have been more logical to go from left to right; I suppose that the Arabs, a Semitic people whose language is read from right to left, read their numbers in the same direction.
None of these procedures in any way necessitates the use of the zero; the number 2003, for instance, could just as well be written 2..3 or even 2 3. The '0's simply serve to indicate the serial position of the significant digits, and play no part in any calculation. I know that when multiplying such a number we are apt to say in our heads "4 × 0 = 0", but this is quite purposeless except perhaps for the purpose of maintaining a certain rhythm in our operations.
What, then, do the '0's signify? Ask a schoolboy, and he will tell you that it means that the number contains no hundreds and no tens. This, however, is manifest nonsense. It contains 20 hundreds and 200 tens. The fact is that the Arabs' procedure conventionally annihilates all those members of any class which can be entirely subsumed in a higher class. The zero symbolises the occurrence of this annihilation.
I suggest that there is much to be said for regarding it as an operator, analogous to +, -, and ×, rather than as a number. If so regarded, I think that many of the rules of mathematics which have to make it an exception when it is given numerical status, would thereby gain in generality, and it is difficult to see what would be lost by the procedure.
As things stand, the operations ± 0 are void, × 0 stands for annihilation, ÷ 0 simply produces the symbol oo, rational infinity, which has always seemed to be particularly useless and is only needed as the product of this procedure. If x' is defined as 1 multiplied once by x, then xº is simply 1 not multiplied by x at all.
An equation such as (a + b)(a - b) = 0 means that under some circumsances the procedure symbolised by the left-hand side of the equation will be equivalent to annihilation.
I cannot see that anything is lost by depriving '0' of its numerical status, except for the symbol oo. Peano's primitive propositions for arithmetic will work perfectly well if 1 is substituted for 0. And the definition of '0' as the class whose only member is the "null class" creates a metaphysical monstrosity. Its description by means of a propositional function is "Ø(x) never", and this implies a view of 'existence' which corresponds to nothing in our thought or experience.
Surely if any property Ø exists, there must be an x which can serve as its argument, otherwise, how could we name or symbolize it? If a predicate can be named or denoted, it must be applicable to at least one case.
I am maintaining here that anything which can be intentionally denoted by a single word has an existence of some kind, either in the physical world or in the world of thought. Denoting words include concrete nouns, adjectives, verbs and prepositions and conjunctions - the last two representing relations which can ultimately be demonstrated ostensively. They do not include adverbs of quality, which are simply adjectives conventionally modifed, in some languages, when they describe adjectives and verbs. More importantly, they do not include abstract nouns. These nearly always betray their adjectival or verbal origin by their etymology, and are linguistic devices which make it possible to use adjectives and verbs in the grammatical rôle of substantives.
Non-existence, then, can only be predicated of entities denoted by more than one simple descriptive term. If Ø(x) and Y(x) are simple propositional functions, it can be shown that there is no term c such that x = c. In other words, some simple sets have no members in common.
I suppose it would be possible to make a 'null' class out of these non-existent 'c's. This would include round squares, golden mountains and the like, and the procedure is curious, to say the least. '0' as the class of such classes would be equivalent to meaningless - which, in my contention, is just what it seems to be!
There can be little doubt that the structure of our thinking is powerfully influenced by the peculiarities of the symbolism by which we try to express it for purposes of communication. The philosophical method of logical analysis is entirely concerned with showing how mistakes arise from the uncritical interpretation of linguistic structure, and has succeeded in discovering the source of much muddled thinking.
In this essay I am concerned to maintain that an analogous misinterpretation of the syntactical function of "zero" has given rise to a great deal of nonsense in mathematical and logical philosophy itself.
Nobody, I assume, will wish to deny that "zero" is an arithmetical symbol. Arithmetical symbols are of two types; the integers, on which operations can be performed, and which we can refer to as "operands", and the "operators", such as +, -, etc., which symbolise the operations that we actually perform on them.
Now in the Arabic numeral notation which is in general use to-day, the position occupied by the symbol"0" is one which is normally occupied by an operand. It does not look in the least like an operator. But this appearance can be shown to be due to a peculiar characteristic of the Arabic notation, which requires the performance of a type of operation which is not usually employed during the process of symbolic description. The nature of this operation can most easily be shown by comparing the Arabic method with those which preceded it.
In the most primitive method of symbolising number, the multiplicity of the group under discussion was represented by the multiplicity of the marks on the writing material employed. For example, "four", in the early Roman notation, was symbolised as "I I I I".
This method is obviously clumsy and of very limited applicability; the eye can take in low multiplicities at a glance, but anything above six or seven is already getting out of range.
It was almost certainly the practice of counting with the fingers which led to the method adopted by the Romans to overcome these disadvantages; the first five units were subsumed into a single group, no longer represented, but expressed, by the symbol V. And since we only have two hands, the process had to start again once we reached ten, so that the first two fives were subsumed into the single groups of ten, symbolised by X. A similar procedure was later employed in order to symbolise multiplicities of tens and hundreds, giving rise to the symbols L, C, D and M.
In the Roman notation, every repetition of a given symbol stands for the same multiplicity; in CXXXIII, for example, each X stand for "ten" and each I, for "one". In the earlier form of the notation, it would have been possible to work out the multiplicity symbolised no matter in which order the symbols were written, but in practice, those referring to the larger groups always preceded those referring to the smaller ones. In the later versions, fourfold repetitions were eliminated, presumably in the interests of abbreviation, by reversals of the normal order, so that "four" became "IV", the number before five, and "ninety", "XC", the group of tens which precedes "one hundred".
This method was clearly incapable of dealing with very large multiplicities, since it involved the incorporation of two new symbols every time that a power of ten was exceeded. It was economical inasmuch as it only required four symbols in order to refer to any number up to "ninety-nine", but the expressions were lengthy; in the older version, "ninety-nine" was symbolised as LXXXXVIIII; this was later abbreviated to XCIX.
"Zero" played no part in this notation, any more than it does in the verbal expression of numbers. The Roman "MI" is closely equivalent to the verbal expression "A thousand and one".
The Arabic notation, however, necessitates a radical departure from the verbal form of expression. "1001", expressed verbally, would run "One thousand, zero hundreds, zero tens and one unit". And it is this appearance of similarity between the syntactical functions of "zero" and "one" which has given rise to the illusion that "zero" is a number.
The Arabic notation differed from the Roman one in two important respects, both of which facilitated calculation. In the first place, the subsumption at the level of five was omitted, so that subsumptions were performed, on a uniform basis, whenever a tenfold multiplicity was attained. This involved the use of a separate symbol for each integer up to nine.
But whereas in the Roman notation a given symbol always stood for the same multiplicity, any integer might now indicate a multiplicity of units, tens, hundred or any power of ten. The nature of the members of the group whose multiplicity was symbolised by the integer was indicated solely by the position of the symbol within the complete expression. It was therefore vital that the position should be clearly defined and easily recognisable.
In theory, this could be achieved without the use of any additional symbol; "one thousand and one" could be expressed as "1 1". But a reader could hardly be expected to see at a glance that the space between the"1"s was wide enough to accommodate precisely two integer-symbols, and in any case the method would fail whenever the unit column was empty, since the end of the expression would not be properly defined.
The Arabs hit on the expedient of inventing a symbol for the sole purpose of defining the position of the integers in all cases where their method of symbolic representation would otherwise leave a gap in the row of symbols. Its primary function was visual rather than symbolic.
Nevertheless, it performs the function of denying that the space which it renders "visible" would otherwise be occupied by an integer. In this respect it differs radically from the integers, each of which asserts the presence, while simultaneously defining the multiplicity, of members of the class whose symbol occupies the appropriate position in the expression of the number concerned.
The denial is only necessary because it is expected that a space will generally be filled; the chances are clearly nine to one in favour. It is certainly not an assertion of the presence of a member of the "null" class. In fact, since it is now generally agreed that any proposition containing the expression "some A" asserts the existence of A, it is plain that the proposition containing "no A", which contradicts it, must do likewise.
In formulating an ordinary symbolic description, we rarely go to the trouble of explicitly denying the absence of any characteristic except when there is a lively expectation of its presence, as, for example, of that of the tail in the case of Manx cats. Otherwise we only do so when confronted with a questionnaire which lists the attributes in which the questioner is interested, and when the instructions for answering forbid us to leave a space blank, by the inclusion of some such expression as "If none, say 'none' ". It is clear that just such an expression is implicit in the Arabic method of symbolising numbers.
It is instructive to note that it does not apply in the case of the algebraic power-series which generalise this method. The general form of any four-figure number is expressed by ax³ + bx² + cd + d, where a to d are integers, and x = 10. The form of "1001" is x³ + 1, and it would never occur to us to deny the presence of any member of the classes x² and x; such a denial would be redundant, since the only power of x which we require to symbolise is clearly indicated, without reference to the spacing; the mode of expression has reverted to the Roman-verbal form "one thousand one", or MI.
It is now clear that zero, in the Arabic notation, is an operator, not an operand, and symbolises the empirical denial of the presence of a member of a class, which is not logically null, within the breakdown of the number expressed by the series of symbols employed. There ought to be an empty space where it stands, and the number-like symbol occupies it for the purely practical purpose of visually defining the position of the occupied space.
The insertion of any other numerical symbol into one of the spaces in the expression "1 1" is a semantic operation which would alter the meaning. "Zero" plays the rôle of a part participle, denying that its own insertion has altered the meaning.
The actual operation of "conventional annihilation" can also be expressed by the use of zero, or its verbal equivalent "no", in conjunction with an operand.
"A thousand and one" can be written as "One thousand, no hundred, no ten and one unit", or as "1 thousand, 0 (hundred), 0 (ten) and 1 unit". In this case the actual operation of denial is symbolised by the transition from the symbol for the operator to that of the operand.
It is not necessary to have a number as operand; "no cheese" and "0 (cheese)" both make good sense. "0 × cheese" adds a superfluous symbol, in a manner which incorrectly assimilates zero to a number.
On the other hand, "4 (cheese)" or "4 × cheese" is incorrect; while the zero can be employed alike with singular and plural forms, this is not permissible in the case of the integers, except in the special case of 1, which will be explained later.
This is because "cheese" stands for a class-concept, while "a cheese" and "cheeses" stand for members of the class to which it has given rise.
It was only by confusing the two that Russell was led to assert the existence of "logically null classes". He needed these for his system of mathematical philosophy, since he had followed Peano in regarding zero as a "number", and his definition of "number" as "The class of all classes similar to (i.e. showing one-one correspondence with) a given class"1 made it necessary to exemplify a type of class with which all members of the "zero" class could show a one-one correspondence.
1. This could be formulated, much more accurately, as "The class-concept differentiating the method of comparison of classes, in respect of the presence or absence of one-one correspondence between their members, from all others".
Here I think that his "robust sense of reality" deserted him. Symbols are intended to communicate, but they are also visible and audible phenomena which have an existence independently of anything that we use them to express. Admittedly, if I write "square circle" - an expression of one of Russell's "logically null" classes - the result looks like a symbol, which asserts the performance of a semantic operation. But the judgment "There is no square circle" or 0 (square circle) is likely to be universal and "eternal", although it is perhaps worth pointing out that the expression might be used, in a modern idiom which is hardly "English", to denote "a coterie of intellectuals".
The judgment is in fact empirical, not logical, and denies that the apparent symbol can effect any semantic operation within the field of "shape". There can be no "class" that it can be used to define.
The reason for this is clear. We can all be confident that the visual difference between "having corners" and "having no corners", which is an important diagnostic feature in the difference between squares and circles, was of considerable importance to every child well before speech began, on account of its tactile connotations. These concepts are therefore mutually exclusive, and their conjunction thus excludes the whole field of the concept "visual shape", within which their differentiation has taken place. Their conjunction can therefore form no part of the language of geometry.
"Logical" contradictions are of two distinct types; the more obvious type is of post-verbal origin, and is created whenever a verbal difference is employed in helping to point out a difference which is generally considered important; such contradictions as "isoceles scalene triangle" exemplify this type. We actually learn the empirical contradiction from an accepted linguistic one.
In the case of "square circles", however, the conjoined concepts have each resulted from considerable differentiation within the field of the wider concepts "having corners" and "having no corners". We are confident that the contradiction between these two will have been judged empirically to be important early in childhood; but there is little chance of our ever remembering the occasion on whcih we first noticed the importance of the difference between them.
It was this original experience which created the separate concepts. This is why "one cheese" is correct, but "four cheese" is not; there must always have been a first time. The concepts are reinforced by the associative action of memory; and the exigencies of linguistic communication, which is a kind of "public" extension of memory, tend to encourage the grouping together of as many acts of awareness as possible, simply for reasons of economy. "Occam's Razor" expresses the common distaste of mankind for having to commit too many symbols to memory.
We cannot possibly invent a fresh symbol to express every little difference that we notice; linguistic description can provide no more than a précis of experience, although we can greatly widen its range by varying the order in which we use the symbols.2 Grammar, syntax and logic formalise the rules which we adopt in order to express observed differences in an approximately agreed order of importance.
2. This is why writing is used in preference to speech, and symbols in preference to words, for the expression of complex variations of structure. The tempo of reading is much faster than that of speech, and the visual field is two-dimensional, as against the single time-dimension of hearing, so that much more variety can be assimilated within one "specious present".
One task of logic is to exclude from the language every conjunction of predicates which would annul any distinction, whether obvious or covert, which we can confidently assume will be of major importance to everybody who uses the words. By clarifying the contrast between the function of zero and that of the integers, I hope I have demonstrated that unless we agree to annul the difference between "assertion" and "denial", "The number zero" is no part of the English language.
We can now proceed to examine the function of "zero" when it no longer forms part of a series of numerals, but stands on its own, as it often does in algebraic generalisations of arithmetical propositions.
It is most frequently used to furnish an alternative method of formulating "equations". A primitive example, which is quite adequate to illustrate the procedure, is the reformulation of the "tautological" equation "1 = 1", as "1 - 1 = 0".
There are, however, two grounds for regarding the second form of expression as, semantically speaking, seriously misleading. In the first place, if the "=" sign is intended to symbolise a symmetrical relationship, it should be permissible to write the "0" on the left-hand side of the equation. The reason why this is never done is that it is realised, perhaps subconsciously, that its original function is one of denial, and that it is not possible to deny until something has been, at least tacitly, asserted.
The transition is thus always a one-way affair, and the "=" sign should be replaced by a directional arrow, "->". An "equals" sign should symbolise the permissibility of the alternative transitions "->/<-" [in the author's typescript the right-arrow appears vertically above the left-arrow] (the "and/or" of logic).
In the second place, it is most misleading to write the initial "1" without a "+" sign. Mathematical symbolisation begins with the writing of the first "1", not after it; the initial "operand" is the blank sheet of paper, or the empty blackboard.
When the first term of any arithmetical or algebraic expression is taken into account in subsequent computations, its "plusness" is always taken for granted, since its initial appearance - an abrupt event, which is meant to be noticed - must surely have effected a modification of the pre-existing, relatively invariant background. The relatively invariant state of "blackboard + 1", which succeds it, can now become the operand for further operations.
If we not proceed to delete the "1", the appearance of the blackboard, subsequent to the operation, will not differ appreciably from our memory of what it looked like before the first operation was performed. Expressing the deletion as "-1", the whole sequence can be expresses as " +1 -1 -> ", where the spaces to the right and left represent the "empty" states of the blackboard.
The emptiness of the blackboard before the operation can, in general, be taken for granted. But most numerical operations are intended to express actual ones, either mental or physical, which are succeeded by states which differ to some extent from those which preceded them. The concluding term of a mathematical expression is generally intended to indicate the nature and extent of that difference.
Whenever we judge it to be the case that no difference has resulted from the operations symbolised, it is therefore expedient to employ a symbol to deny its occurrence.
The "zero" of the Arabic notation is semantically suitable for the purpose. Its function here is somewhat more extensive than that which it fulfils in its original context, where it denies that a semantic operation has been performed within the space in which it stands. Its algebraic use does not, in fact, deny that semantic operations have been performed, but only that there is any difference between the states which precede and succeed them.
The "space" within which semantic alteration is denied includes the whole area of the symbolic expression. A use of the "zero" symbol, more closely analogous to the function which it fulfils in a sequence of numerals, would be:-
The existing notation, however, provided that a direction arrow replaces the "=" sign, has the advantage of indicating that the denial refers to the sequel to the operations rather than to their occurrence.
The validity of a "tautological" type of equation clearly requires that the first operation symbolised must be conceptually reversible, since the successive performance of that operation and its successor must result in the "annihilation" of any difference from the initial state of affairs which may have followed the performance of the first operation.
It follows that such "absence of difference" can only be "absence of important difference", since our final state of awareness is bound to differ from the initial one in so far as we retain any memory of the operations themselves!3 The symbolisation of "absolute identity" is impossible; the repeated use of symbols which are intended not to differ importantly in appearance simply expresses the substantial degree of "gap-indifference" which very often prevails between successive acts of awareness in our everyday experience.
3. It is impossible to annul an "event", but frequently possible to annul certain changes of state which succeed it.
Yet the initial assumption of all mathematical types of logic is "same symbol, same reference". Its primary assertions are of "identity", or "zero" difference, and the "truth" of a proposition is "proved" by substituting for it a sequence of linked propositions, each of which approximates as closely as possible to a "tautology".
The archetype of such a sequence is the conventionally agreed naming of a succession of events by accompanying them with the utterance of the words "one", "two", "three", etc., in the approved order.
A "class" has a "number" only as long as we can assume that the stages of its one-by-one breakdown and its one-by-one reconstruction (which need only be conceptual) will continue to show one-one correspondence with the same multiplicity of this series of events. The termination of either process must coincide with the utterance of the "same" member of the approved series of names - in which "zero" does not occur. Its only possible function in this context is to deny any difference of multiplicity between the two series of associative and dissociative events. "+n -n -> 0". One-one correspondence denies numerical difference. Should we really regard it as an assertion of "zero" difference?
This question is an ethical one, since our answer to it is likely to colour the whole of our outlook on life. It will depend on whether we attach the greater importance to change or to invariance.
Most philosophers, profoundly impressed by the efficiency of language for communicating "facts", have been inclined to regard its progressive improvement for this purpose as the principal task of their subject. They have tended to look upon language4 as an end in itself, rather than as a means for achieving certain limited objectives.
4. The chief use of arithmetic is not that of denoting multiplicities, but of providing a language for the comparison of magnitudes.
Now there can be little doubt that a "good" symbol is one which, judged by pragmatic tests, shows the least variation in its references. This is the case even within individual experience; each of us, in thought as well as in speech, tends to make use of those symbols which we judge to show the greatest measure of "gap-indifference". They function as media of communication between "me-then" and "me-now", as well as between "me-just-now" and "you-now".
But if we contemplate any actual experience that we are trying to communicate, we find that its importance varies directly with the amount of difference that we are experiencing. Invariance is experiential "nullity", while extremes of variation, whether of structure or intensity, are occasions of pain and/or displeasure for all of us. Enjoyment comes somewhere between the two, but the tempo and/or intensity which occasions it tends to be closer to that associated with pain rather than to nullity.
It is only unthinking pessimists who equate "change" with "decay", and "invariance" with "perfection", though we can agree that there have been periods of history, and times within the lives of many individuals, when such an attitude of despair is, at least, intelligible.
The "virtue" of invariance is intrinsic only to language. Even in language, it is mainly the negative one of avoiding errors of expression which may lead to unpleasant experiences - including disappointments occasioned by the missing of pleasant ones. There are no grounds for supposing that the value of "truth" is more than coextensive with the disvalue of "falsehood"; "factual" truths, apart from the occasional beauty of their linguistic expression, communicate no enjoyment. Indeed, tautologies, which are the "truest" of all statements, make no psychological impact whatsoever!
On the other hand, those whose main purpose is to communicate enjoyment, such as poets, deliberately make use of verbal expressions showing the greatest possible ambiguity of reference. All verbal humour likewise results from ambiguities, generally less refined and more obvious than those used in poetry.
My conclusion is that the over-valuation of "truth", and the consequent over-emphasis on the desirability of internal, "logical" consistency in language, has mislead many of the world's leading thinkers into treating "zero" variation as a positive ideal. They have thereby reinforced the pessimistic attitude to change which characterises many religions, and unthinkingly given support to dubious ideals such as that of "equality", i.e. "zero" difference.
For this reason it seems to me important to try to refute the attachment of positive value to "invariance" by demonstrating the negative function of the symbol "zero", which constitutes its expression in mathematics, the ultimate stronghold of symbolic logic.
In constructing a logic that really corresponds to experience, the optimum "truth-value" should be represented, not by "one" but by "zero" - the minimum difference between our interpretation of a statement and our memories and/or expectations; the "ideal" is best exemplified by the tautology, which most closely approximates to psychological "zero".
The positive function of such a logic must be to help us to express differences in an agreed order of importance. This is to a large extent achieved by classificatory and syllogistic logic which, apart from questions of the equivalence of certain alternative formulations of its propositions, is mainly concerned with asymmetrical relations.
The main function of "identity-logic" is in fact a remedial one. It helps us to make sure that serious semantic ambiguities are not being concealed by using one symbolic formula to express two ideas which are mutually contradictory, i.e., differ intolerably. This is precisely the type of treatment which, in the present essay, I have applied to the inclusion of "zero" within the class generated by the class-concept "number".
|
|
|
129-31. Noam Chomsky's letters of 5 March and 9 & 28 April 1977 to J. D. Solomon.
132-3. John Clapham's letters of 24 June and 31 October 1977 to J. D. Solomon.
134. Dorothy Emmet's letter of 10 May 1986 to J. D. Solomon.
135-7. John Ferguson's letter of 5 July 1987 to J. D. Solomon with copies of John Ferguson's “Was Gallio an Epicurean?” (Palæologia, vol. VII, no. 2, Osaka, 1958, pp. 111-4) and "Epicurean language-theory and Lucretian practice" (LCM, 12, 7, July 1987, pp.1-6).
138-9. Nicholas Greaves' letter to J. D. Solomon of 19 June 1989 together with his 8 August 1988 brief synopsis of “Duplication Theory - An Observational Theory for the Operation of Morphic Resonance, to explain certain aspects of Memory, Intuition, some Paranomormal, and other Phenomena”.
145. John Haffenden's letter to J. D. Solomon of 26 February 1984.
146-8. D. W. Hamlyn's letters to J. D. Solomon of 29 May 1973 and of 6 October & 30 December 1975.
149-154. Peter Hewitt's “Reconciling Physics and Metaphysics”, April's “But How Do We Know?”, and other Metaphysics Working Group papers from Nicholas Hagger, David Lorimer, etc., as well as Frederick Copleston's “Dialogue with Ayer” (The Tablet, 15 July 1989, p. 808).
155. Douglas R. Hofstadter's letter to J. D. Solomon of 22 August 1988.
156. Professor Sir Fred Hoyle's letter to J. D. Solomon of 14 June 1985.
157. A. Jaffe's letter to J. D. Solomon of 9 September 1982.
158. Rosamund Jenkinson's letter of J. D. Solomon of 30 June
159. David Lorimer's letter to J. D. Solomon of 5 October 1987.
162*. Video-tape of Richard Wagner's Das Rheingold with the Metropolitan Opera Orchestra conducted by James Levine and stage production by Otto Schenk: © 1990 Metropolitan Opera Association, Inc., New York; © 1991 Deutsche Grammophon GmbH, Hamburg.
163-4. Copies from The Times (20 October 1986) of four Letters re. "The relative values and lure of Einstein", one being from A. P. Miodownik, together with his letter of J. D. Solomon of 3 November 1986.
176-7. Geoffrey Read's 1990 “Science and Survival - a Cosmic Reconciliation” together with his 1991 “The Fatal Trap”.
178. Letter from Kiel to J. D. Solomon dated 14 April 1983.
179. Colin Wilson's letter of 17 May 1983 to J. D. Solomon.
181. Jim's letter of 21 September 1979 to J. D. Solomon.
182-189*. Various printed books relating to J. D. Solomon's life and work, including several extensively annotated in his own hand, including:
244. "I The Problem: ONE - INTRODUCTION" - 7-page typescript of unknown provenance.
245. Colin James Hamer's letter of 25 April 1996 to J. D. Solomon.
246-8. Two letters from Professor Peter Crossley-Holland relating to no. 66 and no. 90* above, together with an original copy of Professor Crossley-Holland's manuscript of his Introït for Organ annotated in his own hand.
249-251. Colin James Hamer's Letter of 21 February 2000 to Andrew Wallace-Hadrill referring to nos. 41 & 160-62 above together with related letters to Austen & Lala Winkley and one of 7 March 2000 from Josephine Pollard, the Archbishop of Canterbury's General Correspondence Secretary.
251*. The Geologists' Association Circular, no. 859 (January 1987).
251†. Extract from The Weekly Law Reports - 1978, Vol. 1, pp.770-78.
- Shalom & Welcome! - 
© The Neith Network Library 2002
Webmaster: H.B. ExtraReverendDoctorColinJames Hamer, The Rainbow Programme
Creativity House, 9 Oxford Street, St. Thomas, EXETER, Devon EX2 9AG, U.K.
Updated 00:01 1/8/2002.
