Psaná a mluvená matematika
In contradiction to common communication, in which the attempt to record the spoken word represented the original impulse for a genesis of the written text, in mathematics the characteristic situation is the reverse. The traces of originality can be found in the written form, and the spoken version should be considered as a derived one or as a copied one. It seems to me this can be shown not only about mathematics of the present day and not only about the pure mathematical text without comments in ordinary language, but it holds true in general.
Mathematics as a branch of science has been based upon records which were probably even older than the first recorded words. Already the neolithic incisions and knots can be considered as records of an abstract conception of numbers. In the spoken form in each natural language the anthropological inspiration can be identified. It confirms that spoken numbers are of a very old origin, but even fingers which are held or crossed represent a record in fact, and so, the spoken language clearly shows that the colloquial form of numbers is derivative.
As to higher numbers, already within the Old Egyptian hieroglyphics the higher quantities have been represented by additive combination of signs, because they were different from simple multiplicative spoken items as ”ten of tens” or ”dozen of dozens” and therefore they could not be easily expressed in speech. The colloquial form of such numbers of any spoken form of them, thus, was correspondingly complicated.
An important innovation into the spoken form of numbers can be found in the Indian and Old Babylonian position system of writing numbers, which we know in modern times as numbers written in Arabic style. It should be emphasized that such position systems lead to an important discovery of the concept of zero, which as a number could never enter the language, before the necessity to express a non–presence of units at a particular position was required.
A quite evident example of the importance of a written record for the development of mathematical thinking can be given in pre–antique tables of compiled fractions of natural numbers (cf. Vygodskij, 1969). Such old arithmetic tables, without any doubt, are to be considered as written texts. Its spoken form, however, if the written form cannot be seen, can never be realized. Moreover, even if being read, such a table is no longer understandable and because of that, it is of no further use, unless new recording is done.
The general acceptance of the written form of numbers in Arabic style formed a basis for introduction of mathematical problems with unknown quantities and variables, which brought a real revolution into mathematics.
Many examples can be given to show that the attempts to express and understand general problems with unknown quantities and variables – even more to solve them – by means of ordinary language exclusively, are far from reality.
Just one example quoted from a papyrus studied by Neugebauer (Neugebauer, 1960). The formulation in ordinary language is near the word equivalent: ”Look for the quantity, of which the half and the quarter added to itself will give the number ten.” Because this task is easy in fact and it can be faced frequently in cases of family heritage, already the Ancient Egyptians tried to write it in a symbolized form. Today, we can understand it and solve it much more easily in the comprimed written equation
1/2x + 1/4x + x = 10
and only such (or a similar) written form enables us to generalize this example and its solution for different constants. Within ordinary language, however, such a problem is generally unsoluble, and the same can be said about any only slightly complicated case. (A certain paradox can be found here too, because the way of solution can be communicated also by reading the written sequence of formulae, and if the person concerned is educated and trained enough he can follow it, understand it and reproduce it in his own words. However such a paradox vanishes immediately if any problem of a higher level is being solved, as, e. g., a set of equations with several variables.)
Thus, without the written form, many mathematical methods become unclear but even more they become unrealizable. They would not even exist without the written form.
This concerns already those cases, in which the sequence of basic operations, as, e. g., addition, subtraction, multiplication and division, is used. Any arithmetical function can be introduced as a good example, because it can be easily described by words as a sequence of basic operations. Furthermore we know that nearly any other function, if special conditions are fulfilled, according to Taylor’s theorem, can be transformed into such arithmetical form. It is however absurd to think of a verbal description of such transformation.
There is no doubt that the written form of a mathematical text needs the accurate knowledge about the way in which it can be read (cf. Nebeský, 1970). If it is compared with the written form of text in ordinary language, the mathematical system of writing is much more sophisticated.
As to mathematical relations and elementary functions, only some of these can be named in ordinary language terms. Any more advanced construction needs a special approach. The mathematicians turn to graphic representation first also here, and in these cases some decisions became a custom already. It is practical to accept them, but it is not strictly necessary. Nonetheless, any deviation from any custom complicates the communication. Therefore, in order to avoid complications as often as possible, the written mathematical text preserves the same form even when expressed in different languages. (This is partly caused by its special origin and itself it represents what Gnedenko assigned as logical stenography, cf. Gnedenko, 1980.) Not only this, logically is the reason of permanent differences between the written and spoken mathematics in different languages. It is evident, that the graphic symbols of relations, functions or operations are in every language read – or spoken – differently. However, within one language too, different ways of reading of the same graphic symbol can be observed, e. g., the written number 1 000 000 000 not only has its different language variants, but it is read differently even in the same language, by an Englishman and by an American.
Mathematics tries to wave any associations of its graphic symbols. The exceptions have their practical reasons in the definition or in the origin of the used symbol, as, e. g., c for constant (constans) and the neighbouring letters in alphabet b and a, or f for function (functio) followed logically by g and h, or x, y, z, for unknown quantity (a new computational custom confirms the independent choice by using ? or *), in physics we find v for the speed (velocitas) or g for the change of speed (gravitatio) etc. This in no case is a dogma. It is a custom only, which leads the mathematician to write
f (x) = c instead of c (f) = x,
although these writings are equivalent in general.
Another inverse difference between mathematics and natural communication can be seen just in this last example: the differently written or read representations correspond with the rôle of synonyms. However, the measure of closeness of the corresponding equivalence is unknown from the ordinary language. In mathematics, the synonymity is absolute. And yet another inverse property of mathematics: the already mentioned synonymity arises every time when different ways of reading of the same graphic record are used, and this is many times more frequent than in ordinary language.
Other examples of differences in written and spoken mathematics can be found at every further level, as, e. g., in formulae, in written sequences of logical postulates, in graphically expressed predicative constructions and in their confluences, in using the successful succession of steps or in converting new proofs into the already known algorithms, or even in using the personal name quotation for a whole theorem or for a set of axioms, etc. In all of these cases, the written form reduces very long explanations and postulates, not only those of quantificators, functionals, operators, automats, etc., but even those which represent the whole theories too (cf. Králík, 1990). From some point of view, we could speak about a wide and in many cases very effective univerbization, which is much deeper than in any ordinary language.
The language of mathematics, in other words, is based on the use of the written form and on a special graphic and, derivatively, word representation of concepts, operations and higher processes, the exact description of which by spoken language would make the whole mathematical communication incomprehensible.
So, having left the elementary level of abstraction, mathematics has to differ from ordinary spoken language, because the ordinary spoken language does not possess sufficiently exact expressions and predicates and therefore it can not aspire to the required level of exact thinking.
The spoken form of written mathematical text, thus is a read form usually, or its substitution only. Fully spoken mathematics did not develop, as it could not be exactly deductive and it would never allow us to reach the present day level of abstraction.
Gnedenko, B. V.: Preface to Dialogues on mathematics by Rényi. In: A. Rényi, Dialogy o matematice. Mladá fronta, Praha 1980.
Králík, J.: Pravděpodobnost metafory (The probability of a metaphor). In: J. Stachová (ed.), Úloha metafory ve vědeckém poznávání a vyjadřování. Filozofický ústav ČSAV, Praha 1990, s. 79–84.
Nebeský, L.: Note on the mathematical metaphor. Revue Roumaine de Linguistique, XV, No. 6, 1970, p. 569–570.
Neugebauer, O.: Arithmetik und Rechnentechnik der Aegypter. In: Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Vol B, part I. Berlin 1960.
Vygodskij, M. Ja.: Arifmetika i algebra v drevnem mire. Nauka, Moskva 1967.
R É S U M É
Zatímco v běžné komunikaci je prvotním impulzem ke vzniku zápisu a psaného textu snaha zaznamenat vyslovené sdělení, mluvený text, v matematice je charakteristická situace právě opačná: záznam a psaná podoba sdělení jsou tu prvotní, mluvená podoba je pouze jeho čtením. Matematika jako obor se konstituovala právě zápisem, a to zápisem starším, než je zápis slov: za matematický zápis je třeba považovat již neolitické doklady o abstrakci čísel reprezentovaných určitým počtem zářezů či uzlů. Nejde tedy jen o dnešní stav, ani výhradně o čistý matematický text zbavený komentářů, ale o obecný jev.
Ústav pro jazyk český AV ČR
Slovo a slovesnost, volume 54 (1993), number 3, pp. 191-193
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