Chapter 4 Mathematics as a Language Peculiarities of Mathematical Language in the Texts of Pure Mathematics What is mathematics? Is mathematics, represented in all its modern variety, a single science? The answer to this clear-cut question was at- tempted by the French mathematicians who sign their papers with the name Nicolas Bourbaki. In their article "Architecture of Mathematics" published in Russian as an appendix to The History of Mathematics' (Bourbaki, 1948), they stated that mathematics (and certainly it is only pure mathematics that is referred to) is a uniform science. Its uniformity is given by the system of its logical constructions. A chracteristic feature of mathematics is the explicit axiomatico-deductive method of construct- ing judgments. Any mathematical paper is first of all characterized by containing a long chain of logical conclusions. But the Bourbaki say that such chaining of syllogisms is no more than a transforming mechanism. It may be applied to any system of premises. It is just an outer sign of a system, its dressing; it does not yet display the essential system of logical constructions given by the postulates. The system of postulates in mathematics is not in the least a colorful mosaic of separate initial statements. The peculiarity of mathematics lies in the ability of the system of postulates to form special concepts, mathematical structures, rich with logical consequences which may be derived from them deductively. Mathematics is principally an axioma- ' This is one of the volumes of the unique tractatus TheElernenrs of Marhernorrcs, which wos ro give the reader the fullest impression of modern mathematics, organized from the standpoint of one of the largest modern schools. 115 116 In the Labyrinths of Language tized field of knowledge, and it is in this sense that mathematics is a unified science. Its uniformity is given by the peculiarity of its logical structure. This is the Bourbaki's central idea which mirrors their mathematical outlook. Below, I try to give an idea of mathematical language. This language is a certain system of rules of operations on signs. To introduce a calculus, we must construct an alphabet of the initial elements, signs, to give the initial words of the calculus and to construct the rules for making new words out of the initial words. They are built upon a set of elements, the physical nature of which remains unknown. In order to give the struc- ture, it is sufficient to define the relation between these elements in a cer- tain system of axioms. The system of judgments in mathematics is built without turning to vaguely implicit assumptions, common sense, or free associations. The task lies in verification of the fact that the results ob- tained really follow from the initial assumptions. It is the question itself about verification of the correctness of the initial axioms in a certain physical sense that is pointless. Mathematicians care only about the logical consistency of axioms; they must contain no inner contradictions. But, again, the system of axioms must be constructed in such a way that it will be rich in its logical consequences. The idea of a universal symbolism and logical calculation can be traced to Leibniz, though modern clear-cut definition of mathematics as strictly formalized calculus became possible only after the works by Frege, Russell, and Hilbert. In Kleene's work (1952) we find the following char- acteristic philosophical views of Hilbert. Those symbols, etc. are themselves the ultimate objects, and are not being used to refer to something other than themselves. The metamathematician looks at them not through or beyond them; thus they are objects without interpretation or meaning. (p. 64) Chess playing is often regarded as a model of mathematics (Weyl, 1927) or, if you like, as a parody of mathematics. Chess figures and the squares on the playboard are the signs of the system, and the rules of the game are the rules of inference; the initial position of the game is the system of axioms, and the subsequent positions are formulae deduced from the ax- ioms. The initial position and the rules of playing prove to be exceedingly rich: in skillful hands they create a variety of interesting games. While the aim of a chess game lies in check-mating the adversary, the aim of mathematical reasoning is the obtaining of certain theorems. In both cases it is important not only to achieve the goal, but also to do it beauti- fully and, of course, without contradictions: in mathematics some situa- tions will be regarded as contradictory in the same way as, for example, the existence of ten queens of the same color would contradict the chess calculus. The most fruitful feature of such a comparison is that in chess, Mathematics as a Language 117 as in mathematics, logical operations are performed without any inter- pretation in terms of the phenomena of the external world: for example, it is not at all important for us to know to what element of reality pawns correspond or whether the limitations imposed upon the rules for moving the bishop are rational. Still, it would not be correct to state that mathematics is a fully for- malized branch of knowledge. Hilbert failed in his attempt to build a strictly formalized system of reasoning out of the absolute consistency of arithmetic. There are also some difficulties in defining formally the no- tion of proof in mathematics in general. In the process of the develop- ment of mathematics, new, previously unknown techniques of reasoning have appeared. (In particular, from this follows the irrationality of state- ments that the proof of mathematical theorems may be fully handed over to computers.) In the above-mentioned book Kleene (1952) states this idea as follows: We can imagine an omniscient member-theorist. We should expect that his ability to see infinitely many facts at once would enable him to recognize as correct some principle of deduction which we could not discover ourselves. But any correct formal system . which he could reveal to us, telling us how it works, without telling us why, would still be incomplete . (p. 303) Sometimes it is said that all mathematical knowledge is implicit in those short statements which are traditionally called mathematical struc- tures and that the proof of theorems is no more than an explication of the content of the structures. This statement would be quite correct if the process of reasoning were strictly formalized. But unless it is so, the proof of theorems themselves already contains some essentially novel in- formation, which is not intrinsic to the structures they serve to elucidate. There is one more reason that we cannot speak of a full formalization of mathematics: the reason is that in mathematics, together with deduc- tive reasoning, plausible reasoning [in the sense of (Polya, 1954)l is also used; the conclusions built on analogy may serve as a good example. True, nobody can estimate the role they play in mathematical judgments. One final remark: mathematical papers still must use ordinary language as a kind of auxiliary means. Mathematical Theory of Language in the Concept of Context-Free Languages The American linguist Chomsky in the late 1950s tried to build a mathematical model of ordinary languages. Formal grammar of the con- text-free languages is built as a calculus for generating the variety of cor- 118 In the LabyrinthsofLanguage rect sentences of the natural language. As with any calculus, here we speak about the grammar, which must consist of some finite alphabet, that is, a variety of initial symbols, of some finite set of inference rules that generate chains, and of the initial chains, axioms. The chains generated by the inference rules are interpreted as sentences. The whole set of sentences is called language. Grammar, if it is properly formu- lated, must unambiguously define the whole set of correct sentences in language. Here, syntactic description is performed in terms of the so- called analysis of immediate constituents. Sentences are divided into fewer and fewer constituents down to the smallest ones (Chomsky, 1956). The theory of context-free languages, as becomes obvious from the statement of the problem itself, is to be built as a purely mathematical discipline. This theory and the theory of finite automata are closely and deeply interrelated. In this book I cannot dwell in detail on the theory of context-free languages; to do so, I would have to write this paragraph in a language different from that of the rest of the book. A short and very popular rendering of this theory can be found in books by Shreider (1971) and Ginsburg (1966). In one of the first papers dealing with the theory of context-free lan- guages, Chomsky tried to establish the justification of his approach. He posed the following questions: Is it possible to formulate simple gram- mars for all the languages that we are interested in? Do such grammars possess any explanatory power? Are there any interesting languages which lie outside this theory? Is not, for example, English such a language? Before long, it appeared that grammars of context-free languages pro- vided very convenient means for the study of programming languages for computers. It would be interesting to pose a broader question: to try to find out to what degree this theory becomes useful for the description and understanding of natural languages which, unlike programming lan- guages, are still non-Godelian systems-at least from our standpoint. This question can be answered in the affirmative if we introduce, after Chomsky (1956), a set of grammatical transformations which transfer sentences with one structure of immediate constituents into new sen- tences with another structure.