Does Mathematics Need Something other than Logic? Katuzi ONO Mathematical Institute, Nagoya University Allow me to reason freely in the primitive logic. Then, I will show you how to develop mathematical theories without assuming anything in it. Introduction "Does mathematics really need something other than logic?" This is a question I have asked myself over and over again. Something in my mind has tempted me very strongly to answer "No. Mathematics needs nothing other than logic. If mathematics at present needs something other than logic, mathematics in the ideal form should be established purely logically." Most people would agree to the answer idealistically. Anyone would like to establish mathematics purely logically, if he could. But, how can he know that we can establish mathematics purely logically without being able to do so? I have always believed mathematics is absolutely applicable. If something is concluded mathematically, I believe it absolutely. If there arises something which contradicts a mathematical conclusion, I would only seek after illusional perceptions of the matter or mis-reasonings leading to the mathematical conclusion. I have to admit that illusional perceptions or mis-reasonings are really possible, but I can not even imagine such a situation that some correct perceptions contradict correct mathematical conclusions. Why can we believe in mathematics so absolutely? The reason seems to be that mathematics can be established purely logically. I do not know whether all the theories of mathematics are really established purely logically, but some thing in my mind forces me to say that all mathematical theories should be es tablished purely logically. In reality, however, we live in the tradition of mathematics, in the long lived tradition from the era of Euclid, which is to develop all mathematical theories starting from respective systems of axioms. I do not assert that we have really developed every mathematical theory starting from a certain system of its axioms , but I must admit that, if not, we are strongly tempted to try so . In the good old days, mathematicians could believe that axioms are self-evi dent truths. Nowadays, we can not agree even that axioms should be self-evident , nor that they could be so. However, if axioms are not self-evident , how can we believe so confidently in the absolute applicability of mathematics? -93- 8 K. ONO Vol. 3 Anyway, axioms are something which are not purely logical. If axioms are really necessary for developing mathematical theories, can we still say that mathematics needs nothing other than logic? If we stick to the axiomatic formali zation of mathematics, I think, we can not assert that mathematics needs nothing other than logic. The axiomatic approach has been really effective for clearly disclosing the logical structure of mathematical theories. However, it has not been the sole approach in the logical study of mathematics. In fact, there has been once an epoch when mathematicians seem to have been trying (successfully or not) to construct fundamental mathematical notions such as "real numbers", exclusively on a logical basis. I am thinking that set theory in the beginning was going to be developed in such an atmosphere. Someone seemed to believe that we needed natural numbers even at the beginning of mathematical theories but we can con struct all the others purely logically. Some mathematicians, however, seem to have been trying to construct even the natural numbers "set-theoretically", this perhaps being synonymous with "logically". I think that the category-theoretical approach of these days must be on the same line. A typical example lying along this line in traditional mathematics must be the definition of real numbers by cuts of all rational numbers. We need not regard real numbers always as cuts of all rational numbers, but cuts of all rational numbers really behave just as real numbers. This is why real numbers can be represented nicely by cuts of all rational numbers. A similar situation arises when we introduce all logical constants in terms of "implication" (notation:"•¨","_??_•¨_??_ is read "_??_ implies _??_") , "universal quanti fication" (notation: "( )", "(ƒÔ)_??_(ƒÔ)" is read "For every ƒÔ,_??_(ƒÔ) holds"), and "nega tion" (notation:"_??_","_??__??_is read "Not _??_"). In classical logic, for instance, "_??_ or _??_" (notation' "_??__??__??_") can be paraphrased as "the denial of W implies 3" (in notation:"_??__??__??__??_") and "There exists an x which satisfies _??_(ƒÔ)" (notation: "(_??_ƒÔ)_??_(ƒÔ)")can be paraphrased as "It does not hold that _??_(ƒÔ) does not hold for every ƒÔ" (in notation: "- _??_(ƒÔ)_??___??_(ƒÔ)"). We need not regard existence notion always as being defined in terms of universal quantification and negation, but the existence notion thus defined behaves just as the real existence notion in classical logic. Therefore, the existence notion can be defined nicely in this way in classical logic. These two examples are really similar in some respects. They differ, however, in the respect that, in the former example, mathematical notions only are paraphrased in a system without the notions but, in the latter example, logical constants are paraphrased in a system without the notions. In constructive theories of mathematics, some mathematical notions are introduced, but we keep usually logical constants unchanged. In our study of logical structure of mathematical theories, we are also in -94- No. 3 Does Mathematics Need Something other than Logic ? 9 terested in mathematical theories standing on the intuitionistic logic, the positive logic, or furthermore, other various logics. In these cases, it seems impossible to keep every logical constant unchanged. In fact, I have been able to develop every finitely axiomatizable formal theory in a very simple logic called primitive logic, which has been introduced in my former papers [2] and [3]. Technical treatment of the matter has been described in details in my former papers [5] and [6]. In the present paper, I will describe our study on the same subject, disclosing my schedules, my failures, and my successes frankly. (1) Reduction of a Formal Theory to Another by Paraphrasing Every Word of the Former in the Latter To speak of the development of one formal theory in another formal theory or to speak of reduction of one formal theory to another formal theory clearly, I would like to study at first how to define these notions. In mathematics, a typical example of reduction of a formal theory to another would be the reduction of geometry to analysis by introducing coordinates. For reducing plane geometry to analysis, any point is paraphrased by its coordinates, i.e. a pair of real numbers, a straight line is paraphrased as a condition expressed by a linear equation with respect to two variables, and "a straight line passes through a point" is paraphrased as "the coordinates of the point satisfies the equation expressing the straight line" . For plane geometry, "point", "straight line", and "a straight line passes through a point" are basic notions in the sense that other plane geometrical notions can be defined in terms of these basic notions. More generally, every formal theory has a system of its basic notions, in terms of which all the other notions of the theory can be defined. Accordingly, if we give the translations of all the basic notions of a formal theory into another, we would have a complete translation of the former theory into the latter. When every proposition holds in the former theory if and only if its translation holds in the latter theory, the former theory can be duly called "reduced to the latter theory." Reductions in mathematics are usually reductions of this kind. Such a kind of reductions rely on the paraphrasing of their basic notions. These basic notions of a theory are usually called the primitive notions of the formal theory. Any primitive notion of a formal theory is usually denoted by a technical term (maybe a word or a phrase) of the theory, and it is denoted by a special predicate symbol when the theory is formalized. If it is denoted by a phrase, the phrase is translated as a whole as if it were a single word. So, the paraphrasing is essenti ally something like a word-for-word translation. Going more into details, any formal theory has a limited number of its primitive notions. Any other notion in the formal theory is paraphrased in terms of these primitive notions. The paraphrasing is usually called the definition of the - 95- 10 K. ONO Vol. 3 notion in the theory. Anyway, any sentence in a formal theory can be expressed in principle in terms of its primitive notions (practicable or not). Accordingly, any sentence in a formal theory is completely interpreted in another theory, if every primitive notion of the former theory is paraphrased in terms of admitted notions in the latter theory. Even if this can not be called "word-for-word" in the strict sense, we can easily modulate it as if it is a word-for-word translation. It is enough, 1) to introduce a word in the latter theory corresponding to each primitive notion in the former theory, and 2) instead of paraphrasing every primitive notion of the former theory in the latter, to translate the primitive notion by the corresponding notion just introduced and regard the paraphrasing of the primitive notion of the former in the latter as a paraphrasing of the word in terms of already admitted notions exclusively in the latter theory. If every sentence in a language can be translated literally into another language without any change of meaning and nuance, both languages could be called equivalent in the most strict sense of the word.