ROBERTO SCOLAThe mathematician and logician Newton da Costa shares with other researchers the same passion for what they do. He often becomes emotional when speaking of subjects that seem strange to those who are unfamiliar with his passion. Some geologists have a soft spot for stones that tell stories of other eras and entomologists have a great deal of affection for repugnant insects. Costa sees beauty in intricate calculations, unsolvable problems and theories that are so abstract that they are only understood by a small group of people.

Newton Carneiro Affonso da Costa (who was born in the city of Curitiba in the state of Paraná 78 years ago, and is married with two sons and a daughter, as well as having two granddaughters) has more reasons than the other researchers, perhaps, to get enthusiastic when talking about his own work. He is recognized both in Brazil as well as abroad – probably more so abroad – as the author of a original theory that he first came up with in 1958, but which began to be widely mentioned and applied from 1976 onwards, when it finally gained the name under which it became known: paraconsistent logic. This is a theory which makes it possible to work with contradictory situations and opinions. It is not by mere chance that Costa is referred to by his pupils and colleagues as the “contradiction thinker.”

Costa graduated with a degree in engineering from the Federal University of Paraná (UFPR) in 1952 and spent a year working in his father-in-law’s construction company. But he finally gave in to his calling and did a degree in mathematics, following this with teaching degree in the same area. He then became a full-time teacher and researcher at UFPR, earning less than half what he had been making at the construction company. While there, he studied for his doctorate and became a university professor. In the 1960’s, he migrated to the Mathematics and Statistics Institute at the University of São Paulo (IME/USP) and spent two years at the State University of Campinas (Unicamp). In both he held the position of head teacher.

He spent time at a number of institutions in Australia, France, the United States, Poland, Italy, Argentina, Mexico and Peru as a visiting professor or researcher. More than 200 of his works have been published, including articles, book chapters and complete books. Among other prizes, he has won the Moinho Santista Prize and the Jabuti Prize in Exact Sciences. In the second half of this month, the publisher Hucitec will relaunch three of his books that have been out of print for many years: “*Introdução aos fundamentos da matemática*” (Introduction to the foundations of mathematics), from 1961, “*Ensaio sobre os fundamentos da lógica*” (Essay on the foundations of logic), from 1979, and “*Lógica indutiva e probabilidade*” (Inductive Logic and Probability), from 1990.

Upon retiring from IME/USP, Newton da Costa became head professor at the School of Philosophy, Letters and Humanities at the University of São Paulo and began to study and teach the philosophy of science. Four years ago he decided to go and live close to his two sons in Florianópolis and to teach philosophy at the Federal University of Santa Catarina (UFSC). His passion for research and teaching remains intact. When I went to interview him at his apartment in downtown Florianopolis, he handed me an article on logic, which he had written especially for the magazine. Further down on this page, we have published the part relating to paraconsistent logic. The reader can read the full article on the magazine’s website: www.revistapesquisa2.fapesp.br.

Newton da Costa prefers to write by hand and admits that he is very reluctant to use a computer, which makes the title of one of his most recent papers “How to build a hypercomputer” seem particularly strange. This paper, which is yet to be published, is an investigation of the limits of computer science. The main passages from the interview can be found below.

**You graduated with a degree in engineering, had a career in mathematics and ended up in philosophy. How did this happen? **When I was about 15 years old, two things happened that had a major impact upon me. First, I read “

*Discourse on the Method*“, by Descartes, which became my Bible. Second, there was frequent contact with my uncle Milton Carneiro, a professor at the Federal University of Paraná. We talked a great deal about philosophy and science. He gave me two books that I never forgot: “

*O Sentido da Nova Lógica*” (The Meaning of New Logic), by W.O. Quine, 1944, which was published at that time in Brazil, and Logique, by L. Liard, a truly classic book on logic, although it contains a section about scientific methodology.

**So one might say that your greatest work, the one on paraconsistent logic, had its roots in that period? **I think it took a bit longer. The conversations with my uncle and reading Descartes obviously helped. My main problem has always been thinking systematically about what knowledge is. Especially what scientific knowledge is. To this day, I still think about this. Then I clearly realized that I’d have to study logic, mathematics and some science, such as physics. Shortly thereafter I started reading Bertrand Russell at my mother’s suggestion. Russell motivates anyone to study questions of this type. This was when I realized that I also needed to learn about the applications of mathematics, not just mathematics itself. Therefore, studying engineering would be interesting. But I especially needed to get a better grasp of mathematics. And so I studied mathematics. Then finally I realized that, at the end of the day, all of this relates to philosophy – which in fact is the thing I most enjoyed.

**More than mathematics? **Oh, much more. For me, mathematics and logic are just tools for understanding what scientific knowledge is. Which leads afterwards to what knowledge general is and whether there is metaphysical knowledge. That in turn led to the need to immerse myself in philosophy. I haven’t got to metaphysics yet because I still need to fully understand scientific knowledge.

**I’d like to talk about paraconsistent logic. How would you explain it to someone who doesn’t understand logic or mathematics? **In 1874, a Russian mathematician called Georg Cantor created set theory. In a short time people realized that all of standard mathematics could be constructed over set theory and it became for all intents and purposes the foundation of mathematics. However, one should note that the concept of sets is an extremely abstract thing that mustn’t be confused with the system of objects or totalities of daily life. But about 30 years later paradoxes began to appear in this theory, including Russell’s paradox, the Burali-Forti paradox and several others, which can’t be explained here because it would take too long. These questions became a philosophically incredible problem: how were paradoxes possible in traditional mathematics and logic, which up until then had been the most perfect example of knowledge? This was terrifying, completely strange and nobody could explain it, so it caused an uproar. This was regarded as the third major crisis in the history of mathematics. The first was caused by the Pythagoreans, when they discovered irrational numbers. The second was caused by differential and integral calculus, a field entirely devoid of logical foundation; but this too was overcome. And, finally, the third major crisis was the Cantorian one, when it transpired that set theory was inconsistent and contradictory, that it couldn’t stand. An attempt was then made to solve the issue by maintaining classical logic and imagining which modifications we could make in set theory in order to overcome the paradoxes. Classical logic is essentially the logic that began with Aristotle and was given its current formulation by Gottlob Brags and Russell around 1870 and 1914, respectively. The problem of contradiction is absolutely fundamental to classical logic, which simply does not admit it.

**So the idea was to correct set theory without destroying it or giving it up? **Exactly. In the middle of these studies and analyses something very interesting appeared. It became clear that there were other ways of overcoming these difficulties, which no equivalent to each other. In other words, there were several theories of possible sets based on classical logic. The basic aim when these subjects began to be studied was to maintain classical logic in the usual solutions as these paradoxes and to change the principles of naive set theory. Based on a sentence uttered by Cantor himself, “the essence of mathematics lies in its complete freedom”, I thought to myself, “Why not do the opposite?” I want to maintain as many as possible of the principles of set theory, but change the underlying classical logic.

**What does that mean? **It means that this logic has to support contradiction. In the case of classical logic, the basic reason why it doesn’t accept contradiction, from the technical point of view, is that the simplest contradiction in a theory destroys it, because everything becomes theory. It was necessary to change and I started building various different types of logic. I showed that there are infinite types of logic that satisfy those conditions and infinite corresponding set theories. I began to develop and apply the logic to other things. But, actually, the solution, the original kickoff, was a purely mathematical point concerning the foundations of Cantor’s set theory.

**Didn’t they accuse you of destroying classical logic? **Everybody said that, especially at the beginning, when I presented my theory here in Brazil. It’s one of the things that annoyed me the most.

**ROBERTO SCOLAWhy? **I’d be an idiot if I thought that classical logic is wrong. What I believe is that it has a multitude of applications, but that under certain circumstances it doesn’t apply. I’ll give you a single example: the general theory of relativity and quantum mechanics are two of the most amazing theories that have appeared in the history of mankind up to now – due to the applications, the precision of the measures, in short, to everything. What they explain is insane. For instance, quantum mechanics explain lasers, masers, chemical structure… However, if you look at them very closely, these two theories are logically incompatible. There is only one way to join the two and physicists do this frequently, although they don’t know how this is done, from the logical point of view.

**What you mean is that they combine the two theories naturally in order to solve problems that arise, without knowing that they are using a different type of logic? **Exactly. This logic is paraconsistent logic. I’m working on this at present, explaining that the logic of physics has to be paraconsistent logic. It is locally classical, but globally paraconsistent. Physics, as it stands at present, works with a combination of incompatible theories and is only possible because of the existence of paraconsistent logic. For instance, plasma theory contains a lot of applications and involves three other theories: classical mechanics, electromagnetism and quantum theory. Two by two, they are contradictory. However, they are used. All the research that I’m doing at the present time resorts to quantum field theory, quantum mechanics, relativity and other theories to systematize science. This is one of the tasks that the philosophers of science face: to systematize various sciences and compare them. There is no solution to this problem if we don’t do this using a type of logic other than the traditional one. Not in today’s world.

**And what about the applications of paraconsistent logic? **For about 30 years I developed paraconsistent logic from a purely abstract point of view. I was only interested in the mathematical beauty that it implies. You can imagine my surprise when I began to get news from abroad, mainly from the United States, about applications in economics, in computer science, in robotics, and in specialist systems… In Brazil, Jair Abe’s group, at Paulista University (Unip), has achieved some very interesting results in the field of artificial intelligence. Recently, a Japanese friend, Kazumi Nakamatsu, stayed with me and showed me the applications of a certain type of paraconsistent logic for controlling the movement of trains, in Japan.

**You can’t get more practical than that. **It’s already well known that you can also use paraconsistent logic for air traffic control. When you have a lot of aircraft that cannot land, due to bad weather for example, the flight controller receives and sends information. The information is never exact because you don’t know at precisely what height the aircraft is. The height is always off by a little. Therefore, it must be correctly interpreted by the air traffic controller’s computer in order to avoid accidents. Paraconsistent logic is one of the ways considered in order to solve the problem.

**So is paraconsistent logic, therefore, a theory that accepts and accommodates contradictory situations? **Contradictory situations and opinions. At present there are hundreds of people all over the world who devote themselves to paraconsistent logic. Some of them are fundamentalists. They think that paraconsistent logic is the only true logic and that classical logic is nothing more than a load of nonsense. One of my best friends, who was a professor at the National Australian University and who visited Brazil on a number of occasions, was Richard Routley, who when he used to meet me every morning in Canberra and even when he was in São Paulo, would greet me by saying “Classical logic is finished.” I always used to say no, that both of them have their uses. Classical logic is the mother of paraconsistent logic.

**Could paraconsistent logic also be used in other fields, such as in psychoanalysis? **According to several psychoanalysts, particularly Lacanians, it has many applications in this area. There are already a great many papers on the subject in psychoanalysis.

**It seems that the repercussions of paraconsistent logic haven’t yet cooled down even after all these years. **This is something that I still find hard to believe. I thought about this when I was very young, in 1949, 1950, I began to work on my first projects in 1958, but my work only started being published in 1963 in France. Then, around the mid-1970’s, I wrote a letter to a great friend, the philosopher of science Francisco Miró Quesada, former Minister of Education in Peru. I said to him, “I need a name for my logic.” Quesada was one of the first ones to advocate the theory in the outside world, when he was an ambassador. He came up with “paraconsistent”, “ultraconsistent” and “metaconsistent.” I chose paraconsistent. After I began to write using this name, less than a year went past before everybody in the field of logic began to speak of paraconsistent logic. Everywhere from France to the former Soviet Union, and from the United States to Japan, articles began to appear which mentioned paraconsistent logic in one way or another. This is one of those things that is unlikely to ever happen again. Quesada began to pull my leg saying, “Actually Newton, I’m the one who created paraconsistent logic, because a thing only exists after it’s given a name. In the Bible it says “In the beginning was the Word…”

**What exactly was it that attracted you to the word paraconsistent? **The word “Para” means “alongside”. I never wanted to destroy classical logic. It is “alongside” it, it “complements it.” Likewise, the general theory of relativity didn’t destroy Newtonian mechanics. Neither did quantum mechanics put an end to Newtonian mechanics. And they don’t exist without Newtonian mechanics.

**What was the name of this logic before it was christened by Quesada? **Theory of inconsistent formal systems. Much too long.

**Do all the applications of your theory mean that you have made some money from it? **I have traveled a great deal, all over the world, and I have never had to spend a cent. But as for earning any real money from it, no. There are no patents for theories. But when I went to the former Soviet Union, for example, I was given an automobile with a driver, and an interpreter, and there were people available to help me with everything.

**At the age of 78, it seems that you’re still energetically keeping up with all your research activities. **Doing what I do is so much fun for me that I might even pay to keep on doing it. When I get to the point where I can’t study what I like and give my classes, then I’d rather just die. In fact, the story goes that for Einstein it seemed that the difference between being alive and being dead was that while he was alive he was certain he could carry on working in the field of physics. He was not sure that he would be able to carry on doing so after he was dead.

**Why did you leave UFPR? **I never wanted to leave Paraná. My whole family is from there and everything was fine for me at UFPR. However, I wanted to set up a group to study logic and the fundamentals of science. Little by little, I came to the conclusion that there was no way in the 1950’s and 1960’s that I could achieve this there, no matter how hard I worked.

**Why? **I think that apart from USP, there was no other university in Brazil that provided the right conditions to produce work of an international level in the field of logic and mathematics. Inviting professors from abroad, spending periods overseas, sending young people to study in other countries. I was given a chair at UFPR, but, although they went out of their way to accommodate me, I felt that I was going through the motions, without actually getting anywhere.

**You went to USP, but first you spent some time at Unicamp, didn’t you? **For a short while, yes. I have a very strong relationship with Unicamp. When I was a professor at IME (the Military Institute of Engineering) I was allowed to spend two years working full time at USP and part time at Unicamp, provided that there was a valid reason for this. I spent time at both of them and, surprisingly, managed to put together a much larger research group at Unicamp. Later on I donated my library and files to Unicamp’s Center of Logic, Epistemology and History of Science.

**Are you one of those scientists who believe that mathematics and physics are harder to understand than the other sciences? **I don’t know if they’re harder. I do know that for some work in these two areas you must have a great sense of abstraction, especially in mathematical physics and theoretical physics. It must be said that there is a sense of beauty in those theories. Edgar Allan Poe used to say that beauty is that which resists familiarity. The more we go back to it, the more we are drawn to return. And whenever we go back we notice new things. Bach’s music is eternal because you can hear it millions of times without getting tired of it. We will always find something new in it. If we hear an ordinary song it doesn’t spark any new ideas, all you need to do is to listen to it three or four times and there is nothing else left. But Bach, Beethoven and Brahms – You never tire of them. If you read a trivial article on mathematics you will soon lose interest. Now, in the case of a good article, we may go back maybe dozens or even hundreds of times. There’s always something else, another idea, or another aspect that we didn’t notice before. I always tell my students that it’s precisely for this reason that mathematics has a supreme beauty. This is even true in the case of works such as those of Isaac Newton, which nobody uses any more to study mechanics or astronomy, because its principles are so well known and in some instances even outdated. But if we go back there and go into the details of the work then we’ll see that there is no end. It is like a symphony in the style of Bach. And guess what? It doesn’t matter what the size of the work is. The doctoral thesis of American mathematician John Nash, who won the Nobel Prize in Economic Sciences, was only five pages long. It’s brilliant. I used to carry copies of it with me in my briefcase to hand out to students and show that size means nothing. If Nash had written this thesis at USP, it would not have been approved because nowadays it seems that they demand something that is at least a hundred pages long.

**How do you feel about the low level of teaching and learning mathematics in Brazil? **It is barbaric. I had some experience of high school tuition in the United States, at the public school at Berkeley. There they have what they refer to as honors courses. Students who want to do technical courses, such as automobile mechanics, have a minimum of English and history classes, etc. Then, if they want, they can make up their credits with the other courses. But only those who want go to university do the honors courses, which consist of small groups of 10 to 12 students, with full-time teachers. The teaching involves differential calculus, integral calculus, computer science, analytical geometry… People join the course of their own free will and undertake not to get low scores. If they don’t manage to keep up, they leave. After they finish the course, all they need to enter university are two letters of recommendation from the teachers. If the student performs well on these courses, he or she is guaranteed a place at university. On a number of occasions I have suggested that we should do something similar here, but I always get told that it is not democratic, that it is elitist…

**Are you against this sort of social expectation in Brazil that everyone should go to university, even those who lack the slightest desire to do so or are not at all qualified? **You can’t put everyone on the same level. There’s no way. The honors courses and the other courses available are a way of including all the interested parties. Those who want to, take them. I also saw, in Berkeley, a great course on automobile mechanics. The students got an automobile and took it apart, screw by screw, and then put the whole thing back together again, without leaving a single piece over. The student leaves with an understanding of cars, becomes an excellent mechanic and may well be just as happy in his work as somebody who spends his entire life studying something very theoretical and abstract. There was a plumber on the campus at the University of California, when I worked there, who was so competent and efficient that he earned more than one of my most brilliant friends, the Polish professor Alfred Tarski, a great logician and the person with the highest salary in the department.

**I would like you to talk about the philosophy of science. What is the quasi-true or partially true concept? **I don’t think that science nowadays is something that tries to portray what is real. When a proposition wants to reflect reality as it really is, we call this the correspondence theory of truth. What I mean is that the thought corresponds to the truth. I don’t think that science is like this, it only partially reflects reality. It is a quasi-truth. Why do quantum mechanics work? Because they say that, under certain circumstances, if I tighten a screw, I get the right result. The great propositions, the great theories, everything takes place in the Universe as if it were true. I formalized this notion of truth – it’s a generalization of the classical notion of truth. It’s a generalization of Tarski’s classical definition of truth. This logician produced a remarkable definition in order to be able to deal with the notion of truth in mathematics, which is where it works. In the case of physics, you need something more elastic. For this task I put forward the concept of quasi-truth or partial truth. But I think that strictly speaking, my conception of truth, which is a mathematical one, more or less reflects the ideas of Charles Sanders Peirce [1839-1914], one of the greatest philosophers of all times. And I think that all the great theories, such as quantum field theory, quantum mechanics, Newton’s classical mechanics, are quasi-true, for instance. It is common to say that relativity has supplanted Newtonian mechanics. That is false. An airplane or a bridge, for instance, is calculated in accordance with Newtonian mechanics. And quantum mechanics and relativity need Newtonian mechanics. Otherwise they don’t work. How can something that is false be used in science? Exactly because, although it is false, it is almost true between certain limits.

**Because it works for certain things under certain circumstances. **Exactly, under certain circumstances everything occurs as if it were true. It is the “as if.”

**And this is expressed mathematically. **Mathematically. I systematized the theory of current science as being almost true. All the great theories of physics are not literally true; they’re almost true. If we compare the theory of relativity exactly with reality, we get divergences. And, even if it did exactly reflect reality, how would we know that it reflected it? Strictly speaking, there is no way to compare theory with reality.

**When did you come up with this theory? **In the 1980’s, quite a long time ago. And, note the following, for the same almost true theory there are infinite other almost true theories, and I can prove this. And those infinite almost true theories are incompatible with each other. Therefore, the logic of the quasi truth is paraconsistent logic.?

**To conclude, what is scientific knowledge? **It’s my opinion that scientific knowledge is an almost true and justified belief. This is my version of the classical conception of knowledge that dates back to Plato’s time. In this, strictly speaking, knowledge should be true; what I did was to replace true by almost true.

**Regarding paraconsistent logic**** **Newton da Costa

Classical logic, as well as various other types of logic, is not suitable for the manipulation of systems of assumptions or of theories that contain contradictions (in which a proposition and its negation are both theorems of the theory or consequences of the systems of assumptions). However, in the sciences, they represent contradictions that are difficult or impossible to eliminate (which occurs, for instance, in physics, where the theory of general relativity and quantum mechanics are logically incompatible, and in the law, where judicial codices always exhibit inconsistencies, etc.). Therefore, it became imperative to create types of logic that could stand contradictions: that is the essence of paraconsistency. In general, paraconsistent logic does not imply that classical logic is wrong, but generalizes it. Paraconsistent logic includes fuzzy logic and has found a very wide range of applications, both theoretical as well as practical. In particular, it inspired a new philosophy of science and extended the field of reason.