The following interview between Federica LaNave and Uri Margolin was conducted on July 20, 2007. The interview as it appears here was transcribed and edited only slightly for clarity.
Margolin: Let me start with a classical question everybody asks: why is the history of science very active in the last thirty years or so while the history of mathematics seems to be dormant or marginal?
LaNave: To begin with, there is a social factor: a student goes to a department of history of science and very rarely, if ever, finds a professor that teaches history of mathematics. What he will normally find are historians of medicine, biology and genetics, and sometimes, but less often, physics and chemistry. Should a student decide to focus on the history of mathematics he/she would be strongly discouraged: there are no career opportunities, and departments do not want to create future unemployed scholars. But why are there no career opportunities for historians of mathematics? One reason is that the vast majority of people, I would say, are not comfortable with mathematics. In elementary school they are taught that mathematics is all about memorizing and calculating fast, so when they are introduced to more complex concepts they find themselves without the mental attitude necessary to understand the subject. The result is that people are afraid of mathematics.
Margolin: May I interrupt? People are much interested in the history of physics, and you cannot do history of physics without good knowledge of mathematics.
LaNave: And that is the other point I wanted to make: many mathematicians focus strongly on mathematics as a discipline outside history. Sometimes during their career there will be some interest in the history of mathematics, but they will focus on a very technical approach, an internal history of mathematics, which will not be accepted in the historians’ environment. So, history of mathematics is mostly left to mathematicians that do this internal history. But if you are a historian of science with a background in mathematics what you want to do is to work on both the mathematics and the social/historical environment around it. You would like to study the historical context, and also speak about mathematics. But if you do the mathematical side you have a public of historians of science who will not listen (and professional journals will not publish your papers with too many formulas in them). Conversely, if you put in the context without the mathematics, then you have a public of mathematicians that will not listen. Physicists are more likely to be willing to listen to their science put into a different context.
Margolin: Well, I think there may be another explanation. (To be honest, I’m just guessing because I’m not in this field). Amongst physicists there is a broad agreement that physical theory is a human construct and its specific nature is very much context- dependent, whereas many mathematicians are ultimately Platonists who believe that what they do isn’t constructing something which is good for a certain period of time but discovering eternal truths. Am I right?
Margolin: And if it’s a question of discovering an eternal truth, then who cares when, who and what, the main thing is that it has been discovered. So in mathematics the status of your claims is not that of a hypothesis and a construct but of a discovery and a proof. And therefore the history and the context of reaching them are less important. Would you think this is an explanation?
LaNave: Yes. Assuming we have a straightforward definition of what is science. Is mathematics a science? People aren’t comfortable when you refer to mathematics as a science because the word “science” carries this idea of a form of knowledge that applies, refers, is related to nature and so mathematics would not look like a science. At least following a Platonic definition of mathematics.
Margolin: Probably because mathematics has no experimental basis.
LaNave: I do not think so. I think that mathematics is experimental. People judge mathematics from the way in which results are presented in its system of knowledge: there is the theorem and the proof and all appears to work purely in virtue of deduction. However, this is only the final narration, the piece as it is presented to the world. Mathematicians do not work that way but follow a path in which deduction, induction, intuition and many other factors interplay with each other just as in any other science. Above all, once we stop thinking of the objects of mathematics as belonging to some unidentified spiritual world then mathematics is not so different from physics. Unfortunately, given the official fable told about what mathematics is, it’s probably closer in people’s minds to theology than to science.
Margolin: Fair enough. So let us go now to the other major question arising from your paper. There are in fact two issues here. One is beliefs, the change of beliefs and the causes for changes of beliefs. The other is the role of belief in theory formation, especially when theory formation is concerned with innovation, with production of new knowledge and the transition from problem to solution. Your particular case is that of Bombelli and thinking about imaginary numbers, but the general problem is the dynamics of belief and the role of belief in theory formation.
LaNave: Yes. That’s well put.
Margolin: In Greek Philosophy truth is defined as correct belief with reasons or grounds. This suggests a sort of belief-knowledge dichotomy and a continuum from one to the other. So we can take the totality of the propositions that are held or asserted by a certain person at a certain time and divide it into two subsets. One would be the set of knowledge claims: those propositions for which the person thinks there exists sufficient evidence or support. In the case of mathematics it would be a proof. And the other subset is the subset of belief claims, namely, those propositions for which there does not yet exist sufficient evidence or support or, in the case of mathematics, a proof. And the sum total of these two will represent the epistemic profile of this individual.
LaNave: Let’s go back to beliefs and their role in the general way of knowledge. The paper on Bombelli and the imaginary numbers is part of a more general study in which I consider different mathematicians in different historical periods and their process of belief formation prior to proof. In dealing with this, I was putting together a historical question and a philosophical question. The philosophical issue is how do beliefs work in the process of knowing, especially in mathematics. I wanted to analyze this philosophical question in history and focus on different cases and periods, on mathematicians from different cultural contexts and with different ideas. What I wanted to know was what constituted the knowledge they had and with what methods they justified it. The result is a process of mapping the beliefs of the mathematician at work. There are various kinds of beliefs involved, and one can really see how much this building up of beliefs is going to create a group of pieces of evidence sustaining and maintaining the person’s conviction, creating what a person perceives as a justification of his or her claims to knowledge. What I could see was that belief had a particular function in the process prior to proof. In philosophy, when we speak of knowledge or of belief we really can have many different positions and definitions, and this process of interconnected beliefs constituting pieces of evidence for knowledge claims doesn’t necessarily match the classical concept of justified true belief. However, it does constitute a justificatory structure for knowledge claims. After all, Bombelli does not find a visual representation for these numbers, yet his belief in the practical evidence for their true nature seems to be a justification for his acceptance of them. This is part of the scientific process of knowing, although we may want to be careful in calling it knowledge. Furthermore, this justificatory structure, this belief, is working as a kind of engine creating new research. Thus, there is a precise function that belief seems to have in the process of knowledge.
Margolin: So that belief is an engine for knowledge production.
LaNave: Yes exactly. This system made out of beliefs justifying knowledge claims is a set in relation with the set of accepted knowledge. But the borders of these sets are quite blurred. It is very hard to say what will be in one and what will be in the other.
Margolin: So, for some claims it will be easy to place them in the knowledge subset because a full proof for them exists.
LaNave: Yes. If there is a proof satisfying the accepted standards of proof in a particular period (these standards have not been always the same).
Margolin: While other ones are definitely just beliefs or hunches or intuitions
Margolin: And some of them just straddle a kind of a fuzzy line between the one and the other.
LaNave: Yes, and these are the beliefs having this particular function of knowledge production.
Margolin: And I guess that if in the course of an intellectual activity you find a proof for a given claim, then it is moved from the belief subset to the knowledge subset.
LaNave: Yes, exactly.
Margolin: As for a hypothesis in the natural sciences, once you’ve had enough experimental support or evidence for it, then it’s no longer a mere hypothesis, but rather a well supported theory.
LaNave: Yes. And, as for mathematics, that is obviously in relation with the particular historical period in which a mathematician lives. What counts as a proof may be different in different periods. The system constituted by what is deemed enough for considering something justified is what puts claims into the knowledge set. But the situation in which Bombelli finds himself is not merely intuitional. The belief in question is different from a mere intuition because there are supporting pieces of evidence for it. Bombelli says these are numbers, and although he does not have a geometrical representation for them, he has a proof of their existence. Thus, there is something increasing the confidence in the justification of the belief in question. This kind of belief will be, as we said earlier, in an intersection set between a set of claims constituting knowledge and a set of mere beliefs. As I said, this belief is in this border situation because Bombelli has a proof of the existence of these numbers and, although he does not have a geometric construction for them, he is capable of visualizing an application for them in the trisection of the angle. The fact that Bombelli can think of an application for these objects (whose existence he has proved) makes him feel more confident in the nature of this objects, in their being real objects and not fictitious entities. He does not know where exactly these objects are and does not have a spatial way to imagine them, yet he has a theoretical way of imagining them. This theoretical glimpse Bombelli has of their visualization constitutes a hook for his belief that these numbers have a representation somewhere, a position in some kind of space, although he doesn’t know which one. So it’s something a little bit more than simply believing and less than having an exact formalized justification for a belief.
Margolin: So it’s a continuum, not a binary.
LaNave: In a way yes, although it can be going up and down…Are you referring to a continuum in the mind of a person? That is, in this case, in Bombelli’s mind?
Margolin: No. I mean that in general propositions can be ranged on a belief-knowledge continuum. But it seems that as you said the movement is not always one way. Sometimes you’re thinking of moving from belief to knowledge and then something doesn’t work and the claim goes back to being just a belief.
LaNave: Yes, that may happen.
Margolin: But if you think it’s knowledge and then you’re proven to be wrong, then the claim is not just downgraded to a belief ; it’s eliminated altogether.
LaNave: The strength of a belief changes and the more things you find that can give strength to this belief, the stronger this belief becomes. Sometimes a belief that seems to be quite well supported can indeed find its strength weakened if related claims lose their supporting strength.
Margolin: So do you think there are degrees of support?
LaNave: I think one can map the degree of this support of belief (and in a relatively precise quantitative way). Think of Bombelli: first of all he has this open status of belief because he wants all the equations of the third degree to be solvable. This puts him in a situation in which he is willing to change his belief (the belief that imaginary numbers are not real numbers) because there is some strong stimulus behind this idea that it is already opening the belief status to change. And then he finds in Barbaro’s “Commentaries” on Vitruvius the construction for the duplication of the cube attributed to Plato. This is the geometric construction that he will use in the proof. He sees some similarities between this construction and the irreducible case of the cubic equations. This is a visual way to build a bridge between the two. This strategy, this building of a network of interconnected justified claims, makes the belief stronger and stronger. The more similarities and supporting situations and propositions a mathematician finds for his/her belief, the stronger the belief gets. This is why I think we can speak about mapping, a map in which the degree of confidence in a belief increases proportionately to the increase of supporting propositions (that is to say pieces of evidence) related to the belief in question.
Margolin: But, once again, as you have said before, it may also go down, as when you discover counter evidence or if you hit a boundary of some kind.
Margolin: And then you go oops, I believed this and it was getting stronger and I thought this is almost knowledge but no, it isn’t.
LaNave: Yes, but the situation is a little more complex. Bombelli’s belief fluctuates in its strength. Yet, once the confidence mechanism was triggered, Bombelli looked for more and more supporting pieces of evidence (despite the doubts). Bombelli did not give up for twenty two years: he thinks he is right, despite everybody else disagreeing with him.
Margolin: But at what point does belief turn into dogma?
LaNave: I would think it would be unreasonable for a mathematician/scientist to give up his/her beliefs at the first non supporting evidence. However, I agree with you, the dividing line between a strong belief and a dogma is a dangerous boundary. Yet, if the belief is not stronger than the experimental evidence, then the danger of turning into a dogma, although still present, is somehow controlled. Particularly in Bombelli’s case the danger of sticking irrationally to a belief was very low given that he had an amazingly practical mind.
Margolin: In general, I think belief turns into dogma when you have evidence that goes against it, and you don’t want to see this evidence or you’re trying to explain it away.
LaNave: That’s true, although, as I mentioned earlier, it seems to me quite rational for a scientist not to give up on a belief if there are only some pieces of evidence against it. In this particular case, Bombelli didn’t really have any strong counter evidence. What was against him was the accepted opinion about the nature of numbers coming from mathematicians before him and in his community. The other mathematicians would tell him: ‘Look, numbers need geometrical representations and you cannot represent roots of negative numbers. What are you going to do with this? They are not numbers.’ So, the only thing stopping him was the traditional concept of numbers. For Bombelli, it was more a question of putting this concept of number under discussion than not adding real counter-evidence. So, once he abandons this state of mind in which he ‘cannot look any farther’ (because it is not allowed) he is capable of opening new possibilities of research. Cardano too played with these numbers, but for him they were “sophistic” entities, nothing more than the result of a pleasant mathematical game. For him, the nature of these numbers (their being real or not) was solely related to the accepted concept of number as a geometrical/quantitative entity. So, the simple fact that he could not find a geometrical place for these numbers was enough to make them not real, but a bizarre mathematical joke. Cardano (and the contemporary community of mathematicians) had an assumption so strong that they were not going to see any other possibilities because this assumption was like a dogma. Bombelli was in a more relaxed mental state in relation to this dogma. He said “What if…?” Bombelli was an engineer after all and had a practical way of approaching mathematics, while Cardano had a rigid position regarding the relations between symbols and reality.
Margolin: I was thinking of a different typology of beliefs. Every scientist, every mathematician has a belief system. Some of these beliefs are more specific and some more universal. For example: some of them would concern the very nature of mathematical objects (Platonism, Intuitionism) while some others would be narrower, being about specific mathematical issues. So there are various kinds of beliefs depending on how fundamental and how general they are. One could also think of the degree to which beliefs are shared. Some beliefs are held by just one individual, some are held by part of the community and some others are held by most of the community. One can thus speak of degrees of sharing, asking how widespread certain beliefs are within a community.
LaNave: I think this is very important. Look for instance at the Riemann Hypothesis, which is something I am working on now and will be the next step in this study of the process of knowledge formation in mathematics, of which Bombelli’s case is part. Most mathematicians feel confident that the hypothesis is provable. The more they tackle it, the more they prove similar related propositions, propositions that make the Riemann Hypothesis more likely to be true, the more they believe it. There are some doubts about the Riemann Hypothesis, but the vast majority of the community believes in its provability. It is a matter of approximation: the more similar things are proved, the stronger the probability of this hypotheses, and the belief in its soundness gets stronger and stronger.
Margolin: Let me repeat something you have already mentioned: the strength of a belief can also be ranked on some kind of a scale, where the minimum would be an initial hunch or an intuition and then, as we have more and more claims to support it, the belief becomes stronger. Still another question would concern the originality of the belief. Is a belief held by a given individual an original one or is it taken from others? Because some of the beliefs we hold are simply opinions taken from others. Belief need not be original. It may be traditional and so on. We can also speak about the degree of novelty of an original belief held by a given individual. How radically does it deviate from the opinio communis, from the generally held system of beliefs about the subject?
LaNave: I think that it is truly rare that such a belief will be far away from the shared beliefs of the scientific community. A more likely situation is one in which some parts of the system of shared beliefs are abandoned by one or more mathematicians. Let us look at Bombelli’s case again. What he is saying is that these new roots are real numbers because, beside proving their existence, he can calculate with them (which, for him, gives them their status of being real). He is not making the revolutionary claim that numbers are abstract/formal entities. Bombelli does not detach himself completely from the contemporary concept of number as a geometrical/quantitative entity, and so for him these new roots are real because they are quantitatively valid entities. However, accepting these new roots as numbers despite the lack of a rigidly formal background, Bombelli starts breaking parts of the shared belief of the community of mathematicians about the nature of numbers. He is not claiming a conscious and elaborate break from the tradition (as it would be to claim that the defining nature of numbers is their being abstract symbols). He’s just saying that there is some kind of numbers with a strange nature. These numbers are, for him, not the same as regular numbers (or, at least, not yet). Nevertheless, he says, one can use them. It is possible to calculate with them. It’s a less radical change in belief than saying: “We made a mistake about the nature of numbers, their real nature is completely different.” It would be quite rare for someone to have such a complete awareness at a stage where things just start to change. Such a radical change in belief would be more likely after years have passed and many mathematicians have continued researching these new ideas. At that point one could find more of a common awareness about the change in the nature of the particular problem, the opening of possibilities and the consequent radical change in the beliefs shared by the community of mathematicians. But when a new question is first opened, it is all very primitive. Thus, I would say, the change in belief status is not so radical.
Margolin: One can also speak about degrees of “entrenchment” of beliefs: How reluctant you are to change or bend your belief when the new evidence goes against you.
LaNave: Inside communities of mathematicians there are strong forces that influence this capacity for abandoning beliefs. For instance, looking again at the Riemann Hypothesis, one can see that the community believes that the possibility of its provability is so strong that abandoning this belief will require a lot of work. The more you are part of a community that shares a belief, the harder it is to abandon that belief because the belief gets strength by being justified somehow by the standards of acceptance.
Margolin: You pointed out that belief has a positive role. It may act as a catalyst to the formulation of new theories, new knowledge and so on. But, even in the case of Einstein, a belief in excess is a sort of obstruction. Einstein said: “God does not play dice.” This prevented him from accepting quantum theory, even though there is a lot of evidence to support it. The belief in the determinism of the universe was so strongly entrenched in him that, in spite of the greatness and the radically innovative nature of his mind, at this point he simply refused to listen. For him, quantum mechanics may look right on the surface and function as a temporary solution to some problems, but it cannot be the final answer.
LaNave: That’s the situation of Cardano. He thinks there is no correspondence between these new roots and reality, while every single symbol must, for him, have correspondence to reality. This strong belief about the nature of numbers prevents him from accepting the validity of these new roots.
Margolin: And if there is no such correspondence, then it cannot be accepted. So, even though it works in practice and even though it is useful, I refuse to accept it as more than just a temporary trick.
LaNave: Despite the problematic nature of the excessive strength of some beliefs in the scientific community, I think this is unavoidable. It is a necessary part of the way in which scientists, in general, think. It would be indeed hard to work without strongly fixed assumptions. This is the “dark side” of the nature of belief. It has a positive role but it also has a very negative one.
Margolin: It may also influence what route you will take and what routes you refuse to take. ‘This route I shall not take because it goes against my belief that the universe is deterministic’. There is a further issue here: Scientists and mathematicians have certain strategies, certain methods or procedures they employ in order to tackle a problem. And these would be kind of methodological “do” and “don’t do” which have to do with how to proceed. How do these methodological norms relate to the knowledge-belief continuum?
LaNave: This is a very complex situation- at least as I see it. We have a community of mathematicians. We have standards of knowledge that determine what kinds of questions are asked and what kinds of results can be accepted. These results are going to shape what doing science or mathematics is all about. When some parts of accepted beliefs start to be abandoned (like Bombelli accepting these new numbers) some of this structure of questions that can be asked and of answers breaks down, and then new research starts which will bring to the formulation of new kinds of questions. This is the situation opened by Bombelli: if mathematicians after Bombelli want to go on accepting these new numbers, they have to come up with more formalized results. In so doing they will have to change the rules of the game and to come up with things that will not look quite correct according to the previously accepted standard of knowledge. Obviously this can be very productive in the long term. It is important that the scientists’ shared system of knowledge gets broken by beliefs that initially look not justified (or, at most, only partially justified), like Bombelli’s.
Margolin: You have this picture in your paper of cuts in a piece of wood. This was one kind of procedure or method or strategy for proving claims that had been employed for a long time. At this point, Bombelli’s beliefs apparently make this particular method not very practical.
LaNave: The first instrument he used (the one with cubes) did not work for his proof. Thus, he goes to the one using two moving L-squares and that one works.
Margolin: I guess there is also some kind of interplay between one’s beliefs and the methods one employs.
LaNave: You either have to come up with new methods or use old methods in a different way, as he does. This will be the way out in a situation in which the methods are not helping you. He was there trying again and again with the cubes and he could not prove the existence of these numbers with this instrument and that is when he changes to the instrument using L-squares.
Margolin: So in this particular case the belief was stronger than the methodological norm. If the old method doesn’t work, I will stick to my belief and I’ll try to find a new way of doing things.
LaNave: Well, stronger, but not so strong that he was ready to leave behind all the accepted standards of which the instrument using the two moving L-squares is part. After all, in the first version of his work, the one in the manuscript of “L’Algebra,” Bombelli agrees with Cardano: these numbers appear “sophistic” to him as well. However, he also did open a possibility for them by saying that should they be actual numbers, then it will be possible to apply them to the trisection of the angle. Bombelli’s imagination was going around quite wildly. He had already the capacity of thinking: if they are numbers, then such and such. This is what Cardano did not have. This really creates an open status of belief. And then, in the course of more than two decades between the manuscript and its publication, he reads Barbaro’s work, and the more he thinks about the method/instrument using L-squares, the more he sees a connection between the duplication of the cube and the solution of the irreducible case. When he borrows the method attributed to Plato for his proof and builds up an instrument, it turns out to work. It is really nice to look at this instrument: you can see the proof much more clearly when you can move the actual L-squares in relation to each other.
Margolin: I believe I have by now exhausted the questions and observations I had in mind. Thank you very much.
LaNave: Thank you.