Baking a mathematical pudding: the role of proof & experimentation?

0:00 But perhaps not 100% what I had originally planned. I've also had to cut down the talk from an hour and 40 minutes, so you will see how I calculate the length of the round. As a mathematics educator, my story goes back to the 1970s when I read Richard Skemp's article about instrumental and relational understanding, which I found very... It was very insightful at that time. One aspect that was missing from the Richards-Kentz model, however, distinguishing between relational and instrumental understanding, was distinguishing between what I call functional understanding, understanding the function or the purpose of a particular procedure or process in mathematics. And I have, over time, done analysis, some naïve, some more in-depth, about different processes in mathematics. I've done an analysis of the role of axiomatization, which is available on my website. I've done an analysis of the role of classifications in mathematics and the role it can play in mathematics education. And of course, I've done an analysis of the role of functions proof, and I have several articles on my website on that, and that article of mine has been translated into several different languages, Dutch... Hebrew, Spanish, and Japanese. Yeah, there are a few languages, of course, that have to be translated. So, you're welcome to... Anyway, now, what am I going to talk about today? I'm going to talk about the role of experimentation, which is, as some people have previously have said, it might also work in progress, but To some extent I have published on this already and it's part of all the work and obviously I'm continuing with my analysis of axiomatization classification proof at the same time. I'm going to give some historical examples and some personal examples. And with experimentation or quasi-empirical methods, I'm meaning they're all non-deductive methods, which includes whatever we can bunch up under intuition, inductive or analogical reasoning, and specifically...

2:30 When it comes to evaluating mathematical conjectures numerically, visually, by whatever means, or making conjectures and generalizations and so on on the basis of intuition, etc. A very vague definition, I'm sure there was a bit of a cringe there, because vagueness like that sometimes deals better with my manners. What are some of the purposes or the roles that experimentation plays in mathematics? Certainly, conjecturing I think is an important one and several speakers before me have alluded to the fact that experimenting, playing around with an idea helps one to do that and connected with the previous talk where mention was made of dynamic geometry, I'm just going to very quickly, well I might as well do it within this screen. One can, for example, in dynamic geometry, draw a triangle, and unlike paper and pencil, this configuration can now be dragged. And now you can do constructions on this dynamic triangle, which will then remain true as you drag, and you can then examine which properties remain invariant. And in this way, one can make conjectures about underlying geometric properties. For example, if we just construct the midpoints there, and we construct the perpendicular bisector there, one here, then one may ask, well, what will happen if I construct the third perpendicular bisector here? Thank you. Much to students' surprise if they have prior experience of three lines not meeting in a point, otherwise they won't be surprised. So what? However, if you have designer experiences where they know that it is highly unusual that three lines meet in a point, they are usually quite surprised by this and can elicit a curiosity amongst them which I would say can be used to introduce them to proof.

5:00 As a means of explanation. I just thought I'd show you this very quickly because as I said there are a number of people here that I've never seen an academic geometry. Obviously using a kind of tool like this enables one to investigate things in a different way from paper and pencil much easier. One can ask questions, when is this pointer from currency outside and which conditions is it on the side of the triangle and when is it inside. Simple things like that can be explored in a different way and I think in many ways math educators Including myself, I don't think we've really come to grips with thinking in a different way about how to deal with Dianne and John, just by the way that we've done. Of course, we can also verify statements by means of experimentation. In the example that I showed, by dragging the triangle around, one can immediately generate many examples, and because it is a kind of continuous transformation, it is quite convincing that the conjecture or the invariant property that you are observing is always true. And I think this is a really important aspect that I focus on in the role of proof, where I've argued in my papers that in many ways the role of proof within a dynamic geometry context has changed from one of exclusively verification to perhaps one where we should focus more on explanatory function of proof, trying to understand why it is true, because we can verify things. To allow us to agree within the Miami Dome deadline. Obviously, using experimentation is useful to disproving false statements. I will give examples of that. It's also useful for heuristic reputation, which doesn't refute a statement but helps one to reformulate it and refine it. This is Lakatos' idea of local counter-examples. I believe that using experimentation, playing around with ideas, generating examples,

7:30 The first example that I want to clarify a little bit is the function of conjecture and refer to historical examples that are probably known to everyone here about the distribution of primes. If we add up to 10, we have 4 primes, and if we calculate the ratio there, we get 2.5, and then similarly if we continue the process up to 100 counter numbers, primes calculate the ratio between n and pi over n, the first thing we notice obviously is that as n increases, this increases, but how does it increase of course is mathematical. The first thing one would notice is that the difference between these ratios is approximately 2.3 And that should ring a bell for most people here, that would probably ring a bell that would remind you of the Lindman's 10, which then leads one to conjecture that pi of n is approximately equal to n over the Lindman's n. And in fact this is called the prime number theorem that most people here know, or probably have heard of. Now if we plug in numbers here we can see, if we take up to a thousand, plug in a thousand there we get 144, now the actual value is 168. And I've got it, it's active here, I can maybe try a few numbers, maybe let's just try a hundred. We can get a feel for the fact that this seems reasonable.

10:00 When we go to higher values, although I might, I think when I go to 100,000, I'll actually run out of the ability of sketch paper, because it's only worth five or six hundred digits. Now, this result was discovered by Carl Gauss when he was age 15, and it appears that he actually did these calculations by hand. And it was on the basis of an empirical investigation that he made this conjecture. And he didn't prove it. It was only proved in 1896 by Hadamard and de la Vallée-Foussin Maybe somebody would like to make a conjecture here at the beginning. There are a few examples, I've only shown three examples, and maybe three examples is too little to notice, that in the examples that I've shown, every time pi of n has given a value smaller than the actual number of primes. The prime number formula always underestimated the actual number of primes. However, that has been disproved in the last 20 years or so, but that was a conjecture that was believed to be true. Let's look at Leon Oiler, who wrote extensively long before Lakatos about quasi-chemical methods, if you like. He wrote, as we must refer to the numbers to the peer intellect to learn, we can hardly understand how observations and quasi-experiments can be of use in investigating the nature of numbers. Yet in fact, the properties of the numbers known today have been mostly discovered by observation, but discovered long before the truth has been verified by rigid illustrations. I think since the time of Euler, we must realise that computers have become very important tools for making an objection. Not only do we have things like dynamic geometry, we have Mathematica, Label, all sorts of other computer software that can make objectors, in some cases even in reduced groups.

12:30 I know a Chinese program that is apparently so powerful I've not worked with it or seen it in action, but I've seen an article on it, that is powerful enough to solve some of the hardest... International Math Olympiad problems that you can find. I don't know how many people engage in IMO problems, but those problems are usually quite hard. So there are even these computer softwares around that can produce clues, and in some cases they differ substantially from computer clues. The style of exploration is entirely modern. It's a kind of experimental mathematics in which the digital computer plays the role of the gelled ship, the astronomer's telescope, the physicist's accelerator. Just as ships, telescopes, and accelerators must be ever larger, more powerful, and more expensive in order to probe even more hidden regions of nature, so will indeed computers of even greater size, speed, and accuracy in order to explore the remoter regions of mathematical space. I think in particular in chaos theory, fractals, there are many examples of people using computers to generate results that become conjectured long before they have rigorous proofs in place. The point that I want to make at this stage, and I don't have time, is that I take it axiomatically, and I don't think this is anything we can improve, but this nebulous idea of intuition. Sure, there are some people that are born with much more natural intuition, mathematical intuition, but I would argue from an educational perspective, intuition is something that is developed over time through immersing in the micro-world, as the previous speaker was talking about, and it's acquired through experience, the memory force that you talk about. It is not something that you're either born with or you're not. And I think experimentation is something that can play a big role in developing intuition. Verification and conviction. I believe we can get a lot of Apariwuri conviction, maybe not 100%, but certainly very close to 100% from experimental evidence.

15:00 Leonid Euler writes, follows about one of his investigations, suffice it to undertake this multiplication. And to continue it as far as it is deemed proper to become convinced of the truth of the series. Yet I have no other evidence for this, except a long induction which I have carried out so far that I cannot in any way doubt the law governing the formation of these terms and their exponents. I have long searched in vain for a rigorous demonstration of the equation between the series and above even the product. And I have proposed the same product to some of my friends whose abilities in these matters I am familiar with. But all have agreed with me on the truth of this transformation of the product in this series without being able to utter any tune of the illustration and someone no less than Whedon say admitting something like this. Paul Amamis in an interview in 1984 Also said something similar, the mathematician at work arranges and rearranges his or her ideas and he or she becomes convinced of their proof long before he or she can write down the logical proof. The conviction is not likely to come early, it usually comes after many attempts, many failures, many discouragements, many false signs. Experimental work is needed, thought experiments. I think all of these convinces one that the result is true, and that creates a need for proof, but maybe not proof as verification, as I've said earlier, proof maybe in another context, either systematizing or maybe trying to understand why it is true and other reasons. Experimental testing, I would argue, also often consists in continued efforts to prove a result. To give you a very simple example from high school geometry, perhaps knowing the properties of an isosceles trapezoid, and this has happened to me in situations where I have given this to both high school students and university students, asking them to come up with a definition for an isosceles trapezoid, some of them may come up with a definition saying that if one with at least one pair of opposite sides parallel,

17:30 As well as equal diagonals will be an associative trapezoid. And maybe if you ask them, well, can you show that this definition suffices by deducing the other properties, they may get stuck. And at this point it may be valuable to investigate using something like Sketchpad or with paper and pencil by drawing. And this is where dynamic geometry is useful and one can construct. A figure where you have two lines parallel, AC to BC, and you can construct diagonals which will remain equal no matter how I transform this figure. And one can just look at it dynamically without... There are a number of ways in which you can do this, and I'm not doing sophisticated measurements, but just looking at the opposite sides, where AB and CD are equal, and they remain equal, I think something like this is fairly convincing that it must be true, and maybe, let's go back, I should be able to prove this, if in other words it is true, and that might give one a new conviction that it is something worthy of proof. This is a generalization of the 9-point circle to a 9-point conne, in fact a 11-point conne, but that's in the projective domain and I'm not going to deal with that. It's the following. If we construct a pair of concurrent cilinders just as we would construct the altitudes of a triangle. Which are dynamic. So I'm constructing a pair of concurrent series which are just lines drawn from a vertex to the opposite side. And I construct from the center of concurrency the midpoint of the segment from this point of concurrency to each vertex.

20:00 So I construct L the midpoint of H, K the midpoint of HC, and I construct the midpoint of HB. Well, at this point, how I actually discovered it was that I looked at the sex points determined by the feet of these serians, D, E, and F, and J, K, and L. And when I drew a conic through any five of them, it passed through the sixth point. Moreover, I was surprised that it passed through the midpoints of the sides. So it is a direct generalization of the 9-point plane. I believe most people care, and of course I can move it until it becomes a circle, and then we have a nine-point circle. Associated with it is a generalization of the Euler line, where the center of concurrency, the center of the conic, at the moment it's just an ellipse, One which is the same ratio as in the normal case. Now, if these points of intersection remain on the sides of the triangle, we have an affine geometry result and then it remains an ellipse. However, if I move it out, it may turn into a hyperbola. Of course, you can have a pair of straight lines and you can also turn it into a parabola by... I'm not going to attempt to do that. The point that I want to make here is that I had no proof at this stage. I discovered all of this by doing construction and measurement and it was this, using this tool, that convinced me that this result was true and my reason for looking for proof was not at all to make sure that the result was true. The purpose of the proof was to, well, it's an intellectual challenge on the one hand, can I prove it? And secondly could I understand why this was true? And in fact in my proof I managed to discover a further generalization on the Euler line.

22:30 So understanding why it was true helped me to generalize it further. Okay, that's the point that I wanted to make there. By the way the result is not new. As I mentioned at the beginning, I rediscovered the result. The result was known in the 1880s, and John Readme, who is retired from Cardiff, when I wrote to him... After speaking to many mathematicians who didn't know this result, he didn't know the result, he came back to me and said, well, it's so great to see an old friend from my youth, when I sat down for my doctoral entrance exams at Cambridge, this was one of the results I had to do the study for my doctoral entrance exams. So he wrote me a much shorter, more elegant proof, but that is another story. Global refrigeration. I think in everyday life people are accustomed to what I would call somewhat fuzzy logic where things lie on a scale from 0 to 100% and we may be 60% sure that South Africa will win the soccer world cup when it is held in South Africa, although many of you will probably think it's only 10%. However, in mathematics, a theorem is only a theorem if there are no exceptions, and I think for students it is certainly very hard to make the transition from everyday fuzzy logic, using fuzzy logic as a separate discipline nowadays in computer science. But certainly this distinction exists. In my opinion, most counter examples are produced through experimental testing. Not exclusively. There are many examples in the history of mathematics where counter examples have been produced.

25:00 Theoretical methods that I will perhaps show later on, but I think certainly at school level much of that sort of thing occurs. Suppose students in the context of trying to develop a definition for a type, or maybe asking them how could you... How many different ways could you construct a kite? And the student might think, well, if I constructed it with a raffle of the particular diagonals, it would give me a kite. And that can be very easily tested by constructing, starting with the diagonals, constructing them perpendicular so that you can see they are perpendicular, and then checking to see if by dragging whether the figure actually remains a kite, and then very quickly you can see that it falls apart. So you very quickly produce a counter example. This is very primitive. But in much the same way, many of the examples that we get in the elementary number theory, like 6x-1 is prime for all x, 1, 2, 3, 4, It works for 1, 2, 3, 4 until you come to x is what I think 6, then it doesn't work anymore. A better example to give to students is n squared minus n plus 41 which holds for all n up to 40 as soon as n is 41 and of course 41 is a common factor. And the last one I think is interesting. n squared minus 79n plus 1601 is prime for all n from 1 to 1600, but for 1601 we obviously have a common factor and it will no longer be prime. What is interesting about it is the birth date of Ammar, the birth year of 1601, so that's a nice one to remember. So you just said, did he die at 79? He got a 79. Did I have a 79? No, he didn't make it, though. He was about 44, so exactly. What was that? Yeah, the 79 is more difficult to remember.

27:30 In 500 BC there was a Chinese conjecture that if 2 to the power n minus 2 is divisible by n, then n would be a prime number. And if you investigate that assertion, if we take n is equal to 2, we have 2 to the power 2 minus 2. Which gives us 4 minus 2 divided by 2 is 1, and 2 is a prime number. We take 3, we get 8. In this case, we substitute 4 in there, it doesn't divide into it exactly, but that's okay, 4 is a composite number, and continuing this way, we find confirmation for this, and I could actually substitute, what I would normally do is get people to nominate from the floor. Examples, particularly if I have a more powerful calculating machine like Mathematica or Maple, where I can work with large numbers, Sketchpad is limited, so I wouldn't use Sketchpad in particular, because it can only work with five significant digits. I'm not sure of the reading, probably nine digits? Calculation? Nine significant digits? Whatever? Ninety-nine? What typically happens when I use this example with students is that they... After generating many examples and randomly nominating examples, they start believing it is true, and it comes to them as a huge surprise when I say, well, it's actually not true. A counter example was found in 1819, a bi-theoretical method that I won't go into here, that in fact, 2 to the power of 341 minus 2 divided by 341 is a whole number, but 341 is not prime, since 341 is equal to 11 times 31. So, using numerical examples to gain conviction is good, but it has limitations, because to do this calculation is limited, so having some theoretical understanding sometimes is good.

30:00 A more contemporary example refers to Lord Kelvin's long-standing conjecture from about 1850 that the optimal partition of space into equal volumes with minimal total surface area is obtained by warping the tiling of space by truncated octahedrons. Everybody seems satisfied with Kelvin's solution and many believe that it was only a matter of time before proof of optimality was produced. And it so happened that, well, this was his conjecture, that if you essentially tile a space, then this is the way to get an optimal partition into equal volumes with minimal total surface. However, this was disproved by a computer program. Listening to the computer, how it has changed mathematics, called surface evolver, used by two physicists, Gray and Phelan, in 1994, produced a space partition of equal volumes with considerably smaller surface area than Kelvin's solution. However, the problem is still not unsolved. We don't know if this is the best possible solution, so the Kelvin problem is still open. But the point that I want to make is, using computer software, Essentially playing around, I know this program, Surface Evolver, is available as a free download from the internet, and I know some artists who use Surface Evolver to create some very beautiful mathematical art, and you might want to look at that if you're finding it. It's a fascinating program. Well, basically, you can produce soap bubbles and study soap bubble geometry with it. Another example, Malfatti's problem. In 1803, Malfatti posed the problem of constructing three circles inside a triangle with maximum possible area, and he proposed a solution at the time that to do that, you needed to construct the three circles in such a way that each circle touched two of the sides as well as the other two circles.

32:30 And nobody seemed to look at the problem much, maybe it wasn't interesting at the time. Everybody was satisfied, until somebody, as recent in the 1900s, looked at a special case, which we as mathematicians often do, in some sense, as a thought experiment, looked at the case of the equilateral triangle, and every... Using Malfatti's proposed solution, we calculate that we get the surface area covered is 0.729, whereas if we use this approach we get a higher coverage of the equilateral triangle of 0.739. So clearly something was wrong here. In 1965, Howard Hughes, the geometer, also observed the following curiosity, that if you look at another extreme case, if you drag up the triangle into this kind of scenario, then clearly, just looking at it, you don't even need to calculate, clearly, this is covering more area than that one we just visually see. Of course, you can do a calculation if you trust your visual intuition. And, in fact, the problem was only solved in 1967 by Michael Goldberg, who demonstrated that mathematics' solution is never correct. Why haven't I showed you the triangle? And in fact, you would need one of these solutions. Either one of these would provide the optimal solution if you arrange the circles in this way or in this way you will get the optimal solution. So that's an interesting curiosity. Well, maybe, I mean, let's be real. I mean, if a lot of mathematics depended on this... If a lot of mathematics depended on this result, mathematicians would have scrutinized this result much more cautiously and would have looked at providing proof for it. It's kind of incidental, but it's still a good example to show the role of experimentation.

35:00 Okay, maybe I'll skip over this example. Or maybe I'll just mention it very quickly. There are all sorts of examples like this. One can construct a proof that a quadrennial with one pair of opposite sides equal and one pair of opposite angles equal is a parallelogram. Which seems fine and you can go through the proof. However, the proof turns out to be false because it's based on certain assumptions. You only realize that the proof is based on wrong assumptions until you actually make a construction of something that complies to this and you begin to understand why this proof breaks down, what is wrong with these assumptions. I'm just going to skip over that, I'm not even halfway through. Very quickly, heuristic reputation, maybe just a contrast to the previous one, of global reputation. Most of us know Lakatos saying that mathematics is not empirical science, but it grows in a similar way to the natural sciences. It's a consequence of quasi-empirical testing of theorem concepts and definitions. And new counter-examples would necessitate the re-examining of old proofs. And this is a diagram that I've used, but many other people have similar diagrams. Of course, that was done within the context of polyhedra. Examples that I've experienced myself are the theorem of Ferdinand Hatch's series. I think the important point that I want to make here is that heuristic refutation, in contrast to global refutation, usually leaves the original theorem relatively intact. Although it may be modified and as reformulated, the original theorem is not really disproved at all. The process, Lakatosian's process of non-surviving in defense of the theorem, and here's an excerpt from his book where people argue about should we accept the counter-example in defense of the theorem, etc. And I would say that implicit within the Juristic refutation of Lakatos is the idea that when a counter-example like this arises, you will start talking about definitions and arguing about definitions.

37:30 What do we mean by polyhedron? What do we mean by quantum natural? What do we mean by continuous function? What do we mean by this or that concept? I think my time is virtually up. I can continue talking if people want to listen more, but I will stop at this point. I'm not done. I haven't got too big to actually conclude, but maybe I can show things. Thank you very much. And some things will come out of the discussion. So who would like to present? Michael, in your view, one of the aspects that you talked about is... In this case, we have a very nice example of Hoyer saying he is completely convinced after seeing a considerable amount of examples confirming a conjecture. But on the other hand, we talk about this disbelief from these methods because we have learned and experienced in history that there may be this. How do you deal with that? I mean, it seems like in the first case, in the first case, what you're doing is saying, okay, I'm going from, I'm almost convinced, so I'm completely convinced, then in the second one you're realizing that you cannot do it at all. My experience with dynamic geometry, and that's largely what I've done with my own mathematics, is that the experience of transforming and manipulating, it gives me the conviction that And the motivation to start looking for proof. Because if I wasn't convinced in the first place that it was true, I wouldn't spend the time, sometimes days, weeks, months, trying to look for proof.

40:00 So I wouldn't like to say I'm 100% convinced, although I have over time learned to trust. Dynamic geometry fairly well, but I'm a sufficiently mature mathematician to say that, well, maybe I'm only 99% convinced, and I actually have examples that I include in the materials that I have developed where I show that in dynamic geometry there are examples of where you can actually make false conjectures if you are not careful. And I think that vertical aspect is there, but I think if you are 99.9% convinced, then the reason for looking for proof is not so much to make sure, but it's really something else. It's more about understanding, it's more about the intellectual challenge. I cannot prove that most mathematicians believe Riemann hypothesis is true, and there's considerable evidence that it is true. None of it, of course, qualifies as a proof, but most people see it more as an intellectual exercise, to prove it as a breakthrough in the grand systematization of mathematics, rather than just making sure that it's true. I don't know if that answers your... I have a question on your concept of experimentation. You simply defined experimentation in mathematics. And I wonder what this concept of experimentation has to do with experimentation in empirical science. I think the experiment in the empirical sciences is rather difficult to set up, to build the right setting for an experiment. I'm not a physicist, but say for example, already you have what is an experiment and how to interpret an experiment and how to decide whether it is valid or not, etc.

42:30 And all this disappears in these easy examples of mathematical experimentation. I think you have invented your own or mathematicians have invented their own concept of experimentation. Yes, I mean my intention was not at all to relate it to experimentation in the empirical sciences. I think at the beginning, I should have said, maybe, and I think this is the sense that Lakatos uses, quasi, methods, at least the way I interpret it, the way he uses it, is that mathematics does not deal with real objects, as do mathematical science, so the method, I think you make suggestions, so of course you can always introduce... In empirical science I am doing quasars, but this is an excuse, so to speak, that is a bit too simple if you compare the theory of experimentation in empirical and the theory of experimentation in quasar empirical.

45:00 I can very briefly, which is that the phrase quasar experiment arises in a Popperian context. Right, so we're not talking about Duhem's conception of what happens in an experiment, we're talking about Popper's far more simple, right, so all that Duhem stuff wasn't in the context where this phrase arises, so I think that's the answer. But Popper had not much to say about experimentation, he had rather... Precisely, precisely. In addition to induction, they generalize and they are right on the spot. There are a lot of cases it doesn't work at all. And in addition, we also know that in a number of cases, it breaks down at the sizes that are not accessible to the machine. We have a very fine example of a movement series that gives you the first four billion digits of pi. And then it breaks down. If I were to make a naive summary out of that, I would say please do not use that method on me. It's so unreliable, yet mathematicians continue to do it. So my naive question is why? Well, that's a very good question. The answer is simple, is that it is a way of discovering new things. We don't only discover things, I'm just saying the fact. We don't just discover things by writing out proofs. We discover things by playing around, observing relationships, and the proofs often come afterwards. We do sometimes sit down and arrive at things deductively. So even though the method isn't true proof, it is still a useful method for finding things that we might not otherwise. That's why I prefer the term exploration instead of reduction. Okay, I think I was maybe not asking questions, but I don't believe to be asking questions.

47:30 It may be the case that the discovery of better environments, either in geometry or when you make numbers or objectives, opens for mathematicians a new way to practice mathematics, to do mathematics, which might not be just reduced to toy examples. Some of the things that you may have suggested are things which we may use with students, for example, like more concrete learning of mathematics. With these environments, people in geometry or in discrete mathematics, at least in my lab, are really doing, or are they doing experiments in the sense of natural science, or are they doing something? It is clear that they are interacting with the environment with the building of a kind of theory of both the thing they are targeting and the behavior of the environment. Which means that when you are constructing an object within the framework and in the context of a computer, a geometry metaphor, in the context of a computer-based macro world model or so ever, you have both to express the type of object you want to manipulate, to express the relationship, to express the manipulation, but also to be in control of the way it will be handled. And here there is quite difficult problems which are midway between experimentation and the kind of work that we should be doing before. So I don't want to close the question and say that we are right on one side or the other, but it's clear that it's a new type of activity. In which you are something which is quite close to what we could see or consider another field of mathematics as we knew it before. I will continue the discussion. I think it's a very good idea to have a look at the more recent philosophy and history of experimentation in science because it really points to a problem. One of the things that one can learn from this literature is that if you do a sort of serious investigation of experimental and empirical matters is that you need to bring in a third category, namely instruments, that experiments are done with instruments and the role of instrumentation and how you introduce and use and change and what are the kinds of instruments that are used is a crucial point. I wonder actually whether you thought...

50:00 It was much more about an instrument for doing mathematical argumentation than about experiment, probably, if we compare this with this more sophisticated history of experiment. Because actually what you did was you showed us some capacities of certain software programs. And this is an instrument, and I think that most of what you described was very closely tied to the instrument that you used in software packages, so I would wonder if the things, the arguments that you made in your context, for instance, would be easily transferred to the domain of mathematics that cannot use these types of instruments, if your argument, for instance, is about them. Seven-dimensional exotic spheres or some exotic constructions in higher set theories. We don't have a computer program, even in mathematics, but maybe they are completely useless in these contexts. So, therefore, I wonder whether the arguments you make are much more tied to the instruments that you're playing around with than with what is the philosopheric, scientific explanation. Yeah, I wouldn't quite agree with that. I think my experience talking to mathematicians who work in many different fields, they do similar things, even though they know that some of them do. Yeah, I know some don't, but there are many that do. I mean, this is all that I'm trying to say, is that some do, and we do find it a useful method. I'm not saying this is a foolproof method. I think it's very challenging, though, your point. I'm sorry. It's very challenging, but it brings us back to the discussion we had yesterday. Because in this case, you are both an instrument.

52:30 I would agree that also the same environments provide you with a kind, I quote, of natural space within which you are interacting and building and manipulating. So what is strike? Maybe part of the problem, and you might be right to look at that, but before we decide that it's right or wrong, when we observe the activity of the mathematicians or the activity of the learners, they are both manipulating the representation of the objects. And an instrument which gives access to the ocean surface, which is a very classical thing I think in mathematics, since you are not a kind of natural world in which these objects are living, so you are always working with representations, which is what is so at the forefront of the discussion here, but with the computer, in the case of math, you are not only an instrument. And it brings to that, I would say, But at the same time, everything was as if it were the case. So you are not only manipulating the instruments, you are also building and understanding meanings about the behaviors and characteristics of objects. That's true of all instruments. That's what I'm pointing to. We have three hour questions now, and I think they should very well be the last three. Two of them I think are specific to this area, so let's maybe have one of you do that last. I have one question. This is also on this? Okay, well, then you go first. I think it's nice to differentiate what kind of experiments we are doing, so if you take this prime number generative function, so you... We can scan a certain number of cases, which in real life breaks down, or it takes too long to realize, but so now there's a massive, very strongly growing area in experimental mathematics, which does the calculations, the symbolic calculations, and they have the... So far, we have found, by kind of pattern recognition, we have found a very interesting new formula, new combinatorial relationship, and so on.

55:00 It's a very different kind. It's a very different kind. It's not a case. It is something which takes diagrams together with the operation rules and finds new diagrams, yes, all of the new ones. It is a vast and growing area, a very successful area. And you call it, whatever, I am not interested. It is the success, I think, which is very important. You find new mathematical entities and new relationships. Success itself is an inductive argument, you know. It's not inductive. It's not inductive. No. No, if you argue with success, this very argument is inductive. Yeah, I mean, I'm sympathetic to this worry that this isn't really experimentation, but not because it's... It lacks auxiliary hypotheses or that it has lacked Duhemian qualities, not because it lacks instruments, but because, I mean, after all, right, let's say you have this conjecture that, you know, some formula captures something about the primes, and then you go and check by cases. I mean, when you go and check each case, you're not… You're not sort of querying the natural world at all, right? You're just doing another, you're just doing other mathematics, which is just as deductive as, I mean, it's a much simpler deduction, it's just a, it's just a numeric calculation, but it's still a deductive argument. I mean, all of these mathematical arguments are deductive in character at some, at some level, right? They're all... They all come from some assumptions about the relationships between numbers. I mean, you're not querying something outside of mathematics the way when you have a scientific theory. You have the theory of querying the world to compare it with your... I mean, so that whole aspect of it, which is sort of characteristic of the empirical sciences, is lacking, right? Yes, but there is an analogue. I mean, in the sense that you can have your conjecture. Which you then check in a variety of ways by looking at special cases, which to me seems very similar to the empirical sciences where you develop a particular theory of the relationship between volume and temperature or gas and then...

57:30 Maybe on theoretical grounds. And then you go check, does it actually hold? To me it seems very similar. The difference is that in mathematics we are producing these objects, well nowadays it was computer software, but also we're looking at limiting cases. These are thought experiments, we look at extreme cases. In my opinion, to test whether your theory or your conjecture is actually true, so to me there is a very strong parallel between what happens in the first place. Alright, okay, well thanks again. I think we'll be moving right on to the so-called round table. When you start a formulation, you start with some propositions, and the formulation has to be truly preserved and preserved. I don't think calculation is literally deduction, but I think that, look, um... On the one hand, you've got the, let's say, you've got the Boyle's Child's book, Gaslight, and over here you've got some conjecture about the primes. Look, there's a profound difference between performing a calculation, which finds you a counterexample, and conducting an experiment, which is a counterexample. The profound difference is not that there's a difference between the two.