Styles in the philosophy of science

0:00 We have time to everybody every player in this field to have his or her own account of all of these things and how they relate to one another. I think that there's a gap here. Almost nobody talks about truth in connection to model. This has been notified by graphical philosophy, economics, namely a chronic irritation by a feature. They are usually much of the time criticized by students, other social scientists and lay people.

25:00 For enjoying these false models and false assumptions and true models that they are either necessarily false or neither true or false, I'm not responsive and there are two ways I use I would like to be avoided. This entangles some of the myths in the way of talking about these things. They violate the whole truth, they isolate, they cover various tools, they remove from their horizons a lot of stuff in the real world or their target systems. They violate nothing but the truth because the employees are violating the assumptions and they are GM's falsehoods. And then part of this view, to be avoided, that getting to truth or truths requires de-isolation and de-idealization, relaxation of these idealizing assumptions so as to get closer to the truth. I don't think this is a good idea, but it's a very popular idea indeed. And you even find it in the Bosnian school of idealization, Leszek Nowak and his followers. An alternative, we should consider the possibility of a model being true despite containing false assumptions, and also a model being true thanks to containing those false assumptions. So these are some of the ideas that I will set out to defend. Many truths are attainable without de-isolation and de-idealization. Outrageously idealizing and simple, tiny models might contain some truth. This is the simple idea, the intuition that I want to articulate.

27:30 The second view to be avoided, models cannot be true because they are not the sorts of entities that are truth-valuant. They are neither true nor false, perhaps because they are not linguistic. This is Ronald Gearest. An alternative that I will defend, we should focus on the nature and locus of truth-bearers inside models, as well as the truth-makers that make them true. And what I'm attempting to do with the paper is a modification and elaboration of Ronald Gehrig's account so as to relocate truth with respect to models. I want to have a potential truth statement about certain properties or relational properties of models. I want to have truth within models. So this is what I want, the possibility of truth in models or truth of models. And the way I proceed to answer this strategic question, if this is what I want, truth in models or truth of models, how do I get it? And of course, in order to answer this question, I need to give an outline of my own account of models and representations that are involved. So, the concept of models is a key role here. In my account, models are isolated representations. In order to mean by this idea, it's a very, very complex model due to this guy. The farmer, an estate owner, well informed about the economics of his time, who developed a very fancy model of agricultural land use.

30:00 The art is still used taught at universities. There is a long, long tradition, Tunanian tradition, economic geography and economics. He called it the isolated state. The original title of his work was the ideal. But the published version has this title, the Isolierte Stahl, the third part of it, 1826, second part, 1842. And this is a very good case indeed because this is the most unlikely candidate for truth because it's so outrageously false if you look at it. I mean the world just does not look like that at all. So if I can defend my case in relation to this particular example, then it provides a very powerful test of my truth inside this very model that is so obviously false. And this is how Fontunen starts. This is a very fast piece of book. Imagine a very large town. Throughout the plain, the soil is capable of cultivation. In this case, the utility far from the town plane turns into an uncultivated wilderness in this state and the outside world. Von Thünen himself was unable to list all the necessary assumptions that entrenched within the model. But the important thing here, which is very useful to notice, the model is... I'll just give an idea of what kind of assumptions there are. Each of them extremely useful for making this case.

32:30 The area is a plain, no mountains, no valleys, crossed by no navigable freeway or canal, throughout capable of cultivation, homogeneous in fertility, the climate is uniform across the state. Communication between the area and the outside world is uncultivated. At the center of the plain there is a town with no spatial dimensions. No other towns, all industrial activity takes place in the town. All interactions between the producers located in the town, interactions only, marketing interactions, selling and buying, no other interactions. Importantly, transportation costs are directly proportional to distance and to the weight and versatility of the goods. This means there are no roads, no preservation technology, actual deliveries. All prices and transportation costs are fixed, production costs are constant over space, the agents are rationally maximized, the agents are fixed. This is not the full list of those assumptions, but nevertheless 16 assumptions, all of which are outrageously, obviously, just utterly false. The point is isolation by idealization. These idealizing assumptions, 1 to 16, serve the function of eliminating, neutralizing a number of causally relevant factors in the actual world, make an impact on agricultural land use patterns. These assumptions thereby help isolate a major cause and its characteristic way of operation, a major cause of actual land use patterns. And what's theoretically isolated in the model is distance from the market, from that point-like city in the center of isolated state. And because transportation caused the function of distance, it's another way of putting it. So what happens in the model? This is a very characteristic way of talking by economists and others. What happens in the model, not what happens in the real world or in the target system, but what happens in the model.

35:00 This is also a continuous question, what pattern of cultivation will take shape in these conditions? And here is the answer. Near the town will be grown those products which are heavy or bulky in relation to their value and hence so expensive to transport, the remoter districts are unable to supply them. Here too we shall find highly perishable products which must be used very quickly. Timber is heavy to transport, so it must be grown close to the city. Tomatoes, likewise, because otherwise they would survive the transport. With increasing distance from the town, the land will progressively be given up to products cheap to transport in relation to their value, such as grain for bread, lighter than timber and lasts longer than tomatoes, for example. For this reason alone, panelists sharply differentiate concentric rings or belts we form around the town, each with its own particular type of product. And this is how it looks like. Concentric rings. Very neat. This is the pattern of agricultural land use in the state under those assumptions. There are verbal arithmetic in the previous one. There is a lot of verbal description of the model going on in Fontuna's work. Likewise, some arithmetic description of it. But the model is the isolated state, the imagined system. That's the model. The properties of the model are examined by performing inferences among model descriptions, these arithmetic and geometric descriptions. And this is how it goes. I mean, for example, you get, you can derive these concentric rings by using so-called grand gradients as your basis.

37:30 I will have no time for the details here, nevertheless it is influenced with the model description. But actual land use patterns don't look like those concentric rings at all. Here is an example, Nuremberg. You know, there is a river, there are mountains, valleys, etc. Differences in fertility of the soil, etc. You get something very different from those concentric rings. Here is Anis Abeba. Likewise, you know, where are the concentric? Nowhere. The real world This is actually von Thünen's own example above. The upper part is a state under those idealizing assumptions, constantly green, and then he adds just two disturbing factors, two relaxations of those idealizing assumptions. The river added a subsidiary town, and the modification of the original pattern is quite obvious. From the late, I think, 1960s, by a guy, here you see one section here is the fertile area, you have the modification, then you put all those modifications together, and we come back to the idea that de-isolation through de-idealization is the only truth, which I dispute, namely relaxing the idealizing assumptions one by one, thereby let previously excluded...

40:00 Should these causal factors work out their impact on the outcome with truth of the actual land-use pattern, this would indeed result in an improved resemblance between de-isolated models and... But I'm arguing de-isolation of this kind is not needed for some other truths, maybe truths about the pattern of land-use, but there are other truths here that we also might want to capture. And for understanding this I need to give an outline of my account of models as representations. They involve two aspects, representative aspect and resemblance. Models are substitutes or target systems. They are examined instead of the target system in order to acquire information about the target system. But the direct target or subject of examination is the The model itself, its properties, its behavior. This is the phrase. Let's see what happens in the model. Let's check what happens in the model. So it's a representative of something else. And then there is a resemblance aspect between the surrogate and target system, and this indeed has something to do with truth. The representative aspect is voluntary, it's up to us. It involves intentionality. In a sense, anything can serve as a model insofar as we decide to use something as a model of something else. That's okay. But there is a constraint here, and that comes from the issue of resemblance. It's constrained by the target, the target system and its real properties, as it were. This gives us an ontological constraint, which is nice for. But on top of that, there are these pragmatic constraints, and here is my summary of the model. Agent A uses object M as the representative of some target system R for purpose P, addressing audience E, prompting issues of resemblance to the right, while assuming capacity to resemble.

42:30 I'm adding audience, I mean, sorry I mentioned it, but it gives credit to me, that this actually helps make models five-placed instead of the previous record, thereby four-placed only. And then the second novelty is that I'm requiring only prompting the issue of resemblance to the right. The role of resemblance is, of course, an important thing to pay attention to if we are interested in the true inside model. Resemblance is not sufficient. If we drop a pebble in still water, what we get is a concentric ring. There is a high resemblance with tuning rings, but this is not part of tuning's model. Just because of that, this should be part of the model. The real-world agricultural land use pattern may be very different from the pattern in the model as we have noticed in our case. I briefly go to this because what I'm suspecting is that there is a lot of confusion around using this very phrase, misrepresentation.

45:00 That only involve the idea of issue of resemblance to a right. Therefore, this very notion of misrepresentation, counter-scientific representation, must accommodate failures and not be built into our notion of representation. I am suggesting that the notion of prompting issues of resemblance to a right starts taking care of challenge. Simply, successful representation is a matter of building, employing, using a representative that resembles the target, it meets the ontological constraints in ways that also serves intended purposes and reaches the intended audience, being in line with its background beliefs and expectations and all that, all of which means that it meets the pragmatic constraints. And it's these pragmatic constraints that determine the required respect and degrees of the original Ingeris account. Ingeris, there is a role for truth to play, but models are not supposed to be true. There is no chance to be true because there are no sorts of things that could possibly be true. Models are non-linguistic abstract objects and they are devoid of truth value because they are not linguistic. They are linguistically and truly described or defined by assumptions, such as our continuous isolated state, this imagined system described or defined in terms of those 16-plus assumptions, and those assumptions are true of that imagined system, because they define it. And then, in Geary's account, there are theoretical hypotheses that are truth-valued claims about the respects and degrees of similarity between model and real.

47:30 So they are not about the real world or the target system directly, they are about how the model as an abstract object and the real system are related to one another. My modifications are three. What we need to revise is the account of truth barriers. I don't require truth barriers to be linguistic. Reinterpretation. Similarities suitably circumscribed may qualify as a kind of truth. And what we need to add is a higher order, what I call commentary, that isolates, it's kind of second order, higher order isolation going on here, isolates truth barriers in models. Models are very rich, there are all sorts of stuff in models, but some tiny bits of those models need to be isolated. And those isolated tiny bits that serve as truth barriers in models are shaped by fantasies and audiences. So models are not truth-bearers in their entirety. Not models in their entirety and not just any subset of model components, but carefully isolated kind of systems. What about Fontunan's case? What are the possible truth-bearers and truth-makers in Fontunan's isolated space? What are precisely idealizing assumptions and what they are about? That's one possibility. Concentric, this image of concentric rings and then actual and new patterns, another possibility. And then, frankly, and this is my favorite, what drives the model system? The causal mechanism or the causal force that drives the model system and then the prominent causal force or mechanism in the real world. The former being the truth bearer and the latter being the truth and indeed I think von Thunan is on my side here. He's my fan or I'm his fan both ways.

50:00 His essay states, I hope the reader who is willing to spend some time and attention on my work will not take exception to the imaginary assumptions, those sixteen false assumptions I make at the beginning, because they do not correspond to conditions and reality, and that he will not reject these assumptions as arbitrary or pointless. They are a necessary part of my argument, allowing me to establish the operation of a certain effect of whose... All these operations we see but dimly in reality, whereas in incessant conflict with others of its kind. Okay, this is why I say that sometimes falsehood, falsehood of these assumptions, is needed for capturing some important truth. It's not in spite of this falsehood, but thanks to this falsehood, falsehood that we are able to get to the truth. Truth bearers and truth makers. I said something about the assumptions, could say something about the land use patterns. They are truth bearers, but not primary. The images of land use patterns are models of the data. They are truth makers. There is an abstract resemblance here. Land use is patterned in a zone-like manner, but even this can be defeated. Other causes are strong enough. But here, this is the key. The significant truth that can be captured by using those outrageously false idealizing assumptions, the principle that gave the isolated state its shape is also present in reality, but the phenomena which here bring it out manifest themselves in changed forms since they are also influenced by, at the same time, by several other relations and conditions. We may divest an acting force and virk in the crowd. Of all incidental conditions and everything accidental, and only in this way can we recognize its role in producing it. So he is really a realist, and he is of the true causation.

52:30 This is an important notion. The model commentary performs a higher order isolation of relevant respects than the virulent truth barriers amongst the ingredients of the model. This derives from theory. There is a very specific cognitive goals, audience expectations, previous applications, other background attitudes, etc. There is no fixed list of instructions. And the various ontologies of truth barriers, this is very difficult. There are linguistic truth barriers, Fonthunen's assumptions and Fonthunen's commentary, And then the imagined system of many kinds of representing capacities, dependence and interplay. They inform one another, they constrain one another, they facilitate one another, they comment and describe one another, etc. And this is the very final slide I'm brushing now. Two options to go. A revisionary and a conservative way to go. We could consider naturalizing truth. Truth barriers, truth makers, truth making of worldly matters, materials and interactions of which may vary from case to case. And this permits images to bear and imagined systems to bear truth. And this is really what gives me what I wanted inside models. And there is a conservative way. Translate those model ingredients in those imaging systems into statements about real-world systems as part of the model commentary. In fact, the time is over, but maybe we can have an extension, five minutes, possibly?

55:00 Well, just one question. With this concept of ideal time, she writes this book, which she describes the market of what is the model of the town and city. And this is, by the way, elaborated by David Slinger's nice book, the methodology of, Weber's methodology, the unity of culture, by the way, is so exciting. What is the secret of this elaboration? Obviously, Weber, I mean, Weber and Schroeder were part of the same culture. German-speaking world who were influenced by von Thünen's work. Hans Freidinger was, Karl Menger was mentioned here once, and many, many others. Now, Weber's particular notion of ideal type I think captures one aspect of von Thünen's strategy, which I did not actually discuss at all. These are examples of a future line of inquiry, not by himself but his followers, who would actually build similar models of those excluded causal factors, each being modeled having their own particular characteristic way of functioning, and then at the end of putting all these small models together so as to give us a composite picture.

57:30 But the point is, my point here, is that each of those tiny models that only describe the functioning of one causal mechanism only is capable of capturing truth, so no de-idealization, de-isolation is needed. That's my reading of Weber, but he's not the clearest of all thinkers. Many of the assumptions that have been made were that it is impossible to explain all of them. Does anything change when you are done with the questions? On an analogy, suppose you have a model of a plane where you have... You can say, well, this is the perfect calculator for this range of calculated numbers. And a lot of companies argue that we are equal to kind of as any number is. So, is there any, you know, is there more to it than just the instruction? Yeah, it's a very good question. I've actually thought a lot about it over the last couple of decades or so. There is one assumption here that's actually physical, if you want, impossibility, and that's the point-like character of the town in the middle. I mean, no dimensions at all. It's impossible. That's simply impossible.

1:00:00 But the defense of that assumption, of course, would be that those dimensions are negligible for the purpose. So we should conclude here. Thank you very much. I speak of Marion Worms, Julian Rice from Erasmus University in Rotterdam. We'll conclude this session with his talk, Is there a role for clinical expertise? Sorry, Marion sends her apologies, she can't make it. Understanding theories before mathematics. General framework and aims of the paper. In this paper, I, and I of course means Marion, I approach understanding not only as understanding phenomena by means of some theoretical material, which provides an explanation of them, but as understanding the theoretical material that is used to explain the phenomena itself. I will approach the issue from the local perspective of individual scientists with particular skills and background knowledge, who learn and apply that theoretical material to phenomenal framework.

1:02:30 What does understanding a theory mean? Let me first state what I take to be an initial characterization of understanding, or, to put it differently, what requirements an agent has to satisfy in order to claim that he or she understands a theory. First, assuming that a theory is a tool enabling one to make predictions about phenomena, one has to know how it works as a tool, one has to be able to use it, to choose the right piece of it, to make it fit the phenomena, and finally to draw inferences in order to obtain predictions. I would call that the computational aspect of understanding. But a theory is not only machine-making predictions, it is at the same time a predictive and representational tool. Understanding a theory does not merely amount to mastering the functioning of a machine. It also consists in understanding what the theory says concerning phenomena as a representation of them. And that amounts to being able to describe the phenomena in its terms, to mastering its conceptual system, and to being able to see the phenomena within its representational scheme. I will call that the representational aspect of understanding. My hypothesis is that these two aspects that I have distinguished for the sake of analysis are closely connected and interdependent in practice. My paper aims to propose tools to analyze that interdependence. Part 2 method. Until now I have been speaking of theories, but my approach is different from classical approaches to scientific theories in the sense that my main unit of analysis will neither be theories nor models, but the concrete particular devices that are used and manipulated by agents in practice to represent and predict the behaviour of systems they study. Those pieces of theories can be of various kinds, equations, diagrams, graphs, descriptions or any other device. I take them as being at the same time expressions of the theory, representations of the phenomena, and predictive tools. My approach consists in taking them as relatively autonomous tools of representation and prediction and then focusing on the particular format in which they are displayed. Formats. In order to make that clearer, I should make precise what I mean by format of representation. There is an important part of the work which I cannot go into now, so I will only give a sketchy characterization of it and hopefully the examples I discuss will illuminate the idea a little bit more. Let's define a representation as a particular concrete object, most of the time marks on paper, standing for something else and whose perceptual properties are signals carrying information concerning the thing represented.

1:05:00 The format of a representation is the particular way in which the representation conveys the information it contains to the user. Indeed, two representations can contain the same information but convey it in different ways, and sometimes the change in format can make some piece of information accessible to human cognitive agents who would not be able to extract it from a representation in another format. As an extreme example, consider a digital picture and the linguistic coding for each of the pictures. Format can be structurally characterised in terms of syntax, which properties count as signals and how they combine, and semantics, how those syntactic properties relate to the thing represented. My main point is that the format of representation matters to the kind of cognitive operation and to the cognitive cost that is required in order to extract some piece of information from it. I suggest that differences between formats are various degrees and occur along various dimensions and are not only a distinction between visual and linguistic representations. Two formats can think tactically as well as semantically more or less close to another, can be think tactically as well as semantically more or less close to another. For instance, drawing a graph from an equation is not the same kind of operation as projecting a graph onto a different reference frame in order to obtain another graph. Both are changes in format occurring along different dimensions. Those differences between formats are of various degrees. I would say that the relative distance could be measured according to the cognitive cost. It is needed to derive a representation in a format and one format from a representation in another format, which is sometimes impossible without any loss of information. I would say that the more cognitive costly the change in format and the more important the loss of information, the more distant the formats. In consequence, the distance between formats is a relative and contextual matter, depending on the use and interpretation that each individual user makes of them according to the skills, background knowledge, beliefs, and theoretical commitments in the case of scientific representations. My point is to use that notion of the format to analyse the relations between various representations, pieces of theory, used in some scientific fields and the different ways they are used and interpreted by different individual scientists, my main assumption being that the format of the representation that is used is crucial to the double aspect of understanding as I have defined it.

1:07:30 Let me now present without any technical detail three examples. I will draw general conclusions after having presented them. Example number one, versions of classical mechanics. My first example consists in a comparison of the core equation of two different formulations of classical mechanics, namely the Newtonian and the Lagrangian, by considering them as constituting different formats. Newton's second law is expressed mathematically by the following equation, s equal to dp over d2. S stands for the force, P for the momentum, that is the product of mass and velocity. Both are vectorial quantities. Solving a problem in that frame consists first in specifying the forces according to the system at hand as well as its initial conditions in order to write its equations of motion, and second in solving the differential equations thus obtained. Sometimes that task is practically impossible, such as in the case of a constrained system. In those cases, the Lagrangian or analytical formulation is more tractable. The core principle of analytical mechanics, from which one can draw the equations of motions for a system, is the principle of least action, or the Hamiltonian principle, or whatever you can think of. A stands for action. Here's the definition of action, the same equation there. It is the integral of the function L, called the Lagrangian, of the system, which is the difference of the kinetic energy and the potential energy, T and V, which are scalar quantities. From that principle, one can derive the Lagrangian equations of motions, which are the following forms, which can be shown to be equivalent to the Newtonian equations. In order to obtain those equations, one does not need to specify the local equations of the forces acting on the system at any instant. One relies instead on a variational principle, namely the Hamiltonian principle, that establishes a general condition prescribing the integral A of the function L, which concerns the trajectory as a whole as extremal, from T1 to T2, variation being null.

1:10:00 These are two different formats of representation of the motion of a mechanical system. These two formats are mathematically equivalent and can be deduced from each other. So in a certain sense, they contain the same information. They are equivalent in principle. But if one focuses on pieces of theory as tools of representation and prediction, namely on the two principles, and on the way they are applied and manipulated, in practice, and not from the abstract logical point of view, they are cognitively different. They are computationally different. They do not allow for the same inferences. Some information cannot be gathered from the Newtonian form of research. Applying one or the other formulation does not amount to the same cognitive operation because they are not couched in the same language, vectorial language versus variational language. They do not represent the phenomena in the same way. They do not refer to the phenomena in the same way. Newtonian mechanics describes motion by means of the concept of force, which is represented by vectorial quantity, while analytical mechanics represents motion by the concept of action and energies, which are scalar quantities. Newtonian mechanics represents motion as a local and instantaneous phenomenon and analytical mechanics as a global phenomenon taken as a whole. Now the second example, Feynman diagrams. Consider now the diagrams introduced by Richard Feynman in 1948 in order to represent complex interactions of atomic particles. At the same time both Richard Feynman and Julian Schwinger gave a solution to the problem of infinities in quantum electrodynamics or QED. Schwinger gave the mathematical derivation and Feynman proposed his now famous diagrams as mnemonic devices to complete complex high-order calculations without confusing or omitting terms. Here is the simplest Feynman diagram for electron scattering and its corresponding mathematical formula. In 1949, Freeman Dyson demonstrated the equivalence of both approaches and introduced rules of derivation for Feynman diagrams. In terms of formats, we have two different formats, linguistic and diagrammatic, that are mathematically equivalent but computationally different, the diagrammatic format being more easily tractable and manipulable. But I would like to go a step further than this first consideration. Indeed, one can distinguish between various ways of using diagrams themselves that are at the same time different ways of interpreting them and of relating them to other formats.

1:12:30 Since Dyson had shown how to derive the diagrams, they could be used as a second psychological aid to perform complex calculations. From that perspective, they were used to visualize a formula from which they were constructed and then avoid mistakes in complex and sometimes practically intractable calculations. But what we could call Feynman's version of the diagrams was thoroughly different. Indeed, he thought of them as of central importance. And considered them as primary representations of the phenomena from which a mathematical formulation could be constructed. Never gave rules for constructing them on a mathematical basis. Thus, they were not visualisations of the formula, but rather visualisations of the physical processes themselves. Feynman said it had well done. All the mathematical proofs were later discovered, but I don't thoroughly understand myself what the physical ideas are. The relation between diagrams and mathematical formulas as two formats does not go in the same direction for Dyson and for Feynman. And those different ways of deriving diagrams are related to different ways of interpreting them as representations. Indeed, Dyson's and Feynman's theoretical commitments in using the diagrams were different. Feynman's proposition of the diagrams had flown from a path-into-role approach to quantum mechanics. He tried to treat particles rather than fields as the primary ingredients of his theory and finally in two papers in 1949 he explicitly divorced the diagrams from QED and gave them an autonomous status while Dyson thought of them as visualising devices within QED, thus adding nothing to it except cognitive tractability. Let me just add that finally Feynman diagrams were used in various other fields of high energy physics to which they were not initially intended to apply. That applicability to other fields and their evolution in meaning and use suggests that if one does not consider them as mere rule-based derivations from the mathematical formulations of QED, one can consider them as becoming themselves generative like a language. Example 3, genetic mappings. My last example is drawn from the field of Mendelian genetics. It is the new form of representation created by Sturtevant in 1913, namely genetic mapping. In the early 1910s, various exceptions were found to the law of independent assortment, according to which factors responsible for the different inherited characters segregate independently from each other during the formation of germ cells.

1:15:00 Relying on Janssen's observation of the intertwining of chromosomes during meiosis, Morgan hypothesized that homologous chromosomes exchanged parts and that factors that had greater chance to be inherited together when they were closely located on the chromosome thus constituting a linkage group. Thus, he suggested that the frequency of linkage of characters could be taken as a measure of the relative distance of factors responsible for them in chromosomes. At that time, cytologists' observation of chromosomes were not precise enough to precisely describe the structure of chromosomes. The chromosome theory of heredity, according to which genes were located on chromosomes, still had a hypothetical status, which was reinforced by the discovery of linkage groups. In 1913, Sturtevant proposed a technique that enabled him to draw a map of the linear ordering of genes along chromosomes relying on numerical data obtained from medallion breeding experiments on Drosophilia melanogaster. These are the famous maps of linkage groups corresponding to the four chromosomes of Drosophilia melanogaster constructed by Morgan's group and put on the frontispiece of their 1915 mechanism of medallion heredity. Finally, in 1934, the discovery of the giant ciliary gland chromosomes in Drosophila gave microscopically observable banding patterns that enabled geneticists to map linkage maps constructed with Sturtevant's medallion technique onto X chromosome locations in the visible giant chromosomes. How can one characterize Sturtevant's maps in terms of format? Although they are called chromosome maps and although the idea of evaluating the distance between factors had grown from a commitment to the chromosome theory of heredity and was intended to give an insight into the location of genes on chromosomes, some remarks are worth making. Sturtevant insisted on the fact that the distance on the linkage map was not to be taken literally as physical distance because different parts of chromosomes might not be equally breakable. As a consequence, distance was a measure of strength as well as of length. Those linkage maps were obtained by computing numerical data from an early breeding experiment.

1:17:30 So in a way they are not different in time from other symbolic representations of factors like Punnett squares and could be thought of as an extension of Mendel's symbolism which consists of factors as independent units by denotating them with letters. The syntax was adopted to new data, but the latter still referred to factors without unanimous commitment to what those factors are, and the spatial relationships represented numerical results, not physical systems. Initially, the realistic status of the spatial relations was both hypothetical and metaphorical, and was not accepted unanimously among all geneticists. Finally, the change of meaning in the spatial location that acquired a realistic status corresponds to a change at the theoretical level. When linkage maps are mapped onto pictures of chromosomes, strictly speaking, we're not within the field of Mendelian theory anymore, but in the chromosome theory of heredity, which does not rely on the same techniques and which is another level of discretion. The notion of format enables us to compare representations along various dimensions. From the three examples I have given, one can draw this non-exhaustive list. Differences between linguistic representations and diagrams, Feynman diagrams and mathematical formulae, linkage maps and numerical results. Two kinds of two-dimensional representations, linkage maps and pictures of chromosomes. They are very similar. Finally, one is mapped onto the other, but they are not related in the same way to other formats. That enables us to see differences between the use and interpretation of a representation by individual scientists and the way they relate it to other representations. Anima diagrams, as well as linkage maps, provide two different accounts. I would like to emphasize the fact that, from that perspective, there is no clear-cut distinction between visual and linguistic representations. Or, to put it differently, that distinction is not the only one which is relevant. The purely linguistic one, like in the case of classical mechanics, is relevant as well. Linkage maps can be described as a two-dimensional extension of Mendelian symbolism intended to present numerical data. Though they look more like pictorial representations, to which they were finally related.

1:20:00 Second, focusing on particular representations by paying attention to their formats is a good tool for analyzing understanding and for answering my initial question concerning the relation between the two aspects of understanding that I had distinguished to wit of the computational and the representational. In the three examples, constraints of applicability of the theory gave birth to new formats of representation. In the case of analytical mechanics, one could describe the invention of variational calculus as an answer to the problem of mathematical tractability, rather intractable, which were better adapted to the predictive and descriptive tasks. These new formats of representation, once successful, themselves acquired an autonomous status and became standard generative representations.