Mathematics and Scientific Representation


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Asking what makes a representation an accurate representation ipso facto presupposes that inaccurate representations are representations too. And this is how it should be. A related condition concerns models that misrepresent in the sense that they lack target systems. Such models lack actual target systems, and one hopes that an account of scientific representation would allow us to understand how these models work. This need not imply the claim that they are representations in the same sense as models with actual targets, and, as we discuss below, there are accounts that deny targetless models the status of being representations.

A further condition of adequacy for an account of scientific representation is that it must account for the directionality of representation. As Goodman points out 5 , representations are about their targets, but at least in general targets are not about their representations: a photograph represents the cracks in the wing of aeroplane, but the wing does not represent the photograph.

So there is an essential directionality to representations, and an account of scientific, or epistemic, representation has to identify the root of this directionality. We call this the Requirement of Directionality. Some representations, in particular models and theories, are mathematized and their mathematical aspects are crucial to their cognitive and representational function.

This forces us to reconsider a time-honoured philosophical puzzle: the applicability of mathematics in the empirical sciences. The question how a mathematized model represents its target implies the question how mathematics applies to a physical system see Pincock for an explicit discussion of the relationship between scientific representation and the applicability of mathematics. For this reason, our fifth and final condition of adequacy is that an account of representation has to explain how mathematics is applied to the physical world. We call this the Applicability of Mathematics Condition.

In answering the above questions one invariably runs up against a further problem, the Problem of Ontology : what kinds of objects are representations? If representations are material objects the answer is straightforward: photographic plates, pieces of paper covered with ink, elliptical blocks of wood immersed in water, and so on. But not all representations are like this.

The Newtonian model of the solar system, the Lotka-Volterra model of predator-prey interaction and the general theory of relativity are not things you can put on your laboratory table and look at. Contessa , Frigg a,b , Godfrey-Smith , Levy , Thomson-Jones , Weisberg , among others, have drawn attention to this problem in different ways. Any satisfactory answer to these five issues will have to meet the following five conditions of adequacy:. Listing the problems in this way is not to say that these are separate and unrelated issues.

This division is analytical, not factual. It serves to structure the discussion and to assess proposals; it does not imply that an answer to one of these questions can be dissociated from what stance we take on the other issues. Any attempt to tackle these questions faces an immediate methodological problem. As per the problem of style, there are different kinds of representations: scientific models, theories, measurement outcomes, images, graphs, diagrams, and linguistic assertions are all scientific representations, and even within these groups there can be considerable variation.

But every analysis has to start somewhere, and so the problem is where. For such a universalist the problem loses its teeth because any starting point will lead to the same result. Those of particularist bent deny that there is such a theory. Different authors assume different stances in this debate, and we will discuss their positions below. However, there are few, if any, thoroughgoing universalists and so a review like the current one has to discuss different cases. This invariably leads to the neglect of some kinds of representations, and the best we can do about this is to be explicit about our choices.

This is in line both with the more recent literature on scientific representation, which is predominantly concerned with scientific models, and with the prime importance that current philosophy of science attaches to models see the SEP entry on models in science for a survey. It is, however, worth briefly mentioning some of the omissions that this brings with it. Various types of images have their place in science, and so do graphs, diagrams, and drawings. Perini and Elkins provide discussions of visual representation in science.

Measurements also supply representations of processes in nature, sometimes together with the subsequent condensation of measurement results in the form of charts, curves, tables and the like see the SEP entry on measurement in science. Furthermore, theories represent their subject matter. At this point the vexing problem of the nature of theories rears again see the SEP entry on the structure of scientific theories and also Portides forthcoming for an extensive discussion.

Proponents of the semantic view of theories construe theories as families of models, and so for them the question of how theories represent coincides with the question of how models represent. By contrast, those who regard theories as linguistic entities see theoretical representation as a special kind of linguistic representation and focus on the analysis of scientific languages, in particular the semantics of so-called theoretical terms see the SEP entry on theoretical terms in science. Before delving into the discussion a common misconception needs to be dispelled.

The misconception is that a representation is a mirror image, a copy, or an imitation of the thing it represents. On this view representation is ipso facto realistic representation. This is a mistake. Representations can be realistic, but they need not. And representations certainly need not be copies of the real thing an observation exploited by Lewis Carroll and Jorge Luis Borges in their satires, Sylvie and Bruno and On Exactitude in Science respectively, about cartographers who produce maps as large as the country itself only to see them abandoned.

Throughout this review we encounter positions that make room for non-realistic representation and hence testify to the fact that representation is a much broader notion than mirroring. Callender and Cohen give a radical answer to the demarcation problem: there is no difference between scientific representations and other kinds of representations, not even between scientific and artistic representation.

The core of GG is the reductive claim that all representations owe their status as representations to a privileged core of fundamental representations. GG then comes with a practical prescription about how to proceed with the analysis of a representation:. First, it explains the representational powers of derivative representations in terms of those of fundamental representations; second, it offers some other story to explain representation for the fundamental bearers of content.

Of these stages only the second requires serious philosophical work, and this work is done in the philosophy of mind because the fundamental form of representation is mental representation. Scientific representation is a derivative kind of representation 71, 75 and hence falls under the first stage of the above recipe. It is reduced to mental representation by an act of stipulation. This supplies an answer to the ER-problem:. The first problem facing Stipulative Fiat is whether or not stipulation, or the bare intentions of language users, suffice to establish representational relationships.

We ignore the difference between meaning and denotation here. And appealing to additional facts about the salt shaker the salt shaker being to the right of the pepper mill might allow us to infer that Madagascar is to the east of Mozambique in order to answer this objection goes beyond Stipulative Fiat. Callender and Cohen do admit some representations are more useful than others, but claim that.

GG only requires that there be some explanation of how derivative representations relate to fundamental representations; it does not require that this explanation be of a particular kind, much less that it consist in nothing but an act of stipulation Toon 77— As Callender and Cohen note, all that it requires is that there is a privileged class of representations and that other types of representations owe their representational capacities to their relationship with the primitive ones.

So philosophers need an account of how members of this privileged class of representations represent, and how derivative representations, which includes scientific models, relate to this class. When stated like this, many recent contributors to the debate on scientific representation can be seen as falling under the umbrella of GG.

Indeed, As we will see below, many of the more developed versions of the accounts of scientific representation discussed throughout this entry invoke the intentions of model users, albeit in a more complex manner that Stipulative Fiat. This conception has universal aspirations in that it is taken to account for representation across a broad range of different domains. So the similarity view repudiates the demarcation problem and submits that the same mechanism, namely similarity, underpins different kinds of representation in a broad variety of contexts. The view also offers an elegant account of surrogative reasoning.

Similarities between model and target can be exploited to carry over insights gained in the model to the target. However, appeal to similarity in the context of representation leaves open whether similarity is offered as an answer to the ER-Problem, the Problem of Style, or whether it is meant to set Standards of Accuracy.

Proponents of the similarity conception typically have offered little guidance on this issue. So we examine each option in turn and ask whether similarity offers a viable answer. We then turn to the question of how the similarity view deals with the Problem of Ontology. A well-known objection to this account is that similarity has the wrong logical properties.

Everything is similar to itself, but most things do not represent themselves. So this account does not meet our fourth condition of adequacy for an account of scientific representation insofar as it does not provide a direction to representation. However, there are accounts of similarity under which similarity is not a symmetric relation see Tversky ; Weisberg , ch. This raises the question of how to analyse similarity.

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We turn to this issue in the next subsection. The most significant problem facing Similarity 1 is that without constraints on what counts as similar, any two things can be considered similar Aronson et al. This, however, has the unfortunate consequence that anything represents anything else. This idea can be moulded into the following definition:. Moreover, Similarity 2 faces three further problems. Firstly, similarity, even restricted to relevant similarities, is too inclusive a concept to account for representation.

In many cases neither one of a pair of similar objects represents the other. This point has been brought home in now-classical thought experiment due to Putnam 1—3. An ant is crawling on a patch of sand and leaves a trace that happens to resemble Winston Churchill. Has the ant produced a picture, a representation, of Churchill? Although someone else might see the trace as a depiction of Churchill, the trace itself does not represent Churchill.

The fact that the trace is similar to Churchill does not suffice to establish that the trace represents him. And what is true of the trace and Churchill is true of every other pair of similar items: even relevant similarity on its own does not establish representation. Secondly, as noted in Section 1 , allowing for the possibility of misrepresentation is a key desiderata required of any account of scientific representation.

In the context of a similarity conception it would seem that a misrepresentation is one that portrays its target as having properties that are not similar in the relevant respects and to the relevant degree to the true properties of the target. Thirdly, there may simply be nothing to be similar to because some representations represent no actual object. Some paintings represent elves and dragons, and some models represent phlogiston and the ether. None of these exist. This is a problem for the similarity view because models without targets cannot represent what they seem to represent because in order for two things to be similar to each other both have to exist.

If there is no ether, then an ether model cannot be similar to the ether. At least some of these problems can be resolved by taking the very act of asserting a specific similarity between a model and a target as constitutive of the scientific representation. Analysing representation in these terms amounts to analysing schemes like. So agents specify which similarities are intended and for what purpose. This leads to the following definition:. If they are, then the representation is accurate or the representation is accurate to the extent that they hold. If they do not, then the representation is a misrepresentation.

However, it fails to resolve the problem with representation without a target. If there is no ether, no hypotheses can be asserted about it, at least in any straightforward way. Similarity 3 , by invoking an active role for the purposes and actions of scientists in constituting scientific representation, marks a significant change in emphasis for similarity-based accounts. By building the purposes of model users directly into an answer to the ER-problem, Similarity 3 is explicitly not a naturalistic account in contrast to Similarity 1. Even though Similarity 3 resolves a number of issues that beset simpler versions, it does not seem to be a successful similarity-based solution to the ER-Problem.

A closer look at Similarity 3 reveals that the role of similarity has shifted. As far as offering a solution to the ER-Problem is concerned, all the heavy lifting in Similarity 3 is done by the appeal to agents and their intentions. But if similarity is not the only way in which a model can be used as a representation, then similarity has become otiose in a reply to the ER-problem.

In fact, being similar in the relevant respects to the relevant degree now plays the role either of a representational style or of a normative criterion for accurate representation, rather than constituting representation per se. We assess in the next section whether similarity offers a cogent reply to the issues of style and accuracy and raise a further problem for any account of scientific representation that relies on the idea that models, specifically non-concrete models, are similar to their targets.

The fact that relevant properties can be delineated in different ways could potentially provide an answer to the Problem of Style. A first step in the direction of such an understanding of styles is the explicit analysis of the notion of similarity. The standard way of cashing out what it means for an object to be similar to another object is to require that they co-instantiate properties. In fact, this is the idea that Quine — and Goodman had in mind in their influential critiques of similarity.

The two most prominent formal frameworks that develop this idea are the geometric and contrast accounts see Decock and Douven for a discussion. The geometric account, associated with Shepard , assigns objects a place in a multidimensional space based on values assigned to their properties. This space is then equipped with a metric and the degree of dis similarity between two objects is the distance between the points representing the two objects in that space.

This account is based on the strong assumptions that values can be assigned to all features relevant to similarity judgments, which is deemed unrealistic and to the best of our knowledge no one has developed such an account in the context of scientific representation. This account defines a gradated notion of similarity based on a weighted comparison of properties.

Weisberg has recently introduced this account into the philosophy of science where it serves as the starting point for his weighted feature matching account of model world-relations for details see Weisberg , ch. Two objects are alike to the extent that they co-instantiate similar properties for example, a red phone box and a red London bus might be alike with respect to their colour, despite not instantiating the exact same shade of red.

Two objects are partially identical to the extent that they co-instantiate identical properties. A further question that remains for someone who uses the notion of similarity to answer to the Problem of Style and provide standards of accuracy in the manner under consideration here is whether it truly captures all of scientific practice.

Similarity theorists are committed to the claim that whenever a scientific model represents its target system, this is established in virtue of a model user specifying a relevant similarity, and if the similarity holds, then the representational relationship is accurate. These universal aspirations require that the notion of similarity invoked capture the relationship that holds between diverse entities such as the Phillips-Newlyn machine and an economy, a tube map and an underground train system, the Lotka-Volterra equations or phase space associated with them and predator-prey populations, and so on.

Whether all of these relationships can be captured in terms of similarity remains an open question. Another problem facing similarity based approaches concerns their treatment of the ontology of models. If models are supposed to be similar to their targets in the ways specified by theoretical hypotheses, then they must be the kind of things that can be so similar. For material models like the San Francisco Bay model Weisberg , ball and stick models of molecules Toon , the Phillips-Newlyn machine Morgan and Boumans , or model organisms Ankeny and Leonelli this seems straightforward because they are of the same ontological kind as their respective targets.

Following Thomson-Jones we call such models non-concrete models. The question then is how such models can be similar to their targets.

Scientific Representation

But if so, then it remains unclear how they can instantiate the sorts of properties specified by theoretical hypotheses, since these properties are typically physical , and presumably being located in space and time is a necessary condition on instantiating such properties. For further discussion of this objection, and proposed solutions, see Teller , Thomson-Jones , and Giere The structuralist conception of model-representation originated in the so-called semantic view of theories that came to prominence in the second half of the 20 th century see the SEP entry on the structure of scientific theories for further details.

The semantic view was originally proposed as an account of theory structure rather than scientific representation. The driving idea behind the position is that scientific theories are best thought of as collections of models. This invites the questions: what are these models, and how do they represent their target systems?

Most defenders of the semantic view of theories with the notable exception of Giere, whose views on scientific representation were discussed in the previous section take models to be structures, which represent their target systems in virtue of there being some kind of morphism isomorphism, partial isomorphism, homomorphism, … between the two. This conception has two prima facie advantages.

The first advantage is that it offers a straightforward answer to the ER-Problem or SR-problem if the focus is on scientific representation , and one that accounts for surrogative reasoning: the mappings between the model and the target allow scientists to convert truths found in the model into claims about the target system. The second advantage concerns the applicability of mathematics. There is time-honoured position in the philosophy of mathematics which sees mathematics as the study of structures; see, for instance Resnik and Shapiro It is a natural move for the scientific structuralist to adopt this point of view, which then provides a neat explanation of how mathematics is used in scientific modelling.

So the first task for a structuralist account of representation is to articulate what notion of structure it employs. A number of different notions of structure have been discussed in the literature for a review see Thomson-Jones , but by far the most common is the notion of structure one finds in set theory and mathematical logic.

This definition of structure is widely used in mathematics and logic. As regards objects, all that matters from a structuralist point of view is that there are so and so many of them. Whether the objects are desks or planets is irrelevant. The relation literally is nothing over and above this class. So a structure consists of dummy-objects between which purely extensionally defined relations hold. The first basic posit of the structuralist theory of representation is that models are structures in this sense the second is that models represent their targets by being suitably morphic to them; we discuss morphisms in the next subsection.

So we are presented with a clear answer to the Problem of Ontology: models are structures. The remaining issue is what structures themselves are. Are they Platonic entities, equivalence classes, modal constructs, or yet something else? In the context of a discussion of scientific representation one can push these questions off to the philosophy of mathematics see the SEP entries on the philosophy of mathematics , nominalism in the philosophy of mathematics , and Platonism in the philosophy of mathematics for further details.

The most basic structuralist conception of scientific representation asserts that scientific models, understood as structures, represent their target systems in virtue of being isomorphic to them. Then the model represents the target iff it is isomorphic to the target:. It bears noting that few adherents of the structuralist account of scientific representation, most closely associated with the semantic view of theories, explicitly defend this position although see Ubbink Representation was not the focus of attention in the semantic view, and the attribution of something like Structuralism 1 to its supporters is an extrapolation.

Representation became a much-debated topic in the first decade of the 21 st century, and many proponents of the semantic view then either moved away from Structuralism 1 , or pointed out that they never held such a view. We turn to more advanced positions shortly, but to understand what motivates such positions it is helpful to understand why Structuralism 1 fails. This problem could be addressed by replacing isomorphism with an alternative mapping.

These suggestions solve some, but not all problems. While many of these mappings are not symmetrical, they are all still reflexive. But even if these formal issues could be resolved in one way or another, a view based on structural mappings would still face other serious problems. For ease of presentation we discuss these problems in the context of the isomorphism view; mutatis mutandis other formal mappings suffer from the same difficulties.

Like similarity, isomorphism is too inclusive: not all things that are isomorphic represent each other. Many mathematical structures were discovered and discussed long before they were used in science. Non-Euclidean geometries were studied by mathematicians long before Einstein used them in the context of spacetime theories, and Hilbert spaces were studied by mathematicians prior to their use in quantum theory.

If representation was nothing over and above isomorphism, then we would have to conclude that Riemann discovered general relativity or that that Hilbert invented quantum mechanics. Isomorphism is more restrictive than similarity: not everything is isomorphic to everything else. But isomorphism is still too abundant to correctly identify what a model represents. The root of the difficulties is that the same structures can be instantiated in different kinds of target systems. Certain geometrical structures are instantiated by many different systems; just think about how many spherical things we find in the world.

The mathematical structure of the pendulum is also the structure of an electric circuit with a condenser and a solenoid Kroes However, isomorphism demands identity of structure: the structural properties of the model and the target must correspond to one another exactly. By the lights of Structuralism 1 it is therefore is not a representation at all.

Partial structures can avoid a mismatch due to a target relation being omitted in the model and hence go some way to shoring up the structuralist account Bueno and French It remains unclear, however, how they accounts for distortive representations Pincock Hence models without target cannot represent what they seem to represent. Most of these problems can be resolved by making moves similar to the ones that lead to Similarity 3 : introduce agents and hypothetical reasoning into the account of representation.

Going through the motions one finds:. As in the shift from Similarity 2 to Similarity 3 , this seems like a successful move, with many although not all of the aforementioned concerns being met. But, again, the role of isomorphism has shifted. Let us now assess how well isomorphism fares as a response to these problems, and the others outlined above.

Unlike similarity, which has been widely discussed across different domains, structural mappings are tied closely to the formal framework of set theory, and have been discussed only sparingly outside the context of the mathematized sciences. An exception is French , who discusses isomorphism accounts in the context of pictorial representation. Therefore representation is the perceived isomorphism of structure French — this point is reaffirmed by Bueno and French — ; see Downes — for a critical discussion.

The Problem of Style is to identify representational styles and characterise them. A proposed structural mapping between the model and the target offers an obvious response to this challenge: one can represent a system by coming up with a model that is proposed to be appropriately morphic to it. This delivers the isomorphism-style, the homomorphism-style, the partial-isomorphism style and so on. This is neat answer. Are morphism-styles merely a subgroup of styles or are they privileged? The former is uncontentious. However, the emphasis many structuralists place on structure preserving mappings suggests that they do not regard morphisms as merely one way among others to represent something.

What they seem to have in mind is the stronger claim that a representation must be of that sort, or that morphism-styles are the only acceptable styles. This claim seems to conflict with scientific practice in at least two respects. Firstly, many representations are inaccurate and known to be in some way. Some models distort, deform and twist properties of the target in ways that seem to undercut isomorphism, or indeed any of the proposed structure preserving mappings. Some models in statistical mechanics have an infinite number of particles and the Newtonian model of the solar system represents the sun as a perfect sphere where in reality it is fiery ball with no well-defined surface at all.

It is at best unclear how isomorphism, partial or otherwise, or homomorphism can account for these kinds of idealisations. So it seems that styles of representation other than structure preserving mappings have to be recognised. Secondly, the structuralist view is a rational reconstruction of scientific modelling, and as such it has some distance from the actual practice. Some philosophers have worried that this distance is too large and that the view is too far removed from the actual practice of science to be able to capture what matters to the practice of modelling this is the thrust of many contributions to Morgan and Morrison ; see also Cartwright Although some models used by scientists may be best thought of as set theoretic structures, there are many where this seems to contradict how scientists actually talk about, and reason with, their models.

Obvious examples include physical models like the San Francisco Bay model Weisberg , but also systems such as the idealized pendulum or imaginary populations of interbreeding animals. Such models have the strange property of being concrete-if-real and scientists talk about them as if they were real systems, despite the fact that they are obviously not Godfrey-Smith There remains a final problem to be addressed in the context of structural accounts of scientific representation.

If there is no ether, then an ether model cannot be similar to the ether. At least some of these problems can be resolved by taking the very act of asserting a specific similarity between a model and a target as constitutive of the scientific representation. Analysing representation in these terms amounts to analysing schemes like. So agents specify which similarities are intended and for what purpose. This leads to the following definition:. If they are, then the representation is accurate or the representation is accurate to the extent that they hold.

If they do not, then the representation is a misrepresentation. However, it fails to resolve the problem with representation without a target. If there is no ether, no hypotheses can be asserted about it, at least in any straightforward way. Similarity 3 , by invoking an active role for the purposes and actions of scientists in constituting scientific representation, marks a significant change in emphasis for similarity-based accounts. By building the purposes of model users directly into an answer to the ER-problem, Similarity 3 is explicitly not a naturalistic account in contrast to Similarity 1.

Even though Similarity 3 resolves a number of issues that beset simpler versions, it does not seem to be a successful similarity-based solution to the ER-Problem. A closer look at Similarity 3 reveals that the role of similarity has shifted. As far as offering a solution to the ER-Problem is concerned, all the heavy lifting in Similarity 3 is done by the appeal to agents and their intentions. But if similarity is not the only way in which a model can be used as a representation, then similarity has become otiose in a reply to the ER-problem.

In fact, being similar in the relevant respects to the relevant degree now plays the role either of a representational style or of a normative criterion for accurate representation, rather than constituting representation per se. We assess in the next section whether similarity offers a cogent reply to the issues of style and accuracy and raise a further problem for any account of scientific representation that relies on the idea that models, specifically non-concrete models, are similar to their targets.

The fact that relevant properties can be delineated in different ways could potentially provide an answer to the Problem of Style. A first step in the direction of such an understanding of styles is the explicit analysis of the notion of similarity. The standard way of cashing out what it means for an object to be similar to another object is to require that they co-instantiate properties. In fact, this is the idea that Quine — and Goodman had in mind in their influential critiques of similarity.

The two most prominent formal frameworks that develop this idea are the geometric and contrast accounts see Decock and Douven for a discussion. The geometric account, associated with Shepard , assigns objects a place in a multidimensional space based on values assigned to their properties. This space is then equipped with a metric and the degree of dis similarity between two objects is the distance between the points representing the two objects in that space. This account is based on the strong assumptions that values can be assigned to all features relevant to similarity judgments, which is deemed unrealistic and to the best of our knowledge no one has developed such an account in the context of scientific representation.

This account defines a gradated notion of similarity based on a weighted comparison of properties. Weisberg has recently introduced this account into the philosophy of science where it serves as the starting point for his weighted feature matching account of model world-relations for details see Weisberg , ch. Two objects are alike to the extent that they co-instantiate similar properties for example, a red phone box and a red London bus might be alike with respect to their colour, despite not instantiating the exact same shade of red.

Two objects are partially identical to the extent that they co-instantiate identical properties. A further question that remains for someone who uses the notion of similarity to answer to the Problem of Style and provide standards of accuracy in the manner under consideration here is whether it truly captures all of scientific practice. Similarity theorists are committed to the claim that whenever a scientific model represents its target system, this is established in virtue of a model user specifying a relevant similarity, and if the similarity holds, then the representational relationship is accurate.

These universal aspirations require that the notion of similarity invoked capture the relationship that holds between diverse entities such as the Phillips-Newlyn machine and an economy, a tube map and an underground train system, the Lotka-Volterra equations or phase space associated with them and predator-prey populations, and so on.

Whether all of these relationships can be captured in terms of similarity remains an open question. Another problem facing similarity based approaches concerns their treatment of the ontology of models. If models are supposed to be similar to their targets in the ways specified by theoretical hypotheses, then they must be the kind of things that can be so similar. For material models like the San Francisco Bay model Weisberg , ball and stick models of molecules Toon , the Phillips-Newlyn machine Morgan and Boumans , or model organisms Ankeny and Leonelli this seems straightforward because they are of the same ontological kind as their respective targets.

Following Thomson-Jones we call such models non-concrete models. The question then is how such models can be similar to their targets.


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  • 1. Problems Concerning Scientific Representation?
  • Scientific Notation!

But if so, then it remains unclear how they can instantiate the sorts of properties specified by theoretical hypotheses, since these properties are typically physical , and presumably being located in space and time is a necessary condition on instantiating such properties. For further discussion of this objection, and proposed solutions, see Teller , Thomson-Jones , and Giere The structuralist conception of model-representation originated in the so-called semantic view of theories that came to prominence in the second half of the 20 th century see the SEP entry on the structure of scientific theories for further details.

Scientific Notation - Explained!

The semantic view was originally proposed as an account of theory structure rather than scientific representation. The driving idea behind the position is that scientific theories are best thought of as collections of models. This invites the questions: what are these models, and how do they represent their target systems? Most defenders of the semantic view of theories with the notable exception of Giere, whose views on scientific representation were discussed in the previous section take models to be structures, which represent their target systems in virtue of there being some kind of morphism isomorphism, partial isomorphism, homomorphism, … between the two.

This conception has two prima facie advantages. The first advantage is that it offers a straightforward answer to the ER-Problem or SR-problem if the focus is on scientific representation , and one that accounts for surrogative reasoning: the mappings between the model and the target allow scientists to convert truths found in the model into claims about the target system. The second advantage concerns the applicability of mathematics. There is time-honoured position in the philosophy of mathematics which sees mathematics as the study of structures; see, for instance Resnik and Shapiro It is a natural move for the scientific structuralist to adopt this point of view, which then provides a neat explanation of how mathematics is used in scientific modelling.

So the first task for a structuralist account of representation is to articulate what notion of structure it employs. A number of different notions of structure have been discussed in the literature for a review see Thomson-Jones , but by far the most common is the notion of structure one finds in set theory and mathematical logic.

This definition of structure is widely used in mathematics and logic. As regards objects, all that matters from a structuralist point of view is that there are so and so many of them. Whether the objects are desks or planets is irrelevant. The relation literally is nothing over and above this class. So a structure consists of dummy-objects between which purely extensionally defined relations hold. The first basic posit of the structuralist theory of representation is that models are structures in this sense the second is that models represent their targets by being suitably morphic to them; we discuss morphisms in the next subsection.

So we are presented with a clear answer to the Problem of Ontology: models are structures. The remaining issue is what structures themselves are. Are they Platonic entities, equivalence classes, modal constructs, or yet something else? In the context of a discussion of scientific representation one can push these questions off to the philosophy of mathematics see the SEP entries on the philosophy of mathematics , nominalism in the philosophy of mathematics , and Platonism in the philosophy of mathematics for further details.

The most basic structuralist conception of scientific representation asserts that scientific models, understood as structures, represent their target systems in virtue of being isomorphic to them. Then the model represents the target iff it is isomorphic to the target:. It bears noting that few adherents of the structuralist account of scientific representation, most closely associated with the semantic view of theories, explicitly defend this position although see Ubbink Representation was not the focus of attention in the semantic view, and the attribution of something like Structuralism 1 to its supporters is an extrapolation.

Representation became a much-debated topic in the first decade of the 21 st century, and many proponents of the semantic view then either moved away from Structuralism 1 , or pointed out that they never held such a view. We turn to more advanced positions shortly, but to understand what motivates such positions it is helpful to understand why Structuralism 1 fails. This problem could be addressed by replacing isomorphism with an alternative mapping. These suggestions solve some, but not all problems.

While many of these mappings are not symmetrical, they are all still reflexive. But even if these formal issues could be resolved in one way or another, a view based on structural mappings would still face other serious problems. For ease of presentation we discuss these problems in the context of the isomorphism view; mutatis mutandis other formal mappings suffer from the same difficulties.

Like similarity, isomorphism is too inclusive: not all things that are isomorphic represent each other. Many mathematical structures were discovered and discussed long before they were used in science. Non-Euclidean geometries were studied by mathematicians long before Einstein used them in the context of spacetime theories, and Hilbert spaces were studied by mathematicians prior to their use in quantum theory.

Chapter 2: The Nature of Mathematics

If representation was nothing over and above isomorphism, then we would have to conclude that Riemann discovered general relativity or that that Hilbert invented quantum mechanics. Isomorphism is more restrictive than similarity: not everything is isomorphic to everything else. But isomorphism is still too abundant to correctly identify what a model represents. The root of the difficulties is that the same structures can be instantiated in different kinds of target systems.

Certain geometrical structures are instantiated by many different systems; just think about how many spherical things we find in the world. The mathematical structure of the pendulum is also the structure of an electric circuit with a condenser and a solenoid Kroes However, isomorphism demands identity of structure: the structural properties of the model and the target must correspond to one another exactly.

By the lights of Structuralism 1 it is therefore is not a representation at all. Partial structures can avoid a mismatch due to a target relation being omitted in the model and hence go some way to shoring up the structuralist account Bueno and French It remains unclear, however, how they accounts for distortive representations Pincock Hence models without target cannot represent what they seem to represent. Most of these problems can be resolved by making moves similar to the ones that lead to Similarity 3 : introduce agents and hypothetical reasoning into the account of representation.

Going through the motions one finds:. As in the shift from Similarity 2 to Similarity 3 , this seems like a successful move, with many although not all of the aforementioned concerns being met. But, again, the role of isomorphism has shifted. Let us now assess how well isomorphism fares as a response to these problems, and the others outlined above. Unlike similarity, which has been widely discussed across different domains, structural mappings are tied closely to the formal framework of set theory, and have been discussed only sparingly outside the context of the mathematized sciences.

An exception is French , who discusses isomorphism accounts in the context of pictorial representation. Therefore representation is the perceived isomorphism of structure French — this point is reaffirmed by Bueno and French — ; see Downes — for a critical discussion. The Problem of Style is to identify representational styles and characterise them. A proposed structural mapping between the model and the target offers an obvious response to this challenge: one can represent a system by coming up with a model that is proposed to be appropriately morphic to it.

This delivers the isomorphism-style, the homomorphism-style, the partial-isomorphism style and so on. This is neat answer. Are morphism-styles merely a subgroup of styles or are they privileged? The former is uncontentious. However, the emphasis many structuralists place on structure preserving mappings suggests that they do not regard morphisms as merely one way among others to represent something. What they seem to have in mind is the stronger claim that a representation must be of that sort, or that morphism-styles are the only acceptable styles.

This claim seems to conflict with scientific practice in at least two respects. Firstly, many representations are inaccurate and known to be in some way. Some models distort, deform and twist properties of the target in ways that seem to undercut isomorphism, or indeed any of the proposed structure preserving mappings.

Some models in statistical mechanics have an infinite number of particles and the Newtonian model of the solar system represents the sun as a perfect sphere where in reality it is fiery ball with no well-defined surface at all. It is at best unclear how isomorphism, partial or otherwise, or homomorphism can account for these kinds of idealisations. So it seems that styles of representation other than structure preserving mappings have to be recognised.

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Secondly, the structuralist view is a rational reconstruction of scientific modelling, and as such it has some distance from the actual practice. Some philosophers have worried that this distance is too large and that the view is too far removed from the actual practice of science to be able to capture what matters to the practice of modelling this is the thrust of many contributions to Morgan and Morrison ; see also Cartwright Although some models used by scientists may be best thought of as set theoretic structures, there are many where this seems to contradict how scientists actually talk about, and reason with, their models.

Obvious examples include physical models like the San Francisco Bay model Weisberg , but also systems such as the idealized pendulum or imaginary populations of interbreeding animals. Such models have the strange property of being concrete-if-real and scientists talk about them as if they were real systems, despite the fact that they are obviously not Godfrey-Smith There remains a final problem to be addressed in the context of structural accounts of scientific representation.

Target systems are physical objects: atoms, planets, populations of rabbits, economic agents, etc. Isomorphism is a relation that holds between two structures and claiming that a set theoretic structure is isomorphic to a piece of the physical world is prima facie a category mistake. By definition, a morphism can only hold between two structures.

But what does it mean for a target system—a part of the physical world—to possess a structure, and where in the target system is the structure located? There are two prominent suggestions in the literature. The first, originally suggested by Suppes [] , is that data models are the target-end structures represented by models.

This approach faces a question whether we should be satisfied with an account of scientific representation that precludes phenomena being represented see Bogen and Woodward for a discussion of the distinction between data and phenomena, and Brading and Landry for a discussion of the distinction in the context of scientific representation.

Van Fraassen has addressed this problem and argues for a pragmatic resolution: in the context of use, there is no pragmatic difference between representing phenomena and data extracted from it see Nguyen for a critical discussion. The alternative approach locates the target-end structure in the target system itself.

One version of this approach sees structures as being instantiated in target systems. This view seems to be implicit in many versions of the semantic view, and it is explicitly held by authors arguing for a structuralist answer to the problem of the applicability of mathematics Resnik ; Shapiro This approach faces underdetermination issues in that the same target can instantiate different structures.

A more radical version simply identifies targets with structures Tegmark This approach is highly revisionary in particular when considering target systems like populations of breeding rabbits or economies. So the question remains for any structuralist account of scientific representation: where are the required target-end structures to be found? The core idea of the inferential conception is to analyse scientific representation in terms of the inferential function of scientific models.

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The accounts discussed in this section reverse this order and explain scientific representation directly in terms of surrogative reasoning. The last step is necessary because demonstrations establish results about the model itself, and in interpreting these results the model user draws inferences about the target from the model Unfortunately Hughes has little to say about what it means to interpret a result of a demonstration on a model in terms of its target system, and so one has to retreat to an intuitive and unanalysed notion of drawing inferences about the target based on the model.

Hughes is explicit that he is not attempting to answer the ER-problem, and that he does not offer denotation, demonstration, and interpretation as individually necessary and jointly sufficient conditions for scientific representation. He prefers the more. This is unsatisfactory because it ultimately remains unclear what allows scientists to use a model to draw inferences about the target, and it raises the question of what would have to be added to the DDI conditions to turn them into a full-fledged response to the ER-problem.

If, alternatively, the conditions were taken to be necessary and sufficient, then the account would require further elaboration on what establishes the conditions. Two different notions of deflationism are in operation in his account. The first is to abandon the aim of seeking necessary and sufficient conditions; necessary conditions will be good enough On might worry that explaining representation in terms of representational force sheds little light on the matter as long as no analysis of representational force is offered.

The second condition is in fact just the Surrogative Reasoning Condition, now taken as a necessary condition on scientific representation. But Contessa 61 points out that it remains mysterious how these inferences are generated. So the tenability of Inferentialism in effect depends on the tenability of deflationism about scientific representation. In as far as one accepts representational force as a cogent concept, targetless models are dealt with successfully because representational force unlike denotation does not require the existence of a target Inferentialism repudiates the Representational Demarcation Problem and aims to offer an account of representation that also works in other domains such as painting The account is ontologically non-committal because anything that has an internal structure that allows an agent to draw inferences can be a representation.

Relatedly, since the account is supposed to apply to a wide variety of entities including equations and mathematical structures, the account implies that mathematics is successfully applied in the sciences, but in keeping with the spirit of deflationism no explanation is offered about how this is possible. The account does not directly address the Problem of Style. Contessa introduces the notion of an interpretation of a model, in terms of its target system, as a necessary and sufficient condition on epistemic representation see also Ducheyne for a related account :.

The leading idea of an interpretation is that the model user first identifies sets of relevant objects in the model and the target, and then pins down sets of properties and relations these objects instantiate both in the model and the target. Interpretation offers a neat answer to the ER-problem. The account also explains the directionality of representation: interpreting a model in terms of a target does not entail interpreting a target in terms of a model. Contessa does not comment on the applicability of mathematics but since his account shares with the structuralist account an emphasis on relations and one-to-one model-target correspondence, Contessa can appeal to the same account of the applicability of mathematics as the structuralist.

But it remains unclear how Interpretation addresses the Problem of Style. As we have seen earlier, in particular visual representations fall into different categories. It is a question for future research how these can be classified within the interpretational framework. With respect to the Question of Ontology, Interpretation itself places few constraints on what scientific models are, ontologically speaking. All it requires is that they consist of objects, properties, relations, and functions but see Contessa for further discussion of what he takes models to be, ontologically speaking.

A recent family of approaches analyses models by drawing an analogy between models and literary fiction. This analogy can be used in two ways, yielding two different version of the fiction view. The first is primarily motivated by ontological considerations rather than the question of scientific representation per se.

Scientific discourse is rife with passages that appear to be descriptions of systems in a particular discipline, and the pages of textbooks and journals are filled with discussions of the properties and the behaviour of those systems. In mechanics, for instance, the dynamical properties of a system consisting of three spinning spheres with homogenous mass distributions are the focus of attention, in biology infinite populations are investigated, and in economics perfectly rational agents with access to perfect information exchange goods.

Their surface structure notwithstanding, no one would mistake descriptions of such systems as descriptions of an actual system: we know very well that there are no such systems. The face-value practice raises a number of questions. What account should be given of these descriptions and what sort of objects, if any, do they describe?

Are we putting forward truth-evaluable claims when putting forward descriptions of missing systems? The fiction view of models provides an answer: models are akin to places and characters in literary fiction and claims about them are true or false in the same way in which claims about these places and characters are true or false. Such a position has been recently defended explicitly by some authors Frigg a,b; Godfrey-Smith , but not without opposition Giere ; Magnani It does bear noting that the analogy has been around for a while Cartwright ; McCloskey ; Vaihinger [].

This leaves the thorny issue of how to analyse fictional places and characters. Here philosophers of science can draw on discussions from aesthetics to fill in the details about these questions Friend and Salis provide useful reviews. The second version of the fiction view explicitly focuses on representation. Most theories of representation we have encountered so far posit that there are model systems and construe scientific representation as a relation between two entities, the model system and the target system.

Toon calls this the indirect view of representation Indeed, Weisberg views this indirectness as the defining feature of modelling This view contrasts with what Toon 43 and Levy call a direct view of representation. This view does not recognise model systems and aims instead to explain representation as a form of direct description. At the heart of this theory is the notion of a game of make-believe see the SEP entry on imagination for further discussion.

We play such a game if, for instance, when walking through a forest we imagine that stumps are bears and if we spot a stump we imagine that we spot a bear. Together a prop and principle of generation prescribe what is to be imagined. Walton considers a vast variety of different props, including statues and works of literary fiction. Toon focuses on the particular kind of game in which we are prescribed to imagine something of a real world object. A statue showing Napoleon on horseback Toon 37 is a prop mandating us to imagine, for instance, that Napoleon has a certain physiognomy and certain facial expressions.

The crucial move is to say that models are props in games of make believe. Specifically, material models are like the statue of Napoleon and theoretical models are like the text of The War of the Worlds : both prescribe, in their own way, to imagine something about a real object. A ball-and-stick model of a methane molecule prescribes us to imagine particular things about methane, and a model description describing a point mass bob bouncing on a perfectly elastic spring represents the real ball and spring system by prescribing imaginings about the real system.

This provides the following answer to the ER-problem Toon 62 :. This account solves some of the problems posed in Section 1 : Direct Representation is asymmetrical, makes room for misrepresentation, and, given its roots in aesthetics, it renounces the Demarcation Problem. The view absolves the Problem of Ontology since models are either physical objects or descriptions, neither or which are problematic in this context.

Toon remains silent on both the Problem of Style, and the applicability of mathematics. Important questions remain. According to Direct Representation models prescribe us to imagine certain things about their target system. The account remains silent, however, on the relationship between what a model prescribes us to imagine and what a model user should actually infer about the target system, and so it offers no answer to the ER-problem. A further worry is how Direct Representation deals with targetless models.

If there is no target system, then what does the model prescribe imaginings about? Toon is well aware of such models and suggests the following solution: if a model has no target it prescribes imaginings about a fictional character This solution, however, comes with ontological costs, and one of the declared aims of the direct view was to avoid such costs by removing model systems from the picture. Levy aims to salvage ontological parsimony and proposes a radical move: there are no targetless models.

If a purported model has no target then it is not a model. There remains a question, however, how this view can be squared with scientific practice where targetless models are not only common but also clearly acknowledged as such. Elgin further developed this account and, crucially, suggested that it also applies to scientific representations.

Caricatures are paradigmatic examples: Churchill is represented as a bulldog and Thatcher is represented as a boxer. The leading idea of the views discussed in this section is that scientific representation works in much the same way. A model of the solar system represents it as consisting of perfect spheres; the logistic model of growth represents the population as reproducing at fixed intervals of time; and so on.

In each instance, models can be used to attempt to learn about their targets by determining what the former represent the latter as being. The question then is what establishes this sort of representational relationship. The answer requires introducing some of the concepts Goodman and Elgin use to develop their account of representation-as. A painting of a unicorn is a unicorn-representation because it shows a unicorn, but it is not a representation of a unicorn because there are no unicorns. Being a representation of something is established by denotation; it is a binary relation that holds between a symbol and the object which it denotes.

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