Chapter 1: The Underdetermination of Theories

Transcription

1 Chapter 1: The Underdetermination of Theories Our lives depend on what we can predict. When we eat, we assume that we will be nourished. When we walk forward, we assume that the ground will support us. Engineers design bridges and buildings, and we assume that they will not give way suddenly for no apparent reason. Physicians treat common diseases on the assumption that will work in the future as they have worked in the past. Our lives depend on our ability to predict what will happen in the future from what has happened in the past. How do we predict the future from the past, the unseen from the seen? The quick answer is by induction. The problem is that nobody has said very precisely what induction is. Philosophers often talk about particular patterns of inductive inference, such as this metal ball sinks in water, that metal ball sinks, and all metal balls observed so far have sunk in water, therefore all metal balls sink in water. This is known as simple enumerative induction. But we know that it is not the only pattern of scientific induction because its conclusion is formulated in the same vocabulary as the premises, whereas scientific theories often invent new vocabulary, such as atom, gene, gravity, spacetime curvature and so on. Philosophers only agree on what scientific induction is not. Induction is not deduction that much we know. An argument is deductive when the conclusion follows deductively from its premises. In the case of simple enumerative induction, the conclusion that all metal balls sink in water does not follow deductively from any enumeration of instances, no matter how numerous or varied. Some nondeductive inferences are universally accepted as bad arguments, such as snow is white, therefore grass is green. But what makes some non-deductive arguments better than others? An argument consists of premises (assumptions) and a conclusion. Arguments are either deductively valid or invalid. All inductive arguments are deductively invalid, but some are better than others. Inference is the action taken when someone draws a conclusion from premises. Inductive inference is therefore refers to the psychological process of inferring a conclusion from premises when the conclusion does not follow deductively. In the kind of inductive inference of interest to us, the premises are taken to be true statements of observed facts, otherwise known as data, or evidence. Simple enumerative induction has this form. But here we need to ask the right questions, because we are not interested in questions of psychological practice. Whether scientists draw conclusions directly from observed data is irrelevant. The issue is, no matter how they arrive at an inductive conclusion, what makes the conclusion reasonable? It concerns the relationship between the conclusion and the premises; therefore, the motivation for studying inductive arguments is independent of the secondary issue about whether inductive inference is something that scientists do. The aim of general philosophy of science is say general things about the relationship between theory and evidence. The hope is that studying patterns of inductive argument will help achieve this goal. The hope is encouraged by the success of this strategy in deductive logic. For example, any argument of the form of modus ponens (If A then B; A, therefore B) is deductively valid. The question of whether the conclusion B should be believed then reduces to question whether the premises should be believed. In the case of Page 2

2 inductive arguments, the believability of statements of evidence is not an issue. So, the study of inductive argument forms appears to be an especially worthwhile enterprise. Statistical methods define a variety of formal inductive arguments, which are widely implemented in science as it is practiced today. David Hume made the logical problem of induction famous long before the development of mathematical statistics. His use of simple enumerative induction was sufficient for the purpose of demonstrating the (trivial) fact that inductive inferences are not deductive, but it is wholly inadequate for the more positive purpose of studying inductive arguments in science. Even more unfortunate is the fact that many important discoveries in statistics, such as the law of error and the central limit theorem, were discovered at a time when philosophers were distracted by breakthroughs in logic. Even today, relatively little has been said about the connection between statistical inference and the philosophical problem of understanding inductive arguments in science. At first glance, this may appear to be wrong. For, the notion of probability is central to statistical thinking, and probability has been applied to problem of induction. But not, I believe, in the right way. The idea has been to weaken the conclusion of an inductive argument by asserting it with some degree of probability, which strengthens the connection between the premise and the conclusion. For example, given the fact that all solid metal balls in the past have sunk in water, it is more reasonable to conclude that the next solid metal ball will probably sink than saying that it will sink. But it does little more than redefine reasonable in terms of probability, and it s not clear that this is the right way to redefine the problem. We need to understand the relationship between theory and evidence that makes the conclusion reasonable. The probabilistic approach to the problem of induction is called Bayesianism. In Chapter 4, I will argue that the theory is inadequate. Another traditional approach to the problem of induction is to strengthen an inductive argument by adding a missing premise so that the conclusion follows deductively. In fact, any inductive argument A therefore B, can be transformed into the a deductively valid argument of the form A and X therefore B, if X implies the proposition if A then B. B then follows deductively from A and X. But now the questions is: Why believe X? X is not a statement of evidence, nor is it implied by the evidence. We began by asking why we should believe B given A, and end up asking why we should believe B if A. This not a solution to the problem. It is merely asks the original question in a different way. It is often said that the missing premise should assert that Nature is uniform that the future is like the past. The idea is that if all solid metal balls observed so far have sunk in water, and Nature is uniform, then all solid metal balls whatsoever will sink in water. Or to put it in an equivalent way, if solid metal balls sometimes sink and sometimes float, then Nature is not uniform. This proposal fails for more interesting reasons. One reason is that the premise is ambiguous. Does it mean that Nature is uniform in all respects? In that case, the premise is plainly false nature is not uniform in all respects. Some things stay the same and some things change. Or maybe the premise is that Nature is uniform in some respects? Then it is true, but not strong enough to allow the conclusion to follow from the premises. Nature has to be uniform in relevant respects, but then we are back to the problem of vagueness. What does relevant mean? Nevertheless, the idea is interesting, for it points the way towards identifying a common feature of good inductive arguments. Constancies of nature have always been Page 3

3 postulated by successful scientific theories. These constancies are well confirmed by the agreement of independent measurements of the constant in question. The empirical success involves two sides of an evidential equation. On the subjective side, the theory must allow for the overdetermination of the constant in question, for otherwise there can are no multiple measurements. On the objective side, nature provides the outcomes of the multiple measurements, and determines whether they agree or disagree. To discover constancies of nature these the two sides needs to come together to form the right kind of relationship. The process can succeed at a lower level before being extended to grander, more unified, theories. At the earlier stages, postulated constancies will not need to be anything as grandiose, or as universal, as the constancy of the speed of light, although that is an exemplary example of what I have in mind. At its more elementary stages, the process is stoked by an overdetermination of more mundane properties of everyday objects, such as mass, volume, size, and shape; which is then extended to theoretically postulated properties. Constancies need not be universal; the two sides of the evidential equation can fit together quite well in restricted domains of inquiry. When the constancies are pushed past their limit, the exceptions point to new theories, and constancies of greater generality. 2.1 An Example I ve already said enough to prove that philosophical theories of confirmation are very abstract. Unfortunately, example from real science are not always very accessible to nonspecialists, while the more accessible stories told by historians of science tend to include irrelevant details, while excluding details that are relevant. At the same time, made-up examples are often designed to illustrate a methodology, rather than to test it. Some kind of compromise is needed. The examples in this book are made-up and simplified, but they are intended to resemble a wide class of real examples in relevant ways. Imagine being in a time when homo sapiens first articulated some rudimentary hypotheses about the world. Suppose that they observed that pieces of wood float on water, and stones sink, without known exceptions, and on the basis of this evidence, they form the universal hypothesis that all pieces of wood float and all stones sink. The weaker inductive conclusion, that some wood floats and some stones sink, is too weak to make precise predictions. Of course, we know now that the strong version of the hypothesis is false some kinds of wood are so dense that they sink in water, and pumice stones float (at least some do). Nevertheless, there are many environments in which exceptions will not be found, and when they are found, they point the way towards better theories. Notice that even in these pre-scientific times, hypotheses postulate elementary constancies of Nature for example, the disposition to sink in water is constant in time. The stone that sinks today also sank yesterday and the day before yesterday. Objects do not change from stones to wood, for from wood to stones (the existence of petrified forests duly noted). There is always a sense in which the predictive content of any hypothesis can be viewed as arising from postulated constancies. The question is whether this point of view is particularly useful. Now imagine ourselves at a time in an environment in which we have conjoined many hypotheses stones sink, wood floats, metals sink, glass sinks, and cork floats. We Page 4

4 may even have found some exceptions the stone bowl floats if its placed on the water in the right way, suppose we may have modified our beliefs to accommodate all the known facts. How do our theories advance beyond this point? How do we discover deeper theories of the world that explain the predictive success of these low level generalizations, to the extent that they have been successful? An important step in this direction is to find a way of quantifying some properties. First and foremost, there is volume. Volume can be quantified by defining the length of some object, such as a rigid rod, as having unit length, and then determining the volume of solid objects shaped like a box by multiplying the lengths of the three sides. 1 But how do we measure the volume of irregular-shaped objects? One solution is to measure the volume of water displaced by an object when it is totally submersed under water. Liquids have the property of taking on the shape of the vessel into which they are poured. So, construct a rectangular vessel, and measure the volume of the displaced water by the usual method. This new mode of measurement can be applied to solid rectangular objects in order to check that the old and the new method yield agreeing measurements. The agreement is then extended to composite objects constructed from a number of different rectangular objects together. The law that the volume of a composite object is that sum of the volumes of its parts, previously assumed to be a self-evident truth, is now a well confirmed proposition. And this leads us to postulate that the new method is a reliable method of measuring the volume of irregular-shaped objects made of stones, wood, metal, glass, or cork. Weight, like length, can be quantified by first selecting something as a unit weight (e.g., a stone ) and using one of several different methods to compare the object with the unit weight. One way is to measure the distance that the object stretches a spring, and compare that to the distance that the unit weight stretches the same spring by computing the ratio of the two quantities. If a different spring is used, the distances will be different, but the ratio will be the same. That is, we observe the agreement of repeated measurements on the same spring and different springs. With this experimental knowledge in hand, it is almost instinctive that we should hypothesize that all objects whosoever have a time invariant property called weight. Weight is different from volume because two objects with the same volume can have different weights, and two objects with the same weight can have different volumes. At this stage of the story, there is no way of making a distinction between weight and mass. How are concepts made more precise? Suppose that we next discover a systematic disagreement of weight measurements using the spring method. We use a spring to compare the weight of a stone fully immersed in water with its weight in the air. Now we discover two quantitatively different weight properties weight-in-air and weight-underwater. We quickly accommodate the old experimental facts by substituting the term weight-in-air in all our hypotheses, in place of weight. Objects that sink have a weight-under-water that is confirmed to be invariant, and weight-under-water is always less than weight-in-air. But there are no other regularities found to hold between them. What about objects that don t sink in water? When we attach a spring to a piece of wood while it is floating in water, and pull slowly upwards, the spring will stretch and 1 Convenience has traditionally dictated that hands, feet, and paces, be used to gauge length, but the limitations of such standards are soon revealed by their failure to produce a very precise agreement of measurements. Page 5

5 eventually lift the object out of water, at which time the spring will measure its in-air weight. But what s its weight under water? What does that even mean? If we use the spring to push the wood under water, the spring will compress. That is to say, it will be stretched by a negative amount. It s weight-under-water is represented by a negative number! Yet, the measurements are repeatable, and they always agree. We now have three quantifiable and invariant properties of objects, volume, weightin-air and weight-under-water, with no apparent connection amongst them. What if such a regularity is found? In particular, suppose that Weight-in-air = Weight-under-water + (Constant Volume), which can be rewritten as Constant = (Weight-in-air Weight-under-water)/Volume. (1.1) The right-hand side of the equation involves numbers that vary for every object, so each object provides an independent determination of the Constant on the left-hand side. And, let s suppose that all of these measurements agree, out of those that are observed and calculated. The conjectured constancy for all objects, seen or unseen, is what gives the hypothesis its predictive content. And the Constant represents a new invariant quantity. The new inductive conclusion predicts each of the three quantities from the other two. For example, if we find a new stone, or a new piece of wood, or any new object, and measure its weight-in-air and its weight-under-water, then we can predict its volume. Or we could measure its volume and weight-under-water and then predict its weight-in-air. Previously, we could only predict the outcomes of repeated measurements of a quantity from a prior measurement of the same quantity. The new regularity, (1.1), does not lead to any novel predictions if the weights of every known objects have already been measured, which may show that the value of novel predictions is overrated. In either case, we have gained very little by way of understanding, or explanation. What does the Constant really represent? What is it a property of? Or is it a universal constant of nature, a property of everything, or nothing in particular? Is it possible to answer such questions? The physical regularities I m describing are familiar from high school physics, but not easily recognized in this form. The way I am describing them is meant to illustrate the sense of puzzlement that scientists sometimes experience. It s like a jigsaw puzzle. Some pieces are in place, but other pieces are missing. In this case, the missing pieces involve both empirical and conceptual elements. On the conceptual side, I will introduce the concepts of density and buoyancy, and show how the new quantities enhance our understanding. Given the three properties possessed by every object volume, weight-in-air and weight-under-water we can define new quantities in terms of these. So, let s just add two definitions, which seem quite arbitrary at first. First, the density of an object to be its weight-in-air divided by its volume. Density Weight-in-air/Volume. (1.2) Density values depend on the units chosen for weight and distance. Density is weight per unit volume. Notice that the Constant is also a weight per unit volume, which will be important in what follows. Page 6

6 The content of all previous hypotheses have presupposed that we have the ability to identify and re-identify individual objects, for otherwise the hypotheses would have no testable content. But it is equally true also have a pre-scientific ability to classify objects according to the kind of material of which they are composed. Two different objects can be made of oak, glass, or iron. These properties are qualitative. Science never succeeds in quantifying everything and qualitative properties always play an essential role. The weights of objects composed of the same material will vary from object to object, as will their volumes, but the densities are always the same. Thus, density is confirmed to be an invariant property not merely of individual objects, but of kinds of objects. This conjecture has new empirical content because it enables us to predict the weight-in-air of an object from a measurement of its volume, or its volume from its weight, and even the kind of object from its density. The last prediction is one made famous by the story of Archimedes when he was asked to determine whether a crown was made of solid gold (gold being the heaviest of all natural materials known cannot be replaced by a composition of other metals to produce the same density). So, the definition of density is accompanied by the conjecture that the densities of certain list of materials are invariant over time and between objects composed of the same material. There is an intuitive sense that the density is ontologically more fundamental because it has broader invariance properties. The weight of an object then becomes a derived quantity, determined from its density and volume. Recall that the law of the composition of weights says that the weight of a composite object is the sum of the weights of its parts. It follows that the density of a composite object can be predicted from the densities and volumes of its component parts. Returning to the regularity described in (1.1), notice that the Constant has the dimensions of density, that is weight per unit volume. Perhaps it is the density of some material? Here s another clue. The formula involves the difference in the two kinds of weight. This motivates a second definition. Buoyancy = Weight-in-air Weight-under-water. (1.3) Weight-in-air is always greater in magnitude than weight-under-water, so buoyancy is always a positive quantity. It s worth noting that buoyancy is a vector notion in modernday physics, with a magnitude and a direction. A force vector, in other words. In fact, weight and buoyancy are all forces. (1.3) is the definition of the magnitude of buoyancy because the notion of force has not been introduced. It should be remembered that buoyancy forces an object upwards. Whether the body moves upwards depends of whether its buoyancy is greater than its weight. Here s another clue. A piece of water, a given volume of water, has a definite boundary, even though it is not marked any physical way. No imagine that this piece of water is under water. The boundaries of the piece of water may change over time, but the magnitude of its volume will not. What is the buoyancy of water? The weight of water under water is zero (because water just floats under the water), so the buoyancy of water is equal to the weight of the water in air. Now apply equation (1.1) to this experiment. It says that the Constant is equal to the density of water. The thought experiment not only leads to a testable prediction, but provides a physical interpretation of the Constant itself. Suppose that the prediction tested. The density of water is not only found to be a new invariant quantity (yielding the same number for any piece of water), but it is also found to be equal to the previously measured Constant. Suddenly, the scientific mind is racing. Page 7

7 Now think about the stone under water held up by a spring. What accounts for fact that the spring is less stretched in this situation than when its out of the water? The regularity in (1.1) tells us that the stone s weight-under-water is equal to its weight-in-air minus the magnitude of its buoyancy. The minus sign tells that the buoyancy has the effect of reducing the objects weight by an amount equal to the weight of the water displaced by the object. The spring is measured the magnitude of that reduced weight. For a piece of wood, the situation is similar. When forced under water, its weight is reduced by an amount equal to the weight of the water it displaces. In this case, the weight of water is greater than the weight of the object itself because the density of water is greater than the density of wood. That is, the weight is reduced to a negative value, which is why the immersed piece of wood pushes up on the springs. Suddenly we are back full circle to the facts about which objects float and which sink in water. The new conjecture is that all objects that have a density less than water will float, and all objects that have a density greater than water will sink. The story is striking. We began with the false inductive generalization that all wood floats and all stones sink. They are false because some rare kinds of wood do sink, and some rare kinds of stones float. Yet we have succeeded in correcting these false generalizations without ever seeing any counterexamples to them. We still don t know, in the story, whether there are counterexamples, but we have learned that if there is wood that has a density greater than water, it will sink, and if there are stones that have a density less that water then they will float. We withdraw these conjectures from our body of beliefs because wood and stones vary in the density property, and we can see no principled reason why the densities of all wood and all stones must fall within some bounded range of values. The new generalization requires us to measure the density of a new material before we predict whether it floats or sinks. It therefore predicts less in advance about new kinds of wood and stones, but makes up for it by its greater generality. It applies to every kind of material. The new hypothesis sticks its neck out in different direction. This instance of scientific learning illustrates a point that will recur throughout this book. The process begins with everyday knowledge of kinds of materials, wood, stones, and metal, as well an ability to identify individual objects; this is the same object, the same spring, and so on. Everything else is built upon our ability to attach numbers to distance quantities, the distance that a spring is stretched and the length, width, breadth of a rectangular object, and so on. Let me refer to these as observational quantities. Then we introduce three new quantities, which we called volume, weight-in-air and weightunder-water. They were tentatively introduced by specifying at least one method for assigning them numbers based on distance measurements. The resulting measurements of volume accord well with our qualitative volume comparison; those two objects have the same size, but that one is clearly bigger. The agreement between qualitative and quantitative judgments of volume is really the lack of any clear-cut disagreement. It is sufficient to justify the use of the same word, but we should not be mislead into thinking that volume is a purely observational quantity. As soon as we discover that the method of multiplying the dimensions of rectangular objects produces the same number as the water displacement method, this equivalence becomes an integral part of the notion itself. But this is a theoretical assumption it could be false in some untried instances. Even the law that the volume of a composite Page 8

8 object is the sum of the volumes of its parts could be falsified empirically. If it were found to be false, then we would find a way of adjusting these assumptions so that there were no known counter-instances. That is exactly the point. As soon as volume quantities are operationally defined so that they are successfully overdetermined by what s observed, they begin to take on a life of their own. So much so, that we are willing to adjust the operational definitions without saying that we are defining a new quantity. Volume is being thought of as a real autonomous property of objects, in spite of its theoretical nature. Other theoretical quantities, such as weight, density, and buoyancy, acquire the same status for the same reasons. The main lesson of this chapter is that scientific induction is not merely a process of extrapolating observational facts from the past to the future. It also involves a process of conceptual innovation. New theoretical quantities are introduced, each of which is tied to a network of operational procedures for assigning them numbers. Over time, the networks is strengthened and refined so that previous overdeterminations are preserved and extended as new data is accumulated. This process of concept development is inextricably tied to the process of hypothesis formation and testing. Induction is multidimensional, and while the logical problem of induction still applies, the problem of saying what makes one induction more reasonable than another should take both dimensions into account. Page 9