What is a scientific theory? An Introduction.

Statement of purpose and preliminaries.
Objectives.
This is intended to give a readable introduction to recent developments surrounding the question "what is a scientific theory?" Its audience is students who are interested in philosophy, but beyond that very little will be assumed. I don't even presuppose that students have had a course in logic. It will be helpful to have some interest in the natural sciences, but an elementary college level knowledge is all that is needed. A good high school level of exposure will probably be enough.
A second objective, more ambitious, is to make the topic at least slightly interesting. The material is somewhat technical, and for a good reason. Philosophers are interested in what a scientific theory is because they want to use to answer to go on to look at further questions. Intuitively interesting questions, such as whether there is a difference between science and religion, or how the sciences explain things, turn out to need an answer to the question of what a theory is before they can be adequately discussed and answers proposed. But our best examples of scientific theories are often explicitly mathematical and always are carefully phrased. We wouldn't dignify a topic with the name of science unless it was systematic and attempted to avoid ambiguity.
So I'll proceed slowly, systematically explaining the various views that have been proposed, I hope in a comprehensible way. I'll try to constantly relate the issues to examples, and I'll try to propose things to do to illustrate some of the points I'll be trying to make. For example, some people (some of my students for example), think that the emphasis on systematic theories and the use of mathematics is constricting in the sciences. You might usefully try to think of theories that have been influential that lack this feature. (Does Darwin's original statement of the theory of evolution count?) Perhaps the emphasis on formal and mathematical theories in the sciences is misplaced, but it can hardly be denied that it exists. Things are unlikely to change unless paradigmatic examples of scientific theories can be proposed that lack this feature.
I do hold a position on these topics, and will state it as it arises. I will do my best, though, to present the issues in a balanced way.

Preliminary considerations.
The first point to make is that the word 'theory' in this context does not mean 'conjecture' or 'speculation'. In ordinary language, a discussion of why it is that the President of the University unexpectedly resigned can often include the proposal of various 'theories' with virtually no evidence in their favor. (Alcoholism, affairs with students etc.) We sometimes say we have a theory about something when we mean we have a proposed explanation, or a link to something else we know. But every explanation or interconnected view that scientists use is called a theory and this need not indicate that anyone is uncertain about it or lacks good evidence in its favor. Things that we now accept without question, such as the fact that illnesses are caused by specific kinds of diseases, were once called "theories", and some still are. (When you think about it, illnesses might form a continuous spectrum, rather than being classifiable into different discontinuous kinds.) We still talk about the theory of relativity, although no one is in any doubt about it (well, almost no one). It's a theory that the earth goes around the sun, and a theory that there are organisms too small to see. Many contemporary philosophers believe that our ordinary view of the world in everyday life is a theory, because there is an interconnected set of expectations about the things we see around us. It does sound odd to think about a 'supermarket theory', but when you try to program a computer to answer simple questions about supermarkets, its surprising how much you have to tell it.
A second simple point concerns the practice of working scientists. It is not unusual to find scientists distinguishing between a hypothesis and a theory. A hypothesis, as they understand it, is a claim or conjecture that has no evidence in its favor. When we get evidence in its favor, it's called a theory. Once again, this isn't what the word 'theory' means in the philosophy of science.
Very loosely defined, in the philosophy of science a scientific theory is an interconnected group of claims that are capable of being supported or refuted by possible observations and which is capable of providing scientific explanations. There need not be any actual evidence in support or contrary to it, so long as there could be, and similarly, no-one need actually appeal to it to explain anything, as long as they could do so.
It might seem that just any set of claims will satisfy this loose definition, so that we'll have to count political speeches and children's stories as scientific theories. Or it might seem that the definition is completely uninformative until we get a theory of what it is to give a scientific explanation and how scientific confirmation by evidence works. It turns out, though, that disputes about what it is to confirm or explain in science depend upon a view of what scientific theories are. We have two rival theories of what it is to be a connected series of claims that are capable of explaining and facing the evidence, which are independent of other rival theories of what it is to explain or confirm scientific theories. We'll be looking at these two rivals in detail.

A brief history of the problem.

There are two rival views of what a scientific theory is, the syntactic, or received view, and the semantic view. The syntactic view developed first, and views a scientific theory as a set of sentences. The semantic view of theories views them as a class of models. In the next two sections we'll look at these in detail, and then at the various points that might be made in favor of one or the other.

The syntactic view has origins in antiquity. Euclid's Elements of Geometry begins with a few simple definitions and a set of five axioms. These axioms supposedly represented a set of basic truths about geometry, for example, "we can draw a straight line between any two points". From these axioms a sequence of indubitable inferences or rules of inference enabled the construction of proofs of various results. Each result is called a theorem of Euclidian geometry.

Euclid's great achievement was greatly admired and often emulated. Although originally a mathematical theory, it could easily be adapted to a scientific theory, and is clearly Aristotle's ideal of a scientific theory in the Posterior Analytics. Probably the most famous example of it is Newton's mechanics. Newton's three laws of motion are probably familiar to those of you with some high school physics. Together with some initial observations, usually called initial conditions and Newton's law of gravitation, Newton was able to prove theorems that predicted when planets would occupy positions in the sky, just as Euclid proved his theorems.

Around the turn of the century the great philosophical and mathematical genius Gottlob Frege succeeded in formulating a set of rigorously defined rules of inference that are usually called predicate logic and which are taught in every basic logic course. Instead of this term, we will use the term first order logic for any system of inference rules like these that always preserve truth, and can be rigorously defined in a way we'll look at shortly. When the laws of a scientific theory are used as axioms, and the theorems are the result of operating on these laws and initial conditions, we will say that this is a syntactic view of the scientific theory. The syntactic view was extremely popular, and virtually everyone used it for several decades.

Popular and powerful though it was, the syntactic view suffered from several disadvantages. First, although the laws were rigorously defined, they were extremely cumbersome to use, and often difficult to apply to the use of actual scientific theories by actual scientists, who tended to use less well defined, but more convenient, forms of inference. Second, because the rules of inference operated upon sentences, the syntactic view could not avoid becoming embroiled in debates about language, which were often difficult, and tended to distract philosophers of science from the practice of science itself. (For example, Newtonian mechanics in French seems to be a different theory from Newtonian mechanics in English according to the syntactic view, because the sentences are different. This is absurd. But to say they are really the same theory means giving an account of when French words mean the same as English words - which is very difficult to do, and nothing to do with Newtonian mechanics.)
Another criticism was rather technical. We ordinarily assume that points in space can be any distance from a given location. That is, given a straight line in space we normally assume that whatever is true of any number we can think of will be true of the number of meters along that line. This is part of what we intend to do when we measure space by numbers of meters. Well, it turns out that whatever set of axioms you use for your scientific theory, first order logic will always fail to give you some truth about numbers, and hence will always fail to give you some truth about some number of meters from a fixed point. But, the objection goes, scientists always intend all the truths about numbers to be truths about numbers of meters. So first order logic and the syntactic view of theories will always omit to say something that scientists intend to be true when they use numbers of meters to describe space. This is a consequence of Gödel's proof, which is one of the great achievements of twentieth century mathematical logic. The proof show that no axiom system and system of proof can prove all and only the truths of arithmetic. Kurt Gödel first proved it in 1931.

Because of all this a new view, the semantic view of theories came to prominence in the nineteen seventies and eighties. In order to get an intuitive grasp on this, it is necessary first to understand the rather odd idea of a mathematical structure or model.

Think of a chessboard at the start of a game. All games of chess begin with the same arrangement of pieces, and so every game looks the same at the beginning. Imagine this 'start point' of the chessboard as a location, perhaps as a photograph pinned to the top of a billboard. There are exactly twenty legal first moves in chess, so take a photograph of the way the board looks after each of these and line them up as twenty 'first move' photographs each connected by a piece of string to the 'start point' photograph. To each of these there are twenty legal responses, so take four hundred photographs of the chess board after two moves and connect each legal move to the photograph of the board that it precedes by a piece of string. After two moves, there are four hundred possible states of legal chess.

Obviously we could continue the exercise. The billboard would rapidly become unmanageable, but that need not stop us. We could store the legal chess moves in a computer and program it to generate all the legal moves for us. Or we could store them in a vast set of index cards, or on a CD (actually, we'd soon need a lot of CDs). If we kept going, we'd be constructing a list of every possible game of chess, most of them very silly.
If you're familiar with the rules of chess, you'll know that if we keep going, eventually every game has to end. Obviously, storing this huge amount of information in a physical object like a computer is impossible, but that doesn't stop us talking about it or thinking about it. We can, for example, get very rough estimates of the number of possible states of the game, and we can sometimes show that a particular photograph of a chessboard could not be the result of any sequence of legal moves.

What we're thinking about is really an abstract mathematical object. It's actually called a 'tree' because new states of the chessboard arise out of old ones, but there are no circular paths along the pieces of string. The tree of legal chess games doesn't exist physically, but it's an abstract object. There are facts about it; some sentences about it are true and some false, so it's not quite an imaginary object. Its more like a set, or a direction, it isn't a physical thing, but we can reason about it and use it.
In the same way, we can conceive of future courses of events for physical objects in the universe. When we state a scientific theory we are forbidding some courses of events from a given starting point and allowing others. We can think of the path of a billiard ball along a straight line, for example, as a line sloping upwards, with the time dimension as the vertical one. If the ball is slowing or speeding up appreciably, the line will be a curve. We can simply think, intuitively, as a scientific theory as a tree of possible future states of the universe. Actually, there is a forest. A tree grows from any possible initial state of the universe.

If you like, you can identify these courses of events with sets of numbers. These sets will have a rather complicated nested structure. The various set-theoretic representations of models are a mathematical way of representing models, and presumably different scientific theories will be more conveniently represented as different mathematical representations. Included in these complicated pieces of set theory will be, for example, ordered pairs of numbers that scientists regard as the energy of a photon in electron volts followed by its frequency per second. Scientists can then compare these numbers to those discovered by experiment to discover whether the theory accurately matches reality.
(An important word of caution is in order here for students who have had logic. The word 'model' is also used for an interpretation that makes all of a specified set of sentences true. This is not the kind of model we are talking about here. The consistency and completeness proofs of first order logic demonstrate that, given a set of sentences, the interpretations that make that set true are just those in which the sentences derived from that set by the classical proofs are true. That is, if we can prove a sentence from members of this set using classical logic, that sentence is true if and only if the premises of the proof are true. But the interpretations that make, for example, the sentences of Newtonian mechanics true will include non-standard models of the real numbers, which are not included in the models of Newtonian mechanics that the semantic view of theories allows. And if you didn't understand this paragraph, you can safely ignore it.)

Now let's look at the two views in more detail.

The syntactic view of theories.

1. What is a proof anyway?

Since antiquity, mathematicians have constructed proofs. These proofs started with certain apparently undeniable truths, and explicitly drew out their consequences. These consequences were as explicit and obvious as possible, so that anyone who doubted the theorem would be led by indisputable truths from the indisputable premises to acknowledge, eventually, the truth of the conclusion. But what made the original truths undeniable, and what made the inferences so certain? Historically, philosophers and mathematicians have appealed to the light of human reason, instilled by God or simply stated to be a fact about human nature. The German philosopher Immanuel Kant proposed a particularly brilliant and sophisticated explanation that held sway for over a hundred years, until the early twentieth century.

By then, mathematical developments had thrown previous accounts of proof into disarray. For one thing, the most conspicuous success of a proof system, Euclid's Elements of Geometry was under challenge from mathematicians. Euclid's apparently undeniable axioms had been challenged, and various very odd new geometries had been suggested. These were certainly unfamiliar, and unexpected, but if the traditional account of proof were correct, they had to be more than that. For if the traditional account were correct, Euclid's axioms were undeniable truths, so the alternatives had to have something profoundly wrong with them; they couldn't be true. The new geometries were peculiar, but not that peculiar. Even if they weren't true, it looked as if they could have been if the universe had been different. (As it happens, one of them actually is true, but even before that fact was known, the traditional view of proof was abandoned.)

First order logic provided a nice clear account of what it was for a set of sentences to be impossible. Let p be any sentence (e.g. 'the sky is blue' or 'sugar is poisonous' or 'all cats are mammals' - each of these could be p). Now let not p be the denial of p (e.g. 'the sky is not blue', etc.). A set of sentences is impossible if, were it true, both p and not p would have to be true too. In such a case, we say the set is inconsistent.
But what does it mean to say that if a set of sentences were true, other sentences would have to be true too? Again, first order logic had an answer. In elementary logic, students learn a set of inference rules. If the course is sophisticated, students also get to prove that, if some other sentence must be true if a set of sentences is, then we can get that sentence as a conclusion if we apply these inference rules in the right way.
And hence, we get a very nice account of a proof. Something is a proof if and only if it is a string of sentences each of which we get from the previous ones by the application of the inference rules. A set of sentences is possibly true if and only if you cannot prove both a sentence and its negation from that set using the inference rules. (A sentence is necessarily true if and only if it can be proved using no assumptions.)

The rules of inference had a very nice property. Intuitively, we can say that once a very simple set of words are defined, we can tell whether a rule can be applied to some sentences just by looking at whether they contain these simple words in the right order. For example, if we only define 'all … are…' then we can tell that from 'all cats are mammals' we can infer 'if Winnie is a cat, then Winnie is a mammal'. The words '… is a cat' and '…is a mammal' could mean anything, and 'Winnie' could refer to anything, as long as it is a name.

Put another way, whether a sentence can be inferred by the rules from another depends only on the syntax of the premise. That is, it depends only upon which words occur in which order. A computer that knew nothing of the world, and was programmed only to recognize certain words, could be programmed to apply the rules of inference to sentences fed as input. The 'shape' of the sentences alone determines whether the rules of inference apply, not what they mean. Of course, what we're trying to capture does depend upon meaning, for if an inference rule applies to a sentence, then the conclusion we draw using that inference rule must be true if the original premise was, and whether sentences are true or not depends upon what they mean. The useful thing about first order logic is that we can tell something about the relations between sentences so far as truth is concerned purely by looking at their syntax.

The syntactic view of theories uses the inference rules of first order logic to account for proofs from statements of scientific laws and initial conditions. That is where the name 'syntactic view' comes from.

Now lets look at a simple example of the syntactic view in action.

2. A simple example

Start with a simple statement of two laws: 'The density of an object is equal to its mass divided by its volume' and 'An object floats on water if and only if its density is less that that of water'.
Then add some initial conditions: 'Fred is an object and Fred's volume is 8 cm3', 'Fred weighs 4 g', 'the density of water is 1 g/cm3'.
It seems to follow immediately that 'Fred floats on water'. However, if you've done some logic and try to prove it using your inference rules, you'll find you need an additional premise or two: '4 divided by 8 is 0.5' and '0.5 is less than 1'. It's an important point that these rules need to be explicitly added. The advantage of first order logic is its systematic, explicit, and syntactical nature. Its disadvantage is that it is unbelievably cumbersome and tedious to actually apply. Although the overwhelming majority of inferences that mathematicians draw can be spelled out in terms of first order logic, mathematicians almost never go through each tedious step. Rather, their familiarity with mathematics allows them to jump directly to the mathematically significant conclusion without spelling everything out in the tortuous detail that first order logic would require.
The conclusion we proved from this trivial theory, 'Fred floats on water', is something we can check to be true or false. Hence, according to the syntactic view, predications issue from a theory simply by being proved from its axioms, which are the laws of the scientific theory and the initial conditions.

3. Empiricism

There is an important historical note to add to the story as I have told it so far. In the heyday of the syntactic view of theories, a position known as empiricism was very popular. Consequently, the syntactic view of theories was often associated, in its statements, with the attempt to articulate empiricism. Empiricism is the view that observation is the sole source of knowledge of the physical world about us, and that its range is very limited. So while we could observe whether something was water or not, and whether something else floated in it, we could not directly observe things like densities or masses of objects. These quantities had to be inferred from things that we could directly observe, such as whether a scale balanced, or which parts of an object were in contact with gradations on a ruler. For empiricists therefore, the initial conditions stating Fred's mass and volume were themselves conclusions drawn by a theory from other pieces of data. The importance of this was that virtually no significant sentence of a scientific theory could be tested by itself. Some link was required between the data from which Fred's length and mass were inferred, and the length and mass. An example would be: 'if a scale with a body labeled "8 grams" balances with a body, then that body weighs 8 grams'. Coupled with the sentence 'Fred is a body, which balances on a scale with a body that is labeled "8 grams"' we can infer using first order logic again that Fred weighs 8 grams.
But obviously even this is too simple. Not every body labeled "8 grams" actually weighs that much and we have to know what a scale is, and when one is accurate. Hence, more scientific theories entered the picture, and these in turn introduced yet others. Only the entirety of scientific knowledge at a time, it appeared, actually faced the data. We almost invariably needed to introduce a sentence that linked directly observable and indirectly observable objects and quantities whenever we tested a sentence of a scientific theory. Such bridge principles, as they were often called, then needed further evidence to establish them, and an infinite regress started.

Not only that, but the empiricists of the day argued that only words that referred to directly observable objects and properties had any immediately ascertainable meaning. The meanings of other words, like 'density' and 'mass' were inferred from our understanding of words that referred to directly observable things, like 'this floats' or 'this pointer and the numeral "8" are touching'. What our words mean is, after all, a matter of the way in which we use them, and constitutes a theory. So that theory, the theory of meaning, is known only because of the connections between observed utterances of words and the observable circumstances that obtain when the utterances are made. It seemed to follow that we must learn words for directly observable objects and properties first and then infer from these the meanings of words for other things.

4. Pragmatism.

This emphasis on language as a central concern for philosophy waned with the work of Willard van Orman Quine which was very influential in the nineteen sixties and seventies. Philosophers in general, and philosophers of science in particular, began to see the concerns of their subject as much more intimately connected with the study of the way scientists actually worked and what they discovered about the way humans think. Consequently, the issues in the philosophy of language that the earlier protagonists of the syntactic view had introduced came to seem irrelevant and distracting. Moreover, Quine's work also shifted philosophy away from empiricism and towards pragmatism. It is difficult to state exactly what pragmatism is in a brief way, but very roughly it includes the view that the only way in which a true theory may be recognized is through its practical success, especially success with the data. It is possible to be both a pragmatist and an empiricist simultaneously, but those pragmatists who were not also empiricists usually emphasized that various incompatible theories might be equally practically successful. Because the data did not allow us to distinguish between these rivals, and because we had no other means of identifying which was true, we selected from the theories by simply using those that were most convenient for us. We prefer simple and powerful theories, for example, because they are easy to use in a variety of circumstances. Unlike empiricists, pragmatists were generally not worried about believing in the existence of objects that were indirectly observable. So long as these objects were helpful to us, they earned their keep and were perfectly acceptable.

The rise of pragmatism meant that the syntactic view of scientific theories dropped many of its connections with specifically empiricist concerns. The desire to avoid commitments to anything but directly observable entities ceased to be of central interest, and the desire to explain the meanings of all words in terms of those that referred only to directly observable entities vanished almost entirely.

But the syntactic view itself continued to be the dominant view of scientific theories. This is an important point to note, because some of the criticisms of the syntactic view have focused upon those aspects of it that were closely linked to the earlier empiricism. It is true that the original proponents of the syntactic view were virtually universally empiricists, and that their views on meaning and observation have come under such sustained attack that no-one now can embrace them without a lengthy and detailed defense. Nonetheless, Quine himself continues to hold the syntactic view, and the collapse of early twentieth century empiricism occurred prior to the widespread acceptance of the semantic view. The view that there is no genuine distinction between directly and indirectly observable entities and that problems of meaning can be safely ignored in philosophy is quite consistent with the syntactic view of scientific theories.

The Semantic View.

What do real scientists do with actual theories?
One of the most attractive features of the semantic view of scientific theories is that it seems to focus directly on the things that actual scientists say and do in practice.

This gets more obvious as the theories get more sophisticated. Even for simple theories, though, it can be made plausible. Consider the example just given of calculating the density of something. One of the problems we encountered was that under the syntactic view, all the mathematical premises had to be made explicit. We had to add the sentences '1/2 is less than 1' for example. This seems to be a very unnatural interpretation of what we actually do in practice. What we seem to do is identify masses with numbers, volumes with numbers, and densities with numbers. We then use the properties of numbers and their relations to each other to get the number we've identified with the density from the numbers we've identified with the masses and volumes.

The important point here is that the sentences, just by themselves, need not mean anything specific. This symbol: '1', for example, need not refer to the number one. Instead, it could be the name of the Eiffel tower or the Pope, or Mahler's fifth symphony. The big advantage of first order logic, as an account of proof, was that we did not have to worry about what symbols meant, by their positions relative to one another alone we could discover whether something was a proof or not. But the big disadvantage of this is that some symbols that are not simply matters of logic, but of mathematics, seem to be regarded as having specific mathematical meanings by scientists in every case.

The semantic view acknowledges from the start that mathematics will be regarded as privileged. What scientists actually do when they propose or apply a scientific theory, the semantic theorists say, is to create a mathematical object. They then identify parts of this object with parts of the world. This enables them to design experiments and make predictions. The 'proofs' that scientists perform in this process are simply a means of developing predictions from a mathematical structure that they have postulated beforehand.

There are a variety of accounts of how scientists do this. Instead of investigating them directly, I will try to develop the simple example already given.

The Semantic View: A simple example.

I'll begin with some elementary properties of the real numbers. The first order of business is to say what 'real number' means. Children learn early how to count things ('one, two, three…'etc.) There are obviously things that they count (ducks, blocks etc.), but the numbers themselves are not these things. (What thing corresponds to zero? How does the same card manage to be both three when we're counting aces and five when we're counting hearts, at the same time?) Some people have concluded that there are no such things as numbers, but this view turns out to be surprisingly difficult to sustain. (If there are no such things as numbers, how come there's a right and a wrong way to balance a checkbook? How come there's a right and a wrong way to solve equations and differentiate?) So lets suppose there are such things as numbers. Obviously they don't have a location; it would be absurd to look for the number three. We can't see them, at least not without considerable chemical assistance. They don't get any older; there is no birthday for the cube root of two. Still, that need not prevent them from existing. One might say the same things about space or the constitution of the Iroquois (or God, or singularities of space-time).

Logicians call the numbers 0, 1, 2, 3, 4, … the natural numbers. (Mathematicians tend to omit zero in their use of this term.) If we add the negative numbers … -3, -2, -1, 0, 1, 2, 3 … we get the integers. Obviously both are infinite in number, but then so might be points in space or time. Now add all the numbers that are fractions, e.g. ½, 4/5, 1289764/-1 etc. This set is called the rational numbers.
You might think that the rational numbers are all the numbers there are, but in fact numbers like pi and the square root of two cannot be represented as a fraction. If you find this surprising, you're not the first person to do so. The numbers were once called 'surds', and it is possible we get our word 'absurd' from this root ('ab' means 'from'). Today, we call these numbers irrational numbers. If you add the irrational numbers to the rational numbers you get the real numbers which are all the numbers of any unit of length (meters, for example) in a straight line of infinite length and arbitrary zero point.
(Proof that there are irrational numbers. I will prove that the square root of two is irrational. Suppose it is rational, that is, suppose it can be expressed as a fraction. I will now reduce this assumption to absurdity. If it's a fraction, it must have a lowest form, let m/n be that form. (m/n)2 = 2, because it's the square root of two. So m2 = 2n2, so m2 is even. Therefore, m must be even, because all odd numbers have odd squares. So m2/2 must also be even, because half of the square of an even number is always even. But m2/2 = n2, so n2 is even. So n must be even. But then both n and m are even and so n/m cannot be the square root of two in its lowest form. This contradicts the stipulation that it is. Hence, the assumption upon which that stipulation is based, that the square root of two is a fraction, must be false. Neat huh? The Pythagoreans proved this about five hundred years before Christ was born.)

The semantic view of theories allows considerably greater mathematical sophistication than just the real numbers, but for simplicity's sake I'll stick to them. The major issues arise with respect to these numbers, and I think they are evil sufficient unto the day.

Now think about how we actually calculate the density of something. We identify non-negative real numbers with the mass and volume of the object and divide them. The operation of division is defined for the real numbers, so this is quite straightforward. If you like, you can think of mass as the x-axis of a graph and volume as the y-axis, then imagine a third axis at right angles to the other two, the z-axis, as the volume. Then the simple theory states that any object will fall somewhere on the surface defined by z = x/y. Not only that, but different samples of the same pure substance will always fall on the same point on the surface, and any sample of any solid will float on any liquid that has a lower density. If necessary, this simple structure could be elaborated, for example by adding another dimension for temperature and relating changes of density to temperature.

This is what the semantic view means by saying that we construct mathematical models when we construct theories. We simply identify quantities like mass and volume with real numbers and use relationships between those numbers (for example, multiplication and division) to get numbers which we then identify with other quantities. There are different views about how this is done, proposed by different thinkers, but this is the core idea.

The semantic view of theories really does well with mathematically sophisticated theories such as space-time theories and quantum mechanics. Here we need to integrate and differentiate vectors in spaces, and the mathematical structure of the theory is quite sophisticated. One shouldn't think it is only in these areas that the semantic view is useful. Models of genetics can also be constructed. Games like chess, and relationships like those found in the periodic table are also amenable to this treatment.

The important distinction between the semantic and syntactic views of scientific theories is that the semantic view proposes a specific, privileged interpretation for scientific theories. Usually, though not invariably, this is proposed to be the one that the scientists themselves are using. For example, in the case of density, we could use only the rational numbers to represent the densities of objects, because every measurement we actually make can be represented as a rational number. (To measure something to possess an irrational value for a quantity we'd need to have infinitely precise measuring instruments, and this we obviously don't have.) But scientists themselves speak, calculate and write as though the real numbers are what they mean, and so this is preferable to represent the model. We take the scientists at their word, and suppose that the models they are talking about are the ones that they say they are.

I have said that we "identify quantities with real numbers" but how do we do this? The literature on the semantic view is not always clear here, but I think the answer is something like this. It is just a fact that certain operations we can perform with objects can successfully be predicted based on relationships between numbers. An example will help demonstrate this. Suppose you have a simple measuring device such as the bathroom scales. Now suppose you put a couple of objects on it, yourself holding the dog for example. Note the numeral that appears on the scales. Now put each object singly on the scales and note the numeral that appears. The relationship between the total weight of objects on the scales at a time and the numeral indicated at that time can be predicted by the following procedure. Take the number that the numeral names according to our ordinary conventions and suppose that that is the total weight of the objects on the scales at that time. We can predict, for example that you and the dog together will indicate the numeral that names the sum of the two numbers indicated by the numerals that appear when you and the dog are on the scale separately.

Put another way, it's just a fact that mathematical objects and relations are 'like' physical objects and relations. We discover general relations between physical objects that obey this 'likeness' relation, and then we can use mathematics to predict their future courses of events, or design experiments. We can even give an account of what the 'likeness' relation is in terms of mathematical models, and hence model the thing we are doing when we use mathematics to make predictions.

A final point, which I find interesting. Whether one is a syntactic or a semantic theorist, one faces a difficulty if one is a realist. At even moderately advanced levels of physics these days, there is a very deep involvement of physical theory with mathematical objects. Obviously, this is something that the semantic theorists emphasize. But according to any sort of realist, there ought to be a distinction between a physical object - one that is physically real - and a mathematical object - one that is part of the mathematical structure of the theory.

Bas van Fraassen has pointed out that it is very difficult to draw this distinction. It's difficult to say whether the wave function in quantum mechanics is a physical or a mathematical object. It's even quite difficult to say whether its square is a physical object, although it has a direct physical interpretation as a probability. What properties do physical objects and properties have that make them different from mathematical objects and properties? The question is quite difficult even for some 'easy' cases. For example, suppose we propose that a sufficient condition for being a physical object is that an object transmits energy. That makes photons physical objects. But what about virtual photons? These odd beings have been proposed as carriers of electrical energy. It's difficult now to see whether they are a mathematical device or a real carrier of energy. A realist is forced to draw a distinction between the mathematical and physically real parts of a theory when neither can be directly observed. If the proposal is that physically real things can be indirectly observed, then the question immediately arises whether we can indirectly observe virtual photons, wave functions etc. and the answer appears to be that we can do so if these are real. Clearly, that will not do as an explanation just by itself. The problem strikes me as very difficult and very interesting, and seems to me to have received too little attention.

Syntactic or Semantic view?
A dialog.

Syn: Let's begin by stating the advantages and disadvantages we see in each other's positions.
Sem: O.K
The main advantages I see in my position fall under three headings:
First, it characterizes scientific theories directly in terms of what scientists seem to be talking about when they discuss and use their theories. It easily handles their statements directly, in terms of what they seem to be talking about. Consequently, it is very easy to apply to real examples.
Second, it avoids problems in the philosophy of language, which aren't helpful to philosophy of science, where we examine the activities of scientists themselves, (which is what science is, after all).
Third, your position uses axioms and first order logic to derive results from them, but this isn't how we reason, and your position ignores the sophisticated mathematical structures we require to formulate scientific theories.
Syn: Well, that seems to me to cover both your view of the disadvantages of my position as well as the advantages of your own!
Let me begin by replying to the claim that I worry too much about language. It seems to me that the only way to propose a scientific theory is to say something, and the only way for someone else to know what your theory is is to understand what it is you say. We're not gods, we can only smear dyes and shake the air to communicate, and so we shouldn't ignore language. In fact, that is precisely what is wrong with your approach; it ignores language.
Sem: Well my reply to that is that the philosophy of science is not about language. We're trying to examine what it is that scientists actually do, to solve the problems that arise for scientists themselves, or arise naturally from their disciplines. Can you name a single debate scientists have ever engaged in that depended upon the solution to some problem in the philosophy of language?
Syn: I suppose not, not directly. Maybe some of their problems have actually turned on the philosophy of language though, and they didn't realize it. Newton and Leibniz on space and time, for example.
But I don't agree with your view of philosophy of science. Its true that part of philosophy of science is examining the problems you state, but the subject isn't just a chambermaid in the great palace of science, it has problems and interests of its own, and one of them is how scientific theories are communicated.
Sem: That seems to me to be exactly the reverse of the way things really are. How we communicate is a subject for empirical investigation, not philosophical speculation. It's a matter for psychology, not philosophy, and psychology is a science, or ought to be anyway. We need to presume the scientific method works in order to get anywhere understanding language, so we're stuck with the assumption that scientists know what they're talking about, and the semantic view is the embodiment of that assumption.
Syn: I think I could agree with the view that empirical investigations are useful in understanding human use of language, but still disagree that philosophy of language is irrelevant to the philosophy of science.
It seems to me that there are philosophical problems that arise concerning science that are not scientific problems. For example, the debate between realists and empiricists. Find out as much as you like about the physical world, you'll never be able to find out whether the things you can't see are really there or not.
Sem: I don't believe that. There are arguments on that topic that stem from the actual details of what scientists do.
But I'll accept your point, at least for the sake of the argument. Suppose that there are independent philosophical problems concerning science. How would it follow that your view of what scientific theories are is any better off?
Syn: Well some of those problems might be helpfully addressed in terms of language. The philosophy of language might be helpful in solving them, or they might just involve linguistic issues.
Sem: The trouble is, there aren't any examples of such problems.
Syn: Mainstream philosophy of science doesn't concentrate on them now, perhaps because of the work of Quine, and certainly because of the fruitfulness of historical case studies in the philosophy of science. Still, it doesn't follow that those problems have been successfully addressed. Lawrence Sklar's work contains lots of examples of puzzles in actual science that contemporary approaches seem to ignore. Historically, these were addressed in terms of language.
Sem: Yeah, and which the philosophy of language notoriously failed to solve.
Look, we're using the approach that is dominant today because right now that looks like the most promising approach. If you want to propose some other approach, it's up to you to show that it actually useful, not to just suggest that it might be. That's completely unconvincing. All kinds of new approaches might be useful in addressing problems that the present approach doesn't.
Syn. I suppose you're right. At the present state of play, it's very hard to claim that introducing linguistic issues into the philosophy of science is much of an advantage. If we want to do that, we'd have to show the productivity of the linguistic approach, and I don't suppose there are many philosophers who believe that has been done.
Still, I don't think the syntactic view is at a disadvantage compared to the semantic view. What we ought to do, to secure mathematics, is simply to include the axioms of arithmetic as a part of every theory. Then we can get all the advantages you have.
Sem: The semantic view does better there too.
You want a set of axioms, the fundamental statements of mathematics, and then you want to derive the real numbers and their properties from those statements. But what rules of inference are you going to use? Your view requires that we use the methods of first order logic. But you know as well as I do that no set of fundamental statements will enable you to draw all the consequences that are true of the real numbers themselves.
Syn: Yes, that's a remarkable result. It turns out that the ordinary mathematical relations we all learned up to high school tell us about a set of objects, the real numbers. These possess properties that simply cannot be proved from any set of axioms, but which are expressed by sentences of first order logic, and therefore ought to be provable if the axioms and their consequences are to capture what it is we learned about.
You draw the consequence that scientists my view must see this as a disadvantage compared to your view. But that is a double edged sword. Why not instead draw the conclusion that your view is at a disadvantage compared to mine, for I have a nice clear theory of proof, and you do not?
Sem: Oh that's completely specious. If you're going to claim that your view can get the advantage that my own view has, that of being able to make use of the real numbers, then you must provide some account of how you're able to do this. When I go on to point out that you can't give this account, you turn around and say that I can't give an account of proof so we're even.
But if you want to make use of the real numbers, with all their properties, you can't give an account of proof either. Your account of proof and scientific theories means that every such theory must omit at least one mathematical truth about the real numbers that you acknowledge ought to be provable. If you want to keep first order logic and the axioms, you've got to give up on the real numbers, and the real numbers are what scientists intend when they talk about their theories. Or are you going to deny that too?
Syn: I'm not sure whether I have to. Let me see what happens if I think things through.
When they propose a theory, scientists have to say and write things. When they prove things, if the proof is a genuine one, they have to make each step explicit, and the proof must be effectively checkable. Now Gödel's proof shows that whatever we interpret them as meaning by their words, the interpretations will always include some that are not the real numbers, but something different.
But conversely, these possible interpretations of their words always will include the real numbers. For we are interpreting a scientist's words, and a scientist will never deny the sentence that ought to be provable from his statements if he intends the real numbers by them. That's obviously because he does intend the real numbers by his statements. So among the mathematical structures he intends is one that is privileged, the intended interpretation. And so I can say that it is this interpretation that I take to be the intended one, and all his subsequent statements will bear me out (as long as he doesn't make a mistake).
That point is not special to mathematics, it's one I would want to make anyway. Suppose a scientist says, "That's a neutron star". Since this is just a sentence, I could interpret as true if and only if that is a piece of green cheese he will never come near to. But that would be silly. Obviously, the best interpretation of his words is that by '…is a neutron star' he means the property of being a neutron star. Similarly, the best interpretation of his mathematical language is that he is referring to numbers, specifically the real numbers.
Sem: It seems to me that you've secured the advantages of my view only by abandoning your own. What you're saying is that you accept the usefulness of my way of seeing scientific theories, and that what you really meant all along was the linguistic view with the addition that we're interested only in some ways of interpreting the language.
But I don't see why talking about language isn't all an enormous irrelevant detour. Why not just stick with your preferred interpretations, which are what is doing all the work of analysis of scientific theories? Why bother with sentences and words at all?
Syn: Well, as I said, partly to give an account of proofs, and partly because scientists must speak or write when they propose a theory.
Sem: The way they speak and write gives no support to the additional structure you include. Scientists frequently say things that rely on inferences that first order logic cannot capture. They draw graphs and diagrams as well as speaking and writing.
Now perhaps you will reply that these words, pictures, and apparent inferences can all be interpreted using all kinds of models other than the one I prefer. But how does that illuminate the practice of science itself? It seems like an irrelevant philosophical grace note to me. As you know, some of the ways of interpreting the languages of first order logic are clearly unintentional and bizarre. Since you yourself admit that you want to make use of the intended interpretations that I do, why not just cut to the chase and talk about the mathematical models scientists themselves do?
Syn: I think you're right. If we're only interested in the philosophy of science as a way of studying scientists and their practices from their own perspective, then the semantic view is the better one. One might claim that there are other philosophical problems involved, which perhaps require taking a philosophical perspective rather than that of the scientists themselves. But even if one believes that, I would still have to show that the study of language and the restriction of inferences to those sanctioned by first order logic would be helpful in solving these philosophical problems.

A personal conclusion.

This debate represents the way I see the dispute between the two views of scientific theories. The semantic view seems to me to have the best case in its favor.
Nonetheless, I do not favor the semantic view myself. The reason is that I remain firmly convinced that the most serious philosophical problems we face today concern language, although they may also be phrased in terms of the confirmation of scientific theories.
In the debate above, both the semantic and syntactic theorists assumed that there was simply no problem about us intending the mathematician's view of the real numbers by our words. Nearly everyone believes this.
Nonetheless there are a few thinkers, originally mathematicians, who do think there is a problem here. Originally, these problems concerned a specific view of what numbers were. Numbers, these thinkers believed, were constructed out of human subjective experience, especially our experience of time. These constructions, while very wide ranging, weren't adequate to construct the real numbers as they are usually viewed.
Recently however, the philosopher Michael Dummett has re-phrased this view, called Intuitionism, from a different foundation. It is not our ability to construct numbers that matters, he has argued, so much as it is our ability to communicate, that leads to Intuitionism. The things we can say and do when we practice mathematics, he argues, are simply insufficient to indicate that it is the real numbers as ordinary mathematics construes them that we intend by our words and actions. Only the much poorer view of numbers that the Intuitionists proposed can be communicated to one another. It isn't that we intend the usual real numbers mentally, and are unable to adequately describe our intention. Rather, because we cannot understand each other in such a way as to establish that we intend the usual real numbers, we cannot even speak meaningfully about them. That is, we can't even meaningfully say "I mean the usual real numbers" or even "I am thinking about the usual real numbers", nor even "I really am thinking about those numbers that I think I'm thinking about, even though I can't communicate to you what they are!" (I say 'the usual real numbers' because Intuitionists also have a theory of what the real numbers are, it's just not as rich as the one most mathematicians would say they mean.)
It would certainly seem absurd for anyone in the philosophy department to go around correcting the mathematics department about mathematics, but I really cannot see any way of evading Dummett's arguments. My own work is that of arguing that we cannot confirm, of one another, that we understand the usual real numbers. That is, I take the business of finding out what we mean by our words to be that of finding evidence for scientific hypotheses, that is, hypotheses about what we mean. So of course, for me, the sentences scientists utter are very important, and I cannot presume that they intend the usual real numbers by these utterances.

And that is why, despite the argument above, I am a syntactic theorist. I realize that to some readers, it will be irritating to read this final, very superficial, section. If I am going to argue something, I ought to actually argue it, and not vaguely gesture in its direction and then finish. But still, this piece has been written as an introduction to students and other philosophers who are not in the philosophy of science, and I do think there is an advantage in indicating to students what a writer thinks, even if it is done very briefly.