The Knight’s Song, or What is a [scientific] theory?

“Or else it doesn’t, you know. The name of the song is called ‘Haddocks’ Eyes.’”
“Oh, that’s the name of the song, is it?” Alice said, trying to feel interested.
“No, you don’t understand,” the Knight said, looking a little vexed. “That’s what the name is called. The name really is ‘The Aged Aged Man.'”
“Then I ought to have said ‘That’s what the song is called’?” Alice corrected herself.
“No, you oughtn’t: that’s quite another thing! The song is called ‘Ways and Means’: but that’s only what it’s called, you know!”
“Well, what is the song, then?” said Alice, who was by this time completely bewildered.
“I was coming to that,” the Knight said. “The song really is ‘A-sitting on a Gate’: and the tune’s my own invention.”
[Through the Looking-Glass, chapter 8]

Few words carry the weight that the Greek word theoria carries today. It is a word that applies to everything from politics to philosophy to mathematics, art, economics, psychology and of course, the natural sciences. Usually contrasted to another Greek word praxis, meaning practice or doing, it is the mental view one has of some aspect of the world. Beyond that, there is almost nothing in common with all uses of the term, especially now that it has been extended to apply to all of our general dispositions to observe and learn.

It comes as something of a surprise to me that the philosophy of science does not have a neat discussion I can locate of what counts as a theory in science. There are two general metalevel views – the traditional or “syntactic” notion and a structural or “semantic” notion – but these do not tell us what theories are, only what they entail in a more general philosophical sense. The syntactic notion holds that theories are logical theorems and their derivations, while the semantic view holds that theories are models that have real world interpretations. The slogan for the semantic view is

SEM: A theory is a collection of models (cf. McEwan unpublished)

There seems not to be an equivalent slogan for the syntactic view, but we can invent one:

SYN: A theory is an axiomatic system in some formal language

However, neither of these slogans make clear either what the difference between the two actually is, or what scientific theories, the things scientists take themselves to be using, making and testing, are. Instead we are dealing with logical objects – sentences, sets, classes and axioms. This is a dispute between philosophers (indeed, between schools of philosophy, the logical positivists supposedly holding SYN, following Carnap, and the post-positivists holding SEM (two originators of this view are Frederick Suppe and Bas van Fraassen).

SEM is about “model-theoretic models”, which is to say “an interpretation which satisfies some set of sentences (or sentential formulae). An interpretation specifies a set of individuals (the domain or universe of discourse) and defines of all the appropriate symbols (i.e., constant, function and predicate symbols) of the language on that set” (McEwan). SYN is a set of sentences in a formal (that is a logico-mathematical) language. Very roughly, a SEM theory is a structural representation of the world, while a SYN theory is a set of formal statements, each of which has some truth value. Both of these are highly abstract objects.

At the other end of the scale, we find scientists talking about theories as laws, prediction techniques, intellectual schemes, descriptions (sometimes) and even extended hypotheses. The Folk, on the other hand, treat “theory” as a special kind of guess. Creationists and other dissemblers regarding some science or other they object to play very strongly on these ambiguities and polysemies.

So I am left to wonder what a theory is (as opposed to how philosophers explicate theories in philosophy). We can consider a few senses for a theory in some domain of investigation:

  • The Mathematical Sense: A theory is some class of mathematical axioms and all their formal implications. This is sometimes called the axiomatic sense of theory – a theory, the real thing, is a class of mathematical theorems taken to be axioms.
  • The Interpretation Sense: A theory is a set of mathematical statements or structures together with rules for interpreting those statements.  If you have a mathematical equation that purports to describe how things fall, you need to know what to interpret the variables in that equation as referring to.
  • The Representation Sense: A theory is a description of the way things are in some domain. It allows the theorist to explain, predict, and manipulate those things.
  • The Worldview Sense: A theory is the set of beliefs that enables the theorist to engage with the world, structuring observation, action, reasoning and expression. This is effectively Thomas Kuhn’s notion of a “paradigm” or his later explication “exemplar”, some conceptual scheme that guides everything in the discipline.
  • The Practical Sense: It seems a bit odd to call a theory “practical” when it is usually contrasted with practice (praxis), but in this sense a theory is the set of conceptual commitments that the theorist employs to do things in the domain. So having a theory enables one to identify the relevant objects under study.
  • The Causal Sense: A theory is an explanation in terms of the causes of objects in the domain

Each of these senses appeals to some intellectual activity or state, but they vary greatly: the mathematical and interpretation senses involve mathematical equations or statements, the representation and worldview senses appeal to linguistic objects such as statements, sentences or logical formulae. The worldview and practical senses involve action-guiding stances. The causal sense is common but not universal. At best we can say they all involve beliefs (in the sense of mental stances, not faith statements necessarily). Of course, an actual theory may exhibit many or even all of these senses.

When we look at actual theories, they typically involve formal models – equations, simulations, algorithms – but this is not true of some older theories prior to the flowering of mathematisation of all science. It is also often not true of theories in domains that lack well-elaborated accounts. For example, theories in psychology are often verbal. Darwin’s theories (I count seven of them) were not mathematical at all, although he clearly intended them to be filled out later (as they were, apart from his theory of heredity). Freud’s theories remain unmathematised. So it doesn’t follow that for something to be a theory, it must be a mathematical structure, even if the philosophical analysis of theories develops a mathematical view. The reason is that “theory” in philosophy is a different beast to “theory” in science, and the relation is more like the relation between “concept” in philosophy and my concept of a television, for example.

So we might entertain the heresy that there is actually no “natural kind” in science that answers to “theory”, or, if you like, there’s no such thing as a theory, just lots of individual and particular intellectual constructs that get called theories. P. D. Magnus has argued that the term “theory” is a “family resemblance predicate” in which there are multiple meanings that overlap and cluster, but which have no necessary and sufficient definienda for all theories (Magnus unpublished), in an analogy with my favourite term of science, “species”. I think that this is correct, but I would go one further, and say that “theory” (unlike “species”) is a term that lacks any substantial meaning in science, and is really an assertion of the sociological status that some ideas have attained in a discipline, which can be for programmatic, political, or educational reasons as well as explanatory. A good theory will exhibit the majority of the cluster properties, but it doesn’t follow that theory is a category that stands alone, as it were, from the psychological, historical and social aspects of science.

What gets called a theory depends on no unique set of inherent properties of the theories themselves. This has some deep implications for thinking about science, if correct. Let’s consider some of them: the scientific process, domain specificity, and theory-dependence of observation.

The scientific process

Scientists are introduced to their disciplines in various ways, but nearly all of them are taught at some point that there is a scientific method. This methodism is, however, itself indefinable. Some accounts introduce a cycle of conjecture, testing, formulation, further  testing and then publication as a law or generalisation. Others focus on the use of statistical adequacy. Yet others make consilience (abductive reasoning) a key virtue – many lines of investigation must coincide.

Nearly all of them work on conceptual elements: statements in ordinary, formal or mathematical languages. Some domains or disciplines are more formalisable than others, but independently of this, scientists work with “hypotheses”, which, when sufficiently well established, become “theories”. In short, a theory is what a hypothesis wants to be when it grows up.

If there is no common property for theories, apart from properties held by things that aren’t theories by any estimation, then this picture of science, while not false, is misleading. There can be no singular method, because there is no singular destination for scientific ideas (see Magnus’ paper for a good discussion about what count as scientific ideas). And as famously expressed by Feyerabend:

It is clear, then, that the idea of a fixed method, or of a fixed theory of rationality, rests on too naive a view of man and his social surroundings. To those who look at the rich material provided by history, and who are not intent on impoverishing it in order to please their lower instincts, their craving for intellectual security in the form of clarity, precision, ‘objectivity’, ‘truth’, it will become clear that there is only one principle that can be defended under all circumstances and in all stages of human development. It is the principle: anything goes. [Against Method pp27-28]

If there are no essential features for theories, then there are no essential methods for attaining them. On the other hand, I think that the simple view that anything is or can be science is equally mistaken. Feyerabend did not actually argue that anything goes, but instead that there is no fixed method; this is a reductio. He knew perfectly well that disciplines have canon of reasoning and methods, and that some methods are inadequate or unfruitful (no aid and comfort to creationists in this argument, at any rate).

Domain specificity

One assumption often made about science is that it is divided into fields of inquiry that are themselves more or less natural. We think, for example, of biology as a natural subset of natural processes where we do not think of medicine that way (because medicine uses techniques and ideas that also apply to veterinary science). One standard view about domains in science is that they are effectively defined by the best attested theory of the phenomena in the domain. As theories develop, and as some parts of a domain are explained by theories in other domains, the scope of the domain is refined and revised (consider how much biology has been relegated to organic chemistry, or psychology to neurobiology).

If, however, we find that there are no such privileged conceptual constructs as theories, what does this mean for domains? How do we anchor domains (like biology) in natural ways? We can always give institutional arguments for domains, like saying that there is a Biology faculty in universities, or a course of educational requirements in schools. Or we can argue that it is easier to teach techniques when bundled together (microscopy, field observation, etc.) but it may just as easily have turned out a different way. Arguments about “what is “life”, for example, presuppose the naturalness of that domain (as do the NASA attempts to locate evidence of “life” elsewhere than earth), but if there is nothing that ties life together but human practical considerations and a collection of theories that are not entirely connected or commensurate, why bother? Why not just further divide the domains into groupings that are natural? Why, for example, does biology consider both evolution by natural selection and biochemistry, the Krebs cycle and ecology, behaviour and biogeography?

Attempts to formulate ontologies of domains also typically derive from the theoretical commitments of the domain (atoms are part of the domain of physics, while pain sensations aren’t); so if the theoretical commitments are sui generis to the domain because the nature of “theory” in that domain is unique also, we have a problem of ontological relativity, which may or may not be a problem, depending on how you think ontologies should be handled.

This is, in effect, an argument for a descriptive pluralism. Pluralisms are often thought of as some kind of failure or postmodern relativism, but I see them rather differently. We start our investigations of things based on the phenomena that present themselves to our inspection. Since we have prior sensory, social and conceptual commitments which may or may not be reliable guides to the structure of the world, we very often have to revise our concepts to fit what we learn by investigation. So, “fish” no longer means any thing that lives in water and moves of its own accord, and humans are now apes. Pluralism is a necessary aspect of discovering that the world wasn’t structured the way we naively thought it was. It is a recognition that words matter less than the world they describe.

But this indicates something about science that is so obvious as to almost not need saying: evidence – observation, measurement and experiment – takes priority over theory. That is perhaps a dumb thing to say, or perhaps it is so dangerous as to be obviously false, depending on what you think about our ways of knowing and explaining (theoretical constructionists would take the latter tack). But I think that theories, and domains demarcated by theories, are definable solely in terms of their being something other than evidence. In short, a theory is what evidence isn’t. That leads naturally to the next point.

Theory dependence of observation revisited

I have previously discussed the “theory-dependence of observation thesis (TDOT)” in detail, so I will be brief. If theory itself is not a natural kind, the claim that observation relies upon it falters. Of course our conceptual furniture affects how we observe – this follows from the mere existence of trained observers – but the nonexistence of Theory (that is, as a natural kind) means that the sting of the TDOT is largely removed. It resolves down to the view that we observe things that we have learned to observe. This is not, I think, so deep as the TDOT5 claim defended by Kuhnians at one time. It certainly does not license the sorts of claims that are sometimes made that science is a self-contained hermeneutic bubble just like any other world view.


I think this is significant in part because it helps up to understand how science really proceeds. The notion of “law” in science has been deprecated recently amongst philosophers (e.g., Cartwright et al. 2005); it is time to deprecate “theory” also.

It also means that what we call a theory and how we talk about the notion of theory is not unlike the Knight’s song. We often refer to philosophical accounts of representation, explanation and formalisation when we discuss “theory” (excluding the terminological arguments amongst other philosophical traditions, like Marxist or phenomenological schools) when we should be talking about the ways scientists use the terms, and then, and only then, consider the philosophical implications. And if “theory” lacks the sort of reality it is sometimes held to have, if it simply is whatever in science is not evidentiary or probative, then we should be more empiricist in our philosophy.

I take this line, of course, to defend the claim that one thing science often does is, independently of theory, classify the world as a way to investigate it. If theory is not a natural kind, then it becomes clear that we can do this, only what we rely upon, our conceptual commitments that make a trained observer a better systematist, is more complex than “theory” suggests it would be.

Cartwright, Nancy, Anna Alexandrova, and Sophia Efstathiou. 2005. Laws. In The Oxford Handbook of Contemporary Philosophy, edited by F. Jackson. New York: Oxford Univ Press:792-818.

Feyerabend, PK. 1975. Against Method. New York: Verso Editions.

Magnus, P. D. Unpublished. What SPECIES can teach us about THEORY.

McEwan, Micheal. Unpublished. The Semantic View of Theories:Models and Misconceptions.

32 thoughts on “The Knight’s Song, or What is a [scientific] theory?

  1. Interesting stuff (as always). But I sometimes wonder, when faced with a conclusionike yours, if there is a nice neat definition but we’re just not intelligent enough to see what it is.

    1. Then we would learn something about science, if it were discovered. But we’ve been at this for about 150 years. I’d hope something as basic as “theory” would be well understood by now.

  2. As I understand it a scientific theory is a collection of rather good ideas — as opposed to like “the atomic hypothesis”, which is just one idea.

    A model (not model-theoretic model but like model) would be “Here’s a useful viewpoint which does not cover 100% of everything but makes intuitive puzzle X less puzzling”. Known in advance to be strictly false.

    That we can’t come up with a good definition of a scientific theory, doesn’t mean there’s no natural kind. I would just guess that model theory, predicates, etc, are not the clearest starting places for thinking about it, and neither is a mathematical model. Like defining ‘tree’, ‘pornography’, or ‘definition’, it’s much easier to give an example than to prescribe the correct ontology. What’s the need to do so, anyway?

    1. The Krugman model you mention here is more like a thought experiment. It doesn’t really predict or explain any real world case, which is not, I think, an objection to it so much as identifying its conceptual limitations. Gedankensexperiments have a long history in science, but they aren’t, quite, theories.

      More generally, though: do such (mathematical) models and simulations act as explanations or predictors of real world cases? For instance, in ecology (a field not unlike economics, but more real) such models are often used to explain or predict outcomes, but they never actually are accurate in their predictions or retrodictions. This, of course, is because the models simplify and the real cases are multivariate in ways that make a difference, so the question is: how do we interpret these models? Subjectively? Is it because they seem to us to resemble the phenomena in the real case? Or is there a principled way we can interpret them? And so we get back to asking questions about theories…

  3. Secondary question: how could there be a concrete definition of ‘theory’ in a non-teleological universe? Without a goal there is no purpose; without a purpose there is no predefined narrative to discover, and so ‘theories’ must be defined by post hoc fuzzy observations. Like the definition of ‘species’. I suspect more work needs to be done on this argument!

    To misquote Jean-Paul: “Abduction before Essentialism”. Or to put it another way: “Fuzzy definition of ‘theory’ therefore no God”.

      1. In the case of those Mathematical “theories”. I think theory just means the collection of work that has been done in that discipline. More a collection of theorems. Scientific theories are created to try to describe the world. Pure math isn’t necessarily created for any other purpose. This article made me think of Gödel’s incompleteness theorem and how it stopped the effort to formalize mathematics in the early 20th century.

  4. I might later give a more lengthy response on my blog. For now, a few quick comments. In practice, our use of the word “theory” is all over the map. And I guess that was one of the points you were making.

    In mathematics, a theory introduces abstract objects and axioms about those abstract objects. For example, group theory invents groups as abstract objects, and binary operators that act on elements in a group. Then it axiomatizes those operations. Likewise, topology introduces topological spaces (as abstract objects) and open and closed sets within those spaces. Then it axiomatizes the way we talk about open and closed sets. So it seems to me that the mathematical use of “theory” includes aspects of both SEM and SYN.

    So we might entertain the heresy that there is actually no “natural kind” in science that answers to “theory”, …

    My heresy goes a bit farther. I say that there are no natural kinds.

    Scientists are introduced to their disciplines in various ways, but nearly all of them are taught at some point that there is a scientific method.

    I share Feyerabend’s skepticism about that.

    I remember a talk, many years ago, by the chairman of the chemistry department at a midwest campus. He said that a scientist is somebody who retains the curiosity he had as a child. I mostly agree with that. If there can be said to be a scientific method, then that method is to be curious, and to follow that curiosity wherever it leads.

    One assumption often made about science is that it is divided into fields of inquiry that are themselves more or less natural.

    I am inclined to think that division into fields of inquiry is socially constructed, and is mostly an artifact of happenstances in the history of science.

    But this indicates something about science that is so obvious as to almost not need saying: evidence – observation, measurement and experiment – takes priority over theory.

    That cannot be right, given that much data is theory laden (I’m aware you disagree with that).

  5. Bloody marvelous post. You are willing to tackle questions that lots of others who write about science consider almost heretical, and I have to salute you for that!

    I mean: I have the sense that any entity which is primarily a linguistic phenomenon cannot be a natural kind. One important thing about language is that it seems to be essentially creative: one can always combine its elements in novel ways, and logically speaking, one will not always be able to know that that the new combination ‘fits’ some prior categorization.

    By comparison, If you were staggeringly bored and feeling intellectually masochistic, you could look at the old debate over ‘subject’ and ‘predicate’: every time some philosopher comes along and defines those terms, someone produces a sentence that seems to defy the definition.

    Q: Your ideas are Wittgensteinean here, and I was wondering if this was self-conscious or if you were just coming to him for the first time.

    1. I’ll address the last first. I am a self-conscious post-Wittgensteinian. I read him extensively as an undergraduate (with a student of Stanley Cavell’s no less, and on my own), although I don’t think of him as The Prophet. I base a lot of my philosophical thinking on On Certainty.

      I think that philosophy often reverts to what I call Science By Definition, or SBD. Of course words do not give us information about the structure of the world (apart from the structure in the heads of a certain primate at a certain time and place), but scientific words can, and often do, represent facts and kinds about the world. It is when they fail to do this when we might expect them to succeed that I find most interesting. Why is it that a word like “species” fails to denote a natural kind? Why is it that “theory” fails likewise? These are very interesting questions to ask, as they challenge our comfortable certainties about the world.

      I read about subject and predicate as an undergraduate too…

  6. Your expressed goal of determining what a theory *is* had me expecting to disagree with you, but I kept being frustrated in that expectation until the end when you suggest that the term be “deprecated”. Grasping at this last straw I claim that even though it is used in many different ways the term is not useless – and I would go so far as to assert that there is a common element to all of the usages of “theory” , namely that they all refer to some specific process for making the predictions that I consider the essence of “science” (and I would include eg the “causal” type of theory in this, as any theory explaining why or how something happened also has implications about what further investigation will reveal).

    In the mathematical context, contra Thony C, I would argue that we do have theories – in at least two senses. Analytic Number Theory comprises a set of techniques and results which use results from analysis to enable us to predict previously unproved theorems about numbers, and Analytic Function Theory comprises the body of known and yet to be discovered facts about analytic functions in complex analysis. In its most elementary form, a theory might be the various ways of using a particular theorem but more generally it puts together those of several related theorems along with others (perhaps as yet undiscovered) which arise in a particular axiomatic context.

    1. The question of prediction fails in the case of theories of history. You can try to use retrospective prediction (retrodiction) as a substitute, but even then there are many things we want to call theories that do not make predictions. I grant that prediction is a good thing, and often theories permit it (although not so much – we can’t even say what the state of the solar system will be over periods greater than tens of thousands of years) but it is not universal to theories, and that was my point.

      1. I should have included “theories of history” in the parenthesis where I referred to “causal” theories. What does it really mean to say that I have a theory that something happened in the past other than that I expect all future observations to be consistent with that past event?

  7. Neil Rickert:
    That cannot be right, given that much data is theory laden (I’m aware you disagree with that).

    To paraphrase The Princess Bride, “mostly theoretical means some evidence is not theoretical”. That is all I need to motivate my argument. As you know, I have developed my view on theory-dependence of observation here, as linked in the main article above.

  8. There might well be natural relations in the semantics. But these relations would be in the form of connections and relevancies between various semantic aspects of the world. They could not be law-like, because law-like relations can only be relations between syntactic elements.

  9. Forgive this chattering chimpanzee for leaning on your silver back. A dyslexic thinks in pictures not words. Words with no picture are bad for me. What does a participle look like -particularly one that’s past? Subject and predicate terminated my English ‘o’ levels. Much later, intention and intension nearly blew me away – but I persevere. Writing is a tortuous process for me. Peculiarly though I love reading. A good writer leads me into picture dimensions of wonder. This post will have me singing the Knight’s song for quite a while to digest the wonderment you present. I thank you for it.

    To keep my picture clear, I tend to rely on Richard Feynman’s summing science (and philosophy perhaps) in his three sentences on the conservation of energy.

    “There is a fact – or if you wish, a law, governing all natural phenomena that are known to date. There is no known exception to this law – it is exact so far as we know. The law is called the conservation of energy.”

    I read this as – To become an accepted scientific fact, any supposition must be accountable to all known fact – without exception. If a supposition complies with that condition, then we can say, ‘That seems to be a fact – a law even.’ – That is, until someone finds an exception to it, or an unknown further fact might offer an alternative supposition.

    a+b=d until somebody finds c.

  10. It comes as something of a surprise to me that the philosophy of science does not have a neat discussion I can locate of what counts as a theory in science.

    This indeed is sad. Perhaps a repuatable phiolsopher of science should write an article that summarizes scientific theory for SEP.

  11. I thought a theory represents what will take place when all is taken into consideration, that has come about by equation or putting circumstances together such as, after all has been learned about a situation by tomorrow at a cirtain place such and such will happen, but it is not always right and the out come can be changed upon the assertion of the theory. Sherlock always seemed to get it right.

    1. It would have been easier on my eye if you’d just copied this table:

      theory                        models
      syntax                        semantics

      And I can appreciate this, but it seems to me this is prescriptive rather than descriptive. Not all models are concrete in sciences, though it might clarify things if we adopted this rule.

  12. On Certainty

    Why is it that a word like “species” fails to denote a natural kind? Why is it that “theory” fails likewise? These are very interesting questions to ask, as they challenge our comfortable certainties about the world.

    Whereof one cannot speak, thereof one must be silent (preserve ‘certainty’)

    Perception underlies representation. Without a viable theory of perception (certainty) a person is doomed to silence .

  13. Does a theory aim to tell what is ‘allowable’ or not, in what we perceive as this particular universe?
    The theory of gravity tells us, I presume, that falling down is allowable, whereas falling sideways is not – so falling is restricted to down. – That a free pendulum is limited to its own arc.
    Does the word theory hide the question, ‘What are the limits of what is allowable in the universe?’ in the same way that the word species hides the question, ‘What are the limits of what is allowable in life?’
    Music theory would tell us that in the medium of air, a bugle is limited to play only certain notes at certain frequencies — harmonics in a particular key. – That a piano is more versatile, allowing more notes.

  14. My understanding of the purpose of model making is to make the simplest model which produces adequate results. In other words, the desired result (and resources available) dictates how complete the model needs to be. So, the argument that a model is not sufficiently accurate is valid only in the context of the accuracy wanted.

  15. None of this comes as a revelation to folks who do science in “messy” realms like biology, where concrete, here’s-the-edges definitions are just invitations to exceptions. A LOT of the terms are very loosely defined, which must be frustrating for structured thinkers (the edges are firmer in physics and chemistry), but it’s just the, pardon the expression, Nature of the Beast.

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