hypothesis for scientific theory

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Biology library

Course: biology library   >   unit 1, the scientific method.

• Controlled experiments
• The scientific method and experimental design

Introduction

• Make an observation.
• Ask a question.
• Form a hypothesis , or testable explanation.
• Make a prediction based on the hypothesis.
• Test the prediction.
• Iterate: use the results to make new hypotheses or predictions.

Scientific method example: Failure to toast

1. make an observation..

• Observation: the toaster won't toast.

2. Ask a question.

• Question: Why won't my toaster toast?

3. Propose a hypothesis.

• Hypothesis: Maybe the outlet is broken.

4. Make predictions.

• Prediction: If I plug the toaster into a different outlet, then it will toast the bread.

5. Test the predictions.

• Test of prediction: Plug the toaster into a different outlet and try again.
• If the toaster does toast, then the hypothesis is supported—likely correct.
• If the toaster doesn't toast, then the hypothesis is not supported—likely wrong.

Logical possibility

Practical possibility, building a body of evidence, 6. iterate..

• Iteration time!
• If the hypothesis was supported, we might do additional tests to confirm it, or revise it to be more specific. For instance, we might investigate why the outlet is broken.
• If the hypothesis was not supported, we would come up with a new hypothesis. For instance, the next hypothesis might be that there's a broken wire in the toaster.

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What is a scientific hypothesis?

It's the initial building block in the scientific method.

Hypothesis basics

What makes a hypothesis testable.

• Types of hypotheses
• Hypothesis versus theory

Additional resources

Bibliography.

A scientific hypothesis is a tentative, testable explanation for a phenomenon in the natural world. It's the initial building block in the scientific method . Many describe it as an "educated guess" based on prior knowledge and observation. While this is true, a hypothesis is more informed than a guess. While an "educated guess" suggests a random prediction based on a person's expertise, developing a hypothesis requires active observation and background research. 

The basic idea of a hypothesis is that there is no predetermined outcome. For a solution to be termed a scientific hypothesis, it has to be an idea that can be supported or refuted through carefully crafted experimentation or observation. This concept, called falsifiability and testability, was advanced in the mid-20th century by Austrian-British philosopher Karl Popper in his famous book "The Logic of Scientific Discovery" (Routledge, 1959).

A key function of a hypothesis is to derive predictions about the results of future experiments and then perform those experiments to see whether they support the predictions.

A hypothesis is usually written in the form of an if-then statement, which gives a possibility (if) and explains what may happen because of the possibility (then). The statement could also include "may," according to California State University, Bakersfield .

Here are some examples of hypothesis statements:

• If garlic repels fleas, then a dog that is given garlic every day will not get fleas.
• If sugar causes cavities, then people who eat a lot of candy may be more prone to cavities.
• If ultraviolet light can damage the eyes, then maybe this light can cause blindness.

A useful hypothesis should be testable and falsifiable. That means that it should be possible to prove it wrong. A theory that can't be proved wrong is nonscientific, according to Karl Popper's 1963 book " Conjectures and Refutations ."

An example of an untestable statement is, "Dogs are better than cats." That's because the definition of "better" is vague and subjective. However, an untestable statement can be reworded to make it testable. For example, the previous statement could be changed to this: "Owning a dog is associated with higher levels of physical fitness than owning a cat." With this statement, the researcher can take measures of physical fitness from dog and cat owners and compare the two.

Types of scientific hypotheses

In an experiment, researchers generally state their hypotheses in two ways. The null hypothesis predicts that there will be no relationship between the variables tested, or no difference between the experimental groups. The alternative hypothesis predicts the opposite: that there will be a difference between the experimental groups. This is usually the hypothesis scientists are most interested in, according to the University of Miami .

For example, a null hypothesis might state, "There will be no difference in the rate of muscle growth between people who take a protein supplement and people who don't." The alternative hypothesis would state, "There will be a difference in the rate of muscle growth between people who take a protein supplement and people who don't."

If the results of the experiment show a relationship between the variables, then the null hypothesis has been rejected in favor of the alternative hypothesis, according to the book " Research Methods in Psychology " (​​BCcampus, 2015). 

There are other ways to describe an alternative hypothesis. The alternative hypothesis above does not specify a direction of the effect, only that there will be a difference between the two groups. That type of prediction is called a two-tailed hypothesis. If a hypothesis specifies a certain direction — for example, that people who take a protein supplement will gain more muscle than people who don't — it is called a one-tailed hypothesis, according to William M. K. Trochim , a professor of Policy Analysis and Management at Cornell University.

Sometimes, errors take place during an experiment. These errors can happen in one of two ways. A type I error is when the null hypothesis is rejected when it is true. This is also known as a false positive. A type II error occurs when the null hypothesis is not rejected when it is false. This is also known as a false negative, according to the University of California, Berkeley . 

A hypothesis can be rejected or modified, but it can never be proved correct 100% of the time. For example, a scientist can form a hypothesis stating that if a certain type of tomato has a gene for red pigment, that type of tomato will be red. During research, the scientist then finds that each tomato of this type is red. Though the findings confirm the hypothesis, there may be a tomato of that type somewhere in the world that isn't red. Thus, the hypothesis is true, but it may not be true 100% of the time.

Scientific theory vs. scientific hypothesis

The best hypotheses are simple. They deal with a relatively narrow set of phenomena. But theories are broader; they generally combine multiple hypotheses into a general explanation for a wide range of phenomena, according to the University of California, Berkeley . For example, a hypothesis might state, "If animals adapt to suit their environments, then birds that live on islands with lots of seeds to eat will have differently shaped beaks than birds that live on islands with lots of insects to eat." After testing many hypotheses like these, Charles Darwin formulated an overarching theory: the theory of evolution by natural selection.

"Theories are the ways that we make sense of what we observe in the natural world," Tanner said. "Theories are structures of ideas that explain and interpret facts." 

• Read more about writing a hypothesis, from the American Medical Writers Association.
• Find out why a hypothesis isn't always necessary in science, from The American Biology Teacher.
• Learn about null and alternative hypotheses, from Prof. Essa on YouTube .

Encyclopedia Britannica. Scientific Hypothesis. Jan. 13, 2022. https://www.britannica.com/science/scientific-hypothesis

Karl Popper, "The Logic of Scientific Discovery," Routledge, 1959.

California State University, Bakersfield, "Formatting a testable hypothesis." https://www.csub.edu/~ddodenhoff/Bio100/Bio100sp04/formattingahypothesis.htm  

Karl Popper, "Conjectures and Refutations," Routledge, 1963.

Price, P., Jhangiani, R., & Chiang, I., "Research Methods of Psychology — 2nd Canadian Edition," BCcampus, 2015.‌

University of Miami, "The Scientific Method" http://www.bio.miami.edu/dana/161/evolution/161app1_scimethod.pdf  

William M.K. Trochim, "Research Methods Knowledge Base," https://conjointly.com/kb/hypotheses-explained/  

University of California, Berkeley, "Multiple Hypothesis Testing and False Discovery Rate" https://www.stat.berkeley.edu/~hhuang/STAT141/Lecture-FDR.pdf  

University of California, Berkeley, "Science at multiple levels" https://undsci.berkeley.edu/article/0_0_0/howscienceworks_19

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The Structure of Scientific Theories

Scientific inquiry has led to immense explanatory and technological successes, partly as a result of the pervasiveness of scientific theories. Relativity theory, evolutionary theory, and plate tectonics were, and continue to be, wildly successful families of theories within physics, biology, and geology. Other powerful theory clusters inhabit comparatively recent disciplines such as cognitive science, climate science, molecular biology, microeconomics, and Geographic Information Science (GIS). Effective scientific theories magnify understanding, help supply legitimate explanations, and assist in formulating predictions. Moving from their knowledge-producing representational functions to their interventional roles (Hacking 1983), theories are integral to building technologies used within consumer, industrial, and scientific milieus.

This entry explores the structure of scientific theories from the perspective of the Syntactic, Semantic, and Pragmatic Views. Each of these answers questions such as the following in unique ways. What is the best characterization of the composition and function of scientific theory? How is theory linked with world? Which philosophical tools can and should be employed in describing and reconstructing scientific theory? Is an understanding of practice and application necessary for a comprehension of the core structure of a scientific theory? Finally, and most generally, how are these three views ultimately related?

1.1 Syntactic, Semantic, and Pragmatic Views: The Basics

1.2 two examples: newtonian mechanics and population genetics, 2.1 theory structure per the syntactic view, 2.2 a running example: newtonian mechanics, 2.3 interpreting theory structure per the syntactic view, 2.4 taking stock: syntactic view, 3.1 theory structure per the semantic view, 3.2 a running example: newtonian mechanics, 3.3 interpreting theory structure per the semantic view, 3.4 taking stock: semantic view, 4.1 theory structure per the pragmatic view, 4.2 a running example: newtonian mechanics, 4.3 interpreting theory structure per the pragmatic view, 4.4 taking stock: pragmatic view, 5. population genetics, 6. conclusion, other internet resources, related entries, 1. introduction.

In philosophy, three families of perspectives on scientific theory are operative: the Syntactic View , the Semantic View , and the Pragmatic View. Savage distills these philosophical perspectives thus:

The syntactic view that a theory is an axiomatized collection of sentences has been challenged by the semantic view that a theory is a collection of nonlinguistic models, and both are challenged by the view that a theory is an amorphous entity consisting perhaps of sentences and models, but just as importantly of exemplars, problems, standards, skills, practices and tendencies. (Savage 1990, vii–viii)

Mormann (2007) characterizes the Syntactic and Semantic Views in similar terms, and is among the first to use the term “Pragmatic View” to capture the third view (137). The three views are baptized via a trichotomy from linguistics deriving from the work of Charles Morris, following Charles S. Peirce. In a classic exposition, the logical positivist Carnap writes:

If in an investigation explicit reference is made to the speaker, or, to put it in more general terms, to the user of a language, then we assign it to the field of pragmatics . (Whether in this case reference to designata is made or not makes no difference for this classification.) If we abstract from the user of the language and analyze only the expressions and their designata, we are in the field of semantics . And if, finally, we abstract from the designata also and analyze only the relations between the expressions, we are in (logical) syntax . The whole science of language, consisting of the three parts mentioned, is called semiotic . (1942, 9; see also Carnap 1939, 3–5, 16)

To summarize, syntax concerns grammar and abstract structures; semantics investigates meaning and representation; and pragmatics explores use. Importantly, while no view is oblivious to the syntax, semantics, or pragmatics of theory, the baptism of each is a product of how one of the three aspects of language is perceived to be dominant: theory as syntactic logical reconstruction (Syntactic View); theory as semantically meaningful mathematical modeling (Semantic View); or theory structure as complex and as closely tied to theory pragmatics, i.e., function and context (Pragmatic View). Each of these philosophical perspectives on scientific theory will be reviewed in this entry. Their relations will be briefly considered in the Conclusion.

It will be helpful to pare each perspective down to its essence. Each endorses a substantive thesis about the structure of scientific theories.

For the Syntactic View, the structure of a scientific theory is its reconstruction in terms of sentences cast in a metamathematical language. Metamathematics is the axiomatic machinery for building clear foundations of mathematics, and includes predicate logic, set theory, and model theory (e.g., Zach 2009; Hacking 2014). A central question of the Syntactic View is: in which logical language should we recast scientific theory?

Some defenders of the Semantic View keep important aspects of this reconstructive agenda, moving the metamathematical apparatus from predicate logic to set theory. Other advocates of the Semantic View insist that the structure of scientific theory is solely mathematical. They argue that we should remain at the mathematical level, rather than move up (or down) a level, into foundations of mathematics. A central question for the Semantic View is: which mathematical models are actually used in science?

Finally, for the Pragmatic View, scientific theory is internally and externally complex. Mathematical components, while often present, are neither necessary nor sufficient for characterizing the core structure of scientific theories. Theory also consists of a rich variety of nonformal components (e.g., analogies and natural kinds). Thus, the Pragmatic View argues, a proper analysis of the grammar (syntax) and meaning (semantics) of theory must pay heed to scientific theory complexity, as well as to the multifarious assumptions, purposes, values, and practices informing theory. A central question the Pragmatic View poses is: which theory components and which modes of theorizing are present in scientific theories found across a variety of disciplines?

In adopting a descriptive perspective on the structure of scientific theories, each view also deploys, at least implicitly, a prescriptive characterization of our central topic. In other words, postulating that scientific theory is \(X\) (e.g., \(X\) = a set-theoretic structure, as per Suppes 1960, 1962, 1967, 1968, 2002) also implies that what is not \(X\) (or could not be recast as \(X\)) is not (or could not possibly be) a scientific theory, and would not help us in providing scientific understanding, explanation, prediction, and intervention. For the Syntactic View, what is not (or cannot be) reconstructed axiomatically is not theoretical, while for the Semantic View, what is not (or cannot be) modeled mathematically is not theoretical. In contrast, in part due to its pluralism about what a scientific theory actually (and possibly) is, and because it interprets theory structure as distributed in practices, the Pragmatic View resists the definitional and normative terms set by the other two views. As a result, the Pragmatic View ultimately reforms the very concepts of “theory” and “theory structure.”

This encyclopedia entry will be organized as follows. After presenting this piece’s two sustained examples, immediately below, the three views are reviewed in as many substantive sections. Each section starts with a brief overview before characterizing that perspective’s account of theory structure. Newtonian mechanics is used as a running example within each section. The interpretation of theory structure—viz., how theory “hooks up” with phenomena, experiment, and the world—is also reviewed in each section. In the final section of this entry, we turn to population genetics and an analysis of the Hardy-Weinberg Principle (HWP) to compare and contrast each view. The Conclusion suggests, and remains non-committal about, three kinds of relations among the views: identity , combat , and complementarity . Theory is not a single, static entity that we are seeing from three different perspectives, as we might represent the Earth using three distinct mathematical map projections. Rather, theory itself changes as a consequence of perspective adopted.

Two examples will be used to illustrate differences between the three views: Newtonian mechanics and population genetics. While relativity theory is the preferred theory of the Syntactic View, Newtonian mechanics is more straightforward. Somewhat permissively construed, the theory of Newtonian mechanics employs the basic conceptual machinery of inertial reference frames, centers of mass, Newton’s laws of motion, etc., to describe the dynamics and kinematics of, among other phenomena, point masses acting vis-à-vis gravitational forces (e.g. the solar system) or with respect to forces involved in collisions (e.g., pool balls on a pool table; a closed container filled with gas). Newtonian mechanics is explored in each section.

Population genetics investigates the genetic composition of populations of natural and domesticated species, including the dynamics and causes of changes in gene frequencies in such populations (for overviews, see Lloyd 1994 [1988]; Gould 2002; Pigliucci and Müller 2010; Okasha 2012). Population genetics emerged as a discipline with the early 20 th century work of R.A. Fisher, Sewall Wright, and J.B.S. Haldane, who synthesized Darwinian evolutionary theory and Mendelian genetics. One important part of population genetic theory is the Hardy-Weinberg Principle. HWP is a null model mathematically stating that gene frequencies remain unchanged across generations when there is no selection, migration, random genetic drift, or other evolutionary forces acting in a given population. HWP peppers early chapters of many introductory textbooks (e.g., Crow and Kimura 1970; Hartl and Clark 1989; Bergstrom and Dugatkin 2012). We return to HWP in Section 5 and here merely state questions each view might ask about population genetics.

The Syntactic View focuses on questions regarding the highest axiomatic level of population genetics (e.g., Williams 1970, 1973; Van Valen 1976; Lewis 1980; Tuomi 1981, 1992). Examples of such queries are:

• What would be the most convenient metamathematical axiomatization of evolutionary processes (e.g., natural selection, drift, migration, speciation, competition)? In which formal language(s) would and could such axiomatizations be articulated (e.g., first-order predicate logic, set theory, and category theory)?
• Which single grammars could contain a variety of deep evolutionary principles and concepts, such as HWP, “heritability,” and “competitive exclusion”?
• Which formal and methodological tools would permit a smooth flow from the metamathematical axiomatization to the mathematical theory of population genetics?

Investigations of the axiomatized rational reconstruction of theory shed light on the power and promises, and weaknesses and incompleteness, of the highest-level theoretical edifice of population genetics.

Secondly, the Semantic View primarily examines questions regarding the mathematical structure of population genetics (Lewontin 1974, Beatty 1981; López Beltrán 1987; Thompson 1989, 2007; Lloyd 1994 [1988]). Very generally, this exploration involves the following questions:

• What is the form and content of the directly presented class of mathematical models of evolutionary theory (e.g., HWP)? How could and should we organize the cluster of mathematical models (sensu Levins 1966) of population genetics?
• Which additional models (e.g., diagrammatic, narrative, scale) might be used to enrich our understanding of evolutionary theory?
• What are the relations among theoretical mathematical models, data models, and experimental models? How does theory explain and shape data? How do the data constrain and confirm theory?

The main subject of investigation is mathematical structure, rather than metamathematics or even alternative model types or modeling methods.

Finally, the Pragmatic View asks about the internal complexity of population genetic theory, as well as about the development and context of population genetics. In so doing, it inquires into how purposes and values have influenced the theoretical structure of evolutionary theory, selecting and shaping current population genetics from a wide variety of possible alternative theoretical articulations. The following questions about the structure of population genetic theory might be here addressed:

• What role did R.A. Fisher’s interest in animal husbandry, and his tenure at Rothamsted Experimental Station, play in shaping his influential methodologies of Analysis of Variance (ANOVA) and experimental design involving randomization, blocking, and factorial designs?
• How did the development of computers and computational practices, statistical techniques, and the molecularization of genetics, shape theory and theorizing in population genetics, especially from the 1980s to today?
• How might normative context surrounding the concept of “race” impact the way concepts such as “heritability” and “lineage,” or principles such as HWP, are deployed in population genetics?

As when studying an organism, the structure of theory cannot be understood independently of its history and function.

2. The Syntactic View

According to the Syntactic View, which emerged mainly out of work of the Vienna Circle and Logical Empiricism (see Coffa 1991; Friedman 1999; Creath 2014; Uebel 2014), philosophy most generally practiced is, and should be, the study of the logic of natural science, or Wissenschaftslogik (Carnap 1937, 1966; Hempel 1966). Robust and clear logical languages allow us to axiomatically reconstruct theories, which—by the Syntacticists’ definition—are sets of sentences in a given logical domain language (e.g., Campbell 1920, 122; Hempel 1958, 46; cf. Carnap 1967 [1928], §156, “Theses about the Constructional System”). Domain languages include “the language of physics, the language of anthropology” (Carnap 1966, 58).

This view has been variously baptized as the Received View (Putnam 1962; Hempel 1970), the Syntactic Approach (van Fraassen 1970, 1989), the Syntactic View (Wessels 1976), the Standard Conception (Hempel 1970), the Orthodox View (Feigl 1970), the Statement View (Moulines 1976, 2002; Stegmüller 1976), the Axiomatic Approach (van Fraassen 1989), and the Once Received View (Craver 2002). For historical reasons, and because of the linguistic trichotomy discussed above, the “Syntactic View” shall be the name of choice in this entry.

Some conceptual taxonomy is required in order to understand the logical framework of the structure of scientific theories for the Syntactic View. We shall distinguish terms , sentences , and languages (see Table 1).

2.1.1 Terms

Building upwards from the bottom, let us start with the three kinds of terms or vocabularies contained in a scientific language: theoretical, logical, and observational. Examples of theoretical terms are “molecule,” “atom,” “proton,” and “protein,” and perhaps even macro-level objects and properties such as “proletariat” and “aggregate demand.” Theoretical terms or concepts can be classificatory (e.g., “cat” or “proton”), comparative (e.g., “warmer”), or quantitative (e.g., “temperature”) (Hempel 1952; Carnap 1966, Chapter 5). Moreover, theoretical terms are “theoretical constructs” introduced “jointly” as a “theoretical system” (Hempel 1952, 32). Logical terms include quantifiers (e.g., \(\forall, \exists\)) and connectives (e.g., \(\wedge, \rightarrow\)). Predicates such as “hard,” “blue,” and “hot,” and relations such as “to the left of” and “smoother than,” are observational terms.

2.1.2 Sentences

Terms can be strung together into three kinds of sentences: theoretical, correspondence, and observational. \(T_S\) is the set of theoretical sentences that are the axioms, theorems, and laws of the theory. Theoretical sentences include the laws of Newtonian mechanics and of the Kinetic Theory of Gases, all suitably axiomatized (e.g., Carnap 1966; Hempel 1966). Primitive theoretical sentences (e.g., axioms) can be distinguished from derivative theoretical sentences (e.g., theorems; see Reichenbach 1969 [1924]; Hempel 1958; Feigl 1970). \(C_S\) is the set of correspondence sentences tying theoretical sentences to observable phenomena or “to a ‘piece of reality’” (Reichenbach 1969 [1924], 8; cf. Einstein 1934, 1936 [1936], 351). To simplify, they provide the theoretical syntax with an interpretation and an application, i.e., a semantics. Suitably axiomatized version of the following sentences provide semantics to Boyle’s law, \(PV = nRT\): “\(V\) in Boyle’s law is equivalent to the measurable volume \(xyz\) of a physical container such as a glass cube that is \(x\), \(y\), and \(z\) centimeters in length, width, and height, and in which the gas measured is contained” and “\(T\) in Boyle’s law is equivalent to the temperature indicated on a reliable thermometer or other relevant measuring device properly calibrated, attached to the physical system, and read.” Carnap (1987 [1932], 466) presents two examples of observational sentences, \(O_S\): “Here (in a laboratory on the surface of the earth) is a pendulum of such and such a kind,” and “the length of the pendulum is 245.3 cm.” Importantly, theoretical sentences can only contain theoretical and logical terms; correspondence sentences involve all three kinds of terms; and observational sentences comprise only logical and observational terms.

2.1.3 Languages

The total domain language of science consists of two languages: the theoretical language, \(L_T\), and the observational language, \(L_O\) (e.g., Hempel 1966, Chapter 6; Carnap 1966, Chapter 23; the index entry for “Language,” of Feigl, Scriven, and Maxwell 1958, 548 has three subheadings: “observation,” “theoretical,” and “ordinary”). The theoretical language includes theoretical vocabulary, while the observational language involves observational terms. Both languages contain logical terms. Finally, the theoretical language includes, and is constrained by, the logical calculus, Calc , of the axiomatic system adopted (e.g., Hempel 1958, 46; Suppe 1977, 50-53). This calculus specifies sentence grammaticality as well as appropriate deductive and non-ampliative inference rules (e.g., modus ponens) pertinent to, especially, theoretical sentences. Calc can itself be written in theoretical sentences.

2.1.4 Theory Structure, in General

Table 1 summarizes the Syntactic View’s account of theory structure:

The salient divide is between theory and observation. Building on Table 1, there are three different levels of scientific knowledge, according to the Syntactic View:

\(\{T_S\} =\) The uninterpreted syntactic system of the scientific theory. \(\{T_S, C_S\} =\) The scientific theory structure of a particular domain (e.g., physics, anthropology). \(\{T_S,C_S,O_S\} =\) All of the science of a particular domain.

Scientific theory is thus taken to be a syntactically formulated set of theoretical sentences (axioms, theorems, and laws) together with their interpretation via correspondence sentences. As we have seen, theoretical sentences and correspondence sentences are cleanly distinct, even if both are included in the structure of a scientific theory.

Open questions remain. Is the observation language a sub-language of the theoretical language, or are they both parts of a fuller language including all the vocabulary? Can the theoretical vocabulary or language be eliminated in favor of a purely observational vocabulary or language? Are there other ways of carving up kinds of languages? First, a “dialectical opposition” between “logic and experience,” “form and content,” “constitutive principles and empirical laws,” and “‘from above’… [and] ‘from below’” pervades the work of the syntacticists (Friedman 1999, 34, 63). Whether syntacticists believe that a synthesis or unification of this general opposition between the theoretical (i.e., logic, form) and the observational (i.e., experience, content) is desirable remains a topic of ongoing discussion. Regarding the second question, Hempel 1958 deflates what he calls “the theoretician’s dilemma”—i.e., the putative reduction without remainder of theoretical concepts and sentences to observational concepts and sentences. Finally, other language divisions are possible, as Carnap 1937 argues (see Friedman 1999, Chapter 7). Returning to the main thread of this section, the distinction toolkit of theoretical and observational terms, sentences, and languages (Table 1) permit the syntacticists to render theoretical structure sharply, thereby aiming at the reconstructive “logic of science” ( Wissenschafstlogik ) that they so desire.

Reichenbach 1969 [1924] stands as a canonical attempt by a central developer of the Syntactic View of axiomatizing a physical theory, viz., relativity theory (cf. Friedman 1983, 1999; see also Reichenbach 1965 [1920]). For the purposes of this encyclopedia entry, it is preferable to turn to another syntactic axiomatization effort. In axiomatizing Newtonian mechanics, the mid-20 th century mathematical logician Hans Hermes spent significant energy defining the concept of mass (Hermes 1938, 1959; Jammer 1961). More precisely, he defines the theoretical concept of “mass ratio” of two particles colliding inelastically in an inertial reference frame \(S\). Here is his full definition of mass ratio (1959, 287):

One paraphrase of this definition is, “‘the mass of \(x\) is α times that of \(x_0\)’ is equivalent to ‘there exists a system \(S\), an instant \(t\), momentary mass points \(y\) and \(y_0\), and initial velocities \(v\) and \(v_0\), such that \(y\) and \(y_0\) are genidentical, respectively, with \(x\) and \(x_0\); the joined mass points move with a velocity of 0 with respect to frame \(S\) immediately upon colliding at time \(t\); and \(y\) and \(y_0\) have determinate velocities \(v\) and \(v_0\) before the collision in the ratio α, which could also be 1 if \(x\) and \(x_0\) are themselves genidentical.’” Hermes employs the notion of “genidentical” to describe the relation between two temporal sections of a given particle’s world line (Jammer 1961, 113). Set aside the worry that two distinct particles cannot be genidentical per Hermes’ definition, though they can have identical properties. In short, this definition is syntactically complete and is written in first-order predicate logic, as are the other axioms and definitions in Hermes (1938, 1959). Correspondence rules connecting a postulated mass \(x\) with an actual mass were not articulated by Hermes.

The link between theory structure and the world, under the Syntactic View, is contained in the theory itself: \(C_S\), the set of correspondence rules. The term “correspondence rules” (Margenau 1950; Nagel 1961, 97–105; Carnap 1966, Chapter 24) has a variety of near-synonyms:

• Dictionary (Campbell 1920)
• Operational rules (Bridgman 1927)
• Coordinative definitions (Reichenbach 1969 [1924], 1938)
• Reduction sentences (Carnap 1936/1937; Hempel 1952)
• Correspondence postulates (Carnap 1963)
• Bridge principles (Hempel 1966; Kitcher 1984)
• Reduction functions (Schaffner 1969, 1976)
• Bridge laws (Sarkar 1998)

Important differences among these terms cannot be mapped out here. However, in order to better understand correspondence rules, two of their functions will be considered: (i) theory interpretation (Carnap, Hempel) and (ii) theory reduction (Nagel, Schaffner). The dominant perspective on correspondence rules is that they interpret theoretical terms. Unlike “mathematical theories,” the axiomatic system of physics “cannot have… a splendid isolation from the world” (Carnap 1966, 237). Instead, scientific theories require observational interpretation through correspondence rules. Even so, surplus meaning always remains in the theoretical structure (Hempel 1958, 87; Carnap 1966). Second, correspondence rules are seen as necessary for inter-theoretic reduction (van Riel and Van Gulick 2014). For instance, they connect observation terms such as “temperature” in phenomenological thermodynamics (the reduced theory) to theoretical concepts such as “mean kinetic energy” in statistical mechanics (the reducing theory). Correspondence rules unleash the reducing theory’s epistemic power. Notably, Nagel (1961, Chapter 11; 1979) and Schaffner (1969, 1976, 1993) allow for multiple kinds of correspondence rules, between terms of either vocabulary, in the reducing and the reduced theory (cf. Callender 1999; Winther 2009; Dizadji-Bahmani, Frigg, and Hartmann 2010). Correspondence rules are a core part of the structure of scientific theories and serve as glue between theory and observation.

Finally, while they are not part of the theory structure, and although we saw some examples above, observation sentences are worth briefly reviewing. Correspondence rules attach to the content of observational sentences. Observational sentences were analyzed as (i) protocol sentences or Protokollsätze (e.g., Schlick 1934; Carnap 1987 [1932], 1937, cf. 1963; Neurath 1983 [1932]), and as (ii) experimental laws (e.g., Campbell 1920; Nagel 1961; Carnap 1966; cf. Duhem 1954 [1906]). Although constrained by Calc , the grammar of these sentences is determined primarily by the order of nature, as it were. In general, syntacticists do not consider methods of data acquisition, experiment, and measurement to be philosophically interesting. In contrast, the confirmation relation between (collected) data and theory, especially as developed in inductive logic (e.g., Reichenbach 1938, 1978; Carnap 1962 [1950], 1952), as well as questions about the conventionality, grammaticality, foundationalism, atomism, and content of sense-data and synthetic statements, are considered philosophically important (e.g., Carnap 1987 [1932], 1937, 1966; Neurath 1983 [1932]; Reichenbach 1951; Schlick 1925 [1918], 1934; for contemporary commentary, see, e.g., Creath 1987, 2014; Rutte 1991; Friedman 1999).

To summarize, the Syntactic View holds that there are three kinds of terms or vocabularies: logical, theoretical, and observational; three kinds of sentences: \(T_S\), \(C_S\), and \(O_S\); and two languages: \(L_T\) and \(L_O\). Moreover, the structure of scientific theories could be analyzed using the logical tools of metamathematics. The goal is to reconstruct the logic of science, viz. to articulate an axiomatic system.

Interestingly, this perspective has able and active defenders today, who discuss constitutive and axiomatized principles of the historical “relativized a priori” (Friedman 2001, cf. 2013), argue that “the semantic view, if plausible, is syntactic” (Halvorson 2013), and explore “logicism” for, and in, the philosophy of science (Demopulous 2003, 2013; van Benthem 2012). Furthermore, for purposes of the syntactic reconstruction of scientific theories, some continue espousing—or perhaps plea for the resurrection of—predicate logic (e.g., Lutz 2012, 2014), while other contemporary syntacticists (e.g., Halvorson 2012, 2013, 2019) endorse more recently developed metamathematical and mathematical equipment, such as category theory, which “turns out to be a kind of universal mathematical language like set theory” (Awodey 2006, 2; see Eilenberg and MacLane 1945). Importantly, Halvorson (2019) urges that interlocutors adopt “structured” rather than “flat” views of theories. For the case of the syntactic view this would mean that rather than accept the usual formulation that a theory is a set of sentences, “… [we] might say that a theory consists of both sentences and inferential relations between those sentences” (Halvorson 2019, 277–8). Classical syntacticists such as Rudolf Carnap (Friedman 1999, 2011; Carus 2007; Blatti and Lapointe 2016; Koellner ms. in Other Internet Resources) and Joseph Henry Woodger (Nicholson and Gawne 2014) have recently received increasing attention.

3. The Semantic View

An overarching theme of the Semantic View is that analyzing theory structure requires employing mathematical tools rather than predicate logic. After all, defining scientific concepts within a specific formal language makes any axiomatizing effort dependent on the choice, nature, and idiosyncrasies of that narrowly-defined language. For instance, Suppes understands first-order predicate logic, with its “linguistic” rather than “set-theoretical” entities, as “utterly impractical” for the formalization of “theories with more complicated structures like probability theory” (Suppes 1957, 232, 248–9; cf. Suppes 2002). Van Fraassen, another influential defender of the Semantic View, believes that the logical apparatus of the Syntactic View “had moved us mille milles de toute habitation scientifique , isolated in our own abstract dreams” (van Fraassen 1989, 225). Indeed, what would the appropriate logical language for specific mathematical structures be, especially when such structures could be reconstructed in a variety of formal languages? Why should we imprison mathematics and mathematical scientific theory in syntactically defined language(s) when we could, instead, directly investigate the mathematical objects, relations, and functions of scientific theory?

Consistent with the combat strategy (discussed in the Conclusion), here is a list of grievances against the Syntactic View discussed at length in the work of some semanticists.

• First-Order Predicate Logic Objection . Theoretical structure is intrinsically and invariably tied to the specific choice of a language, \(L_T\), expressed in first-order predicate logic. This places heavy explanatory and representational responsibility on relatively inflexible and limited languages.
• Theory Individuation Objection . Since theories are individuated by their linguistic formulations, every change in high-level syntactic formulations will bring forth a distinct theory. This produces a reductio: if \(T_1 = p \rightarrow q\) and \(T_2 = \neg p \vee q\) then \(T_1\) and \(T_2\), though logically equivalent, have different syntactic formulations and would be distinct theories.
• Theoretical/Observational Languages Objection . Drawing the theoretical/observational distinction in terms of language is inappropriate, as observability pertains to entities rather than to concepts.
• Unintended Models Objection . There is no clear way of distinguishing between intended and unintended models for syntactically characterized theories (e.g., the Löwenheim-Skolem theorem, Bays 2014).
• Confused Correspondence Rules Objection . Correspondence rules are a confused medley of direct meaning relationships between terms and world, means of inter-theoretic reduction, causal relationship claims, and manners of theoretical concept testing.
• Trivially True yet Non-Useful Objection . Presenting scientific theory in a limited axiomatic system, while clearly syntactically correct, is neither useful nor honest, since scientific theories are mathematical structures.
• Practice and History Ignored Objection . Syntactic approaches do not pay sufficient attention to the actual practice and history of scientific theorizing and experimenting.

What, then, does the Semantic View propose to put in the Syntactic View’s place?

Even a minimal description of the Semantic View must acknowledge two distinct strategies of characterizing and comprehending theory structure: the state-space and the set-/model-theoretic approaches.

3.1.1 The State-Space Approach

The state-space approach emphasizes the mathematical models of actual science, and draws a clear line between mathematics and metamathematics. The structure of a scientific theory is identified with the “class,” “family” or “cluster” of mathematical models constituting it, rather than with any metamathematical axioms “yoked to a particular syntax” (van Fraassen 1989, 366). Under this analysis, “the correct tool for philosophy of science is mathematics, not metamathematics”—this is Suppes’ slogan, per van Fraassen (1989, 221; 1980, 65). In particular, a state space or phase space is an \(N\)-dimensional space, where each of the relevant variables of a theory correspond to a single dimension and each point in that space represents a possible state of a real system. An actual, real system can take on, and change, states according to different kinds of laws, viz., laws of succession determining possible trajectories through that space (e.g., Newtonian kinematic laws); laws of co-existence specifying the permitted regions of the total space (e.g., Boyle’s law); and laws of interaction combining multiple laws of succession or co-existence, or both (e.g., population genetic models combining laws of succession for selection and genetic drift, Wright 1969; Lloyd 1994 [1988]; Rice 2004; Clatterbuck, Sober, and Lewontin 2013). Different models of a given theory will share some dimensions of their state space while differing in others. Such models will also partially overlap in laws (for further discussion of state spaces, laws, and models pertinent to the Semantic View, see Suppe 1977, 224–8; Lloyd 1994, Chapter 2; Nolte 2010; Weisberg 2013, 26–9).

Historically, the state-space approach emerged from work by Evert Beth, John von Neumann, and Hermann Weyl, and has important parallels with Przełęcki (1969) and Dalla Chiara Scabia and Toraldo di Francia (1973) (on the history of the approach see: Suppe 1977; van Fraassen 1980, 65–67; Lorenzano 2013; advocates of the approach include: Beatty 1981; Giere 1988, 2004; Giere, Bickle, and Mauldin 2006; Lloyd 1983, 1994 [1988], 2013 In Press; Suppe 1977, 1989; Thompson, 1989, 2007; van Fraassen 1980, 1989, 2008; for alternative early analyses of models see, e.g., Braithwaite 1962; Hesse 1966, 1967). Interestingly, van Fraassen (1967, 1970) provides a potential reconstruction of state spaces via an analysis of “semi-interpreted languages.” Weisberg (2013), building on many insights from Giere’s work, presents a broad view of modeling that includes mathematical structures that are “trajectories in state spaces” (29), but also permits concrete objects and computational structures such as algorithms to be deemed models. Lorenzano (2013) calls Giere’s (and, by extension, Weisberg’s and even Godfrey-Smith’s 2006) approach “model-based,” separating it out from the state-space approach. A more fine-grained classification of the state-space approach is desirable, particularly if we wish to understand important lessons stemming from the Pragmatic View of Theories, as we shall see below.

As an example of a state-space analysis of modeling, consider a capsule traveling in outer space. An empirically and dynamically adequate mathematical model of the capsule’s behavior would capture the position of the capsule (i.e., three dimensions of the formal state space), as well as the velocity and acceleration vectors for each of the three standard spatial dimensions (i.e., six more dimensions in the formal state space). If the mass were unknown or permitted to vary, we would have to add one more dimension. Possible and actual trajectories of our capsule, with known mass, within this abstract 9-dimensional state space could be inferred via Newtonian dynamical laws of motion (example in Lewontin 1974, 6–8; consult Suppe 1989, 4). Importantly, under the state-space approach, the interesting philosophical work of characterizing theory structure (e.g., as classes of models), theory meaning (e.g., data models mapped to theoretical models), and theory function (e.g., explaining and predicting) happens at the level of mathematical models.

3.1.2 The Set-/Model-Theoretic Approach

Lurking in the background of the state-space conception is the fact that mathematics actually includes set theory and model theory—i.e., mathematical logic. Indeed, according to some interlocutors, “metamathematics is part of mathematics” (Halvorson 2012, 204). Historically, a set-/model-theoretic approach emerged from Tarski’s work and was extensively articulated by Suppes and his associates (van Fraassen 1980, 67). Set theory is a general language for formalizing mathematical structures as collections—i.e., sets—of abstract objects (which can themselves be relations or functions; see Krivine 2013 [1971]). Model theory investigates the relations between, on the one hand, the formal axioms, theorems, and laws of a particular theory and, on the other hand, the mathematical structures—the models—that provide an interpretation of that theory, or put differently, that make the theory’s axioms, theorems, and laws true (Hodges 1997, Chapter 2; Jones 2005). Interestingly, model theory often uses set theory (e.g., Marker 2002); set theory can, in turn, be extended to link axiomatic theories and semantic models via “set-theoretical predicates” (e.g., Suppes 1957, 2002). Finally, there are certain hybrids of these two branches of mathematical logic, including “partial structures” (e.g., da Costa and French 1990, 2003; Bueno 1997; French 2017; French and Ladyman 1999, 2003; Vickers 2009; Bueno, French, and Ladyman 2012). Lorenzano (2013) provides a more complex taxonomy of the intellectual landscape of the Semantic View, including a discussion of Structuralism, a kind of set-/model-theoretic perspective. Structuralism involves theses about “theory-nets,” theory-relative theoretical vs. non-theoretical terms, a diversity of intra- and inter-theoretic laws with different degrees of generality, a typology of inter-theoretic relations, and a rich account of correspondence rules in scientific practice (see Moulines 2002; Pereda 2013; Schmidt 2014; Ladyman 2014). On the whole, the set-/model-theoretic approach of the Semantic View insists on the inseparability of metamathematics and mathematics. In preferring to characterize a theory axiomatically in terms of its intension rather than its extension, it shares the Syntactic View’s aims of reconstructive axiomatization (e.g., Sneed 1979; Stegmüller 1979; Frigg and Votsis 2011; Halvorson 2013, 2019; Lutz 2012, 2014, 2017).

An example will help motivate the relation between theory and model. Two qualifications are required: (i) we return to a more standard set-/model-theoretic illustration below, viz., McKinsey, Sugar, and Suppes’ (1953) axiomatization of particle mechanics, and (ii) this motivational example is not from the heartland of model theory (see Hodges 2013). Following van Fraassen’s intuitive case of “seven-point geometry” (1980, 41–44; 1989, 218–220), also known as “the Fano plane” we see how a particular geometric figure, the model , interprets and makes true a set of axioms and theorems, the theory . In topology and geometry there is rich background theory regarding how to close Euclidean planes and spaces to make finite geometries by, for instance, eliminating parallel lines. Consider the axioms of a projective plane:

• For any two points, exactly one line lies on both.
• For any two lines, exactly one point lies on both.
• There exists a set of four points such that no line has more than two of them.

A figure of a geometric model that makes this theory true is:

This is the smallest geometrical model satisfying the three axioms of the projective plane theory. Indeed, this example fits van Fraassen’s succinct characterization of the theory-model relation:

A model is called a model of a theory exactly if the theory is entirely true if considered with respect to this model alone. (Figuratively: the theory would be true if this model was the whole world.) (1989, 218)

That is, if the entire universe consisted solely of these seven points and seven lines, the projective plane theory would be true. Of course, our universe is bigger. Because Euclidean geometry includes parallel lines, the Fano plane is not a model of Euclidean geometry. Even so, by drawing the plane, we have shown it to be isomorphic to parts of the Euclidean plane. In other words, the Fano plane has been embedded in a Euclidean plane. Below we return to the concepts of embedding and isomorphism, but this example shall suffice for now to indicate how a geometric model can provide a semantics for the axioms of a theory.

In short, for the Semantic View the structure of a scientific theory is its class of mathematical models. According to some advocates of this view, the family of models can itself be axiomatized, with those very models (or other models) serving as axiom truth-makers.

Returning to our running example, consider Suppes’ 1957 model-theoretic articulation of particle mechanics, which builds on his 1953 article with J.C.C. McKinsey and A.C. Sugar. Under this analysis, there is a domain of set-theoretic objects of the form \(\{ P, T, s, m, f, g \}\), where \(P\) and \(T\) are themselves sets, \(s\) and \(g\) are binary functions, \(m\) is a unary and \(f\) a ternary function. \(P\) is the set of particles; \(T\) is a set of real numbers measuring elapsed times; \(s(p, t)\) is the position of particle \(p\) at time \(t\); \(m(p)\) is the mass of particle \(p\); \(f(p, q, t)\) is the force particle \(q\) exerts on \(p\) at time \(t\); and \(g(p, t)\) is the total resultant force (by all other particles) on \(p\) at time \(t\). Suppes and his collaborators defined seven axioms—three kinematical and four dynamical—characterizing Newtonian particle mechanics (see also Simon 1954, 1970). Such axioms include Newton’s third law reconstructed in set-theoretic formulation thus (Suppes 1957, 294):

Importantly, the set-theoretic objects are found in more than one of the axioms of the theory, and Newton’s calculus is reconstructed in a novel, set-theoretic form. Set-theoretic predicates such as “is a binary relation” and “is a function” are also involved in axiomatizing particle mechanics (Suppes 1957, 249). Once these axioms are made explicit, their models can be specified and these can, in turn, be applied to actual systems, thereby providing a semantics for the axioms (e.g., as described in Section 3.3.1 below). A particular system satisfying these seven axioms is a particle mechanics system. (For an example of Newtonian mechanics from the state-space approach, recall the space capsule of Section 3.1.1.)

How is the theory structure, described in Section 3.1, applied to empirical phenomena? How do we connect theory and data via observation and experimental and measuring techniques? The Semantic View distinguishes theory individuation from both theory-phenomena and theory-world relations. Three types of analysis of theory interpretation are worth investigating: (i) a hierarchy of models (e.g., Suppes; Suppe), (ii) similarity (e.g., Giere; Weisberg), and (iii) isomorphism (e.g., van Fraassen; French and Ladyman).

3.3.1 A Hierarchy of Models

One way of analyzing theory structure interpretation is through a series of models falling under the highest-level axiomatizations. This series has been called “a hierarchy of models,” though it need not be considered a nested hierarchy. These models include models of theory, models of experiment, and models of data (Suppes 1962, 2002). Here is a summary of important parts of the hierarchy (Suppes 1962, Table 1, 259; cf. Giere 2010, Figure 1, 270):

• Axioms of Theory . Axioms define set-theoretic predicates, and constitute the core structure of scientific theories, as reviewed in Section 3.1.2.
• Models of Theory. “Representation Theorems,” permit us “to discover if an interesting subset of models for the theory may be found such that any model for the theory is isomorphic to some member of this subset” (Suppes 1957, 263). Representation theorem methodology can be extended (i) down the hierarchy, both to models of experiment and models of data, and (ii) from isomorphism to homomorphism (Suppes 2002, p. 57 ff.; Suppe 2000; Cartwright 2008).
• Models of Experiment . Criteria of experimental design motivate choices for how to set up and analyze experiments. There are complex mappings between models of experiment thus specified, and (i) models of theory, (ii) theories of measurement, and (iii) models of data.
• Models of Data . In building models of data, phenomena are organized with respect to statistical goodness-of-fit tests and parameter estimation, in the context of models of theory. Choices about which parameters to represent must be made.

The temptation to place phenomena at the bottom of the hierarchy must be resisted because phenomena permeate all levels. Indeed, the “class of phenomena” pertinent to a scientific theory is its “intended scope” (Suppe 1977, 223; Weisberg 2013, 40). Furthermore, this temptation raises fundamental questions about scientific representation: “there is the more profound issue of the relationship between the lower most representation in the hierarchy—the data model perhaps—and reality itself, but of course this is hardly something that the semantic approach alone can be expected to address” (French and Ladyman 1999, 113; cf. van Fraassen 2008, 257–258, “The ‘link’ to reality”). Borrowing from David Chalmers, the “hard problem” of philosophy of science remains connecting abstract structures to concrete phenomena, data, and world.

3.3.2 Similarity

The similarity analysis of theory interpretation combines semantic and pragmatic dimensions (Giere 1988, 2004, 2010; Giere, Bickle, and Mauldin 2006; Weisberg 2013). According to Giere, interpretation is mediated by theoretical hypotheses positing representational relations between a model and relevant parts of the world. Such relations may be stated as follows:

Here \(S\) is a scientist, research group or community, \(W\) is a part of the world, and \(X\) is, broadly speaking, any one of a variety of models (Giere 2004, 743, 747, 2010). Model-world similarity judgments are conventional and intentional:

Note that I am not saying that the model itself represents an aspect of the world because it is similar to that aspect. …Anything is similar to anything else in countless respects, but not anything represents anything else. It is not the model that is doing the representing; it is the scientist using the model who is doing the representing. (2004, 747)

Relatedly, Weisberg (2013) draws upon Tversky (1977) to develop a similarity metric for model interpretation (equation 8.10, 148). This metric combines (i) model-target semantics (90–97), and (ii) the pragmatics of “context, conceptualization of the target, and the theoretical goals of the scientist” (149). Giere and Weisberg thus endorse an abundance of adequate mapping relations between a given model and the world. From this diversity, scientists and scientific communities must select particularly useful similarity relationships for contextual modeling purposes. Because of semantic pluralism and irreducible intentionality, this similarity analysis of theory interpretation cannot be accommodated within a hierarchy of models approach, interpreted as a neat model nesting based on pre-given semantic relations among models at different levels.

3.3.3 Isomorphism

The term “isomorphism” is a composite of the Greek words for “equal” and “shape” or “form.” Indeed, in mathematics, isomorphism is a perfect one-to-one, bijective mapping between two structures or sets. Figure (2) literally and figuratively captures the term:

Especially in set theory, category theory, algebra, and topology, there are various kinds of “-morphisms,” viz., of mapping relations between two structures or models. Figure (3) indicates five different kinds of homomorphism, arranged in a Venn diagram.

Although philosophers have focused on isomorphism, other morphisms such as monomorphism (i.e., an injective homomorphism where some elements in the co-domain remain unmapped from the domain) might also be interesting to investigate, especially for embedding data (i.e., the domain) into rich theoretical structures (i.e., the co-domain). To complete the visualization above, an epimorphism is a surjective homomorphism, and an endomorphism is a mapping from a structure to itself, although it need not be a symmetrical—i.e., invertible—mapping, which would be an automorph.

Perhaps the most avid supporter of isomorphism and embedding as the way to understand theory interpretation is van Fraassen. In a nutshell, if we distinguish (i) theoretical models, (ii) “empirical substructures” (van Fraassen 1980, 64, 1989, 227; alternatively: “surface models” 2008, 168), and (iii) “observable phenomena” (1989, 227, 2008, 168), then, van Fraassen argues, theory interpretation is a relation of isomorphism between observable phenomena and empirical substructures, which are themselves isomorphic with one or more theoretical models. Moreover, if a relation of isomorphism holds between \(X\) and a richer \(Y\), we say that we have embedded \(X\) in \(Y\). For instance, with respect to the seven-point geometry above (Figure 1), van Fraassen contends that isomorphism gives embeddability, and that the relation of isomorphism “is important because it is also the exact relation a phenomenon bears to some model or theory, if that theory is empirically adequate” (1989, 219–20; this kind of statement seems to be simultaneously descriptive and prescriptive about scientific representation, see Section 1.1 above). In The Scientific Image he is even clearer about fleshing out the empirical adequacy of a theory (with its theoretical models) in terms of isomorphism between “appearances” (i.e., “the structures which can be described in experimental and measurement reports,” 1980, 64, italics removed) and empirical substructures. Speaking metaphorically,

the phenomena are, from a theoretical point of view, small, arbitrary, and chaotic—even nasty, brutish, and short…—but can be understood as embeddable in beautifully simple but much larger mathematical models. (2008, 247; see also van Fraassen 1981, 666 and 1989, 230)

Interestingly, and as a defender of an identity strategy (see Conclusion), Friedman also appeals to embedding and subsumption relations between theory and phenomena in his analyses of theory interpretation (Friedman 1981, 1983). Bueno, da Costa, French, and Ladyman also employ embedding and (partial) isomorphism in the empirical interpretation of partial structures (Bueno 1997; Bueno, French, and Ladyman 2012; da Costa and French 1990, 2003; French 2017; French and Ladyman 1997, 1999, 2003; Ladyman 2004). Suárez discusses complexities in van Fraassen’s analyses of scientific representation and theory interpretation (Suárez 1999, 2011). On the one hand, representation is structural identity between the theoretical and the empirical. On the other hand, “There is no representation except in the sense that some things are used, made, or taken, to represent some things as thus or so” (van Fraassen 2008, 23, italics removed). The reader interested in learning how van Fraassen simultaneously endorses acontextually structural and contextually pragmatic aspects of representation and interpretation should refer to van Fraassen’s (2008) investigations of maps and “the essential indexical.” [To complement the structure vs. function distinction, see van Fraassen 2008, 309–311 for a structure (“structural relations”) vs. history (“the intellectual processes that lead to those models”) distinction; cf. Ladyman et al. 2011] In all of this, embedding via isomorphism is a clear contender for theory interpretation under the Semantic View.

In short, committing to either a state-space or a set-/model-theoretic view on theory structure does not imply any particular perspective on theory interpretation (e.g., hierarchy of models, similarity, embedding). Instead, commitments to the former are logically and actually separable from positions on the latter (e.g., Suppes and Suppe endorse different accounts of theory structure, but share an understanding of theory interpretation in terms of a hierarchy of models). The Semantic View is alive and well as a family of analyses of theory structure, and continues to be developed in interesting ways both in its state-space and set-/model-theoretic approaches.

4. The Pragmatic View

The Pragmatic View recognizes that a number of assumptions about scientific theory seem to be shared by the Syntactic and Semantic Views. Both perspectives agree, very roughly, that theory is (1) explicit, (2) mathematical, (3) abstract, (4) systematic, (5) readily individualizable, (6) distinct from data and experiment, and (7) highly explanatory and predictive (see Flyvbjerg 2001, 38–39; cf. Dreyfus 1986). The Pragmatic View imagines the structure of scientific theories rather differently, arguing for a variety of theses:

• Limitations . Idealized theory structure might be too weak to ground the predictive and explanatory work syntacticists and semanticists expect of it (e.g., Cartwright 1983, 1999a, b, 2019; Morgan and Morrison 1999; Suárez and Cartwright 2008).
• Pluralism . Theory structure is plural and complex both in the sense of internal variegation and of existing in many types. In other words, there is an internal pluralism of theory (and model) components (e.g., mathematical concepts, metaphors, analogies, ontological assumptions, values, natural kinds and classifications, distinctions, and policy views, e.g., Kuhn 1970; Boumans 1999), as well as a broad external pluralism of different types of theory (and models) operative in science (e.g., mechanistic, historical, and mathematical models, e.g., Hacking 2009, Longino 2013). Indeed, it may be better to speak of the structures of scientific theories, in the double-plural.
• Nonformal aspects. The internal pluralism of theory structure (thesis #2) includes many nonformal aspects deserving attention. That is, many components of theory structure, such as metaphors, analogies, values, and policy views have a non-mathematical and “informal” nature, and they lie implicit or hidden (e.g., Bailer-Jones 2002; Craver 2002; Contessa 2006; Morgan 2012). Interestingly, the common understanding of “formal,” which identifies formalization with mathematization, may itself be a conceptual straightjacket; the term could be broadened to include “diagram abstraction” and “principle extraction” (e.g., Griesemer 2013, who explicitly endorses what he also calls a “Pragmatic View of Theories”).
• Function. Characterizations of the nature and dynamics of theory structure should pay attention to the user as well as to purposes and values (e.g., Apostel 1960; Minsky 1965; Morrison 2007; Winther 2012a).
• Practice . Theory structure is continuous with practice and “the experimental life,” making it difficult to neatly dichotomize theory and practice (e.g., Hacking 1983, 2009; Shapin and Schaffer 1985; Galison 1987, 1988, 1997; Suárez and Cartwright 2008, Cartwright 2019).

These are core commitments of the Pragmatic View.

It is important to note at the outset that the Pragmatic View takes its name from the linguistic trichotomy discussed above, in the Introduction. This perspective need not imply commitment to, or association with, American Pragmatism (e.g. the work of Charles S. Peirce, William James, or John Dewey; cf. Hookway 2013; Richardson 2002). For instance, Hacking (2007a) distinguishes his pragmatic attitudes from the school of Pragmatism. He maps out alternative historical routes of influence, in general and on him, vis-à-vis fallibilism (via Imre Lakatos, Karl Popper; Hacking 2007a, §1), historically conditioned truthfulness (via Bernard Williams; Hacking 2007a, §3), and realism as intervening (via Francis Everitt, Melissa Franklin; Hacking 2007a, §4). To borrow a term from phylogenetics, the Pragmatic View is “polyphyletic.” The components of its analytical framework have multiple, independent origins, some of which circumnavigate American Pragmatism.

With this qualification and the five theses above in mind, let us now turn to the Pragmatic View’s analysis of theory structure and theory interpretation.

We should distinguish two strands of the Pragmatic View: the Pragmatic View of Models and a proper Pragmatic View of Theories .

4.1.1 The Pragmatic View of Models

Nancy Cartwright’s How the Laws of Physics Lie crystallized the Pragmatic View of Models. Under Cartwright’s analysis, models are the appropriate level of investigation for philosophers trying to understand science. She argues for significant limitations of theory (thesis #1), claiming that laws of nature are rarely true, and are epistemically weak. Theory as a collection of laws cannot, therefore, support the many kinds of inferences and explanations that we have come to expect it to license. Cartwright urges us to turn to models and modeling, which are central to scientific practice. Moreover, models “lie”—figuratively and literally—between theory and the world (cf. Derman 2011). That is, “to explain a phenomenon is to find a model that fits it into the basic framework of the theory and that thus allows us to derive analogues for the messy and complicated phenomenological laws which are true of it.” A plurality of models exist, and models “serve a variety of purposes” (Cartwright 1983, 152; cf. Suppes 1978). Cartwright is interested in the practices and purposes of scientific models, and asks us to focus on models rather than theories.

Cartwright’s insights into model pluralism and model practices stand as a significant contribution of “The Stanford School” (cf. Cat 2014), and were further developed by the “models as mediators” group, with participants at LSE, University of Amsterdam, and University of Toronto (Morgan and Morrison 1999; Chang 2011; cf. Martínez 2003). This group insisted on the internal pluralism of model components (thesis #2). According to Morgan and Morrison, building a model involves “fitting together… bits which come from disparate sources,” including “stories” (Morgan and Morrison 1999, 15). Boumans (1999) writes:

model building is like baking a cake without a recipe. The ingredients are theoretical ideas, policy views, mathematisations of the cycle, metaphors and empirical facts. (67) Mathematical moulding is shaping the ingredients in such a mathematical form that integration is possible… (90)

In an instructive diagram, Boumans suggests that a variety of factors besides theory and data feed into a model: metaphors, analogies, policy views, stylised facts, mathematical techniques, and mathematical concepts (93). The full range of components involved in a model will likely vary according to discipline, and with respect to explanations and interventions sought (e.g., analogies but not policy views will be important in theoretical physics). In short, model building involves a complex variety of internal nonformal aspects, some of which are implicit (theses #2 and #3).

As one example of a nonformal component of model construction and model structure, consider metaphors and analogies (e.g., Bailer-Jones 2002). Geary (2011) states the “simplest equation” of metaphor thus: “\(X = Y\)” (8, following Aristotle: “Metaphor consists in giving the thing a name that belongs to something else… ,” Poetics , 1457b). The line between metaphor and analogy in science is blurry. Some interlocutors synonymize them (e.g., Hoffman 1980; Brown 2003), others reduce one to the other (analogy is a form of metaphor, Geary 2011; metaphor is a kind of analogy, Gentner 1982, 2003), and yet others bracket one to focus on the other (e.g., Oppenheimer 1956 sets aside metaphor). One way to distinguish them is to reserve “analogy” for concrete comparisons, with clearly identifiable and demarcated source and target domains, and with specific histories, and use “metaphor” for much broader and indeterminate comparisons, with diffuse trajectories across discourses. Analogies include the “lines of force” of electricity and magnetism (Maxwell and Faraday), the atom as a planetary system (Rutherford and Bohr), the benzene ring as a snake biting its own tail (Kekulé), Darwin’s “natural selection” and “entangled bank,” and behavioral “drives” (Tinbergen) (e.g., Hesse 1966, 1967; Bartha 2010). Examples of metaphor are genetic information, superorganism, and networks (e.g., Keller 1995). More could be said about other informal model components, but this discussion of metaphors and analogies shall suffice to hint at how models do not merely lie between theory and world. Models express a rich internal pluralism (see also de Chadarevian and Hopwood 2004; Morgan 2012).

Model complexity can also be seen in the external plurality of models (thesis #2). Not all models are mathematical, or even ideally recast as mathematical. Non-formalized (i.e., non–state-space, non-set-/model-theoretic) models such as physical, diagrammatic, material, historical, “remnant,” and fictional models are ubiquitous across the sciences (e.g., Frigg and Hartmann 2012; for the biological sciences, see Hull 1975; Beatty 1980; Griesemer 1990, 1991 a, b, 2013; Downes 1992; Richards 1992; Winther 2006a; Leonelli 2008; Weisberg 2013). Moreover, computer simulations differ in important respects from more standard analytical mathematical models (e.g., Smith 1996; Winsberg 2010; Weisberg 2013). According to some (e.g., Griesemer 2013; Downes 1992; Godfrey-Smith 2006; Thomson-Jones 2012), this diversity belies claims by semanticists that models can always be cast “into set theoretic terms” (Lloyd 2013 In Press), are “always a mathematical structure” (van Fraassen 1970, 327), or that “formalisation of a theory is an abstract representation of the theory expressed in a formal deductive framework… in first-order predicate logic with identity, in set theory, in matrix algebra and indeed, any branch of mathematics...” (Thompson 2007, 485–6). Even so, internal pluralism has been interpreted as supporting a “deflationary semantic view,” which is minimally committed to the perspective that “model construction is an important part of scientific theorizing” (Downes 1992, 151). Given the formal and mathematical framework of the Semantic View (see above), however, the broad plurality of kinds of models seems to properly belong under a Pragmatic View of Models.

4.1.2 The Pragmatic View of Theories

Interestingly, while critiquing the Syntactic and Semantic Views on most matters, the Pragmatic View of Models construed theory, the process of theorizing, and the structure of scientific theories, according to terms set by the two earlier views. For instance, Cartwright tends to conceive of theory as explicit, mathematical, abstract, and so forth (see the first paragraph of Section 4). She always resisted “the traditional syntactic/semantic view of theory” for its “vending machine” view, in which a theory is a deductive and automated machine that upon receiving empirical input “gurgitates” and then “drops out the sought-for representation” (1999a, 184–5). Rather than reform Syntactic and Semantic accounts of theory and theory structure, however, she invites us, as we just saw, to think of science as modeling, “with theory as one small component” (Cartwright, Shomar, and Suárez 1995, 138; Suárez and Cartwright 2008). Many have followed her. Kitcher’s predilection is also to accept the terms of the Syntactic and Semantic Views. For instance, he defines theories as “axiomatic deductive systems” (1993, 93). In a strategy complementary to Cartwright’s modeling turn, Kitcher encourages us to focus on practice, including practices of modeling and even practices of theorizing. In The Advancement of Science , practice is analyzed as a 7-tuple, with the following highly abbreviated components: (i) a language; (ii) questions; (iii) statements (pictures, diagrams); (iv) explanatory patterns; (v) standard examples; (vi) paradigms of experimentation and observation, plus instruments and tools; and (vii) methodology (Kitcher 1993, 74). Scientific practice is also center stage for those singing the praises of “the experimental life” (e.g., Hacking 1983; Shapin and Schaffer 1985; Galison 1987), and those highlighting the cognitive grounds of science (e.g., Giere 1988; Martínez 2014) and science’s social and normative context (e.g., Kitcher 1993, 2001; Longino 1995, 2002; Ziman 2000; cf. Simon 1957). Indeed, the modeling and practice turns in the philosophy of science were reasonable reactions to the power of axiomatic reconstructive and mathematical modeling analyses of the structure of scientific theories.

Yet, a Pragmatic View of Theories is also afoot, one resisting orthodox characterizations of theory often embraced, at least early on, by Pragmatic View philosophers such as Cartwright, Hacking, Kitcher, and Longino. For instance, Craver (2002) accepts both the Syntactic and Semantic Views, which he humorously and not inaccurately calls “the Once Received View” and the “Model Model View.” But he also observes:

While these analyses have advanced our understanding of some formal aspects of theories and their uses, they have neglected or obscured those aspects dependent upon nonformal patterns in theories. Progress can be made in understanding scientific theories by attending to their diverse nonformal patterns and by identifying the axes along which such patterns might differ from one another. (55)

Craver then turns to mechanistic theory as a third theory type (and a third philosophical analysis of theory structure) that highlights nonformal patterns:

Different types of mechanisms can be distinguished on the basis of recurrent patterns in their organization. Mechanisms may be organized in series, in parallel, or in cycles. They may contain branches and joins, and they often include feedback and feedforward subcomponents. (71)

Consistent with theses #2 and #3 of the Pragmatic View, we must recognize the internal pluralism of theories as including nonformal components. Some of these are used to represent organizational and compositional relations of complex systems (Craver 2007; Wimsatt 2007; Winther 2011; Walsh 2015). While mechanistic analyses such as Craver’s may not wish to follow every aspect of the Pragmatic View of Theories, there are important and deep resonances between the two.

In a review of da Costa and French (2003), Contessa (2006) writes:

Philosophers of science are increasingly realizing that the differences between the syntactic and the semantic view are less significant than semanticists would have it and that, ultimately, neither is a suitable framework within which to think about scientific theories and models. The crucial divide in philosophy of science, I think, is not the one between advocates of the syntactic view and advocates of the semantic view, but the one between those who think that philosophy of science needs a formal framework or other and those who think otherwise. (376)

Again, we are invited to develop a non-formal framework of science and presumably also of scientific theory. (Halvorson 2012, 203 takes Contessa 2006 to task for advocating “informal philosophy of science.”) Moreover, in asking “what should the content of a given theory be taken to be on a given occasion?”, Vickers (2009) answers:

It seems clear that, in addition to theories being vague objects in the way that ‘heaps’ of sand are, there will be fundamentally different ways to put together theoretical assumptions depending on the particular investigation one is undertaking. For example, sometimes it will be more appropriate to focus on the assumptions which were used by scientists, rather than the ones that were believed to be true. (247, footnote suppressed)

A Pragmatic View of Theories helps make explicit nonformal internal components of theory structure.

Key early defenders of the modeling and practice turns have also recently begun to envision theory in a way distinct from the terms set by the Syntactic and Semantic Views. Suárez and Cartwright (2008) extend and distribute theory by arguing that “What we know ‘theoretically’ is recorded in a vast number of places in a vast number of different ways—not just in words and formulae but in machines, techniques, experiments and applications as well” (79). And while her influence lies primarily in the modeling turn, even in characterizing the “vending machine” view, Cartwright calls for a “reasonable philosophical account of theories” that is “much more textured, and… much more laborious” than that adopted by the Syntactic and Semantic Views (1999a, 185). The theory-data and theory-world axes need to be rethought. In her 2019 book on “artful modeling”, Cartwright emphasizes the importance of know-how and creativity in scientific practice, and “praise[s] engineers and cooks and inventors, as well as experimental physicists like Millikan and Melissa Franklin” (Cartwright 2019, 76). Kitcher wishes to transform talk of theories into discussion of “significance graphs” (2001, 78 ff.). These are network diagrams illustrating which (and how) questions are considered significant in the context of particular scientific communities and norms (cf. Brown 2010). Consistently with a Pragmatic View of Theories, Morrison (2007) reconsiders and reforms canonical conceptualizations of “theory.” Finally, Longino (2013) proposes an archaeology of assumptions behind and under different research programs and theories of human behavior such as neurobiological, molecular behavioral genetic, and social-environmental approaches (e.g., Oyama 2000). For instance, two shared or recurring assumptions across programs and theories are:

(1) that the approach in question has methods of measuring both the behavioral outcome that is the object of investigation and the factors whose association with it are the topic of investigation and (2) that the resulting measurements are exportable beyond the confines of the approach within which they are made. (Longino 2013, 117)

A Pragmatic View of Theories expands the notion of theory to include nonformal aspects, which surely must include elements from Boumans’ list above (e.g., metaphors, analogies, policy views), as well as more standard components such as ontological assumptions (e.g., Kuhn 1970; Levins and Lewontin 1985; Winther 2006b), natural kinds (e.g., Hacking 2007b), and conditions of application or scope (e.g., Longino 2013).

In addition to exploring internal theory diversity and in parallel with plurality of modeling, a Pragmatic View of Theories could also explore pluralism of modes of theorizing, and of philosophically analyzing theoretical structure (thesis #2). Craver (2002) provides a start in this direction in that he accepts three kinds of scientific theory and of philosophical analysis of scientific theory. A more synoptic view of the broader pragmatic context in which theories are embedded can be found in the literature on different “styles” of scientific reasoning and theorizing (e.g., Crombie 1994, 1996; Vicedo 1995; Pickstone 2000; Davidson 2001; Hacking 2002, 2009; Winther 2012b; Elwick 2007; Mancosu 2010). While there is no univocal or dominant classification of styles, two lessons are important. First, a rough consensus exists that theoretical investigations of especially historical, mechanistic, and mathematical structures and relations will involve different styles. Second, each style integrates theoretical products and theorizing processes in unique ways, thus inviting an irreducible pragmatic methodological pluralism in our philosophical analysis of the structure of scientific theories. For instance, the structure of theories of mechanisms in molecular biology or neuroscience involves flow charts, and is distinct from the structure of theories of historical processes and patterns as found in systematics and phylogenetics, which involves phylogenetic trees. As Crombie suggests, we need a “comparative historical anthropology of thinking.” (1996, 71; see Hacking 2009) Mathematical theory hardly remains regnant. It gives way to a pluralism of theory forms and theory processes. Indeed, even mathematical theorizing is a pluralistic motley, as Hacking (2014) argues. Although a “deflationary” Semantic View could account for pluralism of theory forms, the Pragmatic View of Theories, drawing on styles, is required to do justice to the immense variety of theorizing processes, and of philosophical accounts of theory and theory structure.

Finally, outstanding work remains in sorting out the philosophical utility of a variety of proposed units in addition to styles, such as Kuhn’s (1970) paradigms, Lakatos’ (1980) research programmes, Laudan’s (1977) research traditions, and Holton’s (1988) themata. A rational comparative historical anthropology of both theorizing and philosophical analyses of theorizing remains mostly unmapped (cf. Matheson and Dallmann 2014). Such a comparative meta-philosophical analysis should also address Davidson’s (1974) worries about “conceptual schemes” and Popper’s (1996 [1976]) critique of “the myth of the framework” (see Hacking 2002; Godfrey-Smith 2003).

Cartwright has done much to develop a Pragmatic View. Start by considering Newton’s second law:

Here \(F\) is the resultant force on a mass \(m\), and \(a\) is the net acceleration of \(m\); both \(F\) and \(a\) are vectors. This law is considered a “general” (Cartwright 1999a, 187) law expressed with “abstract quantities” (Cartwright 1999b, 249). Newton’s second law can be complemented with other laws, such as (i) Hooke’s law for an ideal spring:

Here \(k\) is the force constant of the spring, and \(x\) the distance along the x-axis from the equilibrium position, and (ii) Coulomb’s law modeling the force between two charged particles:

Here \(K\) is Coulomb’s electrical constant, \(q\) and \(q'\) are the charges of the two objects, and \(r\) the distance between the two objects. The picture Cartwright draws for us is that Newton’s, Hooke’s, and Coulomb’s laws are abstract, leaving out many details. They can be used to derive mathematical models of concrete systems. For instance, by combining (1) and (2), the law of gravitation (a “fundamental” law, Cartwright 1983, 58–59), other source laws, and various simplifying assumptions, we might create a model for the orbit of Mars, treating the Sun and Mars as a 2-body system, ignoring the other planets, asteroids, and Mars’ moons. Indeed, the Solar System is a powerful “nomological machine” (Cartwright 1999a, 50–53), which “is a fixed (enough) arrangement of components, or factors, with stable (enough) capacities that in the right sort of stable (enough) environment will, with repeated operation, give rise to the kind of regular behaviour that we represent in our scientific laws” (Cartwright 1999a, 50). Importantly, most natural systems are complex and irregular, and cannot be neatly characterized as nomological machines. For these cases, abstract laws “run out” (Cartwright 1983) and are rarely smoothly “deidealised” (Suárez 1999). In general, abstract laws predict and explain only within a given domain of application, and only under ideal conditions. More concrete laws or models are not directly deduced from them (e.g., Suárez 1999, Suárez and Cartwright 2008), and they can rarely be combined to form effective “super-laws” (Cartwright 1983, 70–73). In short, the move from (1) and (2) or from (1) and (3) to appropriate phenomenological models, is not fully specified by either abstract law pairing. Indeed, Cartwright developed her notion of “capacities” to discuss how “the principles of physics” “are far better rendered as claims about capacities, capacities that can be assembled and reassembled in different nomological machines, unending in their variety, to give rise to different laws” (1999a, 52). Articulating concrete models requires integrating a mix of mathematical and nonformal components. Laws (1), (2), and (3) remain only one component, among many, of the models useful for, e.g., exploring the behavior of the Solar System, balls on a pool table, or the behavior of charges in electrical fields.

Shifting examples but not philosophical research program, Suárez and Cartwright (2008) explains how analogies such as superconductors as diamagnets (as opposed to ferromagnets) were an integral part of the mathematical model of superconductivity developed by Fritz and Heinz London in the 1930s (63; cf. London and London 1935). Suárez and Cartwright gladly accept that this model “is uncontroversially grounded in classic electromagnetic theory” (64). However, contra Semantic View Structuralists such as Bueno, da Costa, French, and Ladyman, they view nonformal aspects as essential to practices of scientific modeling and theorizing: “The analogy [of diamagnets] helps us to understand how the Londons work with their model… which assumptions they add and which not… a formal reconstruction of the model on its own cannot help us to understand that” (69). In short, the running example of Newtonian mechanics, in conjunction with a glimpse into the use of analogies in mathematical modeling, illustrates the Pragmatic View’s account of theory syntax: theory is constituted by a plurality of formal and informal components.

As we have explored throughout this section, models and theories have informal internal components, and there are distinct modes of modeling and theorizing. Because of the Pragmatic View’s attention to practice, function, and application, distinguishing structure from interpretation is more difficult here than under the Syntactic and Semantic Views. Any synchronic analysis of the structure of models and theories must respect intentional diachronic processes of interpreting and using, as we shall now see.

Regarding the import of function in models and theories (thesis #4), already the Belgian philosopher of science Apostel defined modeling thus: “Let then \(R(S,P,M,T)\) indicate the main variables of the modelling relationship. The subject \(S\) takes, in view of the purpose \(P\), the entity \(M\) as a model for the prototype \(T\)” (1960, 128, see also Apostel 1970). Purposes took center-stage in his article title: “Towards the Formal Study of Models in the Non-Formal Sciences.” MIT Artificial Intelligence trailblazer Minsky also provided a pragmatic analysis:

We use the term “model” in the following sense: To an observer \(B\), an object \(A^*\) is a model of an object \(A\) to the extent that \(B\) can use \(A^*\) to answer questions that interest him about \(A\). The model relation is inherently ternary. Any attempt to suppress the role of the intentions of the investigator \(B\) leads to circular definitions or to ambiguities about “essential features” and the like. (1965, 45)

This account is thoroughly intentionalist and anti-essentialist. That is, mapping relations between model and world are left open and overdetermined. Specifying the relevant relations depends on contextual factors such as questions asked, and the kinds of similarities and isomorphisms deemed to be of interest. The appropriate relations are selected from an infinite (or, at least, near-infinite) variety of possible relations (e.g., Rosenblueth and Wiener 1945; Lowry 1965).

Regarding practice (thesis #5), in addition to ample work on the experimental life mentioned above, consider a small example. A full understanding of the content and structure of the London brothers’ model of superconductivity requires attention to informal aspects such as analogies. Even London and London (1935) state in the summary of their paper that “the current [”in a supraconductor“] is characterized as a kind of diamagnetic volume current” (88). They too saw the diamagnetic analogy as central to their theoretical practices. Criteria and practices of theory confirmation also differ from the ones typical of the Syntactic and Semantic Views. While predictive and explanatory power as well as empirical adequacy remain important, the Pragmatic View also insists on a variety of other justificatory criteria, including pragmatic virtues (sensu Kuhn 1977; Longino 1995) such as fruitfulness and utility. In a nutshell, the Pragmatic View argues that scientific theory structure is deeply shaped and constrained by functions and practices, and that theory can be interpreted and applied validly according to many different criteria.

The analytical framework of the Pragmatic View remains under construction. The emphasis is on internal diversity, and on the external pluralism of models and theories, of modeling and theorizing, and of philosophical analyses of scientific theories. The Pragmatic View acknowledges that scientists use and need different kinds of theories for a variety of purposes. There is no one-size-fits-all structure of scientific theories. Notably, although the Pragmatic View does not necessarily endorse the views of the tradition of American Pragmatism, it has important resonances with the latter school’s emphasis on truth and knowledge as processual, purposive, pluralist, and context-dependent, and on the social and cognitive structure of scientific inquiry.

A further qualification in addition to the one above regarding American Pragmatism is in order. The Pragmatic View has important precursors in the historicist or “world view” perspectives of Feyerabend, Hanson, Kuhn, and Toulmin, which were an influential set of critiques of the Syntactic View utterly distinct from the Semantic View. This philosophical tradition focused on themes such as meaning change and incommensurability of terms across world views (e.g., paradigms), scientific change (e.g., revolutionary: Kuhn 1970; evolutionary: Toulmin 1972), the interweaving of context of discovery and context of justification, and scientific rationality (Preston 2012; Bird 2013; Swoyer 2014). The historicists also opposed the idea that theories can secure meaning and empirical support from a theory-neutral and purely observational source, as the Syntactic View had insisted on with its strong distinction between theoretical and observational vocabularies (cf. Galison 1988). Kuhn’s paradigms or, more precisely, “disciplinary matrices” even had an internal anatomy with four components: (i) laws or symbolic generalizations, (ii) ontological assumptions, (iii) values, and (iv) exemplars (Kuhn 1970, postscript; Godfrey-Smith 2003; Hacking 2012). This work was concerned more with theory change than with theory structure and had fewer conceptual resources from sociology of science and history of science than contemporary Pragmatic View work. Moreover, paradigms never quite caught on the way analyses of models and modeling have. Even so, this work did much to convince later scholars, including many of the Pragmatic View, of certain weaknesses in understanding theories as deductive axiomatic structures.

As a final way to contrast the three views, we return to population genetics and, especially, to the Hardy-Weinberg Principle (HWP). Both Woodger (1937, 1959) and Williams (1970, 1973) provide detailed axiomatizations of certain parts of biology, especially genetics, developmental biology, and phylogenetics. For instance, Woodger (1937) constructs an axiomatic system based on ten logical predicates or relations, including \(\bP\) ( part of ), \(\bT\) ( before in time ), \(\bU\) ( reproduced by cell division or cell fusion ), \(\bm\) ( male gamete ), \(\bff\) ( female gamete ), and \(\bgenet\) ( genetic property ) (cf. Nicholson and Gawne 2014). Woodger (1959) elaborates these logical predicates or relations to produce a careful reconstruction of Mendelian genetics. Here are two axioms in his system (which are rewritten in contemporary notation, since Woodger used Russell and Whitehead’s Principia Mathematica notation):

The first axiom should be read thus: “no gamete is both male and female” (1959, 416). In the second axiom, given that \(DLZxyz\) is a primitive relation defined as “\(x\) is a zygote which develops in the environment \(y\) into the life \(z\)” (1959, 415), the translation is “every life develops in one and only one environment from one and only one zygote” (416). Woodger claims that “the whole of Mendel’s work can be expressed…” via this axiomatic system. Woodger briefly mentions that if one assumes that the entire system or population is random with respect to gamete fusions, “then the Pearson-Hardy law is derivable” (1959, 427). This was a reference to HWP. In her explorations of various axiomatizations of Darwinian lineages and “subclans,” and the process of the “expansion of the fitter,” Williams (1970, 1973) also carefully defines concepts, and axiomatizes basic biological principles of reproduction, natural selection, fitness, and so forth. However, she does not address HWP. Of interest is the lack of axiomatization of HWP or other mathematical principles of population genetics in Woodger’s and Williams’ work. Were such principles considered secondary or uninteresting by Woodger and Williams? Might Woodger’s and Williams’ respective axiomatic systems simply lack the power and conceptual resources to axiomatically reconstruct a mathematical edifice actually cast in terms of probability theory? Finally, other friends of the Syntactic View, such as the early Michael Ruse, do not provide an axiomatization of HWP (Ruse 1975, 241).

Proponents of the Semantic View claim that their perspective on scientific theory accurately portrays the theoretical structure of population genetics. Thompson (2007) provides both set-theoretical and state-space renditions of Mendelian genetics. The first involves defining a set-theoretic predicate for the system, viz., \(\{P, A, f, g\}\), where \(P\) and \(A\) are sets representing, respectively, the total collection of alleles and loci in the population, while \(f\) and \(g\) are functions assigning an allele to a specific location in, respectively, the diploid cells of an individual or the haploid gametic cells. Axioms in this set-theoretic formalization include “The sets \(P\) and \(A\) are finite and non empty” (2007, 498). In contrast, the state-space approach of the Semantic View articulates a phase space with each dimension representing allelic (or genotypic) frequencies (e.g., cover and Chapter 3 of Lloyd 1994 [1988]). As an example, “for population genetic theory, a central law of succession is the Hardy-Weinberg law” (Thompson 2007, 499). Mathematically, the diploid version of HWP is written thus:

Here \(p\) and \(q\) are the frequencies of two distinct alleles at a biallelic locus. The left-hand side represents the allele frequencies in the parental generation and a random mating pattern, while the right-hand side captures genotype frequencies in the offspring generation, as predicted from the parental generation. This is a null theoretical model—actual genotypic and allelic frequencies of the offspring generation often deviate from predicted frequencies (e.g., a lethal homozygote recessive would make the \(q^2_{\text{off}}\) term = 0). Indeed, HWP holds strictly only in abstracted and idealized populations with very specific properties (e.g., infinitely large, individuals reproduce randomly) and only when there are no evolutionary forces operating in the population (e.g., no selection, mutation, migration, or drift) (e.g., Hartl and Clark 1989; Winther et al. 2015). HWP is useful also in the way it interacts with laws of succession for selection, mutation, and so forth (e.g., Okasha 2012). This powerful population genetic principle is central to Semantic View analyses of the mathematical articulation of the theoretical structure of population genetics (see also Lorenzano 2014, Ginnobili 2016).

Recall that the Pragmatic View highlights the internal and external pluralism—as well as the purposiveness—of model and theory structure. Consider recent uses of population genetic theory to specify the kinds and amounts of population structure existing in Homo sapiens . In particular, different measures and mathematical modeling methodologies are employed in investigating human genomic diversity (e.g., Jobling et al. 2004; Barbujani et al. 2013; Kaplan and Winther 2013). It is possible to distinguish at least two different research projects, each of which has a unique pragmatic content (e.g., aims, values, and methods). Diversity partitioning assesses genetic variation within and among pre-determined groups using Analysis of Variance (also crucial to estimating heritability, Downes 2014). Clustering analysis uses Bayesian modeling techniques to simultaneously produce clusters and assign individuals to these “unsupervised” cluster classifications. The robust result of the first modeling project is that (approximately) 85% of all genetic variance is found within human subpopulations (e.g., Han Chinese or Sami), 10% across subpopulations within a continental region, and only 5% is found across continents (i.e., “African,” “Asian,” and “European” – Lewontin 1972, 1974). (Recall also that we are all already identical at, on average, 999 out of 1000 nucleotides.) To calculate diversity partitions at these three nested levels, Lewontin (1972) used a Shannon information-theoretic measure closely related to Sewall Wright’s \(F\)-statistic:

Here \(H_T\) is the total heterozygosity of the population assessed, and \(\bar{H}_S\) is the heterozygosity of each subpopulation (group) of the relevant population, averaged across all the subpopulations. \(F_{ST}\) is bounded by 0 and 1, and is a measure of population structure, with higher \(F_{ST}\) values suggesting more structure, viz., more group differentiation. HWP appears implicitly in both \(H_T\) and \(\bar{H}_S\), which take heterozygosity (\(2pq\)) to be equal to the expected proportion of heterozygotes under HWP rather than the actual frequency of heterozygotes. \(H_T\) is computed by using the grand population average of \(p\) and \(q\), whereas calculating \(\bar{H}_S\) involves averaging across the expected heterozygosities of each subpopulation. If random mating occurs—and thus HWP applies—across the entire population without respecting subpopulation borders, then \(H_T\) and \(\bar{H}_S\) will be equal (i.e., \(p\) of the total population and of each individual subpopulation will be the same; likewise for \(q\)). If, instead, HWP applies only within subpopulations but not across the population as a whole, then \(\bar{H}_S\) will be smaller than \(H_T\), and \(F_{ST}\) will be positive (i.e., there will be “excess homozygosity” across subpopulations, which is known as the “Wahlund Principle” in population genetics). This is one way among many to deploy the population-genetic principle of HWP. Thus, the Lewontin-style diversity partitioning result that only roughly 5% of the total genetic variance is among races is equivalent to saying that \(F_{ST}\) across the big three continental populations in Lewontin’s three-level model is 0.05 (e.g., Barbujani et al. 1997). The basic philosophical tendency is to associate the diversity partitioning research project’s (approximately) 85%-10%-5% result with an anti-realist interpretation of biological race.

In contrast, clustering analysis (e.g., Pritchard et al. 2000; Rosenberg et al. 2002; cf. Edwards 2003) can be readily performed even with the small amount of among-continent genetic variance in Homo sapiens . For instance, when the Bayesian modeling computer program STRUCTURE is asked to produce 5 clusters, continental “races” appear—African, Amerindian, Asian, European, and Pacific Islanders. Interestingly, this modeling technique is also intimately linked to HWP: “Our main modeling assumptions are Hardy-Weinberg equilibrium within populations and complete linkage equilibrium between loci within populations” (Pritchard et al. 2000, 946). That is, for a cluster to eventually be robust in the modeling runs, it should meet HWP expectations. Clustering analysis has sometimes been interpreted as a justification for a realist stance towards biological race (see discussions in Hochman 2013; Winther and Kaplan 2013; Edge and Rosenberg 2015; Spencer 2015).

This example of the mathematical modeling of human genomic diversity teaches that basic and simple formal components can be used in different ways to develop and apply theory, both inside and outside of science. In contrast to the Syntactic and Semantic Views, the Pragmatic View foregrounds tensions vis-à-vis ontological assumptions and political consequences regarding the existence (or not) of biological race between diversity partitioning (Lewontin 1972) and clustering analysis (Pritchard et al. 2000) research packages. These ontological ruptures can be identified despite the fact that both research projects assess population structure by examining departures from HWP (i.e., they measure excess homozygosity), and are completely consistent (e.g., Winther 2014; Ludwig 2015; Edge and Rosenberg 2015).

This exploration of how the three views on the structure of scientific theory address population genetics, and in particular HWP, invites a certain meta-pluralism. That is, the Syntactic View carefully breaks down fundamental concepts and principles in genetics and population genetics, articulating definitions and relations among terms. The Semantic View insightfully decomposes and interweaves the complex mathematical edifice of population genetics. The Pragmatic View sheds light on modeling choices and on distinct interpretations and applications of the same theory or model, both within and without science. The three perspectives are hardly mutually exclusive. (N.B., the two running examples concern theory structure in Newtonian mechanics and population genetics, independently considered. While interesting, debates about “evolutionary forces” are beyond the scope of the current entry; see, e.g., Hitchcock and Velasco 2014.)

The structure of scientific theories is a rich topic. Theorizing and modeling are core activities across the sciences, whether old (e.g., relativity theory, evolutionary theory) or new (e.g., climate modeling, cognitive science, and systems biology). Furthermore, theory remains essential to developing multipurpose tools such as statistical models and procedures (e.g., Bayesian models for data analysis, agent-based models for simulation, network theory for systems analysis). Given the strength and relevance of theory and theorizing to the natural sciences, and even to the social sciences (e.g., microeconomics, physical, if not cultural, anthropology), philosophical attention to the structure of scientific theories could and should increase. This piece has focused on a comparison of three major perspectives: Syntactic View, Semantic View, and Pragmatic View. In order to handle these complex debates effectively, we have sidestepped certain key philosophical questions, including questions about scientific realism; scientific explanation and prediction; theoretical and ontological reductionism; knowledge-production and epistemic inference; the distinction between science and technology; and the relationship between science and society. Each of these topics bears further philosophical investigation in light of the three perspectives here explored.

A table helps summarize general aspects of the three views’ analyses of the structure of scientific theories:

Table 2. General aspects of each view’s analysis of the structure of scientific theories.

The Syntactic, Semantic, and Pragmatic views are often taken to be mutually exclusive and, thus, to be in competition with one another. They indeed make distinct claims about the anatomy of scientific theories. But one can also imagine them to be complementary, focusing on different aspects and questions of the structure of scientific theories and the process of scientific theorizing. For instance, in exploring nonformal and implicit components of theory, the Pragmatic View accepts that scientific theories often include mathematical parts, but tends to be less interested in these components. Moreover, there is overlap in questions—e.g., Syntactic and Semantic Views share an interest in formalizing theory; the Semantic and Pragmatic Views both exhibit concern for scientific practice.

How are these three views ultimately related? A standard philosophical move is to generalize and abstract, understanding a situation from a higher level. One “meta” hypothesis is that a given philosophical analysis of theory structure tends to be associated with a perceived relationship among the three views here discussed. The Syntactic View is inclined to interpret the Semantic View’s formal machinery as continuous with its own generalizing axiomatic strategy, and hence diagnoses many standard Semantic View critiques (Section 3) as missing their mark (the strategy of identity ; e.g., Friedman 1982; Worrall 1984; Halvorson 2012, 2013, 2019; Lutz 2012, 2017; cf. Chakravartty 2001). The Semantic View explicitly contrasts its characterization of theory structure with the “linguistic” or “metamathematical” apparatus of the Syntactic View (the strategy of combat ; e.g., Suppe 1977; van Fraassen 1980, 1989; Lloyd 1994 [1988]). Finally, the Pragmatic View, which did not exist as a perspective until relatively recently, imagines theory as pluralistic and can thus ground a holistic philosophical investigation. It envisions a meta-pluralism in which reconstructive axiomatization and mathematical modeling remain important, though not necessary for all theories. This third view endorses a panoply of theoretical structures and theorizing styles, negotiating continuity both between theorizing and “the experimental life,” and among philosophical analyses of the structure of scientific theories (the strategy of complementarity ; e.g., Hacking 1983, 2009; Galison 1988, 1997; Craver 2002; Suárez and Cartwright 2008; Griesemer 2013). Interestingly, Suárez and Pero (2019) explicitly concur with the Pragmatic View as described in this article, but believe that “the semantic conception in its bare minimal expression” is compatible with, if not sufficient for, capturing “pragmatic elements and themes involved in a more flexible and open-ended approach to scientific theory” (Suárez and Pero 2019, 348). By design, the ecumenical meta-pluralism sanctioned by the Pragmatic View does not completely offset identity and combat strategies. Moreover, only “partial acceptance” of the respective views may ultimately be possible. Even so, the complementarity strategy might be worth developing further. Compared to identity and combat meta-perspectives, it provides broader—or at least different—insights into the structure of scientific theories. More generally, exploring the relations among these views is itself a rich topic for future philosophical work, as is investigating their role in, and interpretation of, active scientific fields ripe for further philosophical analysis such as climate change (e.g., Winsberg 2018), model organisms (e.g., Ankeny and Leonelli 2020), and cartography and GIS (e.g., Winther 2020).

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Acknowledgments

The following provided helpful feedback or conversation, or both, Jácome Armas, Nancy Cartwright, Mario Casanueva, Carl Craver, Eugene Earnshaw, Doc Edge, Michael Friedman, Sari Friedman, Fermín Fulda, Ryan Giordano, Ian Hacking, Hervé Kieffel, Elisabeth A. Lloyd, Helen Longino, Carlos López Beltrán, Greg Lusk, Sebastian Lutz, Sergio Martínez, Amir Najmi, Thomas Ryckman, Mette Bannergaard Johansen, Mette Smølz Skau, Bas van Fraassen, Denis Walsh, Ole Wæver, and two anonymous reviewers. Alex Dor, Cory Knudson, and Lucas McGranahan offered expert research assistance.

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1.6: Hypothesis, Theories, and Laws

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  Learning Objectives

• Describe the difference between hypothesis and theory as scientific terms.
• Describe the difference between a theory and scientific law.

Although many have taken science classes throughout the course of their studies, people often have incorrect or misleading ideas about some of the most important and basic principles in science. Most students have heard of hypotheses, theories, and laws, but what do these terms really mean? Prior to reading this section, consider what you have learned about these terms before. What do these terms mean to you? What do you read that contradicts or supports what you thought?

What is a Fact?

A fact is a basic statement established by experiment or observation. All facts are true under the specific conditions of the observation.

What is a Hypothesis?

One of the most common terms used in science classes is a "hypothesis". The word can have many different definitions, depending on the context in which it is being used:

• An educated guess: a scientific hypothesis provides a suggested solution based on evidence.
• Prediction: if you have ever carried out a science experiment, you probably made this type of hypothesis when you predicted the outcome of your experiment.
• Tentative or proposed explanation: hypotheses can be suggestions about why something is observed. In order for it to be scientific, however, a scientist must be able to test the explanation to see if it works and if it is able to correctly predict what will happen in a situation. For example, "if my hypothesis is correct, we should see ___ result when we perform ___ test."

A hypothesis is very tentative; it can be easily changed.

What is a Theory?

The United States National Academy of Sciences describes what a theory is as follows:

"Some scientific explanations are so well established that no new evidence is likely to alter them. The explanation becomes a scientific theory. In everyday language a theory means a hunch or speculation. Not so in science. In science, the word theory refers to a comprehensive explanation of an important feature of nature supported by facts gathered over time. Theories also allow scientists to make predictions about as yet unobserved phenomena."

"A scientific theory is a well-substantiated explanation of some aspect of the natural world, based on a body of facts that have been repeatedly confirmed through observation and experimentation. Such fact-supported theories are not "guesses" but reliable accounts of the real world. The theory of biological evolution is more than "just a theory." It is as factual an explanation of the universe as the atomic theory of matter (stating that everything is made of atoms) or the germ theory of disease (which states that many diseases are caused by germs). Our understanding of gravity is still a work in progress. But the phenomenon of gravity, like evolution, is an accepted fact.

Note some key features of theories that are important to understand from this description:

• Theories are explanations of natural phenomena. They aren't predictions (although we may use theories to make predictions). They are explanations as to why we observe something.
• Theories aren't likely to change. They have a large amount of support and are able to satisfactorily explain numerous observations. Theories can, indeed, be facts. Theories can change, but it is a long and difficult process. In order for a theory to change, there must be many observations or pieces of evidence that the theory cannot explain.
• Theories are not guesses. The phrase "just a theory" has no room in science. To be a scientific theory carries a lot of weight; it is not just one person's idea about something

Theories aren't likely to change.

What is a Law?

Scientific laws are similar to scientific theories in that they are principles that can be used to predict the behavior of the natural world. Both scientific laws and scientific theories are typically well-supported by observations and/or experimental evidence. Usually scientific laws refer to rules for how nature will behave under certain conditions, frequently written as an equation. Scientific theories are more overarching explanations of how nature works and why it exhibits certain characteristics. As a comparison, theories explain why we observe what we do and laws describe what happens.

For example, around the year 1800, Jacques Charles and other scientists were working with gases to, among other reasons, improve the design of the hot air balloon. These scientists found, after many, many tests, that certain patterns existed in the observations on gas behavior. If the temperature of the gas is increased, the volume of the gas increased. This is known as a natural law. A law is a relationship that exists between variables in a group of data. Laws describe the patterns we see in large amounts of data, but do not describe why the patterns exist.

What is a Belief?

A belief is a statement that is not scientifically provable. Beliefs may or may not be incorrect; they just are outside the realm of science to explore.

Laws vs. Theories

A common misconception is that scientific theories are rudimentary ideas that will eventually graduate into scientific laws when enough data and evidence has accumulated. A theory does not change into a scientific law with the accumulation of new or better evidence. Remember, theories are explanations and laws are patterns we see in large amounts of data, frequently written as an equation. A theory will always remain a theory; a law will always remain a law.

Video \(\PageIndex{1}\): What’s the difference between a scientific law and theory?

• A hypothesis is a tentative explanation that can be tested by further investigation.
• A theory is a well-supported explanation of observations.
• A scientific law is a statement that summarizes the relationship between variables.
• An experiment is a controlled method of testing a hypothesis.

Contributions & Attributions

Marisa Alviar-Agnew  ( Sacramento City College )

Henry Agnew (UC Davis)

Incorporate STEM journalism in your classroom

• Exercise type: Discussion
• Topic: Earth
• Category: Research & Design

How a scientific theory is born

• Download Student Worksheet

Directions for teachers:

Use the online Science News article “ How the Earth-shaking theory of plate tectonics was born ,” and the prompts below to have students explore scientific theories and determine the process behind creating theories. A version of the story, “Shaking up Earth,” appears in the January 16, 2021 issue of Science News . As a final exercise, have students discuss the definition of a scientific theory and compare it with hypotheses and scientific laws.

This story is the first installment in a series that celebrates Science News ’ upcoming 100th anniversary by highlighting some of the biggest advancements in science over the last century. For more on the story of plate tectonics, and to see the rest of series as it appears, visit Science News ’ Century of Science site at www.sciencenews.org/century .

Want to make it a virtual lesson? Post the online Science News article“ How the Earth-shaking theory of plate tectonics was born ,” to your learning management system. Pair up students and allow them to connect via virtual breakout rooms in a video conference, over the phone, in a shared document or using another chat system. Have each pair submit its answers to the second set of questions to you.

Thinking about theories

Discuss the following questions with a partner before reading the Science News article.

1. What does it mean to say that you have a theory about something? Think of a theory you’ve had about something outside of science.

Typically, when people say that they have theory, it means that they have an idea or philosophy. Student examples of theories will vary.

2. What is one scientific theory you have learned about this year in science? Explain what you remember about it.

Student answers will vary, but may include the general theory of relativity, evolution, etc.

3. How does the general use of the term theory differ from its use in a scientific context?    

Theories in science are explanations rooted in data. Having a theory outside of the scientific context may be based on observations or data, or the term may be used to state a logical idea.

The theory of plate tectonics

Read the online Science News article “How plate tectonics upended our understanding of Earth,” and answer the following questions individually before discussing them as a class.

1. What is the theory of plate tectonics? Over how many years was it developed?

The theory of plate tectonics states that the Earth’s surface is broken up into various pieces (plates) and describes how and why they are constantly in motion and how that motion is linked to features seen on Earth. The theory was developed over about 50 years.

2. Who helped develop the theory and what did they contribute to it? What types of scientists were they and where were they from?

Meteorologist Alfred Wegner proposed the idea of continental drift in 1912, and geologist Arthur Holmes added to that proposal years later with an explanation for how the continents might drift. These ideas were the precursors to the development of the theory of plate tectonics. From there, seismologists, geophysicists, mathematicians and physicists established the ideas, such as seafloor spreading, and found the data necessary to develop the theory. Notable scientists include Lynn Sykes, Harry Hess, Robert S. Dietz, Robert Parker, W. Jason Morgan and Dan McKenzie.  The researchers were from England and the United States.

3. Before the theory’s development, what were the conflicting lines of thought?

Wegner’s proposal sparked debates between mobilists, who supported the idea that the Earth’s surface was in motion, and fixists, who thought the Earth’s surface was static.

4. What did scientists need to resolve the conflict? Why did the conflict take so long to resolve?

In order to resolve the debate, scientists needed evidence. Wegner made his proposal in the early 1900s, but scientific evidence for why the continents move and how didn’t become available until after World War II, when technological advancements allowed scientists to study Earth’s surface and interior, and particularly the bottom of the oceans, in unprecedented detail.

5. How was evidence communicated to other members of the scientific community? Why was the communication important?

Evidence was communicated at conferences attended by scientists including geologists and geophysicists. By building on each other’s ideas and using each other’s data, the scientists were able to go beyond the idea of continental drift and come up with the unified theory of plate tectonics.

Defining a scientific theory

Discuss the following questions with a classmate.

1. Based on your answers to the questions above, how would you define a scientific theory?

A scientific theory is an explanation for how and why a natural phenomenon occurs based on evidence.

2. Think about a scientific hypothesis that you have written or look up an example of a hypothesis. How would you define a hypothesis? How is it different than a theory?

A hypothesis is a proposed explanation for a scientific question that hasn’t been validated with evidence. A theory relies on evidence to explain phenomena, whereas a hypothesis is proposed before the gathering of evidence. A hypothesis can become a theory once it is proven or disproven with supporting evidence.

Possible Extension

What is a scientific law that you have learned about in school? Explain how a scientific law is different than a scientific theory. For more information, watch this Ted-Ed video called “ What’s the difference between a scientific law and a theory? ” by educator Matt Anticole.

Student answers will vary, but could include Newton’s three laws of motion, Bernoulli’s principle, etc. A scientific law is different than a scientific theory in that it describes and predicts the relationships among variables, whereas a scientific theory describes how or why something happens.

Hypothesis vs. Theory

A hypothesis is either a suggested explanation for an observable phenomenon, or a reasoned prediction of a possible causal correlation among multiple phenomena. In science , a theory is a tested, well-substantiated, unifying explanation for a set of verified, proven factors. A theory is always backed by evidence; a hypothesis is only a suggested possible outcome, and is testable and falsifiable.

Comparison chart

Examples of theory and hypothesis.

Theory: Einstein's theory of relativity is a theory because it has been tested and verified innumerable times, with results consistently verifying Einstein's conclusion. However, simply because Einstein's conclusion has become a theory does not mean testing of this theory has stopped; all science is ongoing. See also the Big Bang theory , germ theory , and climate change .

Hypothesis: One might think that a prisoner who learns a work skill while in prison will be less likely to commit a crime when released. This is a hypothesis, an "educated guess." The scientific method can be used to test this hypothesis, to either prove it is false or prove that it warrants further study. (Note: Simply because a hypothesis is not found to be false does not mean it is true all or even most of the time. If it is consistently true after considerable time and research, it may be on its way to becoming a theory.)

This video further explains the difference between a theory and a hypothesis:

Common Misconception

People often tend to say "theory" when what they're actually talking about is a hypothesis. For instance, "Migraines are caused by drinking coffee after 2 p.m. — well, it's just a theory, not a rule."

This is actually a logically reasoned proposal based on an observation — say 2 instances of drinking coffee after 2 p.m. caused a migraine — but even if this were true, the migraine could have actually been caused by some other factors.

Because this observation is merely a reasoned possibility, it is testable and can be falsified — which makes it a hypothesis, not a theory.

• What is a Scientific Hypothesis? - LiveScience
• Wikipedia:Scientific theory

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Comments: Hypothesis vs Theory

Anonymous comments (2).

October 11, 2013, 1:11pm "In science, a theory is a well-substantiated, unifying explanation for a set of verified, proven hypotheses." But there's no such thing as "proven hypotheses". Hypotheses can be tested/falsified, they can't be "proven". That's just not how science works. Logical deductions based on axioms can be proven, but not scientific hypotheses. On top of that I find it somewhat strange to claim that a theory doesn't have to be testable, if it's built up from hypotheses, which DO have to be testable... — 80.✗.✗.139

May 6, 2014, 11:45pm "Evolution is a theory, not a fact, regarding the origin of living things." this statement is poorly formed because it implies that a thing is a theory until it gets proven and then it is somehow promoted to fact. this is just a misunderstanding of what the words mean, and of how science progresses generally. to say that a theory is inherently dubious because "it isn't a fact" is pretty much a meaningless statement. no expression which qualified as a mere fact could do a very good job of explaining the complicated process by which species have arisen on Earth over the last billion years. in fact, if you claimed that you could come up with such a single fact, now THAT would be dubious! everything we observe in nature supports the theory of evolution, and nothing we observe contradicts it. when you can say this about a theory, it's a pretty fair bet that the theory is correct. — 71.✗.✗.151

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Hypothesis, Model, Theory, and Law

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In common usage, the words hypothesis, model, theory, and law have different interpretations and are at times used without precision, but in science they have very exact meanings.

Perhaps the most difficult and intriguing step is the development of a specific, testable hypothesis. A useful hypothesis enables predictions by applying deductive reasoning, often in the form of mathematical analysis. It is a limited statement regarding the cause and effect in a specific situation, which can be tested by experimentation and observation or by statistical analysis of the probabilities from the data obtained. The outcome of the test hypothesis should be currently unknown, so that the results can provide useful data regarding the validity of the hypothesis.

Sometimes a hypothesis is developed that must wait for new knowledge or technology to be testable. The concept of atoms was proposed by the ancient Greeks , who had no means of testing it. Centuries later, when more knowledge became available, the hypothesis gained support and was eventually accepted by the scientific community, though it has had to be amended many times over the year. Atoms are not indivisible, as the Greeks supposed.

A model is used for situations when it is known that the hypothesis has a limitation on its validity. The Bohr model of the atom , for example, depicts electrons circling the atomic nucleus in a fashion similar to planets in the solar system. This model is useful in determining the energies of the quantum states of the electron in the simple hydrogen atom, but it is by no means represents the true nature of the atom. Scientists (and science students) often use such idealized models  to get an initial grasp on analyzing complex situations.

Theory and Law

A scientific theory or law represents a hypothesis (or group of related hypotheses) which has been confirmed through repeated testing, almost always conducted over a span of many years. Generally, a theory is an explanation for a set of related phenomena, like the theory of evolution or the big bang theory . 

The word "law" is often invoked in reference to a specific mathematical equation that relates the different elements within a theory. Pascal's Law refers an equation that describes differences in pressure based on height. In the overall theory of universal gravitation developed by Sir Isaac Newton , the key equation that describes the gravitational attraction between two objects is called the law of gravity .

These days, physicists rarely apply the word "law" to their ideas. In part, this is because so many of the previous "laws of nature" were found to be not so much laws as guidelines, that work well within certain parameters but not within others.

Scientific Paradigms

Once a scientific theory is established, it is very hard to get the scientific community to discard it. In physics, the concept of ether as a medium for light wave transmission ran into serious opposition in the late 1800s, but it was not disregarded until the early 1900s, when Albert Einstein proposed alternate explanations for the wave nature of light that did not rely upon a medium for transmission.

The science philosopher Thomas Kuhn developed the term scientific paradigm to explain the working set of theories under which science operates. He did extensive work on the scientific revolutions that take place when one paradigm is overturned in favor of a new set of theories. His work suggests that the very nature of science changes when these paradigms are significantly different. The nature of physics prior to relativity and quantum mechanics is fundamentally different from that after their discovery, just as biology prior to Darwin’s Theory of Evolution is fundamentally different from the biology that followed it. The very nature of the inquiry changes.

One consequence of the scientific method is to try to maintain consistency in the inquiry when these revolutions occur and to avoid attempts to overthrow existing paradigms on ideological grounds.

Occam’s Razor

One principle of note in regards to the scientific method is Occam’s Razor (alternately spelled Ockham's Razor), which is named after the 14th century English logician and Franciscan friar William of Ockham. Occam did not create the concept—the work of Thomas Aquinas and even Aristotle referred to some form of it. The name was first attributed to him (to our knowledge) in the 1800s, indicating that he must have espoused the philosophy enough that his name became associated with it.

The Razor is often stated in Latin as:

entia non sunt multiplicanda praeter necessitatem

or, translated to English:

entities should not be multiplied beyond necessity

Occam's Razor indicates that the most simple explanation that fits the available data is the one which is preferable. Assuming that two hypotheses presented have equal predictive power, the one which makes the fewest assumptions and hypothetical entities takes precedence. This appeal to simplicity has been adopted by most of science, and is invoked in this popular quote by Albert Einstein:

Everything should be made as simple as possible, but not simpler.

It is significant to note that Occam's Razor does not prove that the simpler hypothesis is, indeed, the true explanation of how nature behaves. Scientific principles should be as simple as possible, but that's no proof that nature itself is simple.

However, it is generally the case that when a more complex system is at work there is some element of the evidence which doesn't fit the simpler hypothesis, so Occam's Razor is rarely wrong as it deals only with hypotheses of purely equal predictive power. The predictive power is more important than the simplicity.

Edited by Anne Marie Helmenstine, Ph.D.

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How to Write a Great Hypothesis

Hypothesis Definition, Format, Examples, and Tips

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Amy Morin, LCSW, is a psychotherapist and international bestselling author. Her books, including "13 Things Mentally Strong People Don't Do," have been translated into more than 40 languages. Her TEDx talk,  "The Secret of Becoming Mentally Strong," is one of the most viewed talks of all time.

Verywell / Alex Dos Diaz

• The Scientific Method

Hypothesis Format

Falsifiability of a hypothesis.

• Operationalization

Hypothesis Types

Hypotheses examples.

• Collecting Data

A hypothesis is a tentative statement about the relationship between two or more variables. It is a specific, testable prediction about what you expect to happen in a study. It is a preliminary answer to your question that helps guide the research process.

Consider a study designed to examine the relationship between sleep deprivation and test performance. The hypothesis might be: "This study is designed to assess the hypothesis that sleep-deprived people will perform worse on a test than individuals who are not sleep-deprived."

At a Glance

A hypothesis is crucial to scientific research because it offers a clear direction for what the researchers are looking to find. This allows them to design experiments to test their predictions and add to our scientific knowledge about the world. This article explores how a hypothesis is used in psychology research, how to write a good hypothesis, and the different types of hypotheses you might use.

The Hypothesis in the Scientific Method

In the scientific method , whether it involves research in psychology, biology, or some other area, a hypothesis represents what the researchers think will happen in an experiment. The scientific method involves the following steps:

• Forming a question
• Performing background research
• Creating a hypothesis
• Designing an experiment
• Collecting data
• Analyzing the results
• Drawing conclusions
• Communicating the results

The hypothesis is a prediction, but it involves more than a guess. Most of the time, the hypothesis begins with a question which is then explored through background research. At this point, researchers then begin to develop a testable hypothesis.

Unless you are creating an exploratory study, your hypothesis should always explain what you  expect  to happen.

In a study exploring the effects of a particular drug, the hypothesis might be that researchers expect the drug to have some type of effect on the symptoms of a specific illness. In psychology, the hypothesis might focus on how a certain aspect of the environment might influence a particular behavior.

Remember, a hypothesis does not have to be correct. While the hypothesis predicts what the researchers expect to see, the goal of the research is to determine whether this guess is right or wrong. When conducting an experiment, researchers might explore numerous factors to determine which ones might contribute to the ultimate outcome.

In many cases, researchers may find that the results of an experiment  do not  support the original hypothesis. When writing up these results, the researchers might suggest other options that should be explored in future studies.

In many cases, researchers might draw a hypothesis from a specific theory or build on previous research. For example, prior research has shown that stress can impact the immune system. So a researcher might hypothesize: "People with high-stress levels will be more likely to contract a common cold after being exposed to the virus than people who have low-stress levels."

In other instances, researchers might look at commonly held beliefs or folk wisdom. "Birds of a feather flock together" is one example of folk adage that a psychologist might try to investigate. The researcher might pose a specific hypothesis that "People tend to select romantic partners who are similar to them in interests and educational level."

Elements of a Good Hypothesis

So how do you write a good hypothesis? When trying to come up with a hypothesis for your research or experiments, ask yourself the following questions:

• Is your hypothesis based on your research on a topic?
• Can your hypothesis be tested?
• Does your hypothesis include independent and dependent variables?

Before you come up with a specific hypothesis, spend some time doing background research. Once you have completed a literature review, start thinking about potential questions you still have. Pay attention to the discussion section in the  journal articles you read . Many authors will suggest questions that still need to be explored.

How to Formulate a Good Hypothesis

To form a hypothesis, you should take these steps:

• Collect as many observations about a topic or problem as you can.
• Evaluate these observations and look for possible causes of the problem.
• Create a list of possible explanations that you might want to explore.
• After you have developed some possible hypotheses, think of ways that you could confirm or disprove each hypothesis through experimentation. This is known as falsifiability.

In the scientific method ,  falsifiability is an important part of any valid hypothesis. In order to test a claim scientifically, it must be possible that the claim could be proven false.

Students sometimes confuse the idea of falsifiability with the idea that it means that something is false, which is not the case. What falsifiability means is that  if  something was false, then it is possible to demonstrate that it is false.

One of the hallmarks of pseudoscience is that it makes claims that cannot be refuted or proven false.

The Importance of Operational Definitions

A variable is a factor or element that can be changed and manipulated in ways that are observable and measurable. However, the researcher must also define how the variable will be manipulated and measured in the study.

Operational definitions are specific definitions for all relevant factors in a study. This process helps make vague or ambiguous concepts detailed and measurable.

For example, a researcher might operationally define the variable " test anxiety " as the results of a self-report measure of anxiety experienced during an exam. A "study habits" variable might be defined by the amount of studying that actually occurs as measured by time.

These precise descriptions are important because many things can be measured in various ways. Clearly defining these variables and how they are measured helps ensure that other researchers can replicate your results.

Replicability

One of the basic principles of any type of scientific research is that the results must be replicable.

Replication means repeating an experiment in the same way to produce the same results. By clearly detailing the specifics of how the variables were measured and manipulated, other researchers can better understand the results and repeat the study if needed.

Some variables are more difficult than others to define. For example, how would you operationally define a variable such as aggression ? For obvious ethical reasons, researchers cannot create a situation in which a person behaves aggressively toward others.

To measure this variable, the researcher must devise a measurement that assesses aggressive behavior without harming others. The researcher might utilize a simulated task to measure aggressiveness in this situation.

Hypothesis Checklist

• Does your hypothesis focus on something that you can actually test?
• Does your hypothesis include both an independent and dependent variable?
• Can you manipulate the variables?
• Can your hypothesis be tested without violating ethical standards?

The hypothesis you use will depend on what you are investigating and hoping to find. Some of the main types of hypotheses that you might use include:

• Simple hypothesis : This type of hypothesis suggests there is a relationship between one independent variable and one dependent variable.
• Complex hypothesis : This type suggests a relationship between three or more variables, such as two independent and dependent variables.
• Null hypothesis : This hypothesis suggests no relationship exists between two or more variables.
• Alternative hypothesis : This hypothesis states the opposite of the null hypothesis.
• Statistical hypothesis : This hypothesis uses statistical analysis to evaluate a representative population sample and then generalizes the findings to the larger group.
• Logical hypothesis : This hypothesis assumes a relationship between variables without collecting data or evidence.

A hypothesis often follows a basic format of "If {this happens} then {this will happen}." One way to structure your hypothesis is to describe what will happen to the  dependent variable  if you change the  independent variable .

The basic format might be: "If {these changes are made to a certain independent variable}, then we will observe {a change in a specific dependent variable}."

A few examples of simple hypotheses:

• "Students who eat breakfast will perform better on a math exam than students who do not eat breakfast."
• "Students who experience test anxiety before an English exam will get lower scores than students who do not experience test anxiety."​
• "Motorists who talk on the phone while driving will be more likely to make errors on a driving course than those who do not talk on the phone."
• "Children who receive a new reading intervention will have higher reading scores than students who do not receive the intervention."

Examples of a complex hypothesis include:

• "People with high-sugar diets and sedentary activity levels are more likely to develop depression."
• "Younger people who are regularly exposed to green, outdoor areas have better subjective well-being than older adults who have limited exposure to green spaces."

Examples of a null hypothesis include:

• "There is no difference in anxiety levels between people who take St. John's wort supplements and those who do not."
• "There is no difference in scores on a memory recall task between children and adults."
• "There is no difference in aggression levels between children who play first-person shooter games and those who do not."

Examples of an alternative hypothesis:

• "People who take St. John's wort supplements will have less anxiety than those who do not."
• "Adults will perform better on a memory task than children."
• "Children who play first-person shooter games will show higher levels of aggression than children who do not." 

Collecting Data on Your Hypothesis

Once a researcher has formed a testable hypothesis, the next step is to select a research design and start collecting data. The research method depends largely on exactly what they are studying. There are two basic types of research methods: descriptive research and experimental research.

Descriptive Research Methods

Descriptive research such as  case studies ,  naturalistic observations , and surveys are often used when  conducting an experiment is difficult or impossible. These methods are best used to describe different aspects of a behavior or psychological phenomenon.

Once a researcher has collected data using descriptive methods, a  correlational study  can examine how the variables are related. This research method might be used to investigate a hypothesis that is difficult to test experimentally.

Experimental Research Methods

Experimental methods  are used to demonstrate causal relationships between variables. In an experiment, the researcher systematically manipulates a variable of interest (known as the independent variable) and measures the effect on another variable (known as the dependent variable).

Unlike correlational studies, which can only be used to determine if there is a relationship between two variables, experimental methods can be used to determine the actual nature of the relationship—whether changes in one variable actually  cause  another to change.

The hypothesis is a critical part of any scientific exploration. It represents what researchers expect to find in a study or experiment. In situations where the hypothesis is unsupported by the research, the research still has value. Such research helps us better understand how different aspects of the natural world relate to one another. It also helps us develop new hypotheses that can then be tested in the future.

Thompson WH, Skau S. On the scope of scientific hypotheses .  R Soc Open Sci . 2023;10(8):230607. doi:10.1098/rsos.230607

Taran S, Adhikari NKJ, Fan E. Falsifiability in medicine: what clinicians can learn from Karl Popper [published correction appears in Intensive Care Med. 2021 Jun 17;:].  Intensive Care Med . 2021;47(9):1054-1056. doi:10.1007/s00134-021-06432-z

Eyler AA. Research Methods for Public Health . 1st ed. Springer Publishing Company; 2020. doi:10.1891/9780826182067.0004

Nosek BA, Errington TM. What is replication ?  PLoS Biol . 2020;18(3):e3000691. doi:10.1371/journal.pbio.3000691

Aggarwal R, Ranganathan P. Study designs: Part 2 - Descriptive studies .  Perspect Clin Res . 2019;10(1):34-36. doi:10.4103/picr.PICR_154_18

Nevid J. Psychology: Concepts and Applications. Wadworth, 2013.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

This is the Difference Between a Hypothesis and a Theory

What to Know A hypothesis is an assumption made before any research has been done. It is formed so that it can be tested to see if it might be true. A theory is a principle formed to explain the things already shown in data. Because of the rigors of experiment and control, it is much more likely that a theory will be true than a hypothesis.

As anyone who has worked in a laboratory or out in the field can tell you, science is about process: that of observing, making inferences about those observations, and then performing tests to see if the truth value of those inferences holds up. The scientific method is designed to be a rigorous procedure for acquiring knowledge about the world around us.

In scientific reasoning, a hypothesis is constructed before any applicable research has been done. A theory, on the other hand, is supported by evidence: it's a principle formed as an attempt to explain things that have already been substantiated by data.

Toward that end, science employs a particular vocabulary for describing how ideas are proposed, tested, and supported or disproven. And that's where we see the difference between a hypothesis and a theory .

A hypothesis is an assumption, something proposed for the sake of argument so that it can be tested to see if it might be true.

In the scientific method, the hypothesis is constructed before any applicable research has been done, apart from a basic background review. You ask a question, read up on what has been studied before, and then form a hypothesis.

What is a Hypothesis?

A hypothesis is usually tentative, an assumption or suggestion made strictly for the objective of being tested.

When a character which has been lost in a breed, reappears after a great number of generations, the most probable hypothesis is, not that the offspring suddenly takes after an ancestor some hundred generations distant, but that in each successive generation there has been a tendency to reproduce the character in question, which at last, under unknown favourable conditions, gains an ascendancy. Charles Darwin, On the Origin of Species , 1859 According to one widely reported hypothesis , cell-phone transmissions were disrupting the bees' navigational abilities. (Few experts took the cell-phone conjecture seriously; as one scientist said to me, "If that were the case, Dave Hackenberg's hives would have been dead a long time ago.") Elizabeth Kolbert, The New Yorker , 6 Aug. 2007

What is a Theory?

A theory , in contrast, is a principle that has been formed as an attempt to explain things that have already been substantiated by data. It is used in the names of a number of principles accepted in the scientific community, such as the Big Bang Theory . Because of the rigors of experimentation and control, its likelihood as truth is much higher than that of a hypothesis.

It is evident, on our theory , that coasts merely fringed by reefs cannot have subsided to any perceptible amount; and therefore they must, since the growth of their corals, either have remained stationary or have been upheaved. Now, it is remarkable how generally it can be shown, by the presence of upraised organic remains, that the fringed islands have been elevated: and so far, this is indirect evidence in favour of our theory . Charles Darwin, The Voyage of the Beagle , 1839 An example of a fundamental principle in physics, first proposed by Galileo in 1632 and extended by Einstein in 1905, is the following: All observers traveling at constant velocity relative to one another, should witness identical laws of nature. From this principle, Einstein derived his theory of special relativity. Alan Lightman, Harper's , December 2011

Non-Scientific Use

In non-scientific use, however, hypothesis and theory are often used interchangeably to mean simply an idea, speculation, or hunch (though theory is more common in this regard):

The theory of the teacher with all these immigrant kids was that if you spoke English loudly enough they would eventually understand. E. L. Doctorow, Loon Lake , 1979 Chicago is famous for asking questions for which there can be no boilerplate answers. Example: given the probability that the federal tax code, nondairy creamer, Dennis Rodman and the art of mime all came from outer space, name something else that has extraterrestrial origins and defend your hypothesis . John McCormick, Newsweek , 5 Apr. 1999 In his mind's eye, Miller saw his case suddenly taking form: Richard Bailey had Helen Brach killed because she was threatening to sue him over the horses she had purchased. It was, he realized, only a theory , but it was one he felt certain he could, in time, prove. Full of urgency, a man with a mission now that he had a hypothesis to guide him, he issued new orders to his troops: Find out everything you can about Richard Bailey and his crowd. Howard Blum, Vanity Fair , January 1995

And sometimes one term is used as a genus, or a means for defining the other:

Laplace's popular version of his astronomy, the Système du monde , was famous for introducing what came to be known as the nebular hypothesis , the theory that the solar system was formed by the condensation, through gradual cooling, of the gaseous atmosphere (the nebulae) surrounding the sun. Louis Menand, The Metaphysical Club , 2001 Researchers use this information to support the gateway drug theory — the hypothesis that using one intoxicating substance leads to future use of another. Jordy Byrd, The Pacific Northwest Inlander , 6 May 2015 Fox, the business and economics columnist for Time magazine, tells the story of the professors who enabled those abuses under the banner of the financial theory known as the efficient market hypothesis . Paul Krugman, The New York Times Book Review , 9 Aug. 2009

Incorrect Interpretations of "Theory"

Since this casual use does away with the distinctions upheld by the scientific community, hypothesis and theory are prone to being wrongly interpreted even when they are encountered in scientific contexts—or at least, contexts that allude to scientific study without making the critical distinction that scientists employ when weighing hypotheses and theories.

The most common occurrence is when theory is interpreted—and sometimes even gleefully seized upon—to mean something having less truth value than other scientific principles. (The word law applies to principles so firmly established that they are almost never questioned, such as the law of gravity.)

This mistake is one of projection: since we use theory in general use to mean something lightly speculated, then it's implied that scientists must be talking about the same level of uncertainty when they use theory to refer to their well-tested and reasoned principles.

The distinction has come to the forefront particularly on occasions when the content of science curricula in schools has been challenged—notably, when a school board in Georgia put stickers on textbooks stating that evolution was "a theory, not a fact, regarding the origin of living things." As Kenneth R. Miller, a cell biologist at Brown University, has said , a theory "doesn’t mean a hunch or a guess. A theory is a system of explanations that ties together a whole bunch of facts. It not only explains those facts, but predicts what you ought to find from other observations and experiments.”

While theories are never completely infallible, they form the basis of scientific reasoning because, as Miller said "to the best of our ability, we’ve tested them, and they’ve held up."

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All Human Existence May Have Begun in a Black Hole, Some Scientists Believe

There’s an intriguing possibility that the emergence of conscious life is not just a coincidence, but an inevitable outcome of cosmic evolution.

So let’s contemplate something simpler: why does the universe allow us to exist? Yet again, we run into the same problem: if the universe didn’t allow us to exist, we wouldn’t be here to think about it. This is called the “anthropic principle.” For some, it’s the only answer we need to explain, well , everything; but for others, it’s a philosophical thorn in the side. Everything we know about the universe so far—dating back to the 16th-century Polish astronomer Copernicus, who first proposed that Earth travels around the sun rather than the other way around—tells us that we have no special place in the cosmos. We are not at the center. This is the “Copernican principle.”

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The anthropic and Copernican principles are conflicting axioms about the universe’s existence and our place within it. The anthropic principle says the universe depends on our being here. Meanwhile, the Copernican principle says that we are not special, and no law of physics should depend on our existence. Yet, the vast and ancient universe we see in our telescopes appears to balance both principles, like a pin balanced on the edge of a glass.

So why is our universe the way it is, and why do we exist as self-aware beings , tiny in size and minuscule in lifespan, relative to the lonely cosmic vastness mostly devoid of life? If the universe were made just for us, surely it would be small, human sized, perhaps just one planet or solar system or galaxy, not billions. Why should a universe made for us have black holes, for example? They seem to contribute nothing to our welfare.

Some scientists believe the universe wasn’t finely tuned to create intelligent life like us at all. Instead, they say, the universe evolved its own insurance policy by creating as many black holes as possible, which is the universe’s method of reproduction. Following this line of thinking, the universe itself may very well be alive—and the fact that we humans exist at all is just a happy side effect.

A Finely-Tuned Universe

One of the biggest philosophical problems with the universe is that it has to be finely tuned for us to even exist. If the universe were random, things would quickly become messy. If modified only a tiny bit one way or another, physical parameters such as the speed of light ; the mass of the electron, proton, and neutron ; the gravitational constant ; and so on would eliminate all life—possibly all matter itself—and even the universe as a whole would not last long enough to evolve anything. For example, if their masses were slightly different, protons would decay into neutrons instead of the other way around, and as a result, there would be no atoms.

One possible solution to fine tuning is the multiverse . In this speculative theory, our universe is one of many in the same way that the planet Earth is one of many planets. Different universes have different laws of physics and, therefore, that ours supports life is simply a matter of luck. While some theories of the multiverse propose that these universes are essentially random and have no relationship to one another, one particular multiverse theory suggests that universes in fact reproduce like living beings and have ancestors and descendants. This theory is called cosmological natural selection (or CNS for short). First proposed by theoretical physicist Lee Smolin in 1992, the CNS theory is a strong contender for why our universe seems to balance both the Anthropic and Copernican principles.

When we look at the complexity of living things and the sheer number of non-living configurations there are, we’re left to assume that there’s no way species could appear randomly. Hence, some powerful being must have created all types of living creatures individually as a watchmaker builds a watch, the thinking often goes. However, Charles Darwin’s theory of evolution, which he first posited in his 1859 book, On the Origin of Species, provides a mechanism that explains why living things are non-random. Their parameters are not freely chosen; they are the product of natural selection , the process by which members of a species that are better fit to survive and/or reproduce more effectively are more likely to pass on their genes.

The theory of evolution is one of the greatest success stories in the history of science because it provided a mechanism by which a thing that is highly ordered, complex, and finely tuned for its survival could arise from natural processes. The theory was successful not only because it explained how species arise, but also because it generated new predictions that we could then test. For example, the theory of evolution explains why species appear related to one another.

The Beauty in Black Holes

The cosmological natural selection theory solves the pernicious problem of a universe finely tuned for life. That idea may make sense to us, living on a planet full of complex, multicellular organisms, but Earth is surrounded by mostly dead space and, as far as we know, dead planets, and moons and light years of interstellar dust and stray photons.

Earth is finely tuned for life; the universe is not. However, the cosmological natural selection theory says that the universe is finely tuned for something else: its method of reproduction, giving birth to new universes.

Under the CNS theory framework, every black hole becomes a baby universe . Our universe, likewise, started out as a black hole in its mother universe. The theory says that inside every black hole, the central singularity—which is matter highly compressed in space in the mother universe—becomes a highly compressed point in time in the new universe. This point expands, creating new matter and energy. You get a complete universe from even a tiny black hole .

This means that our universe is finely tuned not for life, but for black holes, which typically come from massive stars (although they can have other origins). It turns out that massive star formation depends on an element also important for life on Earth: carbon.

Carbon monoxide is the second-most common molecule in the universe after molecular hydrogen, even more common than water. In the molecular clouds of gas and dust that form from supernovae, massive stars coalesce amid gaseous carbon monoxide molecules, which act as a coolant. This cooling helps matter clump together and form the stars. Carbon is a critical component in all life that we know of. Therefore, life is, in fact, a byproduct of stellar formation, which is itself a byproduct of what the universe evolved to do: create as many black holes as possible.

The cosmological natural selection theory helps explain why our universe is so highly ordered, complex, and self-sustaining like Darwin’s theory explains the same for living things. That leads to the tantalizing, if speculative, conclusion that perhaps, by some definition, our universe itself is alive.

Dr. Tim Andersen is a principal research scientist at Georgia Tech Research Institute. He earned his doctorate in mathematics from Rensselaer Polytechnic Institute in Troy, New York, and his undergraduate degree from the University of Texas at Austin. He has published academic works in statistical mechanics, fluid dynamics (including a monograph on vortex filaments), quantum field theory, and general relativity. He is the author of The Infinite Universe on Medium and Stubstack and a book by the same name. He lives with his wife and two cats, and has a son and daughter at home as well as one grown son.

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May 6, 2024

AI Could Help Find a Solution for String Theory

String theory could provide a theory of everything for our universe—but it entails 10 500 (more than a centillion) possible solutions. AI models could help to find the right one

By Manon Bischoff

String theory provides billions upon billions of possible solutions—and each of them describes its own universe. Is there a solution that corresponds to our world?

fotojog/Getty Images

Tiny little threads whizzing through spacetime and vibrating incessantly: this is roughly how you can imagine the universe, according to string theory. The various vibrations of the threads generate the elementary particles, such as electrons and quarks, and the forces acting among them.

Over the years, however, string theorists have been confronted with increasing difficulties. For the theory to work, additional spatial dimensions are needed, as well as additional particles that have not yet been observed. One of the most serious setbacks, however, is the sheer number of possible solutions that string theory offers: more than 10 500 possible universes tumble out of the equations. And among them, you would have to find the one that fits our world, describing the same elementary particles and forces. But it’s not just the enormous number of solutions that causes problems. From a mathematical point of view, it is also extremely difficult to deduce the observable particles and forces from a particular solution. This is why string theorists are now turning to certain artificial intelligence algorithms to help them with the task. In early 2024 researchers achieved impressive results, which they posted on the preprint server arXiv.org .

When physicists worked out string theory in the 1980s, they quickly realized that the mathematical formalism did not fit into four spacetime dimensions. In order for the theory to work, a 10-dimensional spacetime would be required: one dimension of time and nine dimensions of space. Because we only perceive three spatial dimensions in our world, however, the remaining six would have to be rolled up in dimensions that were so tiny that we would not notice any of them. This means that at every point in space there would be a tiny folded-up entity that existed in six dimensions—so small that no microscope could ever resolve it.

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The way in which the six-dimensional space is rolled up plays an important part in string theory. This is because the exact geometry determines how the strings vibrate—and therefore which elementary particles and forces are created. The ultimate aim is to find the six-dimensional geometry of the coiled space that exactly describes our universe. String theorists formulated some requirements that the rolled-up geometries must fulfill but were still confronted with a huge number of possibilities: far more six dimensional geometries than there are particles in the universe.

And then there was another problem: calculating the exact geometric details of the rolled-up dimensions turned out to be a mammoth task. Describing complicated six-dimensional shapes with mathematical precision is virtually impossible. And even if you had the solution, you would still have to calculate how the tiny threads interact with the six-dimensional rolled-up space. This is the only way to deduce which particles and forces could arise in the universe described. The experts were at an impasse.

Advances in the field of artificial intelligence caught the attention of some string theorists, as they seemingly have for a large fraction of the Earth’s population. AI could make it possible to perform calculations that were previously out of reach. In particular, the string theorists turned their attention to neural networks: algorithms whose structure is based on the functioning of the neural cortex. These networks process information in several layers of artificial neurons, transforming an input (for example, the rough shape of the rolled-up dimensions) into a corresponding output (their detailed geometry). By training the networks with thousands and thousands of pieces of data, scientists can induce them to eventually uncover patterns that most people miss. And it is precisely this property that string theorists have now exploited.

In 2017 researchers used neural networks to determine the exact geometries of six of the simplest ways to roll up the excess dimensions. Then Andrei Constantin of the University of Oxford led the team that posted a preprint paper in early 2024: The physicists used other neural networks to investigate how strings would interact with different rolled-up geometries and what elementary particles would result. In the end, they were able to derive the masses of three different quarks and thus provide a concrete prediction in their preprint.

The results do not match the quark masses that we observe in our universe, however. This is not surprising: after all, the scientists picked out individual geometries from 10 500 possible configurations and tested them. The correct geometry that describes our universe could therefore still be hidden among these 10 500 variants. This is why some string theorists are now aiming to first reduce this monstrous number and only then examine the remaining geometries with neural networks.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

In the race for space metals, companies hope to cash in

Mining asteroids could, in theory, reduce the burden on Earth’s resources. Will it live up to its promise?

By Sarah Scoles/Undark | Published May 10, 2024 8:00 AM EDT

This article was originally featured on Undark .

In April 2023, a satellite the size of a microwave launched to space. Its goal: to get ready to mine asteroids. While the mission, courtesy of a company called AstroForge, ran into problems, it’s part of a new wave of would-be asteroid miners hoping to cash in on cosmic resources.

Potential applications of space-mined material abound: Asteroids contain metals like platinum and cobalt, which are used in electronics and electric vehicle batteries, respectively. Although there’s plenty of these materials on Earth, they can be more concentrated on asteroids than mountainsides, making them easier to scrape out. And scraping in space, advocates say, could cut down on the damaging impacts that  mining  has on this planet. Space-resource advocates also want to explore the potential of other substances. What if space ice could be used for spacecraft and rocket propellant? Space dirt for housing structures for astronauts and radiation shielding?

Previous companies have rocketed toward similar goals before but went bust about a half decade ago. In the years since that first cohort left the stage, though, “the field has exploded in interest,” said Angel Abbud-Madrid, director of the Center for Space Resources at the Colorado School of Mines.

A lot of the attention has focused on the moon, since nations plan to set up outposts there and will need supplies. NASA, for instance, has ambitions to build astronaut  base camps  within the next decade. China, meanwhile, hopes to found an international lunar  research station .

Still, the pull of space rocks remains powerful and the new crop of companies hopeful. The economic picture has improved with the cost of rocket launches decreasing, as has the regulatory environment, with countries creating laws specifically allowing space mining. But only time will tell if this decade’s prospectors will cash in where others have drilled into the red or be buried by their business plans.

An asteroid-mining company needs one major ingredient to get started: optimism. A hope that they could start a new industry, one not of this world. “Not a lot of humans are built to work like that,” said Matt Gialich, co-founder and CEO of AstroForge. Since the company’s April 2023 demo mission, it has yet to come close to mining anything.

What he and colleagues hope to extract, though, are  platinum-group metals , some of which are used in devices like catalytic converters, which reduce gas emissions. Substances like platinum and iridium, meanwhile, are used in electronics. There are also opportunities in green technology, and  new pushes  to produce platinum-based batteries with better storage that could end up in electric vehicles and energy storage systems.

To further the company’s goals, AstroForge’s initial mission was loaded with simulated asteroid material and a refinery system designed to extract platinum from the simulant, to show that metal-processing could happen in space.

“The field has exploded in interest.”

Things didn’t go exactly as planned. After the small craft got to orbit, it was hard to identify and communicate with among the dozens of other newly launched satellites. The solar panels, which provide the spacecraft with power, wouldn’t deploy at first. And the satellite was initially beset with a wobble that prevented communication. They have not been able to do the simulated extraction.

The company will soon embark on a second mission, with a different goal: to slingshot to an asteroid and take a picture—a surveying project which may help the company understand which valuable materials exist on a particular asteroid.

Another company, called TransAstra, is selling a telescope and software designed to detect objects like asteroids moving through the sky; Chinese corporation Origin Space has an asteroid-observing satellite in orbit around Earth, and is testing out its mining-relevant technology there. Meanwhile, Colorado company Karman+ plans to go straight to an asteroid in 2026 and test out excavation equipment.

To achieve the ultimate goal of pulling metals from space rocks, TransAstra, Karman+, and AstroForge have received a combined tens of millions of dollars in venture-capital funding to date.

Another company with similar aims, simply called Asteroid Mining Corporation Ltd., doesn’t want to rely much on outside investment in the long term. Such reliance, in fact, helped sink earlier companies. Instead, founder and CEO Mitch Hunter-Scullion is focusing his company’s early work on terrestrial applications that pay up immediately so he can fund future work in the broader universe. In 2021, the company partnered with Tohoku University Space Robotics Laboratory, based in Japan, to work on space robots.

Together, they have built a six-legged robot called the Space Capable Asteroid Robotic Explorer, or SCAR-E. Designed to operate in microgravity, it can crawl around a rugged surface and take data on, and samples of, what’s there. In 2026, the company plans to do a demonstration mission analyzing soil on the Moon.

For now, though, SCAR-E will stay on Earth and inspect ship hulls. According to one market research platform, this is a nearly  $13 billion dollar market  globally—as compared to the asteroid-mining market, currently $0, as no one has yet mined an asteroid.

Such grounded work may give the company a revenue stream before, and during, their time in space. “I think every asteroid-mining company has this realization that money runs out, investors get tired, and you have to do something,” said Hunter-Scullion.

“My opinion is that unless you’ve built something which makes sense on Earth,” he added, “you’re never going to be able to mine an asteroid.”

Ian Lange sees sideways applications like ship-inspections as the “gin” of space mining: Lots of distilleries want to make whiskey. But making whiskey requires years of maturing alcohol in a cask. “You can make gin right now,” said Lange, an economist at the Colorado School of Mines. The gin can float a company until their whiskey’s ripe.

AstroForge is betting that asteroid mining will happen soon enough that it doesn’t need gin.

Some of the economics are, in fact, better than they were in the 2010s. Rockets are significantly cheaper than when the prior companies started (and then stopped), for instance. But other considerations are more complicated. For one, said Lange, there’s no shortage of terrestrial resources. “It’s not that we don’t have lithium around,” he said, as an example. “The problem is that for a number of reasons, we’re not allowing ourselves to take them out of the ground.” Often, those reasons involve environmental concerns.

For some, extracting materials from space offers a way to reduce the burden on Earth’s resources. That consideration is especially prevalent when it comes to the extraction required for clean energy technologies, like hydrogen fuel production, which uses iridium, and hydrogen fuel cell cars, which require platinum. Certain metals—like nickel, cobalt, and iron—are also more concentrated in asteroids since, unlike Earth’s richest deposits, they haven’t been tapped yet.

“The problem is that for a number of reasons, we’re not allowing ourselves to take them out of the ground.”

But space mining isn’t without its own environmental impacts: Rocket launches, for instance, contribute greenhouse gases to the atmosphere, something Lange’s  research  notes. Extraction on space rocks would generate waste and debris, which would float out into the vacuum of space.

To combat such concerns, researchers  suggested  in 2019 that much of the solar system should be set aside as “wilderness,” like protected land on Earth. Allowing exploitation like mining on only one-eighth of cosmic resources, they wrote, could prevent the kinds of impacts that overzealous extraction has wrought at home. In 2021, researchers at the University of New South Wales in Australia published a proposed  environmental impact framework  to assess how damaging a given space-mining project would be, through factors like how much dust it would stir up.

But the kind of public outcry that accompanies endeavors like  deep-sea mining  isn’t likely to happen for asteroids: A 2022 study  found  that the public is largely in favor of asteroid mining, a finding that held regardless of people’s preconceived ideas about ecological fragility or their political ideology.

“People were much more supportive of mining asteroids than other forms of frontier mining like mining the ocean floor, mining Antarctica, and mining the Alaskan tundra,” Matthew Hornsey, lead author of the paper and a professor at the University of Queensland, wrote in an email to Undark. “They didn’t raise the same ethical objections that they did to other forms of mining, and they reported little anxiety about it.” Even those who typically would object to mining on environmental grounds felt that way, Hornsey says, likely because they saw the trade-off: Scar the Earth or scrape an asteroid.

The same, though, was not true of the moon, where respondents disapproved of mining more—sometimes more than they did ecologically sensitive parts of Earth. “The moon is visible, beautiful, and associated with purity and spirituality,” he wrote. “I can see why people might see the need to preserve the sanctity of the moon more so than asteroids.”

Regardless of environmental pros and cons, making the leap to cosmic extraction will likely require further constraints on Earth—for example, stricter environmental regulations—that make space mining more appealing than digging another hole in the ground at home.

Still, Lange said, “it’s not clear that we will be able to bring costs down to match terrestrial minerals.” It’s simply a bet that the new optimists are making.

A few other things have changed since the 2010s: For one, would-be asteroid miners have more data. Recent NASA missions have revealed more about asteroids’ composition and structure.

The regulatory landscape for space mining has also shifted in recent years, said Melissa de Zwart, a professor who specializes in space law at the University of Adelaide in Australia.  Lack of existing regulation, she says, is part of why the first crop of companies failed. There wasn’t a legal framework that explicitly stated that space mining was allowed.

Today, Japan, the United Arab Emirates, Luxembourg, and the United States all have legislation enshrining their companies’ and countries’ rights to own space material. The U.S. has also established the Artemis Accords, a set of best practices for behavior on the moon. Other nations have signed on to these principles, which include ensuring that lunar digs align with the 1967 Outer Space Treaty, the major international law governing space exploration, including mining (even though it was enacted long before space mining was on the practical horizon).

The Outer Space Treaty doesn’t prohibit space mining—but it lays some ground rules: The “U.S. couldn’t go to the moon, put the U.S. flag on there and say, ‘It belongs to us,’ and start mining,” said de Zwart. But the country could extract material—like ice or metallic rocks—from the moon, or an asteroid, and then own that material. The treaty also says nations are responsible for what their private companies do, and their activities must benefit humanity.

But that framework leaves a lot of gaps—for example  how  exactly mining can be done responsibly. And there is talk toward that more logistical end: The Hague International Space Resources Governance Working Group has put together some  starting points  for developing an international how-to framework, and the United Nations has a working group dedicated to the cause.

That framework is likely to be tested, with the first proverbial pickaxe strike, not on asteroids but on the moon since that’s where humans are set to take their next small steps. “The moment you start talking sustained presence, you’re going to have to start looking at resources,” said Abbud-Madrid.

“I can see why people might see the need to preserve the sanctity of the moon more so than asteroids.”

Lunar surveying is already happening, and as of December 2020, NASA has contracts with four companies: Their task is to gather a small amount of material from the lunar surface—as a proof of concept to show that extraction is possible .

NASA doesn’t have a similar demonstration for mining asteroids. But the space rock seekers nevertheless continue their quest for treasure, even though its potential payoff is farther off. They believe Earth needs, and will pay handily for, what space has to offer.

At least that’s the optimistic take. “This has to happen,” said Gialich. “And I just hope that we’re the first.”

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Computer Science > Computer Vision and Pattern Recognition

Title: visiongraph: leveraging large multimodal models for graph theory problems in visual context.

Abstract: Large Multimodal Models (LMMs) have achieved impressive success in visual understanding and reasoning, remarkably improving the performance of mathematical reasoning in a visual context. Yet, a challenging type of visual math lies in the multimodal graph theory problem, which demands that LMMs understand the graphical structures accurately and perform multi-step reasoning on the visual graph. Additionally, exploring multimodal graph theory problems will lead to more effective strategies in fields like biology, transportation, and robotics planning. To step forward in this direction, we are the first to design a benchmark named VisionGraph, used to explore the capabilities of advanced LMMs in solving multimodal graph theory problems. It encompasses eight complex graph problem tasks, from connectivity to shortest path problems. Subsequently, we present a Description-Program-Reasoning (DPR) chain to enhance the logical accuracy of reasoning processes through graphical structure description generation and algorithm-aware multi-step reasoning. Our extensive study shows that 1) GPT-4V outperforms Gemini Pro in multi-step graph reasoning; 2) All LMMs exhibit inferior perception accuracy for graphical structures, whether in zero/few-shot settings or with supervised fine-tuning (SFT), which further affects problem-solving performance; 3) DPR significantly improves the multi-step graph reasoning capabilities of LMMs and the GPT-4V (DPR) agent achieves SOTA performance.

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The Fermi Paradox and the Berserker Hypothesis: Exploring Cosmic Silence Through Science Fiction

I n the realm of cosmic conundrums, the Fermi Paradox stands out: why, in a universe replete with billions of stars and planets, have we yet to find any signs of extraterrestrial intelligent life? The “berserker hypothesis,” a spine-chilling explanation rooted in science and popularized by science fiction, suggests a grim answer to this enduring mystery.

The concept’s moniker traces back to Fred Saberhagen’s “Berserker” series of novels, and it paints a picture of the cosmos where intelligent life forms are systematically eradicated by self-replicating probes, known as “berserkers.” These probes, initially intended to explore and report back, turn rogue and annihilate any signs of civilizations they encounter. The hypothesis emerges as a rather dark twist on the concept of von Neumann probes—machines capable of self-replication using local resources, which could theoretically colonize the galaxy rapidly.

Diving into the technicalities, the berserker hypothesis operates as a potential solution to the Hart-Tipler conjecture, which posits the lack of detectable probes as evidence that no intelligent life exists outside our solar system. Instead, this hypothesis flips the script: the absence of such probes doesn’t point to a lack of life but rather to the possibility that these probes have become cosmic predators, leaving a trail of silence in their wake.

Astronomer David Brin’s chilling summation underscores the potential severity of the hypothesis: “It need only happen once for the results of this scenario to become the equilibrium conditions in the Galaxy…because all were killed shortly after discovering radio.” If these berserker probes exist and are as efficient as theorized, then humanity’s attempts at communication with extraterrestrial beings could be akin to lighting a beacon for our own destruction.

Despite its foundation in speculative thought, the theory isn’t without its scientific evaluations. Anders Sandberg and Stuart Armstrong from the Future of Humanity Institute speculated that, given the vastness of the universe and even a slow replication rate, these berserker probes—if they existed—would likely have already found and destroyed us. It’s both a chilling and somewhat reassuring analysis that treads the line between fiction and potential reality.

Within the eclectic array of solutions to the Fermi Paradox, the berserker hypothesis stands out for its seamless blend of science fiction inspiration and scientific discourse. It connects with other notions such as the Great Filter, which suggests that life elsewhere in the universe is being systematically snuffed out before it can reach a space-faring stage, and the Dark Forest hypothesis, which posits that civilizations remain silent to avoid detection by such cosmic hunters.

Relevant articles:

– TIL about the berserker hypothesis, a proposed solution to the Fermi paradox stating the reason why we haven’t found other sentient species yet is because those species have been wiped out by self-replicating “berserker” probes.

– The Berserker Hypothesis: The Darkest Explanation Of The Fermi Paradox

– Beyond “Fermi’s Paradox” VI: What is the Berserker Hypothesis?

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Is dark matter’s main rival theory dead, there’s bad news from the cassini spacecraft and other recent tests..

Indranil Banik and Harry Desmond, The Conversation - May 11, 2024 11:25 am UTC

One of the biggest mysteries in astrophysics today is that the forces in galaxies do not seem to add up. Galaxies rotate much faster than predicted by applying Newton’s law of gravity to their visible matter, despite those laws working well everywhere in the Solar System.

To prevent galaxies from flying apart, some additional gravity is needed. This is why the idea of an invisible substance called dark matter was first proposed. But nobody has ever seen the stuff. And there are no particles in the hugely successful Standard Model of particle physics that could be the dark matter—it must be something quite exotic.

This has led to the rival idea that the galactic discrepancies are caused instead by a breakdown of Newton’s laws. The most successful such idea is known as Milgromian dynamics or Mond , proposed by Israeli physicist Mordehai Milgrom in 1982. But our recent research shows this theory is in trouble.

The main postulate of Mond is that gravity starts behaving differently from what Newton expected when it becomes very weak, as at the edges of galaxies. Mond is quite successful at predicting galaxy rotation without any dark matter, and it has a few other successes. But many of these can also be explained with dark matter, preserving Newton’s laws.

So how do we put Mond to a definitive test? We have been pursuing this for many years. The key is that Mond only changes the behavior of gravity at low accelerations, not at a specific distance from an object. You’ll feel lower acceleration on the outskirts of any celestial object—a planet, star, or galaxy—than when you are close to it. But it is the amount of acceleration, rather than the distance, that predicts where gravity should be stronger.

This means that, although Mond effects would typically kick in several thousand light years away from a galaxy, if we look at an individual star, the effects would become highly significant at a tenth of a light year. That is only a few thousand times larger than an astronomical unit (AU)—the distance between the Earth and the Sun. But weaker Mond effects should also be detectable at even smaller scales, such as in the outer Solar System.

This brings us to the Cassini mission , which orbited Saturn between 2004 and its final fiery crash into the planet in 2017. Saturn orbits the Sun at 10 AU. Due to a quirk of Mond, the gravity from the rest of our galaxy should cause Saturn’s orbit to deviate from the Newtonian expectation in a subtle way.

This can be tested by timing radio pulses between Earth and Cassini. Since Cassini was orbiting Saturn, this helped to measure the Earth-Saturn distance and allowed us to precisely track Saturn’s orbit. But Cassini did not find any anomaly of the kind expected in Mond. Newton still works well for Saturn.

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