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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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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.

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The word “hypothesis” is of ancient Greek origin and composed of two parts: “hypo” for “under,” and “thesis” for “to put there”; in Latin, this translated “to suppose” or “supposition”; made up of “sub” [under] and “positum” [put there]. It refers to something that we put there, maybe to start with, maybe to stay with us as an installation. Hence in modern English we say “ let us hypothesize, suppose,” or “let us put it that … .,” and then we start the argument by developing implications and reaching conclusions. The term “hypothesis” marks a space of possibilities in several ways. Firstly, it is the uncertain starting point from which firmer conclusions might be drawn. Public reasoning examines how, from uncertain hypotheses, neither true nor false, we can nevertheless reach useful conclusions. Secondly, the hypothesis is the end point of a logical process of firming up on reality through scientific enquiry. Scientific methodology makes hypothesis testing the gold standard...

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Bauer, M.W. (2022). Hypothesis. In: Glăveanu, V.P. (eds) The Palgrave Encyclopedia of the Possible. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-90913-0_193

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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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1.2: The Process of Science

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Like geology, physics, and chemistry, biology is a science that gathers knowledge about the natural world. Specifically, biology is the study of life. The discoveries of biology are made by a community of researchers who work individually and together using agreed-on methods. In this sense, biology, like all sciences is a social enterprise like politics or the arts.

The methods of science include careful observation, record keeping, logical and mathematical reasoning, experimentation, and submitting conclusions to the scrutiny of others. Science also requires considerable imagination and creativity; a well-designed experiment is commonly described as elegant, or beautiful. Like politics, science has considerable practical implications and some science is dedicated to practical applications, such as the prevention of disease (Figure \(\PageIndex{2}\)). Other science proceeds largely motivated by curiosity. Whatever its goal, there is no doubt that science, including biology, has transformed human existence and will continue to do so.

The Nature of Science

Biology is a science, but what exactly is science? What does the study of biology share with other scientific disciplines? Science (from the Latin scientia, meaning "knowledge") can be defined as knowledge about the natural world.

Science is a very specific way of learning, or knowing, about the world. The history of the past 500 years demonstrates that science is a very powerful way of knowing about the world; it is largely responsible for the technological revolutions that have taken place during this time. There are however, areas of knowledge and human experience that the methods of science cannot be applied to. These include such things as answering purely moral questions, aesthetic questions, or what can be generally categorized as spiritual questions. Science has cannot investigate these areas because they are outside the realm of material phenomena, the phenomena of matter and energy, and cannot be observed and measured.

The scientific method is a method of research with defined steps that include experiments and careful observation. The steps of the scientific method will be examined in detail later, but one of the most important aspects of this method is the testing of hypotheses. A hypothesis is a suggested explanation for an event, which can be tested. Hypotheses, or tentative explanations, are generally produced within the context of a scientific theory. A scientific theory is a generally accepted, thoroughly tested and confirmed explanation for a set of observations or phenomena. Scientific theory is the foundation of scientific knowledge. In addition, in many scientific disciplines (less so in biology) there are scientific laws, often expressed in mathematical formulas, which describe how elements of nature will behave under certain specific conditions. There is not an evolution of hypotheses through theories to laws as if they represented some increase in certainty about the world. Hypotheses are the day-to-day material that scientists work with and they are developed within the context of theories. Laws are concise descriptions of parts of the world that are amenable to formulaic or mathematical description.

Natural Sciences

What would you expect to see in a museum of natural sciences? Frogs? Plants? Dinosaur skeletons? Exhibits about how the brain functions? A planetarium? Gems and minerals? Or maybe all of the above? Science includes such diverse fields as astronomy, biology, computer sciences, geology, logic, physics, chemistry, and mathematics (Figure \(\PageIndex{3}\)). However, those fields of science related to the physical world and its phenomena and processes are considered natural sciences. Thus, a museum of natural sciences might contain any of the items listed above.

There is no complete agreement when it comes to defining what the natural sciences include. For some experts, the natural sciences are astronomy, biology, chemistry, earth science, and physics. Other scholars choose to divide natural sciences into life sciences, which study living things and include biology, and physical sciences, which study nonliving matter and include astronomy, physics, and chemistry. Some disciplines such as biophysics and biochemistry build on two sciences and are interdisciplinary.

Scientific Inquiry

One thing is common to all forms of science: an ultimate goal “to know.” Curiosity and inquiry are the driving forces for the development of science. Scientists seek to understand the world and the way it operates. Two methods of logical thinking are used: inductive reasoning and deductive reasoning.

Inductive reasoning is a form of logical thinking that uses related observations to arrive at a general conclusion. This type of reasoning is common in descriptive science. A life scientist such as a biologist makes observations and records them. These data can be qualitative (descriptive) or quantitative (consisting of numbers), and the raw data can be supplemented with drawings, pictures, photos, or videos. From many observations, the scientist can infer conclusions (inductions) based on evidence. Inductive reasoning involves formulating generalizations inferred from careful observation and the analysis of a large amount of data. Brain studies often work this way. Many brains are observed while people are doing a task. The part of the brain that lights up, indicating activity, is then demonstrated to be the part controlling the response to that task.

Deductive reasoning or deduction is the type of logic used in hypothesis-based science. In deductive reasoning, the pattern of thinking moves in the opposite direction as compared to inductive reasoning. Deductive reasoning is a form of logical thinking that uses a general principle or law to forecast specific results. From those general principles, a scientist can extrapolate and predict the specific results that would be valid as long as the general principles are valid. For example, a prediction would be that if the climate is becoming warmer in a region, the distribution of plants and animals should change. Comparisons have been made between distributions in the past and the present, and the many changes that have been found are consistent with a warming climate. Finding the change in distribution is evidence that the climate change conclusion is a valid one.

Both types of logical thinking are related to the two main pathways of scientific study: descriptive science and hypothesis-based science. Descriptive (or discovery) science aims to observe, explore, and discover, while hypothesis-based science begins with a specific question or problem and a potential answer or solution that can be tested. The boundary between these two forms of study is often blurred, because most scientific endeavors combine both approaches. Observations lead to questions, questions lead to forming a hypothesis as a possible answer to those questions, and then the hypothesis is tested. Thus, descriptive science and hypothesis-based science are in continuous dialogue.

Hypothesis Testing

Biologists study the living world by posing questions about it and seeking science-based responses. This approach is common to other sciences as well and is often referred to as the scientific method. The scientific method was used even in ancient times, but it was first documented by England’s Sir Francis Bacon (1561–1626) (Figure \(\PageIndex{4}\)), who set up inductive methods for scientific inquiry. The scientific method is not exclusively used by biologists but can be applied to almost anything as a logical problem-solving method.

The scientific process typically starts with an observation (often a problem to be solved) that leads to a question. Let’s think about a simple problem that starts with an observation and apply the scientific method to solve the problem. One Monday morning, a student arrives at class and quickly discovers that the classroom is too warm. That is an observation that also describes a problem: the classroom is too warm. The student then asks a question: “Why is the classroom so warm?”

Recall that a hypothesis is a suggested explanation that can be tested. To solve a problem, several hypotheses may be proposed. For example, one hypothesis might be, “The classroom is warm because no one turned on the air conditioning.” But there could be other responses to the question, and therefore other hypotheses may be proposed. A second hypothesis might be, “The classroom is warm because there is a power failure, and so the air conditioning doesn’t work.”

Once a hypothesis has been selected, a prediction may be made. A prediction is similar to a hypothesis but it typically has the format “If . . . then . . . .” For example, the prediction for the first hypothesis might be, “ If the student turns on the air conditioning, then the classroom will no longer be too warm.”

A hypothesis must be testable to ensure that it is valid. For example, a hypothesis that depends on what a bear thinks is not testable, because it can never be known what a bear thinks. It should also be falsifiable, meaning that it can be disproven by experimental results. An example of an unfalsifiable hypothesis is “Botticelli’s Birth of Venus is beautiful.” There is no experiment that might show this statement to be false. To test a hypothesis, a researcher will conduct one or more experiments designed to eliminate one or more of the hypotheses. This is important. A hypothesis can be disproven, or eliminated, but it can never be proven. Science does not deal in proofs like mathematics. If an experiment fails to disprove a hypothesis, then we find support for that explanation, but this is not to say that down the road a better explanation will not be found, or a more carefully designed experiment will be found to falsify the hypothesis.

Each experiment will have one or more variables and one or more controls. A variable is any part of the experiment that can vary or change during the experiment. A control is a part of the experiment that does not change. Look for the variables and controls in the example that follows. As a simple example, an experiment might be conducted to test the hypothesis that phosphate limits the growth of algae in freshwater ponds. A series of artificial ponds are filled with water and half of them are treated by adding phosphate each week, while the other half are treated by adding a salt that is known not to be used by algae. The variable here is the phosphate (or lack of phosphate), the experimental or treatment cases are the ponds with added phosphate and the control ponds are those with something inert added, such as the salt. Just adding something is also a control against the possibility that adding extra matter to the pond has an effect. If the treated ponds show lesser growth of algae, then we have found support for our hypothesis. If they do not, then we reject our hypothesis. Be aware that rejecting one hypothesis does not determine whether or not the other hypotheses can be accepted; it simply eliminates one hypothesis that is not valid (Figure \(\PageIndex{5}\)). Using the scientific method, the hypotheses that are inconsistent with experimental data are rejected.

Example \(\PageIndex{1}\)

In the example below, the scientific method is used to solve an everyday problem. Which part in the example below is the hypothesis? Which is the prediction? Based on the results of the experiment, is the hypothesis supported? If it is not supported, propose some alternative hypotheses.

• My toaster doesn’t toast my bread.
• Why doesn’t my toaster work?
• There is something wrong with the electrical outlet.
• If something is wrong with the outlet, my coffeemaker also won’t work when plugged into it.
• I plug my coffeemaker into the outlet.
• My coffeemaker works.

The hypothesis is #3 (there is something wrong with the electrical outlet), and the prediction is #4 (if something is wrong with the outlet, then the coffeemaker also won’t work when plugged into the outlet). The original hypothesis is not supported, as the coffee maker works when plugged into the outlet. Alternative hypotheses may include (1) the toaster might be broken or (2) the toaster wasn’t turned on.

In practice, the scientific method is not as rigid and structured as it might at first appear. Sometimes an experiment leads to conclusions that favor a change in approach; often, an experiment brings entirely new scientific questions to the puzzle. Many times, science does not operate in a linear fashion; instead, scientists continually draw inferences and make generalizations, finding patterns as their research proceeds. Scientific reasoning is more complex than the scientific method alone suggests.

Basic and Applied Science

The scientific community has been debating for the last few decades about the value of different types of science. Is it valuable to pursue science for the sake of simply gaining knowledge, or does scientific knowledge only have worth if we can apply it to solving a specific problem or bettering our lives? This question focuses on the differences between two types of science: basic science and applied science.

Basic science or “pure” science seeks to expand knowledge regardless of the short-term application of that knowledge. It is not focused on developing a product or a service of immediate public or commercial value. The immediate goal of basic science is knowledge for knowledge’s sake, though this does not mean that in the end it may not result in an application.

In contrast, applied science or “technology,” aims to use science to solve real-world problems, making it possible, for example, to improve a crop yield, find a cure for a particular disease, or save animals threatened by a natural disaster. In applied science, the problem is usually defined for the researcher.

Some individuals may perceive applied science as “useful” and basic science as “useless.” A question these people might pose to a scientist advocating knowledge acquisition would be, “What for?” A careful look at the history of science, however, reveals that basic knowledge has resulted in many remarkable applications of great value. Many scientists think that a basic understanding of science is necessary before an application is developed; therefore, applied science relies on the results generated through basic science. Other scientists think that it is time to move on from basic science and instead to find solutions to actual problems. Both approaches are valid. It is true that there are problems that demand immediate attention; however, few solutions would be found without the help of the knowledge generated through basic science.

One example of how basic and applied science can work together to solve practical problems occurred after the discovery of DNA structure led to an understanding of the molecular mechanisms governing DNA replication. Strands of DNA, unique in every human, are found in our cells, where they provide the instructions necessary for life. During DNA replication, new copies of DNA are made, shortly before a cell divides to form new cells. Understanding the mechanisms of DNA replication enabled scientists to develop laboratory techniques that are now used to identify genetic diseases, pinpoint individuals who were at a crime scene, and determine paternity. Without basic science, it is unlikely that applied science would exist.

Another example of the link between basic and applied research is the Human Genome Project, a study in which each human chromosome was analyzed and mapped to determine the precise sequence of DNA subunits and the exact location of each gene. (The gene is the basic unit of heredity; an individual’s complete collection of genes is his or her genome.) Other organisms have also been studied as part of this project to gain a better understanding of human chromosomes. The Human Genome Project (Figure \(\PageIndex{6}\)) relied on basic research carried out with non-human organisms and, later, with the human genome. An important end goal eventually became using the data for applied research seeking cures for genetically related diseases.

While research efforts in both basic science and applied science are usually carefully planned, it is important to note that some discoveries are made by serendipity, that is, by means of a fortunate accident or a lucky surprise. Penicillin was discovered when biologist Alexander Fleming accidentally left a petri dish of Staphylococcus bacteria open. An unwanted mold grew, killing the bacteria. The mold turned out to be Penicillium , and a new antibiotic was discovered. Even in the highly organized world of science, luck—when combined with an observant, curious mind—can lead to unexpected breakthroughs.

Reporting Scientific Work

Whether scientific research is basic science or applied science, scientists must share their findings for other researchers to expand and build upon their discoveries. Communication and collaboration within and between sub disciplines of science are key to the advancement of knowledge in science. For this reason, an important aspect of a scientist’s work is disseminating results and communicating with peers. Scientists can share results by presenting them at a scientific meeting or conference, but this approach can reach only the limited few who are present. Instead, most scientists present their results in peer-reviewed articles that are published in scientific journals. Peer-reviewed articles are scientific papers that are reviewed, usually anonymously by a scientist’s colleagues, or peers. These colleagues are qualified individuals, often experts in the same research area, who judge whether or not the scientist’s work is suitable for publication. The process of peer review helps to ensure that the research described in a scientific paper or grant proposal is original, significant, logical, and thorough. Grant proposals, which are requests for research funding, are also subject to peer review. Scientists publish their work so other scientists can reproduce their experiments under similar or different conditions to expand on the findings. The experimental results must be consistent with the findings of other scientists.

There are many journals and the popular press that do not use a peer-review system. A large number of online open-access journals, journals with articles available without cost, are now available many of which use rigorous peer-review systems, but some of which do not. Results of any studies published in these forums without peer review are not reliable and should not form the basis for other scientific work. In one exception, journals may allow a researcher to cite a personal communication from another researcher about unpublished results with the cited author’s permission.

Biology is the science that studies living organisms and their interactions with one another and their environments. Science attempts to describe and understand the nature of the universe in whole or in part. Science has many fields; those fields related to the physical world and its phenomena are considered natural sciences.

A hypothesis is a tentative explanation for an observation. A scientific theory is a well-tested and consistently verified explanation for a set of observations or phenomena. A scientific law is a description, often in the form of a mathematical formula, of the behavior of an aspect of nature under certain circumstances. Two types of logical reasoning are used in science. Inductive reasoning uses results to produce general scientific principles. Deductive reasoning is a form of logical thinking that predicts results by applying general principles. The common thread throughout scientific research is the use of the scientific method. Scientists present their results in peer-reviewed scientific papers published in scientific journals.

Science can be basic or applied. The main goal of basic science is to expand knowledge without any expectation of short-term practical application of that knowledge. The primary goal of applied research, however, is to solve practical problems.

Contributors and Attributions

Samantha Fowler (Clayton State University), Rebecca Roush (Sandhills Community College), James Wise (Hampton University). Original content by OpenStax (CC BY 4.0; Access for free at https://cnx.org/contents/b3c1e1d2-83...4-e119a8aafbdd ).

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Chemistry library

Course: chemistry library   >   unit 5.

• The history of atomic chemistry

Dalton's atomic theory

• Discovery of the electron and nucleus
• Rutherford’s gold foil experiment
• Bohr's model of hydrogen

• Dalton's atomic theory was the first complete attempt to describe all matter in terms of atoms and their properties.
• Dalton based his theory on the law of conservation of mass and the law of constant composition .
• The first part of his theory states that all matter is made of atoms, which are indivisible .
• The second part of the theory says all atoms of a given element are identical in mass and properties .
• The third part says compounds are combinations of two or more different types of atoms .
• The fourth part of the theory states that a chemical reaction is a rearrangement of atoms .
• Parts of the theory had to be modified based on the discovery of subatomic particles and isotopes.

Chemists ask questions.

Basis for dalton's theory, part 1: all matter is made of atoms., part 2: all atoms of a given element are identical in mass and properties., part 3: compounds are combinations of two or more different types of atoms., part 4: a chemical reaction is a rearrangement of atoms., what have we learned since dalton proposed his theory, attributions:.

• OpenStax College. "Early Ideas in Atomic Theory." OpenStax CNX. October 2, 2014. http://cnx.org/contents/[email protected]:HdZmYjzP@4/Early-Ideas-in-Atomic-Theory . CC-BY 4.0
• "Atomic Theory." UC Davis ChemWiki. http://chemwiki.ucdavis.edu/Physical_Chemistry/Atomic_Theory/Atomic_Theory . CC-BY-NC-SA 3.0 US
• The first part of his theory states that all matter is made of atoms, which are indivisible.
• The second part of the theory says all atoms of a given element are identical in mass and properties.
• The third part says compounds are combinations of two or more different types of atoms.
• The fourth part of the theory states that a chemical reaction is a rearrangement of atoms.
• Parts of the theory had to be modified based on the existence of subatomic particles and isotopes.

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11.5: de Broglie's Postulate

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Learning Objectives

• To introduce the wave-particle duality of light extends to matter
• To describe how matter (e.g., electrons and protons) can exhibit wavelike properties, e.g., interference and diffraction patterns
• To use algebra to find the de Broglie wavelength or momentum of a particle when either one of these quantities is given

The next real advance in understanding the atom came from an unlikely quarter - a student prince in Paris. Prince Louis de Broglie was a member of an illustrious family, prominent in politics and the military since the 1600's. Louis began his university studies with history, but his elder brother Maurice studied x-rays in his own laboratory, and Louis became interested in physics. After World War I, de Broglie focused his attention on Einstein's two major achievements, the theory of special relativity and the quantization of light waves. He wondered if there could be some connection between them. Perhaps the quantum of radiation really should be thought of as a particle. De Broglie suggested that if waves (photons) could behave as particles, as demonstrated by the photoelectric effect, then the converse, namely that particles could behave as waves, should be true. He associated a wavelength \(\lambda\) to a particle with momentum \(p\) using Planck's constant as the constant of proportionality:

\[\lambda =\dfrac{h}{p} \label{1.6.1} \]

which is called t he de Broglie wavelength . The fact that particles can behave as waves but also as particles, depending on which experiment you perform on them, is known as the wave-particle duality .

Deriving the de Broglie Wavelength

From the discussion of the photoelectric effect, we have the first part of the particle-wave duality, namely, that electromagnetic waves can behave like particles. These particles are known as photons , and they move at the speed of light. Any particle that moves at or near the speed of light has kinetic energy given by Einstein's special theory of relatively . In general, a particle of mass \(m\) and momentum \(p\) has an energy

\[E=\sqrt{p^2 c^2+m^2 c^4} \label{1.6.2} \]

Note that if \(p=0\) , this reduces to the famous rest-energy expression \(E=mc^2\) . However, photons are massless particles (technically rest-massless) that always have a finite momentum \(p\) . In this case, Equation \ref{1.6.2} becomes

\[E=pc. \nonumber \]

From Planck's hypothesis, one quantum of electromagnetic radiation has energy \(E=h\nu\) . Thus, equating these two expressions for the kinetic energy of a photon, we have

\[h\nu =\dfrac{hc}{\lambda}=pc \label{1.6.4} \]

Solving for the wavelength \(\lambda\) gives Equation \ref{1.6.1}:

\[\lambda=\dfrac{h}{p}= \dfrac{h}{mv} \nonumber \]

where \(v\) is the velocity of the particle. Hence , de Broglie argued that if particles can behave as waves, then a relationship like this, which pertains particularly to waves, should also apply to particles.

Equation \ref{1.6.1} allows us to associate a wavelength \(\lambda\) to a particle with momentum \(p\). A s the momentum increases, the wavelength decreases. In both cases, this means the energy becomes larger. i.e., short wavelengths and high momenta correspond to high energies.

It is a common feature of quantum mechanics that particles and waves with short wavelengths correspond to high energies and vice versa.

Having decided that the photon might well be a particle with a rest mass, even if very small, it dawned on de Broglie that in other respects it might not be too different from other particles, especially the very light electron. In particular, maybe the electron also had an associated wave. The obvious objection was that if the electron was wavelike, why had no diffraction or interference effects been observed? But there was an answer. If de Broglie's relation between momentum and wavelength also held for electrons, the wavelength was sufficiently short that these effects would be easy to miss. As de Broglie himself pointed out, the wave nature of light is not very evident in everyday life. As the next section will demonstrate, the validity of de Broglie’s proposal was confirmed by electron diffraction experiments of G.P. Thomson in 1926 and of C. Davisson and L. H. Germer in 1927. In these experiments it was found that electrons were scattered from atoms in a crystal and that these scattered electrons produced an interference pattern. These diffraction patterns are characteristic of wave-like behavior and are exhibited by both electrons (i.e., matter) and electromagnetic radiation (i.e., light).

Example 1.6.1 : Electron Waves

Calculate the de Broglie wavelength for an electron with a kinetic energy of 1000 eV.

To calculate the de Broglie wavelength (Equation \ref{1.6.1}), the momentum of the particle must be established and requires knowledge of both the mass and velocity of the particle. The mass of an electron is \(9.109383 \times 10^{−28}\; g\) and the velocity is obtained from the given kinetic energy of 1000 eV:

\[\begin{align*} KE &= \dfrac{mv^2}{2} \\[4pt] &= \dfrac{p^2}{2m} = 1000 \;eV \end{align*} \nonumber \]

Solve for momentum

\[ p = \sqrt{2 m KE} \nonumber \]

convert to SI units

\[ p = \sqrt{(1000 \; \cancel{eV}) \left( \dfrac{1.6 \times 10^{-19} \; J}{1\; \cancel{ eV}} \right) (2) (9.109383 \times 10^{-31}\; kg)} \nonumber \]

expanding definition of joule into base SI units and cancel

\[\begin{align*} p &= \sqrt{(3.1 \times 10^{-16} \;kg \cdot m^2/s^2 ) (9.109383 \times 10^{-31}\; kg)} \\[4pt] &= \sqrt{ 2.9 \times 10^{-40 }\, kg^2 \;m^2/s^2 } \\[4pt] &= 1.7 \times 10^{-23} kg \cdot m/s \end{align*} \nonumber \]

Now substitute the momentum into the equation for de Broglie's wavelength (Equation \(\ref{1.6.1}\)) with Planck's constant (\(h = 6.626069 \times 10^{−34}\;J \cdot s\)). After expanding units in Plank's constant

\[\begin{align*} \lambda &=\dfrac{h}{p} \\[4pt] &= \dfrac{6.626069 \times 10^{−34}\;kg \cdot m^2/s}{1.7 \times 10^{-23} kg \cdot m/s} \\[4pt] &= 3.87 \times 10^{-11}\; m \\[4pt] &=38.9\; pm \end{align*} \nonumber \]

Exercise 1.6.1 : Baseball Waves

Calculate the de Broglie wavelength for a fast ball thrown at 100 miles per hour and weighing 4 ounces. Comment on whether the wave properties of baseballs could be experimentally observed.

Following the unit conversions below, a 4 oz baseball has a mass of 0.11 kg. The velocity of a fast ball thrown at 100 miles per hour in m/s is 44.7 m/s.

\[ m = \left(4 \; \cancel{oz}\right)\left(\frac{0.0283 \; kg}{1 \; \cancel{oz}}\right) = 0.11 kg \nonumber \]

\[ v = \left(\frac{100 \; \cancel{mi}}{\cancel{hr}}\right) \left(\frac{1609.34 \; m}{\cancel{mi}}\right) \left( \frac{1 \; \cancel{hr}}{3600 \; s}\right) = 44.7 \; m/s \nonumber \]

The de Broglie wavelength of this fast ball is:

\[ \lambda = \frac{h}{mv} = \frac{6.626069 \times 10^{-34}\;kg \cdot m^2/s}{(0.11 \; kg)(44.7 \;m/s)} = 1.3 \times 10^{-34} m \nonumber \]

Exercise 1.6.2 : Electrons vs. Protons

If an electron and a proton have the same velocity, which would have the longer de Broglie wavelength?

• The electron
• They would have the same wavelength

Equation \ref{1.6.1} shows that the de Broglie wavelength of a particle's matter wave is inversely proportional to its momentum (mass times velocity). Therefore the smaller mass particle will have a smaller momentum and longer wavelength. The electron is the lightest and will have the longest wavelength.

This was the prince's Ph.D. thesis, presented in 1924. His thesis advisor was somewhat taken aback, and was not sure if this was sound work. He asked de Broglie for an extra copy of the thesis, which he sent to Einstein. Einstein wrote shortly afterwards: "I believe it is a first feeble ray of light on this worst of our physics enigmas " and the prince got his Ph.D.

Contributors and Attributions

Michael Fowler  (Beams Professor,  Department of Physics ,  University of Virginia)

Mark Tuckerman ( New York University )

David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski (" Quantum States of Atoms and Molecules ")

Mathematical Mysteries

Revealing the mysteries of mathematics

Axiom, Corollary, Lemma, Postulate, Conjectures and Theorems

“Lions and tigers, and bears, oh my!” ~ Dorothy in Wizard of Oz

Or should we say axioms, corollaries, lemmas, postulates, conjectures and theorems, oh my!

There are certain elementary statements, which are self evident and which are accepted without any questions. These are called  axioms.

Axiom 1: Things which are equal to the same thing are equal to one another.

For example:

Draw a line segment AB of length 10cm. Draw a second line CD having length equal to that of AB, using a compass. Measure the length of CD. We see that, CD = 10cm.

We can write it as, CD = AB and AB = 10cm implies CD = 10cm.

Arif, View. 2016. “Axioms, Postulates And Theorems – Class VIII”.  Breath Math . https://breathmath.com/2016/02/18/axioms-postulates-and-theorems-class-viii/ .

A statement that is taken to be true, so that further reasoning can be done.

It is not something we want to prove.

Example: one of Euclid’s axioms (over 2300 years ago!) is: “If A and B are two numbers that are the same, and C and D are also the same, A+C is the same as B+D”

“Definition Of Axiom”. 2021.  mathsisfun.Com . https://www.mathsisfun.com/definitions/axiom.html .

In mathematics an axiom is something which is the starting point for the logical deduction of other theorems. They cannot be proven with a logic derivation unless they are redundant. That means every field in mathematics can be boiled down to a set of axioms. One of the axioms of arithmetic is that a + b = b + a. You can’t prove that, but it is the basis of arithmetic and something we use rather often.

“Theorems, Lemmas And Other Definitions | Mathblog”. 2011.  mathblog.dk . https://www.mathblog.dk/theorems-lemmas/ .

In math it is known that you can’t prove everything. So, in order to lay a ground work for proving things, there is a list of things we “take for granted as true”. These things are either very basic definitions such as “point” “line”, or facts assumed to be true without proof that are very very simple. Then with these an accepted rules, one can prove other statements are true. The assumed facts are called “axioms” or sometimes “postulates”. The most famous are five postulates/axioms that Euclid’s geometry takes for granted. There are the following:

• A straight  line segment  can be drawn joining any two points.
• Any straight  line segment  can be extended indefinitely in a straight  line .
• Given any straight  line segment , a  circle  can be drawn having the segment as  radius  and one endpoint as center.
• All  right angles  are  congruent .
• If two lines are drawn which  intersect  a third in such a way that the sum of the inner angles on one side is less than two  right angles , then the two lines inevitably must  intersect  each other on that side if extended far enough. This postulate is equivalent to what is known as the  parallel postulate .

The fifth postulate is perhaps the most “famous” as it is complex and people wanted to prove it from the first four, but couldn’t, and then it was discovered that there were systems in which the first four were true but the fifth wasn’t. These are called “non-Euclidean” geometries. Of course, here we take for granted what a point, line segment, line, circle, angle, and radius are at least as well.

Farris, Steven. “I don’t understand the concept of an axiom in mathematics. What is an axiom? How would you introduce or explain this concept to a 10-year-old?”. 2023.  Quora . https://qr.ae/pyVTM1 .

An  axiom  is just any concept or statement that we take as being true, without any need for a formal proof. It is usually something very fundamental to a given field, very well-established and/or self-evident. A non-mathematical example might be a simple statement of an observed truth, such as “the Sun rises in the East.” In math, such things as “a line can be extended to infinity” or “a point has no size” might be good examples. An axiom differs from a  postulate  in that an axiom is typically more general and common, while a postulate may apply only to a specific field. For instance, the difference between Euclidean and non-Euclidean geometries are just changes to one or more of the postulates on which they’re based. Another way to look at this is that a postulate is something we assume to be true only within that specific field.

Myers, Bob. “I don’t understand the concept of an axiom in mathematics. What is an axiom? How would you introduce or explain this concept to a 10-year-old?”. 2023.  Quora . https://qr.ae/pyVTwW .

It’s not so much that they don’t  require  proof, it’s that they can’t be proven. Axioms are  starting assumptions .

Everything that is proven is based on axioms, theorems, or definitions. You can’t prove an axiom without already having something to base your proof on, because deductive reasoning always needs a starting place. You have to start with good assumptions, and hope they’re true, or at least useful in the type of math you wish to create. (Don’t forget that math is just a human construct!)

That doesn’t mean that axioms come out of thin air. Some axioms are developed because if they don’t exist, the math doesn’t model the way we want it to. If you put 3 apples in your grocery cart, then put 4 more in, you have 7. But it works the same if you put 3 in, then 4. Now you have the commutative property of addition. You can’t  prove  addition works this way, but you need to set it up so that it does.

Often axioms are demonstrable. Try to draw two non-congruent triangles with sides of length 3, 4, and 5 units. You can’t. But you haven’t  proved  it using deductive reasoning. You’ve made a conjecture using inductive reasoning.

McClung, Carter. “Why don’t axioms require proofs?”. 2023.  Quora . https://qr.ae/pyVTO4 .

The axioms or postulates are the assumptions that are obvious universal truths, they are not proved. Euclid has introduced the geometry fundamentals like geometric shapes and figures in his book elements and has stated 5 main axioms or postulates. Here, we are going to discuss the definition of euclidean geometry, its elements, axioms and five important postulates. [4]

A theorem that  follows on  from another theorem.

Example: there is a  Theorem  that says: two angles that together form a straight line are “supplementary” (they add to 180°).

A  Corollary  to this is the “Vertical Angle Theorem” that says: where two lines intersect, the angles opposite each other are equal (a=c and b=d in the diagram).

Proof that a=c: Angles a and b are on a straight line, so: ⇒ angles a + b = 180° and so a = 180° − b Angles c and b are also on a straight line, so: ⇒ angles c + b = 180° and so c = 180° − b So angle a = angle c

“Corollary Definition (Illustrated Mathematics Dictionary)”. 2021.  mathsisfun.com . https://www.mathsisfun.com/definitions/corollary.html .

A corollary of a theorem or a definition is a statement that can be deduced directly from that theorem or statement. It still needs to be proved, though.

A simple example: Theorem: The sum of the angles of a triangle is pi radians.

Corollary: No angle in a right angled triangle can be obtuse.

Or: Definition: A prime number is one that can be divided without remainder only by 1 and itself.

Corollary: No even number > 2 can be prime.

A corollary is a theorem that can be proved from another theorem. For example: If two angles of a triangle are equal, then the sides opposite them are equal . A corollary would be: If a triangle is equilateral, it is also equiangular.

“What Are The Examples Of Corollary In Math? – Quora”. 2021.  quora.com . https://www.quora.com/What-are-the-examples-of-corollary-in-math .

Lemmas and corollaries are theorems themselves. It’s really not necessary to have different names for them. A corollary is a theorem that “easily” follows from the preceding theorem. For example, after proving the theorem that the sum of the angles in a triangle is 180°, an easy theorem to prove is that the sum of the angles in a quadrilateral is 360°. The proof is just to cut the quadrilateral into two triangles. So that theorem could be called a corollary. [2]

There is not formal difference between a theorem and a lemma.  A lemma is a proven proposition just like a theorem. Usually a lemma is used as a stepping stone for proving something larger. That means the convention is to call the main statement for a theorem and then split the problem into several smaller problems which are stated as lemmas.  Wolfram  suggest that a lemma  is a short theorem used to prove something larger.

Breaking part of the main proof out into lemmas is a good way to create a structure in a proof and sometimes their importance will prove more valuable than the main theorem.

Like a Theorem, but not as important. It is a minor result that has been proved to be true (using facts that were already known). [3]

Lemmas and corollaries are theorems themselves. It’s really not necessary to have different names for them. A lemma is a theorem that’s mentioned primarily because it’s used in one or more following theorems, but it’s not so interesting in itself. Sometimes lemmas are just minor observations, but sometimes they’ve got detailed proofs. [2]

Postulates  in geometry are very similar to axioms, self-evident truths, and beliefs in logic, political philosophy and personal decision-making.

Geometry postulates, or axioms, are accepted statements or facts. Thus, there is no need to prove them.

Postulate 1.1, Through two points, there is exactly 1 line. Line t is the only line passing through E and F.

In geometry, “ Axiom ” and “ Postulate ” are essentially interchangeable. In antiquity, they referred to propositions that were “obviously true” and only had to be stated, and not proven. In modern mathematics there is no longer an assumption that axioms are “obviously true”. Axioms are merely ‘background’ assumptions we make. The best analogy I know is that axioms are the “rules of the game”. In Euclid’s Geometry, the main axioms/postulates are:

• Given any two distinct points, there is a line that contains them.
• Any line segment can be extended to an infinite line.
• Given a point and a radius, there is a circle with center in that point and that radius.
• All right angles are equal to one another.
• If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. (The  parallel postulate ).

A  theorem  is a logical consequence of the axioms. In Geometry, the “propositions” are all theorems: they are derived using the axioms and the valid rules. A “Corollary” is a theorem that is usually considered an “easy consequence” of another theorem. What is or is not a corollary is entirely subjective. Sometimes what an author thinks is a ‘corollary’ is deemed more important than the corresponding theorem. (The same goes for “ Lemma “s, which are theorems that are considered auxiliary to proving some other, more important in the view of the author, theorem).

A “ hypothesis ” is an assumption made. For example, “If xx is an even integer, then x2x2 is an even integer” I am not asserting that x2x2 is even or odd; I am asserting that if  something  happens (namely, if xx happens to be an even integer) then  something else  will also happen. Here, “xx is an even integer” is the hypothesis being made to prove it.

Gordon Gustafson, and Arturo Magidin. 2010. “Difference Between Axioms, Theorems, Postulates, Corollaries, And Hypotheses”.  Mathematics Stack Exchange . https://math.stackexchange.com/questions/7717/difference-between-axioms-theorems-postulates-corollaries-and-hypotheses .

In geometry, a postulate is a statement that is assumed to be true based on basic geometric principles. An example of a postulate is the statement “exactly one line may be drawn through any two points.” A long time ago, postulates were the ideas that were thought to be so obviously true they did not require a proof. [1]

An axiom is a statement, usually considered to be self-evident, that assumed to be true without proof. It is used as a starting point in mathematical proof for deducing other truths.

Classically, axioms were considered different from postulates. An axiom would refer to a self-evident assumption common to many areas of inquiry, while a postulate referred to a hypothesis specific to a certain line of inquiry, that was accepted without proof. As an example, in Euclid’s Elements, you can compare “common notions” (axioms) with postulates.

In much of modern mathematics, however, there is generally no difference between what were classically referred to as “axioms” and “postulates”. Modern mathematics distinguishes between logical axioms and non-logical axioms, with the latter sometimes being referred to as postulates.

Postulates are assumptions which are specific to geometry but axioms are assumptions are used thru’ out mathematics and not specific to geometry.

“What is the difference between an axiom and postulates”. 2023.  BYJUs . https://byjus.com/question-answer/what-is-the-difference-between-an-axiom-and-postulates/ .

Hint: First you need to define both the terms, axiom and postulates. Examples of both can be stated. The main difference is between their application in specific fields in mathematics.

An axiom is a statement or proposition which is regarded as being established, accepted, or self-evidently true on which an abstractly defined structure is based. More precisely an axiom is a statement that is self-evident without any proof which is a starting point for further reasoning and arguments.

Postulate verbally means a fact, or truth of (something) as a basis for reasoning, discussion, or belief. Postulates are the basic structure from which lemmas and theorems are derived.

Nowadays ‘axiom’ and ‘postulate’ are usually interchangeable terms. One key difference between them is that postulates are true assumptions that are specific to geometry. Axioms are true assumptions used throughout mathematics and not specifically linked to geometry.

“What is the difference between an axiom and a postulate?”. 2023. Vedantu . https://www.vedantu.com/question-answer/difference-between-an-axiom-and-a-post-class-10-maths-cbse-5efeafa98c08f1791a1cc34a .

A  conjecture  is a mathematical statement that has not yet been rigorously proved. Conjectures arise when one notices a pattern that holds true for many  cases . However, just because a pattern holds true for many cases does not mean that the pattern will hold true for all cases. Conjectures must be proved for the mathematical observation to be fully accepted. When a conjecture is rigorously proved, it becomes a theorem.

“Conjectures | Brilliant Math & Science Wiki”. 2022.  brilliant.org . https://brilliant.org/wiki/conjectures/ .

“The Subtle Art Of The Mathematical Conjecture | Quanta Magazine”. 2019.  Quanta Magazine . https://www.quantamagazine.org/the-subtle-art-of-the-mathematical-conjecture-20190507/ .

A result that has been  proved to be true  (using operations and facts that were already known).

Example: The “Pythagoras Theorem” proved that a 2  + b 2  = c 2  for a right angled triangle.

A Theorem is a major result, a minor result is called a Lemma.

“Theorem Definition (Illustrated Mathematics Dictionary)”. 2021.  mathsisfun.Com . https://www.mathsisfun.com/definitions/theorem.html .

“Theorems, Corollaries, Lemmas”. 2021.  mathsisfun.com . https://www.mathsisfun.com/algebra/theorems-lemmas.html .

A statement that is proven true using postulates, definitions, and previously proven theorems.

A theorem is a mathematical statement that can and must be proven to be true. You may have been first exposed to the term when learning about the Pythagorean Theorem . Learning different theorems and proving they are true is an important part of Geometry. [1]

[1] “4.1 Theorems and Proofs”. 2022. CK-12 Foundation . https://flexbooks.ck12.org/cbook/ck-12-interactive-geometry-for-ccss/section/4.1/primary/lesson/theorems-and-proofs-geo-ccss/ .

[2] Joyce, David . “Can a theorem be proved by a corollary?”. 2023.  Quora . https://qr.ae/pybAMq .

Yes, a theorem can be proved by a corollary just so long as the corollary is proved first. You might have a sequence of theorems in logical order like this: Theorem 1, Corollary 2, Lemma 3, Theorem 4, Theorem 5. Each one is proved from those that precede it, but Theorem 5 could depend only on Corollary 2 and Lemma 3. Sometimes theorems are presented in a different order than the logical order, and sometimes even in reverse logical order, but whatever order they’re presented, it is necessary that there is no circular logic.

[3] “Definition Of Lemma”. 2021. mathsisfun.com . https://www.mathsisfun.com/definitions/lemma.html .

[4] “Euclidean Geometry (Definition, Facts, Axioms and Postulates)”. 2021. BYJUS . BYJU’S. September 20. https://byjus.com/maths/euclidean-geometry/ .

Additional Reading

“Basic Math Definitions”. 2021.  mathsisfun.com . https://www.mathsisfun.com/basic-math-definitions.html .

Browning, Wes . “Can a theorem be proved by another theorem?”. 2023.  Quora . https://qr.ae/pybAUz .

Sure. Sometimes the second theorem is called a “corollary.” Sometimes the first theorem is called a “lemma” and the second is called a theorem implied by the lemma. Or they’re both called theorems. The choice of names is up to the author of the exposition and is meant to clarify the logical flow. You may occasionally also see the term “ porism ” used. After a theorem has been proved, a porism is another theorem that can be proved by essentially the same proof as the first, usually by obvious modifications. I had a professor in math grad school who loved to trot porisms out after proving a theorem in his classes.

“Byrne’s Euclid”. 2021.  C82.Net . https://www.c82.net/euclid/ .

THE FIRST SIX BOOKS OF THE ELEMENTS OF EUCLID WITH COLOURED DIAGRAMS AND SYMBOLS A reproduction of Oliver Byrne’s celebrated work from 1847 plus interactive diagrams, cross references, and posters designed by Nicholas Rougeux

“Definitions. Postulates. Axioms: First Principles Of Plane Geometry “. 2021.  themathpage.com . https://themathpage.com/aBookI/first.htm#post .

“Geometry Postulates”. 2021.  basic-mathematics.com . https://www.basic-mathematics.com/geometry-postulates.html .

Mystery, Mike the. 2024. “Is George Orwell Right About 2+2=4 in Maths?”  Medium . Medium. March 12. https://medium.com/@Mike_Meng/is-george-orwell-right-about-2-2-4-in-maths-3bb0f6d5dd88 .

Freedom is the freedom to say that two plus two makes four. ——George Orwell, Nineteen Eighty-Four. When I first read George Orwell’s great “1984”, the above sentence left an indelible impact on me. It is worth mentioning that my first reaction to this quote was why Orwell used 2+2=4 instead of 1+1=2. And that’s exactly the first time I realized I was pedantic enough to get a maths degree in future. Ok, so why 2+2=4 is true? Before directly into the topic, i need to introduce some basic rules that we use to calculate numbers every single day. The rule is actually called Peano axioms, which is a logic system about natural numbers proposed by the 19th-century mathematician Giuseppe Peano . And we can establish an arithmetic system by these sets of axioms, which is also known as the Peano arithmetic system.

“Zermelo-Fraenkel Set Theory (ZFC)”. 2023.  Mathematical Mysteries . https://mathematicalmysteries.org/zermelo-fraenkel-set-theory-zfc/ .

Zermelo–Fraenkel set theory  (abbreviated  ZF ) is a system of  axioms  used to describe  set theory . When the  axiom of choice  is added to ZF, the system is called  ZFC . It is the system of axioms used in set theory by most mathematicians today.

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2: The Postulates of Quantum Mechanics

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The entire structure of quantum mechanics (including its relativistic extension) can be formulated in terms of states and operations in Hilbert space. We need rules that map the physical quantities such as states, observables, and measurements to the mathematical structure of vector spaces, vectors and operators. There are several ways in which this can be done, and here we summarize these rules in terms of five postulates.

Postulate 1

A physical system is described by a Hilbert space \(\mathscr{H}\), and the state of the system is represented by a ray with norm 1 in \(\mathscr{H}\).

There are a number of important aspects to this postulate. First, the fact that states are rays, rather than vectors means that an overall phase \(e^{i \varphi}\) of the state does not have any physically observable consequences, and \(e^{i \varphi}|\psi\rangle\) represents the same state as \(|\psi\rangle\). Second, the state contains all information about the system. In particular, there are no hidden variables in this standard formulation of quantum mechanics. Finally, the dimension of \(\mathscr{H}\) may be infinite, which is the case, for example, when \(\mathscr{H}\) is the space of square-integrable functions.

As an example of this postulate, consider a two-level quantum system (a qubit). This system can be described by two orthonormal states \(|0\rangle\) and \(|1\rangle\). Due to linearity of Hilbert space, the superposition \(\alpha|0\rangle+\beta|1\rangle\) is again a state of the system if it has norm 1, or

\[(\alpha ^ { * } \langle0|+\beta^{*}\langle 1|)(\alpha|0\rangle+\beta|1\rangle)=1 \quad \text { or } \quad|\alpha|^{2}+|\beta|^{2}=1\tag{2.1}\]

This is called the superposition principle: any normalised superposition of valid quantum states is again a valid quantum state. It is a direct consequence of the linearity of the vector space, and as we shall see later, this principle has some bizarre consequences that have been corroborated in many experiments.

Postulate 2

Every physical observable \(A\) corresponds to a self-adjoint (Hermitian 1 ) operator \(\hat{A}\) whose eigenvectors form a complete basis.

We use a hat to distinguish between the observable and the operator, but usually this distinction is not necessary. In these notes, we will use hats only when there is a danger of confusion.

As an example, take the operator \(X\):

\[X|0\rangle=|1\rangle \quad \text { and } \quad X|1\rangle=|0\rangle.\tag{2.2}\]

This operator can be interpreted as a bit flip of a qubit. In matrix notation the state vectors can be written as

\[|0\rangle=\left(\begin{array}{l}1 \\ 0 \end{array}\right) \quad \text { and } \quad|1\rangle=\left(\begin{array}{l} 0 \\ 1 \end{array}\right),\tag{2.3}\]

which means that \(X\) is written as

\[X=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right)\tag{2.4}\]

with eigenvalues ±1. The eigenstates of \(X\) are

\[|\pm\rangle=\frac{|0\rangle \pm|1\rangle}{\sqrt{2}}.\tag{2.5}\]

These states form an orthonormal basis.

Postulate 3

The eigenvalues of \(A\) are the possible measurement outcomes, and the probability of finding the outcome \(a_{j}\) in a measurement is given by the Born rule:

\[p\left(a_{j}\right)=\left|\left\langle a_{j} \mid \psi\right\rangle\right|^{2},\tag{2.6}\]

where \(|\psi\rangle\) is the state of the system, and \(\left|a_{j}\right\rangle\) is the eigenvector associated with the eigenvalue \(a_{j}\) via \(A\left|a_{j}\right\rangle=a_{j}\left|a_{j}\right\rangle\). If \(a_{j}\) is \(m\)-fold degenerate, then

\[p(a_{j})=\sum_{l=1}^{m}|\langle a_{j}^{(l)} \mid \psi\rangle|^{2},\tag{2.7}\]

where the \(\left|a_{j}^{(l)}\right\rangle\) span the \(m\)-fold degenerate subspace

The expectation value of \(A\) with respect to the state of the system \(|\psi\rangle\) is denoted by \(\langle A\rangle\), and evaluated as

\[\langle A\rangle=\langle\psi|A| \psi\rangle=\langle\psi|(\sum_{j} a_{j}|a_{j}\rangle\langle a_{j}|)| \psi\rangle=\sum_{j} p(a_{j}) a_{j}\tag{2.8}\]

This is the weighted average of the measurement outcomes. The spread of the measurement outcomes (or the uncertainty) is given by the variance

\[(\Delta A)^{2}=\left\langle(A-\langle A\rangle)^{2}\right\rangle=\left\langle A^{2}\right\rangle-\langle A\rangle^{2}\tag{2.9}\]

So far we mainly dealt with discrete systems on finite-dimensional Hilbert spaces. But what about continuous systems, such as a particle in a box, or a harmonic oscillator? We can still write the spectral decomposition of an operator A but the sum must be replace by an integral:

\[A=\int d a f_{A}(a)|a\rangle\langle a|\tag{2.10}\]

where \(|a\rangle\) is an eigenstate of \(A\). Typically, there are problems with the normalization of \(|a\rangle\), which is related to the impossibility of preparing a system in exactly the state \(|a\rangle\). We will not explore these subtleties further in this course, but you should be aware that they exist. The expectation value of \(A\) is

\[\langle A\rangle=\langle\psi|A| \psi\rangle=\int d a f_{A}(a)\langle\psi \mid a\rangle\langle a \mid \psi\rangle \equiv \int d a f_{A}(a)|\psi(a)|^{2},\tag{2.11}\]

where we defined the wave function \(\psi(a)=\langle a \mid \psi\rangle\), and \(|\psi(a)|^{2}\) is properly interpreted as the probability density that you remember from second-year quantum mechanics.

The probability of finding the eigenvalue of an operator \(A\) in the interval \(a\) and \(a+d a\) given the state \(|\psi\rangle\) is

\[\langle\psi|(|a\rangle\langle a| d a)| \psi\rangle \equiv d p(a),\tag{2.12}\]

since both sides must be infinitesimal. We therefore find that

\[\frac{d p(a)}{d a}=|\psi(a)|^{2}\tag{2.13}\]

Postulate 4

The dynamics of quantum systems is governed by unitary transformations

We can write the state of a system at time \(t\) as \(|\psi(t)\rangle\), and at some time \(t_{0}<t\) as \(\left|\psi\left(t_{0}\right)\right\rangle\). The fourth postulate tells us that there is a unitary operator \(U\left(t, t_{0}\right)\) that transforms the state at time \(t_{0}\) to the state at time \(t\):

\[|\psi(t)\rangle=U\left(t, t_{0}\right)\left|\psi\left(t_{0}\right)\right\rangle\tag{2.14}\]

Since the evolution from time \(t\) to \(t\) is denoted by \(U(t, t)\) and must be equal to the identity, we deduce that \(U\) depends only on time differences: \(U\left(t, t_{0}\right)=U\left(t-t_{0}\right)\), and \(U(0)=\mathbb{I}\).

As an example, let \(U(t)\) be generated by a Hermitian operator \(A\) according to

\(U(t)=\exp \left(-\frac{i}{\hbar} A t\right)\tag{2.15}\)

The argument of the exponential must be dimensionless, so \(A\) must be proportional to \(\hbar\) times an angular frequency (in other words, an energy). Suppose that \(|\psi(t)\rangle\) is the state of a qubit, and that \(A=\hbar \omega X\). If \(|\psi(0)\rangle=|0\rangle\) we want to calculate the state of the system at time \(t\). We can write

\[|\psi(t)\rangle=U(t)|\psi(0)\rangle=\exp (-i \omega t X)|0\rangle=\sum_{n=0}^{\infty} \frac{(-i \omega t)^{n}}{n !} X^{n}\tag{2.16}\]

Observe that \(X^{2}=\mathbb{I}\), so we can separate the power series into even and odd values of n:

\[|\psi(t)\rangle=\sum_{n=0}^{\infty} \frac{(-i \omega t)^{2 n}}{(2 n) !}|0\rangle+\sum_{n=0}^{\infty} \frac{(-i \omega t)^{2 n+1}}{(2 n+1) !} X|0\rangle=\cos (\omega t)|0\rangle-i \sin (\omega t)|1\rangle\tag{2.17}\]

In other words, the state oscillates between \(|0\rangle\) and \(|1\rangle\).

The fourth postulate also leads to the Schrödinger equation. Let’s take the infinitesimal form of Eq. (2.14):

\[|\psi(t+d t)\rangle=U(d t)|\psi(t)\rangle\tag{2.18}\]

We require that \(U(d t)\) is generated by some Hermitian operator \(H\):

\[U(d t)=\exp \left(-\frac{i}{\hbar} H d t\right)\tag{2.19}\]

\(H\) must have the dimensions of energy, so we identify it with the energy operator, or the Hamiltonian. We can now take a Taylor expansion of \(|\psi(t+d t)\rangle\) to first order in dt:

\[|\psi(t+d t)\rangle=|\psi(t)\rangle+d t \frac{d}{d t}|\psi(t)\rangle+\ldots,\tag{2.20}\]

and we expand the unitary operator to first order in dt as well:

\[U(d t)=1-\frac{i}{\hbar} H d t+\ldots\tag{2.21}\]

We combine this into

\[|\psi(t)\rangle+d t \frac{d}{d t}|\psi(t)\rangle=\left(1-\frac{i}{\hbar} H d t\right)|\psi(t)\rangle,\tag{2.22}\]

which can be recast into the Schrödinger equation:

\[i \hbar \frac{d}{d t}|\psi(t)\rangle=H|\psi(t)\rangle\tag{2.23}\]Therefore, the Schrödinger equation follows directly from the postulates!

Postulate 5

If a measurement of an observable \(A\) yields an eigenvalue \(a_{j}\), then immediately after the measurement, the system is in the eigenstate \(\left|a_{j}\right\rangle\) corresponding to the eigenvalue

This is the infamous projection postulate, so named because a measurement “projects” the system to the eigenstate corresponding to the measured value. This postulate has as observable consequence that a second measurement immediately after the first will also find the outcome \(a_{j}\). Each measurement outcome \(a_{j}\) corresponds to a projection operator \(P_{j}\) on the subspace spanned by the eigenvector(s) belonging to \(a_{j}\). A (perfect) measurement can be described by applying a projector to the state, and renormalize:

\[|\psi\rangle \rightarrow \frac{P_{j}|\psi\rangle}{\| P_{j}|\psi\rangle \|}\tag{2.24}\]

This also works for degenerate eigenvalues.

We have established earlier that the expectation value of \(A\) can be written as a trace:

\[\langle A\rangle=\operatorname{Tr}(|\psi\rangle\langle\psi| A)\tag{2.25}\]

Now instead of the full operator \(A\), we calculate the trace of \(P_{j}=\left|a_{j}\right\rangle\left\langle a_{j}\right|\):

\[\left\langle P_{j}\right\rangle=\operatorname{Tr}\left(|\psi\rangle\langle\psi| P_{j}\right)=\operatorname{Tr}\left(|\psi\rangle\left\langle\psi \mid a_{j}\right\rangle\left\langle a_{j}\right|\right)=\left|\left\langle a_{j} \mid \psi\right\rangle\right|^{2}=p\left(a_{j}\right)\tag{2.26}\]

So we can calculate the probability of a measurement outcome by taking the expectation value of the projection operator that corresponds to the eigenstate of the measurement outcome. This is one of the basic calculations in quantum mechanics that you should be able to do.

The Measurement Problem

The projection postulate is somewhat problematic for the interpretation of quantum mechanics, because it leads to the so-called measurement problem : Why does a measurement induce a non-unitary evolution of the system? After all, the measurement apparatus can also be described quantum mechanically 2 and then the system plus the measurement apparatus evolves unitarily. But then we must invoke a new device that measures the combined system and measurement apparatus. However, this in turn can be described quantum mechanically, and so on.

On the other hand, we do see definite measurement outcomes when we do experiments, so at some level the projection postulate is necessary, and somewhere there must be a “collapse of the wave function”. Schrödinger already struggled with this question, and came up with his famous thought experiment about a cat in a box with a poison-filled vial attached to a Geiger counter monitoring a radioactive atom (Figure 1). When the atom decays, it will trigger the Geiger counter, which in turn causes the release of the poison killing the cat. When we do not look inside the box (more precisely: when no information about the atom-counter-vial-cat system escapes from the box), the entire system is in a quantum superposition. However, when we open the box, we do find the cat either dead or alive. One solution of the problem seems to be that the quantum state represents our knowledge of the system, and that looking inside the box merely updates our information about the atom, counter, vial and the cat. So nothing “collapses” except our own state of mind.

However, this cannot be the entire story, because quantum mechanics clearly is not just about our opinions of cats and decaying atoms. In particular, if we prepare an electron in a spin “up” state \(|\uparrow\rangle\), then whenever we measure the spin along the \(z\)-direction we will find the measurement outcome “up”, no matter what we think about electrons and quantum mechanics. So there seems to be some physical property associated with the electron that determines the measurement outcome and is described by the quantum state.

Various interpretations of quantum mechanics attempt to address these (and other) issues. The original interpretation of quantum mechanics was mainly put forward by Niels Bohr, and is called the Copenhagen interpretation . Broadly speaking, it says that the quantum state is a convenient fiction, used to calculate the results of measurement outcomes, and that the system cannot be considered separate from the measurement apparatus. Alternatively, there are interpretations of quantum mechanics, such as the Ghirardi-Rimini-Weber interpretation , that do ascribe some kind of reality to the state of the system, in which case a physical mechanism for the collapse of the wave function must be given. Many of these interpretations can be classified as hidden variable theories, which postulate that there is a deeper physical reality described by some “hidden variables” that we must average over. This in turn explains the probabilistic nature of quantum mechanics. The problem with such theories is that these hidden variables must be quite weird: they can change instantly depending on events light-years away3 , thus violating Einstein’s theory of special relativity. Many physicists do not like this aspect of hidden variable theories.

Alternatively, quantum mechanics can be interpreted in terms of “many worlds”: the Many Worlds interpretation states that there is one state vector for the entire universe, and that each measurement splits the universe into different branches corresponding to the different measurement outcomes. It is attractive since it seems to be a philosophically consistent interpretation, and while it has been acquiring a growing number of supporters over recent years 4 , a lot of physicists have a deep aversion to the idea of parallel universes.

Finally, there is the epistemic interpretation , which is very similar to the Copenhagen interpretation in that it treats the quantum state to a large extent as a measure of our knowledge of the quantum system (and the measurement apparatus). At the same time, it denies a deeper underlying reality (i.e., no hidden variables). The attractive feature of this interpretation is that it requires a minimal amount of fuss, and fits naturally with current research in quantum information theory. The downside is that you have to abandon simple scientific realism that allows you to talk about the properties of electrons and photons, and many physicists are not prepared to do that.

As you can see, quantum mechanics forces us to abandon some deeply held (classical) convictions about Nature. Depending on your preference, you may be drawn to one or other interpretation. It is currently not know which interpretation is the correct one.

• Calculate the eigenvalues and the eigenstates of the bit flip operator \(X\), and show that the eigenstates form an orthonormal basis. Calculate the expectation value of \(X\) for \(|\psi\rangle=1 / \sqrt{3}|0\rangle+i \sqrt{2 / 3}|1\rangle\).
• Show that the variance of \(A\) vanishes when \(|\psi\rangle\) is an eigenstate of \(A\).
• Prove that an operator is Hermitian if and only if it has real eigenvalues.
• Show that a qubit in an unknown state \(|\psi\rangle\) cannot be copied. This is the no-cloning theorem. Hint: start with a state \(|\psi\rangle|i\rangle\) for some initial state \(|i\rangle\), and require that for \(|\psi\rangle=|0\rangle\) and \(|\psi\rangle=|1\rangle\) the cloning procedure is a unitary transformation \(|0\rangle|i\rangle \rightarrow|0\rangle|0\rangle\) and \(|1\rangle|i\rangle \rightarrow|1\rangle|1\rangle\).

\[(\Delta A)^{2}(\Delta B)^{2} \geq \frac{1}{4}|\langle[A, B]\rangle|^{2}\tag{2.27}\]

Hint: define \(|f\rangle=(A-\langle A\rangle)|\psi\rangle\) and \(|g\rangle=i(B-\langle B\rangle)|\psi\rangle\), and use that \(|\langle f \mid g\rangle| \geq \frac{1}{2} \mid\langle f \mid g\rangle+\langle g|f\rangle|\).

• Show that this reduces to Heisenberg’s uncertainty relation when \(A\) and \(B\) are canonically conjugate observables, for example position and momentum.
• Does this method work for deriving the uncertainty principle between energy and time?

\[H=E\left(\begin{array}{ccc} 0 & i & 0 \\ -i & 0 & 0 \\ 0 & 0 & -1 \end{array}\right) \quad \text { and } \quad|\psi\rangle=\frac{1}{\sqrt{5}}\left(\begin{array}{c} 1-i \\ 1-i \\ 1 \end{array}\right)\tag{2.28}\]

where \(E\) is a constant with dimensions of energy. Calculate the energy eigenvalues and the expectation value of the Hamiltonian.

• Show that the momentum and the total energy can be measured simultaneously only when the potential is constant everywhere. What does a constant potential mean in terms of the dynamics of a particle?

1 In Hilbert spaces of infinite dimensionality, there are subtle differences between self-adjoint and Hermitian operators. We ignore these subtleties here, because we will be mostly dealing with finite-dimensional spaces.

2 This is something most people require from a fundamental theory: quantum mechanics should not just break down for macroscopic objects. Indeed, experimental evidence of macroscopic superpositions has been found in the form of “cat states”.

3 . . . even though the averaging over the hidden variables means you can never signal faster than light.

4 There seems to be some evidence that the Many Worlds interpretation fits well with the latest cosmological models based on string theory