Philosophical Essays on the Foundations of Mathematics and the Philosophy of Science
Introduction: The Philosopher’s Engagement with Mathematics and Science
We are at the opening of the volume, where Putnam establishes the scope and method of the essays collected here, and sets forth his central concerns regarding the nature of mathematical knowledge and its relationship to empirical science.
Putnam begins by distinguishing his approach from two dominant positions that he finds inadequate. The first is logical positivism, which held that mathematical truths are merely analytic or tautological, and that scientific theories are to be judged solely by their empirical content. The second is ordinary language philosophy, which treated philosophical problems as confusions arising from misuse of language. Putnam argues that both schools have failed to engage seriously with the actual content of mathematics and science. He writes that “the philosophy of mathematics has been dominated by two great traditions, the *Platonist* and the *nominalist*,” and that each tradition, when pressed, leads to severe difficulties. Platonism treats mathematical objects as real, mind-independent entities, which raises the problem of how we could have knowledge of such abstract objects. Nominalism denies the existence of such objects, but then struggles to explain the apparent truth and applicability of mathematical statements.
The essays in this volume, Putnam explains, attempt to steer a middle course. He is concerned with “the *methodology* of the exact sciences” and with “the *interpretation* of scientific theories.” His central claim is that mathematics and physics cannot be neatly separated. Mathematics is not a purely formal game played with symbols; it has genuine content and makes assertions about the world. At the same time, physical theories are not merely instrumental devices for predicting observations; they make claims about the nature of reality, including claims that involve mathematical objects. Putnam argues that “the problems of the philosophy of mathematics and the philosophy of physics are inseparably intertwined.” A satisfactory account of either domain must take the other seriously.
What this means, as Putnam develops it, is that the philosopher must attend to what working mathematicians and physicists actually do, not to a priori doctrines about what they *ought* to be doing. The essays are organized into two parts: the first, “Mathematics,” deals with the philosophy of mathematics proper, including issues of truth, existence, and the nature of mathematical proof. The second, “Method,” deals with the philosophy of science, including the nature of scientific explanation, confirmation, and the structure of physical theories. The introduction thus alerts the reader that the volume will proceed through detailed engagements with specific problems—the status of set theory, the nature of geometry, the logic of quantum mechanics—rather than through broad system-building.
Part I: Mathematics — The Thesis That Mathematics Is Not Completely A Priori
We enter Part I of the collection, where Putnam presents his opening essay, “Mathematics and the Existence of Abstract Entities,” which argues against the view that mathematical truths are known purely by logical deduction from self-evident axioms, and in favor of a position that ties mathematical knowledge to empirical science.
Putnam opens by confronting the central problem of mathematical ontology: do numbers, sets, and other mathematical objects exist? The standard argument for nominalism holds that we should not posit such objects because they are not needed for science; all we need is the truth of sentences that appear to refer to numbers, but these sentences can be reinterpreted as statements about concrete possibilities. Putnam rejects this. He argues that if we take the language of science seriously, we are committed to the existence of mathematical objects. The physicist speaks of “the set of all solutions to this differential equation”; the biologist speaks of “the probability that a mutation will occur.” These are not optional metaphors but integral parts of the theories’ claims. “The language of science,” Putnam writes, “is not a code for talking about something else; it is the language that we actually use to make assertions about the world.”
However, Putnam does not embrace a simple Platonism in which we possess a mysterious faculty of intellectual intuition that reveals mathematical truths. Instead, he proposes a quasi-empirical view of mathematics. On this view, mathematical principles can be supported by arguments that are not strictly deductive — arguments that involve their success in explaining empirical phenomena. Putnam points to the history of geometry as a key example. Euclidean geometry was once thought to be a body of a priori truths known by reason alone. But the development of non-Euclidean geometries, and their application in Einstein’s General Theory of Relativity, showed that the choice of a geometry for physical space is an empirical matter. “The axioms of geometry,” he states, “are not known a priori; they are postulates that can be confirmed or disconfirmed by experience.”
The significance of this argument is that it collapses the sharp distinction between mathematical and scientific knowledge. Putnam is not claiming that all mathematical truths are empirical — only that the boundary between the a priori and the a posteriori is not fixed in advance. Some mathematical principles may be revised in light of experience, just as scientific principles are. The lesson is that the philosopher of mathematics must attend to the actual role that mathematics plays in scientific practice, rather than legislating from a prior philosophical commitment.
Mathematics and the Existence of Abstract Entities: The Argument from Science
Continuing within the same essay, Putnam develops his core argument that the best explanation for the success of mathematics in science is that mathematical statements are true in a robust sense, and that their truth involves reference to abstract entities.
Putnam presents the argument in the form of a challenge to the nominalist. Suppose a physicist uses a mathematical theory — say, the theory of Hilbert spaces in quantum mechanics — to make predictions that are then confirmed. The nominalist wants to say that the mathematical theory is merely a useful fiction; what is really true are the empirical predictions about observable phenomena. But Putnam asks: what is the *explanatory* relationship here? The mathematical theory does not simply produce predictions; it provides the very *structure* within which physical laws are formulated. The Hamiltonian operator, the eigenvalues, the wave function — these are mathematical objects, and they figure essentially in the explanation of why certain experimental results obtain. “If the mathematical entities are fictions,” Putnam writes, “then the scientific explanation is a fiction as well.”
Putnam elaborates this through what has become known as the indispensability argument. The argument proceeds in two steps. First, we ought to be committed to the existence of whatever entities are indispensable to our best scientific theories. Second, mathematical entities — specifically, numbers, sets, functions, and spaces — are indispensable to our best scientific theories. Therefore, we ought to be committed to the existence of mathematical entities. This is not a metaphysical proof of Platonism, but a methodological principle: if we accept scientific realism about electrons and genes, we should accept realism about numbers and sets as well, because they play the same kind of explanatory role.
Putnam is careful to distinguish his position from naive Platonism. He does not claim that we have a direct intuition of abstract entities, or that mathematical truth is entirely mind-independent in a way that makes it mysterious how we could know it. Rather, he argues that the knowledge we have of mathematical entities is continuous with the knowledge we have of physical entities: both are known through their role in our best theories, and both are subject to revision. The mathematician who proves a theorem about prime numbers is engaged in a rational activity, but that activity is not cut off from the rest of our inquiry about the world. “Mathematics,” Putnam insists, “is not a game played by arbitrary rules; it is a body of knowledge of real structures.”
The Thesis That Mathematics Is Not Completely A Priori: Geometry and Physics
We now move to the essay “The Thesis That Mathematics Is Not Completely A Priori,” where Putnam examines the specific case of geometry to argue that the revision of mathematical principles on empirical grounds is not merely a historical curiosity but a live possibility.
Putnam begins by recalling the conventional view that geometry was the paradigm of a priori knowledge. From Euclid to Kant, it was held that the truths of geometry were known with certainty, independent of experience. The discovery of non-Euclidean geometries by Gauss, Bolyai, and Lobachevsky seemed to show that different geometries were logically possible. But this was still often interpreted as showing that geometry is a matter of convention or definition: we choose a geometry and then derive its consequences. Putnam rejects this conventionalist interpretation. He points out that in Einstein’s General Relativity, the choice of a geometry for physical space is not a matter of convention; it is forced by empirical evidence. The geometry of spacetime is determined by the distribution of mass and energy, and experiments — such as the bending of light by gravity — can confirm or refute a proposed geometry.
Putnam then draws a more general moral. If a mathematical principle like the parallel postulate can be revised in light of experience, then why not other mathematical principles? He considers the case of set theory and the continuum hypothesis. The continuum hypothesis — that there is no set whose cardinality is strictly between that of the integers and that of the real numbers — was shown by Gödel and Cohen to be independent of the standard axioms of set theory. Putnam argues that this independence does not mean the hypothesis has no truth value. Rather, it may be that new mathematical or physical arguments will eventually decide it. “The fact that the continuum hypothesis is not decided by the currently accepted axioms,” he writes, “does not mean that it is not a genuine problem.”
Putnam extends this line of thought to the law of the excluded middle and the validity of classical logic. He notes that in quantum mechanics, some physicists and philosophers have proposed abandoning classical logic in favor of a quantum logic that rejects distributivity. While he does not here endorse such a move, he takes it seriously as an illustration of his central thesis: the principles of mathematics and logic are not immune to revision. They can be changed, if doing so yields a better overall theory of the world. This is not to say that the revision is easy or that it happens often; it is to say that the separation between the a priori and the a posteriori is a matter of degree, not of kind.
What Is Mathematical Truth?
In this essay, Putnam shifts from the question of ontology to the question of truth, examining what it means for a mathematical statement to be true, and arguing against both the formalist and the intuitionist accounts.
Putnam begins by surveying the main competing views of mathematical truth. Formalism, associated with Hilbert, holds that mathematics is a formal game with symbols; a mathematical statement is true if it can be derived from the axioms by the rules of inference. Intuitionism, associated with Brouwer, holds that mathematical truth is grounded in mental constructions; a statement is true only if we have an explicit construction that demonstrates it. Both views, Putnam argues, are inadequate. Formalism fails because it cannot account for the *applicability* of mathematics to the world. Why should a game played with symbols yield useful predictions about electrons? Intuitionism fails because it is too restrictive: it rejects parts of classical mathematics that are both mathematically fruitful and physically applicable.
Putnam proposes instead a realist view of mathematical truth, but one that is tied to the notion of truth as idealized rational acceptability. On this view, a mathematical statement is true if it would be accepted by a sufficiently rational inquirer who had all the relevant evidence. This is not the same as saying that the statement is accepted *now* by the mathematical community, or that it follows from current axioms. It is a regulative ideal: mathematical truth is what we would arrive at if we could extend our knowledge indefinitely. Putnam compares this to the way we think about physical truth: we do not think that current physical theories are perfectly true, but we hold that there is a fact of the matter about what the world is like, and that inquiry can approach it.
Putnam applies this account to the problem of mathematical undecidability. Gödel’s incompleteness theorems show that any consistent axiomatic system that can express arithmetic contains statements that can neither be proved nor disproved within that system. If truth were identical to provability, then these statements would be neither true nor false. But on Putnam’s account, they have a determinate truth value — we just cannot prove it from the given axioms. “The incompleteness theorems,” he writes, “show that we cannot capture all mathematical truths in a single axiomatic system; they do not show that there is no fact of the matter about those truths.” This leads Putnam to a position he later calls internal realism: truth is not a simple correspondence to a mind-independent reality, but it is also not reducible to what is proved. It is what is rationally assertible under ideal epistemic conditions.
Part II: Philosophy of Physical Science — The Nature of Scientific Theories
We move now from the philosophy of mathematics to the philosophy of physical science, where Putnam turns his attention to the logical structure and ontological commitments of scientific theories. This section occupies roughly the middle third of the volume and contains some of his most influential essays.
The opening essay, “What Theories Are Not,” takes aim at the then-dominant received view of scientific theories, associated with logical empiricism. According to this view, a scientific theory consists of two parts: a calculus — a formal axiomatic system expressed in mathematical logic — and a set of correspondence rules that connect some of the terms in the calculus to observable phenomena. The theoretical terms (like “electron” or “electromagnetic field”) are given meaning only indirectly, through the network of axioms and the partial interpretations provided by the correspondence rules. Putnam argues that this picture is fundamentally mistaken. The correspondence rules, he points out, are not given in a separate, observation-language; they are themselves theoretical statements. Take a typical correspondence rule: “If a metal rod is heated, its length increases.” This looks like a statement linking theory to observation, but it already uses the theoretical term “length” in a quantitative, measure-theoretic sense. The line between “theoretical” and “observational” is not a sharp logical boundary but a matter of degree and context.
“The notion that there is a dichotomy between ‘observation terms’ and ‘theoretical terms’ is a mistake.”
Putnam’s positive proposal is that scientific theories should be understood as conceptual systems that organize our experience holistically. A theory does not have a separate “observational part” and “theoretical part”; rather, the entire system — including its mathematical apparatus — is applied to phenomena as a whole. The meaning of a term like “electron” is given not by a definition in terms of observables but by its role in the entire theoretical structure. This anticipates the semantic view of theories that later became influential: a theory is a collection of models, not a set of sentences with a partial interpretation.
What this means within the work is that Putnam is rejecting the empiricist attempt to reduce scientific knowledge to a foundation of sensory experience. If theoretical terms cannot be defined in observational terms, then the ontology of science — what exists according to our best theories — is not eliminable. We cannot “translate” talk of electrons into talk of meter readings; we must take the theoretical entities as real. This sets the stage for his argument for scientific realism in the subsequent essays.
Explanation and Reference: The Causal Theory
The next chapter, “The Meaning of ‘Meaning’,” is one of Putnam’s most famous works, though here presented in a condensed form focused on its implications for science. He introduces the theory of direct reference for natural kind terms. The classical view, from Frege to Carnap, held that the meaning of a term like “water” is given by a concept — a set of necessary and sufficient conditions for something to count as water. Putnam argues that this cannot be right. He imagines a Twin Earth where a substance with the same observable properties as water — it quenches thirst, fills the lakes, falls as rain — has a different chemical composition, say XYZ instead of H₂O. When an Earthling in 1750 says “water” and a Twin Earthling says “water,” they mean different things, even though their concepts are identical. The difference is not in their heads — it is in the world. The reference of “water” is fixed not by a description but by an indexical component: “water” means the stuff that is actually the liquid in our lakes and rivers, whatever its deep structure turns out to be.
“Cut the pie any way you like, ‘meanings’ just ain’t in the head!”
This has profound implications for the philosophy of science. If the meanings of natural kind terms are not fixed by the beliefs of individual speakers, then scientific progress is not merely a change in what people believe — it is sometimes a discovery of what the terms actually referred to all along. When chemists discovered that water is H₂O, they were not changing the meaning of “water”; they were finding out what water really is. This picture supports a realist view of science: later theories can genuinely refer to the same entities as earlier theories, even when the descriptions differ. The term “electron” as used by Bohr and as used by Dirac refers to the same kind of particle, despite the vast differences in their theories.
Within the overall argument of the volume, this essay provides the semantic foundation for scientific realism. If meanings were entirely internal to theories, then different theories might be incommensurable — they would be talking about different things. But Putnam’s account of reference shows how continuity is possible: later theories can be better descriptions of the same entities that earlier theories referred to. This is not a guarantee of realism, but it removes a major philosophical objection to it.
The Analytic and the Synthetic: Revisiting the Distinction
Part II also contains a sustained critique of the analytic-synthetic distinction, which had been a cornerstone of logical empiricism. According to that distinction, some statements are true solely in virtue of their meaning (analytic), while others depend on how the world is (synthetic). Kant’s example: “All bachelors are unmarried” is analytic; “All bachelors are unhappy” is synthetic. The logical empiricists used this distinction to separate the a priori (logic and mathematics) from the a posteriori (empirical science). Putnam argues that the distinction is far less clear than it appears.
He examines the statement “All cats are animals.” Is this analytic? One might think so — part of the meaning of “cat” is that a cat is an animal. But suppose we discover that the things we call “cats” are actually robots from Mars, cleverly disguised to look like animals. Would we say that “cat” meant “robot all along”? Or would we say that we were wrong about cats? Putnam argues that natural kind terms like “cat” have a stereotype — a set of typical features — but this stereotype is not a definition. The meaning of “cat” includes an indexical component: a cat is a member of the natural kind that includes the paradigmatic examples we point to. If those paradigmatic examples turn out to be robots, then “cat” means “robot” after all. But this is not a linguistic decision; it is a discovery about the world. The analytic-synthetic distinction collapses because the line between meaning and empirical fact cannot be drawn sharply for natural kind terms.
“The analytic-synthetic distinction is not a distinction between two kinds of truth; it is a distinction between two kinds of statements that we treat differently in our epistemic practice.”
This matters for the volume’s overall project because it undermines the idea that there is a privileged class of truths — those of logic and mathematics — that are immune from empirical revision. If the analytic-synthetic distinction is not absolute, then even statements that seem to be “true by definition” might be revised in light of new empirical evidence. This reinforces the theme from Part I: the boundaries between the a priori and the a posteriori are not fixed by the nature of the statements themselves but are relative to our evolving conceptual scheme.
Scientific Realism and the No-Miracles Argument
The volume then presents Putnam’s most direct argument for scientific realism, often called the “no-miracles” argument or the ultimate argument. The argument is deceptively simple. Consider the history of science: we have theories that make successful predictions, generate new technologies, and enable us to manipulate the world. If these theories were not at least approximately true — if the entities they postulate did not actually exist — then this success would be a miracle. The fact that our theories work so well is strong evidence that they are tracking something real.
Putnam writes: “The positive argument for realism is that it is the only philosophy that does not make the success of science a miracle.” He contrasts realism with instrumentalism, which holds that theories are merely useful fictions for organizing observations. On the instrumentalist view, a theory’s success is just a matter of predictive adequacy; it does not require the entities of the theory to be real. But Putnam points out that instrumentalism cannot explain why the predictions work. If electrons are just a convenient fiction, why do the predictions based on electron theory succeed so spectacularly? Why do fictions lead to successful interventions in the world? The realist answer is that electrons are not fictions — they are real, and our theories correctly describe (to a first approximation) their behavior.
The argument is not airtight. Putnam acknowledges that a determined anti-realist could always say that the success of science is brute fact — it just happens that our theories work. But the realist explanation is more satisfying and more plausible. Moreover, the argument gains strength when combined with the semantic points from “The Meaning of ‘Meaning'”: if the terms of successful theories refer to real entities, then the continuity of reference across theory change supports the idea that we are progressively converging on the truth about those entities.
Within the work, this chapter represents the culmination of the philosophy of science section. Putnam has argued that theories are not reducible to observation, that meaning is not in the head, and that the analytic-synthetic distinction does not provide a sharp boundary between the a priori and the empirical. The no-miracles argument ties these threads together: the best explanation for the success of science is that our theories are approximately true and refer to real entities. This is not a proof, but it is a powerful abductive inference — the inference to the best explanation.
Part III: Metaphysics — The Nature of Reality and Mind
The Logical Structure of Physical Concepts: Measurement and Reality
We are now in Part III of the work, where Putnam turns from the philosophy of mathematics and physical science to broader metaphysical questions about the nature of reality, causality, and the mind-body problem.
The chapter opens with an examination of how physical concepts are actually structured in scientific practice, particularly through the role of measurement. Putnam argues that the logical structure of physical concepts cannot be understood simply by analyzing their formal definitions or operational criteria. Instead, he presents the view that physical concepts are theory-laden — they derive their meaning from their place within an entire theoretical framework. He writes:
“The meaning of a physical concept is not given by a set of operational definitions, but by the role it plays in a theory.”
Putnam develops this point by considering the concept of temperature. One might think that temperature is defined by the behavior of a mercury thermometer, but this operational definition fails when we consider that the expansion of mercury is itself a physical process that depends on other theoretical assumptions. The concept of temperature, he argues, is actually intertwined with the kinetic theory of gases, the laws of thermodynamics, and the statistical mechanics of molecular motion. To understand what temperature “really means,” one must understand the entire network of laws and principles in which it is embedded. This is what Putnam calls the network theory of meaning: a physical concept is a node in a web of theoretical relationships.
The significance of this argument within the work is that it dissolves the classical empiricist attempt to ground physical concepts in pure observation. Putnam is showing that even the most basic physical magnitudes — length, time, mass, temperature — cannot be reduced to direct sensory experience. They are inherently theoretical, and their logical structure reflects the structure of the theory that employs them. This prepares the ground for the metaphysical arguments that follow, where Putnam will argue that reality itself cannot be divided into “observable” and “theoretical” realms.
Time and Physical Geometry: The Conventionality of Metric
This chapter addresses a classic problem in the philosophy of space and time: whether the metric of spacetime is a matter of convention or a feature of objective reality. Putnam enters this debate by examining the work of Henri Poincaré and Adolf Grünbaum, who argued that the geometry of spacetime is conventional — that there is no fact of the matter which geometry is “really” instantiated by the physical world.
Putnam presents the conventionalist argument clearly: if we can always adjust the laws of physics to accommodate any geometry, then geometry itself is not empirically determined. For example, one could say that space is Euclidean but that measuring rods shrink or expand when moved, or one could say that space is non-Euclidean and measuring rods remain rigid. The two theories are empirically equivalent; they make the same predictions about what we will observe. Therefore, the choice between them is a matter of convention. Putnam quotes the conventionalist position:
“If the geometry is conventional, then there is no fact of the matter as to whether space is Euclidean or non-Euclidean; the decision is a matter of definitional convenience.”
Putnam’s response is nuanced. He agrees that there is an element of convention in the choice of geometry, but he argues that this does not make geometry entirely conventional or non-factual. The key is to recognize that the metric — the standard of length or duration — is not given by nature but is imposed by our choice of measuring procedures. However, once we fix a metric, the geometry becomes an empirical question. Putnam contends that the conventionality of the metric does not entail the conventionality of spacetime structure as a whole. There are invariant features of spacetime — causal structure, for example — that are independent of our choice of metric.
Within the broader work, this chapter serves as a bridge between the philosophy of science and metaphysics. Putnam is showing that the logical structure of physical concepts, including geometric ones, cannot be reduced to pure convention or pure fact. The lesson is that our conceptual framework and the objective world are not separable in the way philosophers have traditionally assumed. This prepares the way for the more radical metaphysical arguments that follow, particularly concerning the nature of time and causality.
The Mind-Body Problem: Functionalism and the Identity Theory
Putnam here turns to one of the central problems in philosophy: the relationship between mind and body. The chapter begins by rejecting the traditional identity theory — the view that mental states are identical to brain states. Putnam argues that this view, while an improvement over dualism, is too narrow. He presents an alternative that he calls functionalism, a position that will become one of the most influential in the philosophy of mind.
The core argument is that mental states are not defined by their physical substrate but by their functional role — the causal relationships they have with sensory inputs, behavioral outputs, and other mental states. Putnam illustrates this with an analogy: what makes something a mousetrap is not the materials it is made of but the function it performs. A mousetrap can be made of wood, metal, or plastic; the same is true of mental states. As he writes:
“The functional organization of a system is what determines its mental states, not the physical composition of the system.”
Putnam develops this point by considering the case of pain. The identity theorist says that pain is identical to the firing of C-fibers in the brain. But Putnam asks: suppose there were beings on another planet, made of silicon rather than carbon, who also avoid harmful stimuli, cry out when damaged, and exhibit all the behavioral and functional correlates of pain. Would we deny that they feel pain? The functionalist answer is no — pain is a functional state that can be realized in multiple physical systems. This is the multiple realizability argument, which becomes the central objection to the identity theory.
Putnam then extends the argument to mental concepts generally. He distinguishes between the functional specification of a mental state (its causal role) and its physical realization (the specific brain state that implements it). The functional specification is what gives the mental state its identity; the physical realization is contingent. This means that psychology and neuroscience are at different levels of explanation: psychology studies functional organization, while neuroscience studies physical implementation.
Within the work, this chapter represents a major metaphysical contribution. Putnam is not merely criticizing the identity theory; he is offering a positive account of what mental states are. Functionalism provides a way to be a physicalist (there is no separate mental substance) while avoiding the reductionism of the identity theory. The mind, for Putnam, is a functional system — a certain kind of causal organization — that can be realized in different physical substrates. This position will remain central in the philosophy of mind for decades, and its articulation here is one of the most important moments in the book.
The Nature of Mental States: Psychological Predicates and Reduction
This chapter deepens the functionalist analysis by examining the logical character of psychological predicates — terms like “believes,” “desires,” “intends,” and “feels pain.” Putnam argues that these predicates are not reducible to physical or biological terms, but they are nonetheless perfectly real and causally efficacious.
Putnam begins by considering the reductionist program in psychology, which held that psychological concepts should be reduced to neurological or behavioral ones. He argues that this program fails because psychological states are functionally defined in terms of their causal roles, and those causal roles can be realized in many different physical systems. This is the multiple realizability argument applied to psychological predicates generally. As he states:
“Psychological predicates are not ‘theoretical terms’ in the sense that they are defined by their place in a reduction to physics. They are defined by their place in a functional organization.”
Putnam then addresses a potential objection: if psychological states are functionally defined, does that make them epiphenomenal — mere by-products of physical processes with no causal power of their own? He argues that functional states are causally real. A functional state like “being in pain” causes behavior (avoidance, crying out) and also causes other mental states (desire for relief). The fact that pain is realized in a physical state does not mean that pain itself has no causal role; on the contrary, its causal role is precisely what defines it.
The chapter also explores the relationship between psychological predicates and concepts like belief and desire. Putnam argues that these are not “inner objects” that we introspect but rather functional states that explain behavior. The intentionality of mental states — their “aboutness” or reference to objects and states of affairs — is not a mysterious property but a feature of their functional role. A belief that it is raining is a state that, in combination with other beliefs and desires, leads to behavior such as taking an umbrella. The content of the belief is determined by its place in this functional network.
Within the work, this chapter consolidates the functionalist position and defends it against standard objections. Putnam is showing that mental states are real, causally efficacious, and irreducible to physics — not because they are non-physical, but because the level of functional description is autonomous. The mind, for Putnam, is a causal system with its own laws and properties, even though it is physically realized. This is a naturalistic but non-reductive account of the mental, and it sets the stage for later debates in the philosophy of mind.
The Nature of Mental States: Further Clarifications and the Problem of Other Minds
We are in the heart of Part III, where Putnam continues to refine his functionalist account by addressing the nature of mental states as functional states of a whole organism, and then considers the epistemological problem of other minds. This section aims to show how functionalism dissolves traditional puzzles about mental state identity and knowledge of other minds.
Putnam begins by reiterating that a mental state, like pain, is not a brain state in the sense of being identical to some specific physical-chemical state of the brain. Instead, pain is a functional state — a state that is probabilistically caused by certain inputs (e.g., injury) and, in conjunction with other functional states, produces certain outputs (e.g., withdrawal behavior, groaning). He emphasizes that this is not a reduction of mental states to behavior; the functional state is internal and real, but it is defined by its causal role, not by its physical composition. The same functional state could be realized by different physical systems — a human, a robot, or a Martian with a different biochemistry.
From this, Putnam draws a key conclusion about the problem of other minds. The traditional skeptic asks: How can I know that other beings have minds, since I cannot directly observe their inner experiences? Putnam argues that if mental states are functional states, then the evidence for other minds is straightforward. If a system exhibits the same functional organization — that is, if it has the same causal structure of inputs, internal states, and outputs — then it is in the same mental states. The fact that the system is made of flesh and blood or of silicon is irrelevant; what matters is the functional pattern.
“If we take the view that mental states are functional states… then the problem of other minds becomes the problem of whether other organisms have the same functional organization as we do.”
He clarifies that this does not mean we must prove that others are functionally identical to us in every detail. Rather, we observe that other humans respond to stimuli, learn, reason, and behave in ways that are causally organized similarly to ourselves. The evidence is empirical and public — we can study their behavior, their neurophysiology (if relevant), and their verbal reports. The functionalist framework thus provides a criterion for ascribing mental states that is not mysterious or private. It demystifies the traditional philosophical problem by showing that the question is not about a hidden inner substance but about observable patterns of causal organization.
Within the work, this section is crucial because it answers a classic objection to mental state theories: the problem of epistemological access. Functionalists are often accused of making mental states unknowable, but Putnam turns this around. He argues that functionalism actually makes mental states more accessible than the identity theory does, because the functional description is based on publicly observable input-output relations. It also explains why we can confidently attribute minds to other humans (they share our functional organization) and why we might hesitate with simple machines (they lack the full causal complexity). The chapter thus ties together the metaphysical claim about the nature of mind with an epistemological claim about how we know minds.
The Meaning of “Meaning”: A Semantic Interlude
This section marks a significant shift from the philosophy of mind into the philosophy of language, as Putnam introduces his famous semantic externalism through the thought experiments that would later become central to his work. The chapter is titled “The Meaning of ‘Meaning'” and lays out a systematic critique of traditional theories of meaning, which held that the meaning of a term is determined by what is in the speaker’s head — that is, by internal mental representations.
Putnam begins by presenting the traditional view, which he calls the “mentalist” theory of meaning. According to this view, to understand a word like “water” is to have a certain concept or mental image associated with it, and the meaning of the word is that concept. The truth conditions of a sentence like “Water is wet” are determined by whether the object in the world fits the mental concept. Putnam challenges this with a series of Twin Earth thought experiments. He asks us to imagine a planet, Twin Earth, which is identical to Earth in every respect except for one: the liquid that falls from the sky, fills the lakes, and is called “water” by the inhabitants is not H₂O, but a different chemical compound with the same macro-properties (e.g., XYZ). On Earth, “water” means H₂O; on Twin Earth, “water” means XYZ.
Putnam then asks: Do the inhabitants of Earth and Twin Earth have the same mental state when they think about “water”? On the mentalist view, they would — after all, their internal concepts, images, and feelings are identical. But Putnam argues that the meanings of their words are different, because the extension of the term (the set of things it refers to) is different. The meaning of “water” depends on the actual chemical nature of the stuff in the environment, not just on what is in the speaker’s mind. He summarizes this with the famous slogan: “Meaning just ain’t in the head.”
“Cut the pie any way you like, ‘meaning’ has been given a faulty interpretation. The traditional view that meaning is a suitable concept… is simply wrong.”
He develops this by introducing the notion of linguistic division of labor. Most speakers do not know the precise chemical composition of water, gold, or elm trees, yet they successfully refer to these things. They rely on experts — chemists, botanists, metallurgists — to fix the reference. The community as a whole, through its experts, has the ability to determine whether something is really water or not. This means that reference is determined in part by the social and physical environment, not solely by individual mental states.
Within the work, this chapter is a revolutionary break with internalist theories of meaning and mind. It shows that the content of mental states themselves — what a person is thinking about — depends on factors external to the thinker. This has profound implications for the philosophy of mind: if meaning is external, then mental content is also external in some sense. The chapter thus connects back to Putnam’s earlier work on the nature of mental states and forward to his later views on realism and truth. It is one of the most cited and debated pieces in twentieth-century philosophy.
The Division of Linguistic Labor and the Indexicality of Natural Kind Terms
Continuing directly from the previous section, Putnam elaborates on the mechanisms that make reference work in a community. He introduces the concept of linguistic division of labor as a social phenomenon. In any community, there are some speakers who are experts — they can distinguish gold from fool’s gold, or elms from beeches — and there are others who are not. The non-experts still successfully refer to gold or elms because their linguistic community has a system of deference: they intend to refer to whatever the experts would identify as gold or elms.
Putnam argues that this is not a peripheral feature of language but central to how natural kind terms function. A term like “gold” does not have its reference fixed by a description like “yellow, malleable metal.” Instead, it is fixed by a causal-historical chain linking the current use to an original act of baptism — a moment when the term was introduced to refer to a certain kind of substance. The actual nature of that substance (its atomic number 79) then determines what counts as gold, regardless of the beliefs speakers have about it. This is why, Putnam says, a person can have a false belief about gold (e.g., that it is a compound) and still succeed in referring to gold.
“The extension of our terms depends upon the actual nature of the things that are pointed to or described at the time of the introduction of the term.”
He then introduces the idea that natural kind terms are indexical — like “I,” “here,” or “now.” Just as “here” refers to the place where the speaker is located, “water” refers to the stuff that has the same underlying nature as the stuff the speaker (or her community) pointed to as samples. This means that the reference of “water” is not fixed by a description in the head but by the actual environment. If the liquid on Twin Earth has a different chemical structure, then “water” on Earth and “water” on Twin Earth are different extensions, even if the speakers’ mental states are identical.
Within the work, this section deepens the externalist thesis by explaining the social and historical mechanisms that underpin reference. It also clarifies the indexical analogy, which is essential to understanding why the Twin Earth argument works. The indexicality of natural kind terms means that reference is tied to the actual world, not to possible worlds or mental concepts. This has implications for metaphysics as well: it suggests that the world itself, not just our concepts, plays a role in determining what our words mean. For Putnam, this is a key plank of his scientific realism — the view that scientific terms refer to real kinds in nature, and that successful reference depends on the actual structure of reality.
Realism and the Nature of Truth: Beyond Verificationism
In this final major section of Part III and the volume, Putnam turns to the nature of truth and realism, critiquing verificationist theories of truth that equate truth with what is verifiable or what would be verified under ideal conditions. He argues that such theories collapse the distinction between truth and justification, and that they cannot account for the fact that scientific theories often refer to unobservable entities.
Putnam begins by presenting the verificationist view in its strongest form: the claim that the truth of a statement consists in its being verified (or verifiable) by the appropriate methods. This view is often associated with logical positivism and with some interpretations of pragmatism. Putnam objects that verificationism leads to idealism or relativism, because it ties truth to human cognitive capacities. Instead, he defends a realist conception of truth: a statement is true if it corresponds to the way the world is, independently of whether we can verify it.
He uses the example of past statements about unobservable entities. In the early twentieth century, scientists postulated the existence of atoms. Could they have verified that atoms exist? Only indirectly, through theoretical inference. On a verificationist view, the statement “atoms exist” would not be true until the technology existed to detect them directly (e.g., with a scanning tunneling microscope). But Putnam argues that atoms did exist before anyone could directly detect them, and the statement “atoms exist” was true in 1900, even though no one could verify it. Truth is not the same as verification.
“Truth is not the same as justified assertibility. A statement can be true even if it is not verifiable, and false even if it is verified.”
He then introduces the notion of idealized epistemic conditions. Some verificationists try to avoid the objection by saying that a statement is true if it would be verified under ideal conditions (e.g., with unlimited time and resources). Putnam is skeptical of this move, because it is not clear how to define “ideal conditions” without already presupposing a notion of truth. If ideal conditions are defined as those in which all relevant facts are known, then the definition is circular. The realist alternative, he argues, is simpler and more plausible: truth is a non-epistemic notion, and our best scientific theories aim to describe a mind-independent reality.
Within the work, this chapter serves as the culmination of Putnam’s realism. It brings together the themes from earlier sections: the rejection of verificationism in mathematics (Part I), the defense of scientific realism against instrumentalism (Part II), and the externalist theory of meaning (Part III). Putnam is arguing for a robust realism that acknowledges the autonomy of the world from our theories, while also recognizing that our theories are fallible and historically conditioned. He is not a naive realist; he knows that scientific theories change. But he insists that there is an objective world that our theories attempt to describe, and that success in reference and prediction gives us reason to believe that we are making progress. The section ends with a defense of metaphysical realism against its critics, setting the stage for Putnam’s later, more complex views.
