singular term, and ‘prime number’ is a common noun. If the surface grammar of this sentence reflects its *logical form, and if ‘there is’ means ‘there exists’, then the sen- tence entails that both the number 1,000,000 and a greater prime number exist. For the realist-in-truth-value, this existence is objective, and so we are led to realism-in- ontology. In sum, if one is a realist-in-truth-value, then realism-in-ontology is the result of taking mathematical assertions at face value. Conversely, realist-in-ontology at least suggests a straightforward, Tarskian semantics. If mathematical objects exist independently of the mathematician, then there is no impediment to a straightforward model- theoretic semantics, which would presumably render assertions true or false objectively. Despite these natural alliances, the literature reveals no consensus on logical relationships between the two var- ieties of realism. Each of the four possible positions is articulated and defended by established and influential philosophers of mathematics. Kurt Gödel and Penelope Maddy adopt both forms of realism; Hartry Field and the traditional intuitionists L. E. J. Brouwer and Arend Heyting reject both forms; Geoffrey Hellman and Charles Chihara defend realism-in-truth-value, anti-realism-in- ontology; and Neil Tennant adopts the reverse, realism- in-ontology, anti-realism-in-truth-value (see the selected bibliography, especially the Benacerraf and Putnam anthology). Typical forms of realism-in-ontology nicely account for the necessity of mathematics and give impetus to the trad- itional view that mathematical knowledge is a priori. If mathematics is about a realm of eternal, abstract objects, then mathematical truth is surely independent of any con- tingencies of the material world around us, and math- ematics is not known through sensory experience. Of course, this alone does not account for the a priori nature of mathematics, until the realist provides an epistemology. It is not clear what these two varieties of realism have to say about the application of mathematics to science. Presum- ably, there is some sort of connection between the realm of abstract objects and the material world. The problem is to articulate it. For the most part, alternatives to realism-in-ontology fall into two groups. First, there are those who agree that mathematics has a distinctive subject-matter, but hold that mathematical objects are not independent of the mind, conventions, or language of the mathematician. The most common views in this camp take mathematical objects to be mental constructions, and so are varieties of *idealism (or applications of idealism to mathematics). Within this group, one unlikely possibility is to hold that mathematics is subjective, and so, presumably, each person has his own mathematics. A problem with this subjective idealism is to account for the fact that for the overwhelming most part, mathematicians agree with each other. It is more common for the idealist to hold that mathematics is both mind-dependent and objective or at least intersubjective in a principled way, perhaps by following *Kant in asserting that mathematics deals with structures common to human minds. The Kantian view accounts for the necessity of mathematics by holding that mathematics represents ways we must think, perceive, and apprehend, if we are to think, perceive, and appre- hend at all. Mathematical knowledge is a priori to the extent that we have a priori access to the structures com- mon to our minds. A Kantian might attempt to account for the application of mathematics to science by relating mathematics to the forms of sense perception. The writ- ings of traditional *intuitionists, such as Brouwer, have both subjective and Kantian themes, mostly the latter (see also *constructivism). The other alternative to realism-in-ontology is to sim- ply deny that mathematics has a subject-matter: there are no numbers, functions, sets, etc. If this view is not to lapse into a general scepticism, denying the truth of ordinary or scientific assertions, the burden is to give an account of mathematics, and its role in the intellectual enterprise, that does not involve a mathematical ontology. One com- mon manœuvre in this direction is to reconstrue math- ematical assertions in modal terms. For example, instead of asserting that there is a natural number with a given property, one asserts that it is possible to construct a cer- tain item with the given property. If the modal assertions have objective truth-values, then the result would be a realism-in-truth value, anti-realism-in-ontology (see the items by Hellman and Chihara in the selected bibliography). Another irrealist programme construes mathematics as fiction, much like what we read in novels. Statements about prime numbers are akin to statements about Miss Marple. At least prima facie, fictional discourse does not have ontological commitments. We do not believe that Miss Marple exists. On this view, mathematical assertions, taken literally, are vacuous. ‘There are at least a million prime numbers’ is false, for the simple reason that there are no numbers, and so no prime numbers. ‘All numbers are prime’ is vacuously true. With fiction, we distinguish literal truth from truth-in-the-story. ‘Miss Marple is a nosy busybody’ is false, but true in the relevant stories (or most of them). In this sense, mathematical assertions get their usual truth-values ‘in the story’. To account for the appli- cations of mathematics, a fictionalist might then try to give an account of the role of fictional mathematics in pre- sumably non-fictional discourses, like science. (See the items by Field and Burgess and Rosen in the selected bibli- ography.) Another view, developed by *logicism and adopted by *Logical Positivism, is to construe mathematical truths as *analytic, true in virtue of the meanings of its terms. The necessity of mathematics is semantic, or linguistic. Math- ematical knowledge is knowledge of meaning, which is presumably a priori. The problem, however, is to square this view with mathematics as practised. One needs to give an explication of the meanings of mathematical ter- minology according to which every mathematical truth is analytic. 570 mathematics, philosophy of, problems of The logicist Gottlob Frege showed how to derive the usual principles of arithmetic from what has come to be known as Hume’s principle: The number of F’s is identical to the number of G’s if and only if the F’s are equinumerous with the G’s. This result has considerable mathematical interest. Who would have thought that so much mathematics can be derived from a simple fact about the application of arith- metic to counting? Contemporary neo-logicists, such as Crispin Wright, propose to make this derivation the foun- dation of arithmetic. Hume’s principle, and *second-order logic, provide an epistemological *foundation of math- ematics. Neo-logicism accounts for the a priori nature of arithmetic, to the extent that we know Hume’s principle a priori, and that second-order derivation preserves a priori knowledge. The neo-logicists are realists-in-ontology, holding that the existence of numbers follows from Hume’s principle (together with some uncontroversial logical truths). Structuralism is a philosophy of mathematics that turns attention away from individual mathematical objects. The subject-matter of arithmetic, for example, is the nat- ural number structure, the form common to any infinite system of objects that has a distinguished initial object, which plays the role of zero, and a successor relation or operation that satisfies the induction principle. The nat- ural number structure is exemplified by the arabic numerals, sequences of characters on an alphabet in lexical order, an infinite sequence of distinct moments of time, etc. Simi- larly, real analysis is about the real number structure; topology is about topological structures; etc. According to the structuralist, the application of mathematics to science occurs, in part, by discovering or postulating that certain structures are exemplified in the material world. Math- ematics is to material reality as pattern is to patterned (see the items by Hellman, Resnik, and Shapiro in the selected bibliography). Most structuralists are realists-in-truth-value, but they differ among themselves on the question of ontology. Ante rem structuralism is a version of realism-in-ontology, taking structures, and their places, to be bona fide objects that exist independently of the mathematician. On this view, mathematical assertions are understood at face value. Against this, eliminative, or modal, structuralism takes statements about structures, and assertions within mathematics generally, to be generalizations over all sys- tems, or possible systems, that exemplify the structure. It is a structuralism without structures. There is a relatively recent tradition, traced to the later Wittgenstein, that attempts to accommodate mathematics in terms of the normative social practice inherent in a linguistic community. This denies that math- ematics is necessary and a priori, but it does account for the perceived necessity of mathematics. We have to accept the basic principles because we cannot imagine living any other way: they are fundamental to our ‘form of life’. To a large extent, epistemology depends on ontology and semantics. How we know mathematics is surely related to what mathematics is about and what its asser- tions mean. The most difficult problem with realism-in- ontology lies in this area. How can we know anything about a realm of eternal, acausal objects? Indeed, how can we have any confidence that what we say about these objects is true? To produce an adequate epistemology, a realist-in-ontology must tell us something about our- selves, the knowers, and something about mathematical objects, the known. A tall order. One route is to postulate a special faculty of mathemat- ical intuition, something that gives the mathematician direct or indirect access to the abstract, eternal, acausal mathematical universe. Supposedly, this mathematical intuition is analogous to sense perception, which gives us access to material objects. Plato and Gödel proposed epi- stemologies like this. Despite the eminence of these thinkers, this view is rejected, almost out of hand, by those who put constraints of *naturalism on epistemology. The idea is that humans must be understood as organisms in the natural world, and, as such, all faculties must be amenable to ordinary scientific scrutiny. For example, many philosophers hold that a person cannot have know- ledge about a certain type of object unless he or she has causal contact with at least samples of the objects. This rules out the sort of mathematical intuition envisioned above, assuming that mathematical objects are acausal. The few varieties of realism-in-ontology that deny that mathematical objects are abstract and eternal do not auto- matically run afoul of the naturalistic constraint. Presum- ably, if mathematical objects are material, then knowledge of them is no more problematic than know- ledge of any other material objects. Of course, this view gives up realism’s prima-facie account of the necessity and apriority of mathematics, but for some, this is a welcome loss. The problem is to square this view with mathematics as practised, and its apparent massively infinite ontology. W. V. O. Quine and Hilary Putnam, among others, have proposed a hypothetico-deductive account of math- ematical epistemology. The view begins with the obser- vation that virtually all of science is formulated in mathematical terms. Moreover, as far as we know, this is the only way to formulate scientific theories. So math- ematics is ‘confirmed’ to the extent that scientific theories are. The argument is that because mathematics is indis- pensable for science, and science is well-confirmed and (approximately) true, mathematics is well-confirmed and true as well. On this view, mathematical objects, like numbers and functions, are theoretical posits. They are the same kind of thing as electrons, and we know about them the same way we know about electrons—via their role in mature, well-confirmed scientific theories. Articu- lations of this view should (but usually do not) provide a careful analysis of the role of mathematics in science, rather than just noting the existence of this role. Such an account would shed some light on the ‘abstract’ nature of mathematical objects and the relationships between mathematics, philosophy of, problems of 571 mathematical objects and scientific or ordinary material objects. Typically, an advocate of the Quine–Putnam indispensability argument denies the necessity and aprior- ity of mathematics. Mathematics is only known through its role in science, which is clearly a contingent, a poster- iori affair. Because mathematics plays a central role in vir- tually every science, its disconfirmation is unlikely, but still possible in principle. Against such accounts, it does not seem that assertions like ‘2 + 3 = 5’ are really of a piece with assertions about small molecules. Indeed, it would seem to follow from the view in question that the math- ematical assertions are less firmly established than asser- tions about molecules, since mathematics is more theoretical—it lies further from sensory experience. Moreover, the view in question does not account for branches of mathematics, like higher set theory, that have not found application in science. In general, mathemat- icians usually do not look for confirmation in science before publishing their results, or otherwise claiming to know them. A typical strategy among both realists and anti-realists (of any stripe) is to relate mathematics to some other area of knowledge. This allows the philosopher to appropriate epistemological gains from the other area or, more often, to claim that mathematical knowledge is no more prob- lematic than knowledge in the other area (while conced- ing that the latter has its own epistemological problems). For example, according to ante rem structuralism, a mathematical structure and a pattern are the same kind of thing. Patterns are at least prima facie abstract, and yet we manage to have knowledge of them. Thus, some struc- turalists attempt to account for some mathematical knowledge via the psychological mechanism of pattern recognition. Patterns themselves are abstract, but we know about them, in part, by ordinary perceptual contact with systems of physical objects that exemplify them. At best, however, this suggestion accounts for knowledge of small, finite structures. One still needs to accommodate knowledge about the infinite structures studied in live mathematics. Along similar lines, modal structuralists and some fictionalists assimilate mathematical knowledge to know- ledge of modal assertions. There is no consensus concern- ing knowledge of what is necessary and possible, but it is generally agreed that we have such knowledge. In relatively recent history, there have been disputes concerning some principles and inferences within math- ematics. These include the law of *excluded middle, the axiom of *choice, the *extensionality of mathematical functions and properties, and *impredicative definitions, linguistic entities that define an item by reference to a class that contains the object being defined (e.g. ‘the least upper bound’). Such principles have been criticized (and defended) on philosophical grounds. For example, if mathematical objects are mental constructions or cre- ations, as the traditional intuitionists contend, then impredicative definitions are circular. One cannot create or construct an object by referring to a class of objects that (already) contains the item being created or constructed. On the other hand, for a realist-in-ontology, a definition does not represent a recipe for creating or constructing a mathematical object. Rather, a definition is a characteriza- tion or description of an object that already exists. From this point of view, there is nothing illicit in definitions that refer to classes containing the item in question. Character- izing ‘the least upper bound’ of a set is no different from defining the ‘elder jurist’ as ‘the oldest member of the Supreme Court’. The meaning of mathematical locutions like ‘There is a number such that . . .’ is related to one’s philosophy of mathematics. Intuitionists, for example, take this locution to mean ‘One can construct a number such that . . .’, while realists-in-ontology take existence to be independent of construction, or any human abilities for that matter. It fol- lows that the proper logic of mathematics is also tied to philosophical considerations. No doubt, how mathemat- ics is done, or should be done, has something to do with what mathematical discourse means. Typically, intuition- ists propose revisions in mathematical practice, based on philosophical considerations. The arguments concern the nature of mathematics and mathematical objects, with Brouwer, or the learnability of mathematical language and the ability of mathematicians to communicate with each other, with Michael Dummett. There is a substantial technical question of whether mathematics could serve the needs of science if the intuitionistic revisions were adopted. That is, if mathematics were changed to con- form to intuitionism, would the rest of the scientific enter- prise suffer? As far as contemporary mathematics is concerned, the aforementioned disputes are substantially over. The law of excluded middle, impredicative definitions, and the like are central parts of the enterprise nowadays. But this battle was not fought on philosophical grounds. Math- ematicians did not temporarily don philosophical hats and decide that numbers, say, really do exist independent of the mathematician and, for that reason, that it is correct to engage in the erstwhile disputed methodologies. Rather, the practices in question were found to be conducive to the practice of mathematics. This raises a meta-question concerning the relationship between mathematics and the philosophy of mathemat- ics. It is a central tenet of the naturalistically minded philosopher that there is no first philosophy that stands prior to science, ready to criticize it. Science should guide philosophy, not the other way around. I presume that the same goes for mathematics. If so, then one must either reject intuitionism or else find some mathematical or sci- entific reasons to revise mathematics, reasons that math- ematicians have overlooked to date, but which they would accept as compelling on mathematical grounds alone. That sort of quest does not seem to be under way— intuitionists are not naturalists, or else they do not extend naturalism to mathematics. s.s. *foundationalism in mathematics; intuitionism. 572 mathematics, philosophy of, problems of Paul Benacerraf and Hilary Putnam (eds.), Philosophy of Math- ematics (Englewood Cliffs, NJ, 1964) (comprehensive anthol- ogy). John Burgess and Gideon Rosen, A Subject with No Object: Strat- egies for Nominalistic Interpretation of Mathematics (Oxford, 1997). Charles Chihara, Constructibility and Mathematical Existence (Oxford, 1990). Hartry Field, Science without Numbers (Princeton, NJ, 1980). Gottlob Frege, Die Grundlagen der Arithmetik (Breslau, 1884), tr. J. L. Austin as The Foundations of Arithmetic, 2nd edn. (New York, 1960). Geoffrey Hellman, Mathematics without Numbers (Oxford, 1989). Penelope Maddy, Realism in Mathematics (Oxford, 1990). —— Naturalism in Mathematics (Oxford, 1997). Michael Resnik, Mathematics as the Science of Patterns (Oxford, 1997). Stewart Shapiro, Philosophy of Mathematics: Structure and Ontology (Oxford, 1997). —— Thinking about Mathematics: The Philosophy of Mathematics (Oxford, 2000) (general text). Neil Tennant, The Taming of the True (Oxford, 1997). Crispin Wright, Frege’s Conception of Numbers as Objects (Aberdeen, 1983). matter. ‘What is matter?—Never mind. What is mind?— No matter’. This Victorian joke has some substance to it, in that it draws attention to the fact that it is easier to dis- tinguish matter by contrasting it with something else than to say what it is. The joke also shows that if *substance is the ultimate ontological category, the fundamental stuff of *being or *existence, then matter is not the only candi- date for substantial status: common-sense *ontology holds that there are two substances, matter and some- thing else, *mind, *soul, or *spirit, the main characteristic of which is that it is non-material! Thus ‘contrast’ accounts of matter, though in some ways illuminating, are also frustrating. Alternative, non-contrasting accounts of matter tend merely to substitute something equally puzzling. Does it help to say that matter is physical substance, the basic raw material from which everything physical is composed? But some help is at hand through the suggestion (or the- ory) that matter is what is preserved during any process of physical change. The search by the *Pre-Socratics for what would later be called matter arose from their adherence to a generalized conservation principle: something cannot be created out of nothing and something cannot disappear into nothing. Thus whatever exists fundamentally can be neither created nor destroyed but persists and is con- served throughout all changes in nature. This doctrine was clearest among the *atomists, who claimed that what exists fundamentally is material atoms and the void, and all change or alteration, such as motion, combustion, or the growth and decay of living things, is merely the rearrangement of atoms in the void. But it was left to Aris- totle to establish matter as a category, by contrasting it with *form. The contrast between matter and mind also has Greek origins, but it was Descartes who elevated it into the metaphysical dualism that has proved so compelling to common sense, in spite of obvious difficulties (how do matter and mind interact?). Descartes equated matter and extension, adhering to the ancient principle that empty space is an impossibility, but his seventeenth-century rivals were reviving Greek atomism, or the ‘corpuscular philosophy’. Thus in Locke, for example, is found the idea that matter consists of microscopic particles, though this idea coexists uneasily with an influential alternative the- ory, that matter is the underlying *substratum that sup- ports the observable properties of things. Another contrast is that between matter and life. If mat- ter is simply inert substance, how can it produce the phe- nomena of *life and *consciousness? It cannot, according to *vitalism: something non-material must be added for living organisms to exist, a vital spark, a soul or spirit. But other theories of matter deny this contrast, and claim that life and consciousness are *emergent properties of matter, exhibited at a sufficiently complex level of organization. A variation is *panpsychism, according to which matter itself has non-physical properties of life and consciousness, in addition to the usual physical properties. What then is matter? From the point of view of science, it is matter as what-is-conserved that matters. But hasn’t modern physics ‘dematerialized’ matter, replacing it with energy or something even more abstract, like variations in the *space-time curvature? It is true that conservation principles—of mass, of momentum, of energy, etc.—are susceptible to the progress of science. Still, matter can be thought of as both what is fundamental in existence and what is conserved in change, granted that ideas about this are dependent on changing scientific theory. So matter persists. But conceptions of it change, sometimes rad- ically, but only for good theoretical reasons. a.bel. *prime matter; materialism; change. Rom Harré (ed.), The Physical Sciences since Antiquity (London, 1986). Thomas A. Holden, The Architecture of Matter: Galileo to Kant (Oxford, 2004). Ernan McMullin (ed.), The Concept of Matter in Modern Philosophy (Notre Dame, Ind., 1978). Stephen Toulmin and June Goodfield, The Architecture of Matter (London, 1962). matter, prime: see prime matter. maximin and minimax. Game- and decision-theoretical strategies which require one to make one’s worst possible outcome as good as possible (that is, to maximize the min- imum). Maximin grounds Rawls’s *‘difference principle’, that political institutions should make the position of the worst-off group as good as possible. ‘Minimax’ is occasion- ally used as a synonym for ‘maximin’. Used more precisely, it refers to maximin in zero-sum games, where the gain to one equals the loss to another. Maximin rationality is criti- cized for its extreme risk-aversion, and maximin justice for its insensitivity to the aggregation of welfare. r.cri. *game theory; decision theory. J. Rawls, A Theory of Justice (Cambridge, Mass., 1971), sect. 26. maximin and minimax 573 Maxwell’s Demon. A *thought experiment created by James Clerk Maxwell in which the second law of thermo- dynamics—that *entropy always increases—is apparently violated. The demon operates a trap-door in a partition which separates two compartments, A and B, of randomly moving particles of gas, letting those of above average velocity pass into A and those of below average velocity into B. The result will be greater temperature and pres- sure in A than B, a differential which can be exploited to run a perpetual motion engine. This conclusion is dis- puted on the grounds that entropy does increase within the whole system of gas plus the demon. s.r.a. *energy; time. Harvey S. Leff and Andrew F. Rex (eds.), Maxwell’s Demon 2: Entropy, Classical and Quantum Information, Computing (Bristol, 2002). McDowell, John (1942– ). Oxford philosopher, subse- quently professor at the University of Pittsburgh. McDowell has developed a conception of mind, language, and morality that derives its main inspiration from the later work of Wittgenstein. McDowell argues that most contemporary philosophy of mind is committed to the Cartesian picture of the subjective realm as something private, essentially detachable from its relations to the world. This picture leads to an intolerable scepticism about the *external world, or to the ‘darkness within’: the inability to explain genuine *intentionality. Much of McDowell’s work is an attempt to free the philosophy of mind from this picture; to this end he has articulated a radically exter- nalist theory of the mind, in which certain thoughts are not thinkable in the absence of the objects they are about (‘Russellian Singular Thoughts’; *Evans). t.c. *externalism; anti-individualism; heredity and environ- ment. J. McDowell, Mind and World (Cambridge, Mass., 1994). —— Mind, Value, and Reality (Cambridge, Mass., 1998). —— Meaning, Knowledge, and Reality (Cambridge, Mass., 1998). McGinn, Colin (1950– ). First studied psychology at Man- chester University and then philosophy at Oxford. He won the John Locke Prize at Oxford in 1973, going on to teach at University College London from 1974 to 1985, whereupon he became Wilde Reader in Mental Philoso- phy at Oxford until 1990, at which time he joined Rutgers University. His early work was mainly in philosophy of language, this giving way to an interest in philosophy of mind and metaphysics. He has written on subjectivity and objectivity, on the content of propositional attitudes, on the later Wittgenstein, and on metaphilosophy. His most recent work is concerned with the solubility or otherwise of philosophical problems, particularly the mind–body problem. He maintains that the deepest metaphysical problems—such as the nature of the self, meaning, free will—have solutions that lie outside the contingent bounds of human cognitive power. n.b. Colin McGinn, The Problem of Consciousness (Oxford, 1991). —— Problems in Philosophy: The Limits of Inquiry (Oxford, 1993). —— Knowledge and Reality: Selected Papers (Oxford, 1998). McTaggart, John McTaggart Ellis (1866–1925). Cam- bridge atheistic idealist now best known for his argument (in Mind (1908)) that *time is unreal. McTaggart distin- guished *‘A-series’ terms like ‘past’, ‘present’, ‘future’, and ‘B-series’ terms like ‘precede’, ‘simultaneous’, ‘follow’. He argued first that the B-series presupposes the A-series (e.g. if X precedes Y, there must be a time at which X is past and Y present), and then that the A-series is incoherent, since any event must have all three A-properties (‘past’, ‘pre- sent’, ‘future’), yet these are inconsistent. Saying it has them at different times, the apparently obvious way out, leads to an infinite regress, he claimed, since then we must raise the same question about these different times them- selves. The coherence of the A-series is still disputed, but so too is the alleged need for it. a.r.l. C. D. Broad, An Examination of McTaggart’s Philosophy, 2 vols. (Cambridge, 1933, 1938). J. M. E. McTaggart, Philosophical Studies, ed. S. V. Keeling (London, 1934). Mead, George Herbert (1863–1931). American philoso- pher of social *pragmatism and pioneer of sociology who taught at the University of Chicago as a prominent mem- ber of the Chicago School. The self, for Mead, ‘arises in the process of social experience and activity’. Essential to this process is the role of language as the form of reflexive communication. ‘It is in addressing himself in the role of an other that his self arises in experience.’ The social in humans generalized as fundamental to all nature is ‘social- ity’, ‘the capacity of being several things at once’. ‘[T]he emergent object belongs to different systems in its passage from the old to the new because of its systematic relation- ship with other structures, and possesses the characters it has because of its membership in these different systems.’ Human minds capable of occupying other systems as well as their own are, Mead said, ‘only the culmination of that sociality which is found throughout the universe’ p.h.h. David L. Miller, George Herbert Mead: Self, Language and the World (Austin, Tex., 1973). mean, doctrine of the. A central doctrine in Aristotle’s account of excellence of *character. Aristotle describes that excellence as concerned with pathe¯, i.e. motivational impulses, chiefly emotions, and with actions (sc. which issue from those motivations), and defines it as ‘a settled state issuing in choice, in a mean determined by a rational principle, viz. the one by which the agent of practical wis- dom would determine it’. This settled state is in a mean in the sense that the virtuous agent is neither excessively given to the various motivations prompting to action (e.g. excessively irascible) nor insufficiently sensitive to them, but responsive to the right extent, so at to choose to act on each motivation to the right degree, on the right occa- sions, for the right reasons, with reference to the right people, etc. The determination of what is right in all these 574 Maxwell’s Demon particular respects cannot be captured in any formula, but has to be the task of the educated judgement of the practic- ally wise agent, responding to the indefinitely variable range of circumstances in which action is required. c.c.w.t. J. O. Urmson, ‘Aristotle’s Doctrine of the Mean’, American Philo- sophical Quarterly (1973); repr. in A. Rorty (ed.), Essays on Aris- totle’s Ethics (Berkeley, Calif., 1980). meaning. Twentieth-century philosophy, in both the ‘analytic’ and ‘contintental’ traditions, was preoccupied with questions about linguistic meaning and the way lan- guage relates to reality. In the analytic tradition, this was largely as a consequence of the revolutions in logic initi- ated by Frege and Russell. Indeed, Michael Dummett has argued that the distinctive feature of *analytic philosophy is its assumption that ‘the philosophy of language is the foundation of the rest of the subject’. Even if one does not accept this claim, it is undeniable that the phenomena of meaning present some of the most intractable problems of philosophy. The meaning of a word looks, as it were, both ‘out- wards’ into the world, and ‘inwards’ to other words. The meaning of the word ‘tiger’, for example, is related both to those things in the world—tigers—to which it applies, and to other words with which it combines to make sentences which can be used to make assertions, ask questions, give warnings, and so on: ‘Tigers are animals’, ‘Is that a tiger?’, ‘Look out! A tiger!’ Whatever else is involved in meaning, it is clear that these two roles are clearly essential: for if one knows the meaning of the word ‘tiger’, one must have some grasp of how it applies to things in the world, and one must also be able to employ the word in an indefinite number of sentences. A theory of meaning—a ‘semantic theory’—is therefore obliged to explain how words can perform this dual function. In the seminal semantic theory of Gottlob Frege these two roles are explained together. Frege associated with each meaningful part of a *language something he called its ‘Bedeutung’, normally translated as ‘reference’. The reference of an expression is, intuitively, what it ‘stands for’: the reference of ‘George Orwell’, for example, is a particular man. Frege’s insight was to see that the refer- ences of the parts of a sentence contribute in a systematic way to the truth or falsehood of sentences in which those parts occur. Thus the truth or falsehood of the sentence ‘George Orwell wrote 1984’ is determined by the refer- ences of the individual words and the way they are put together. The overall significance of the sentence—for Frege, its truth or falsehood—is fixed by what the parts of the sentence ‘stand for’ in the world, and the relations between those parts. It follows from this claim that if you replace a word in a sentence S with a word having the same reference, the truth or falsehood of S will not change. But this gives rise to a notorious problem. Suppose that Alf believes that George Orwell wrote 1984, but does not know that Orwell is Blair. Then while the sentence ‘Alf believes that George Orwell wrote 1984’ will be true, the sentence ‘Alf believes that Eric Blair wrote 1984’ will be false. So if meaning is what determines the truth or falsehood of a sentence, there must be more to the meaning of a sentence than the references of its parts. Frege accounted for this by introducing another notion into the theory of meaning, which he called ‘Sinn’, usually translated ‘sense’. The sense of an expression is, intu- itively, not what is referred to by an expression, but the way it is referred to. Each sense determines one reference, but to one reference there may correspond many senses. (*Sense and reference; *intension and extension.) Central to Frege’s view is that senses are abstract objects, not ideas in people’s minds. (*Psychologism.) Frege’s basic idea is very appealing. However, one nat- ural question arises that Frege’s own work (deliberately) doesn’t address: given that words do refer to things, how do we explain this relation of reference? What makes it the case that any word refers to any object at all? A natural if vague answer is in terms of the psychological capacities of users of a language: words mean what they do because of what speakers of the language do with them. An example of this approach is *Logical Positivism, which held that the meaning of a sentence is given by an account of what it would take to verify the sentence. Here meaning is explained in terms of the psychological and other abilities of speakers to tell whether a sentence is true. The demise of Logical Positivism’s account of meaning was followed by an outbreak of scepticism about the notion of meaning, most influentially expressed in the work of W. V. Quine. Quine followed the positivists in linking meaning to experience, but argued that experience does not relate to individual sentences but to whole the- ories. Since he thinks that meaning must be empirically available, Quine frames the question thus: What evidence determines that someone means something by making certain sounds? Quine thinks that the only acceptable evi- dence is behavioural, and therefore shuns any appeal to introspection or Frege’s senses (the latter are ‘creatures of darkness’ whose criteria of identity are utterly obscure). But no amount of behavioural evidence can determine that a person’s words mean one thing rather another—it is always possible to construct alternative and incompatible ‘translations’ of the evidence. From here Quine moves to his famous claim that *translation is indeterminate, and reference is inscrutable: strictly speaking, there are no facts about what words and sentences mean. This is not an epistemic claim: reference is inscrutable because ‘there is nothing to scrute’. (We also find a very different scepti- cism about philosophical accounts of meaning in the later writings of Wittgenstein.) A significant attempt to explain meaning, with one eye towards Quine’s scepticism, was propounded by Donald Davidson in the 1960s and 1970s. Sharing Quine’s sympa- thies with *extensionality, Davidson attempted to account for meaning in terms of truth, which for some time had seemed more logically tractable than meaning. In particular, the Polish logician Alfred Tarski had defined meaning 575 truth for the sentences of certain *formal languages in terms of the relation of *satisfaction holding between the parts of sentences and sequences of objects. A sentence’s truth is determined systematically by the satisfaction of its parts; thus Tarski could show how to formally derive, from the axioms and rules of the theory, sentences (so-called ‘T-sentences’) which state what might intuti- tively be regarded as the conditions under which any sentence of the language is true. (The apparently banal T-sentence ‘“Snow is white” is true if and only if snow is white’ is a favourite example.) As we saw, the idea of the parts of a sentence making a systematic contribution to the meaning of the whole sen- tence was a key idea in Frege’s work. But Davidson explains meaning without using the troublesome idea of sense. Instead he proposes using a theory of truth in Tarski’s style to ‘serve’ as a theory of meaning. In outline, the idea is this: a theory of meaning for a language should at least entail, for any sentence of the language, a sentence that ‘gives its meaning’. The most obvious sort of case would be the ‘homophonic’ case: to give the meaning of a sentence would just be to give the sentence itself. For example, we recognize immediately that the sentence ‘“Snow is white” means that snow is white’ gives the meaning of ‘Snow is white’. This looks trivial, of course, but that is only because we already know what ‘Snow is white’ means. (The sentence ‘“La neve è bianca” means that snow is white’ does not look so trivial.) The theory must also show how the individual parts make a system- atic contribution to the sentences in which they occur. So now we know what the consequences of a theory of meaning must be—but how can we construct a theory that does actually have these consequences? Davidson’s insight was to see that if we replace ‘means that’ in the above sentence with ‘is true if and only if’, we will get the T-sentences that Tarski showed how to prove. And Tarski did this by showing how the truth of sentences was sys- tematically determined by the semantic properties of their parts. By employing Tarski’s theory of truth as a theory of meaning, Davidson put flesh on the skeletal idea that to give the meaning of a sentence is to give the conditions under which it is true. (*Truth-conditions.) But how does Davidson’s theory account for the phe- nomena Frege explained by employing the notion of sense? Giving truth-conditions alone will not do this— since ‘Orwell wrote 1984’ and ‘Blair wrote 1984’ have the same truth-conditions. Davidson replies that it is one thing to construct a formal theory that shows how the semantic properties of whole sentences are systematically formed from the semantic properties of their parts; it is another thing to establish how such a theory applies to individual speakers. The latter task is to provide an inter- pretative Tarskian truth-theory. In applying a truth-theory to a speaker, we must apply the constraints of a theory of radical interpretation, notably the ‘principle of charity’: assume that on the whole speakers are speaking the truth. Interpretation then proceeds as follows: collect the sentences that a speaker ‘holds true’, and devise a truth-theory that has these sentences as a formal conse- quence. To respect the intensionality of meaning, we need a theory that proves sentences like ‘“Orwell wrote 1984” is true if and only if Orwell wrote 1984’, and not ‘“Orwell wrote 1984” is true if and only if Blair wrote 1984’. But the theory that proves the interpretative T-sentences will be purely extensional. For Davidson, belief and meaning are interdepend- ent—one of the lessons he draws is that nothing can genu- inely have beliefs unless it also has a public language. Many philosophers have recoiled from this, both because they think that it is undeniable that certain non-linguistic creatures—such as dogs and apes—do have beliefs, and because they hope that meaning may yet be explained in terms of, or ultimately reduced to, the contents of mental states. One influential proposal is that of H. P. Grice, who suggested that the meanings of sentences can be reduced to a speaker’s intention to induce a belief in the hearer by means of their recognition of that intention. Although Grice’s programme is not as popular as it once was, the general idea of reducing meaning to the psychological states of speakers is now widely accepted (pace Davidson, Wittgenstein, and their followers). This is illustrated by the fact that, at the time of writing, the phil- osophy of language has to some extent yielded the centre stage to the philosophy of mind—and the problem of meaning has become the problem of intentionality. t.c. *cognitive meaning; communication; emotive and descriptive meaning; focal meaning; implicature; inde- terminacy of meaning; linguistic acts; phrastic and neustic; picture theory of meaning; language, problems of the philosophy of; use and meaning. Donald Davidson, Inquiries into Truth and Interpretation (Oxford, 1984). Gottlob Frege, Collected Papers (Oxford, 1984). H. P. Grice, ‘Meaning’, Philosophical Review (1957). A. W. Moore (ed.), Meaning and Reference (Oxford, 1993). W. V. Quine, Word and Object (Cambridge, Mass., 1960). Ludwig Wittgenstein, Philosophical Investigations (Oxford, 1953). meaning, cognitive: see cognitive meaning. meaning, emotive and descriptive: see emotive and descriptive meaning. meaning, focal: see focal meaning. meaning, picture theory of: see picture theory of meaning. meaning of life: see life, meaning of. means: see ends and means; instrumental value. measurement. An empirical procedure for ascertaining the *magnitude of a given quantitative property possessed by an object. Objects are measured on a scale, which assigns a unique numerical value to each magnitude of the 576 meaning quantitative property. The same quantitative property may be measured on different scales, and by different pro- cedures. Thus length can be measured in either feet or metres, with a ruler or by triangulation. Scales are typ- ically (but not always) defined by selecting a standard whose magnitude becomes the unit, 1. Other objects are then measured by determining how many times greater their magnitude is than that of the standard. An object found to be five times longer than the standard metre measures five metres. Sometimes scales are defined in terms of other scales, as with the cubic metre scale of vol- ume. In that case, the quantitative property is measured by measuring the quantities in terms of which the scale was defined and then calculating. The metre and cubic metre scales are called ratio scales, since numerical ratios among the scale values represent quantitative ratios among the magnitudes represented by those values. If the numerical value assigned to A is twice the numerical value assigned to B, then A is twice as long or twice as large as B. Many important scales of measurement lack this prop- erty, such as the Fahrenheit and Celsius scales of tempera- ture, and the Mohs scale of hardness. A 60° day is not twice as hot as a 30° day. Measurement in general has been crucial to scientific and technological progress, which in turn has increased phenomenally the precision and range of measurement. w.a.d. *number. N. R. Campbell, An Account of the Principles of Measurement and Calculations (London, 1938). B. Ellis, Basic Concepts of Measurement (Cambridge, 1966). mechanism. In the philosophy of mind, the doctrine that we are machines. Descartes held that other animals are machines, but only to emphasize his own view that human beings are not machines because they have *minds, which he supposed to be non-physical. The idea that human beings are machines was later urged by La Mettrie. Some form of this idea is widely accepted today. But decisions must be made about how to understand it, and there is resistance to it for other reasons than a com- mitment to *dualism. Leibniz argued in a famous passage that ‘perception, and what depends on it, cannot be explained mechan- ically’. For ‘if we imagine a machine whose construction ensures that it has thoughts, feelings, and perceptions, we can conceive it to be so enlarged . . . that we could enter it like a mill. On that supposition, when visiting it we shall find inside only components pushing one another, and never anything that could explain perception’ (Monadolo- gie, sect. 17). Such reasoning can seem very persuasive, but it begs the question. Leibniz just assumes that ‘compon- ents pushing one another’ could not amount to the machine’s thinking, feeling, or perceiving. Of course the assumption is easily made. Even mechanists concede that it is difficult to acquire even a faint idea of how a machine might have thoughts and feelings. The difficulty is all the greater if our conception of machinery is restricted to hydraulic and clockwork systems. However, if Leibniz had known about *computers he would at least have wanted to supplement his argument. The mathematical and logical thinking which con- tributed to the design and construction of computers also produced the basis for a definition of mechanism. Accord- ing to the ‘Church–Turing thesis’ any mechanical process can be modelled by means of a certain kind of abstract sys- tem known as a *Turing machine, and therefore by any equivalent system, such as a computer program. So we might define mechanism as the view that the workings of human beings exemplify computer programs. It is now clear that decisions have to be made. We are, after all, very complex systems, whose ‘workings’ may be considered from different points of view and at different levels of description and explanation. Assuming we are purely physical systems, we are composed of swarms of elemen- tary particles. But these particles are organized into atoms, and these into molecules. The molecules in their turn make up the organs and other components of our bodies. You might assume that if our workings exemplify com- puter programs at some level of description and explan- ation, they must do so at the other levels as well. But conceivably the behaviour of the elementary particles does not exemplify a program (as in effect we learn from *quantum mechanics) while the behaviour of the bodily organs does. If so, our workings—like those of computers themselves—are in the relevant sense mechanical at some levels of description but not at others. It seems that any variety of mechanism needs to be relativized to a level of description and explanation. An extreme variety might claim that we instantiate computer programs whose basic data represent the beliefs and desires of everyday psychology. That used to be the claim of some *artificial intelligence enthusiasts. The prac- tical difficulties of that approach are now fairly well known. One interesting theoretical objection, tenaciously developed by J. R. Lucas but widely attacked, has been that *Gödel’s theorem implies that human logicians can do things which would be impossible if we instantiated such programs. r.k. *determinism; determinism, scientific; freedom and determinism; mental reductionism. M. A. Boden (ed.), The Philosophy of Artificial Intelligence (Oxford, 1990). D. R. Hofstadter, Gödel, Escher, Bach (New York, 1979). J. R. Lucas, ‘Minds, Machines, and Gödel’, repr. in R. Anderson (ed.), Minds and Machines (Englewood Cliffs, NJ, 1964). A. M. Turing, ‘Computing Machinery and Intelligence’, repr. in R. Anderson (ed.), Minds and Machines (Englewood Cliffs, NJ, 1964). medical ethics. The study of ethical problems in medicine using the concepts, theories, literature, and techniques of moral philosophy. The phrase is used also to refer to the ethical beliefs or habits of behaviour of doctors and nurses, or to explicit codes governing professional behav- iour, such as the International Code of the World Medical medical ethics 577 Association. The subject has burgeoned since the 1960s into what amounts to an independent discipline, with its own specialists, centres, and journals. Many medical stu- dents are now exposed to at least some medical ethics education. Much medical ethics involves the deployment of philo- sophical moral theories in an attempt to solve medical eth- ical problems. This is useful not only in promoting understanding of the problem and possible solutions, but in elucidating and developing the theories themselves. For example, consider one of the many life-and-death issues which have come to dominate medical ethics: paternalism. A utilitarian philosopher who believes that welfare should be maximized might be tempted to sug- gest that a doctor should do whatever she believes to be in her patient’s best interests. But when she realizes that her theory may allow the doctor completely to ignore the patient’s wishes, she may attempt to incorporate the value of autonomy into her theory, perhaps as part of welfare. Over recent years, antipathy towards ethical theory in an abstract and systematic form (so-called anti-theory) has come to influence some writers in medical ethics, who tend to place weight less on the application of principles than on careful and sensitive appreciation of the particular case. It is worth noting also that pluralistic positions are available, combining, say, a utilitarian principle requiring medical personnel to do the greatest good for their patients with a Kantian principle setting various con- straints on maximization. For example, transplant sur- geons might be required to do the greatest good, but not at the price of forcibly violating the bodily integrity of involuntary ‘donors’. These issues are discussed in great depth in the work of Frances Kamm and Jeff McMahan. The teaching of medical ethics, especially in the USA, has been dominated by a pluralistic position: the four prin- ciples of Beauchamp and Childress—autonomy, non- maleficence, beneficence, and justice. One standard objection to the Beauchamp and Childress position applies to pluralistic views more generally: in cases of con- flict, unless we are to advocate some kind of *incommen- surability, we must make some decision, and can that decision not itself be seen as an articulation of a higher- order principle? And, if so, is there not a highest-order principle? Further, while it is true that the four principles are useful in alerting medical students to the various val- ues at stake in ethical dilemmas, it is equally important that emphasis in teaching be placed on the character or virtues involved in everyday ‘good practice’. Other life-and-death issues commonly discussed include *abortion and *euthanasia. Utilitarians have found it difficult to show why infanticide, on their view, can be any worse than abortion. The abortion debate has revolved around the partly Kantian question whether the foetus is a *person with *rights. Some have argued that the status of the foetus as a potential person is important here. At the other end of life, similar theoretical issues arise. If I have a right to life, can I waive it voluntarily, enabling a doctor to administer a lethal dose if I am in the terminal stages of some illness? Should we administer such doses to decrease overall suffering? These questions have become more urgent as medical life-preserving technology has advanced. Advances in reproductive technology have also raised several significant issues. Many people have seen tech- niques such as in vitro fertilization for infertility as the first step on a slippery slope to a Brave New World. The nature of the family has also been thrown into doubt: why should a homosexual couple or a single person not bring up a child created using reproductive technology? Is it accept- able that a child might have three ‘mothers’ (genetic, birth, nurturing)? Genetics too has thrown up many prob- lems, concerning, for example, genetic screening or the genetic modification of non-human animals for xeno- transplantation. Other topics concern the everyday practice of health care personnel more directly. Positions taken on confi- dentiality, for example, again arise from differences at the level of ethical theory. A Kantian may argue that the promise implied in the contract between doctor and patient forbids any breach of confidentiality for the bene- fit of others or the patient herself. Or a utilitarian may point to the harm which might occur if people were no longer able to trust doctors, thus supplying a welfare- based ground for the practice of respecting confidentiality. Finally, a philosopher attracted to the *virtues might stress the importance of confidentiality in the relationship of trust between the patient and the doctor. The main political issue discussed in medical ethics is the allocation of resources. A popular notion here is the QALY (quality-adjusted life-year), which represents an attempt to make length or quantity of life commensurate with its quality. Thus a year of healthy life is said to be worth 1, while a year of rather poor health might be worth only 0.5. The QALY theory most often used is essentially a health-maximizing version of *utilitarianism. It therefore runs into the same problems as most versions of utilitar- ianism: its conception of what is to be maximized, and how, is dubious, and it ignores fairness. r.cri. *Applied ethics; bioethics. T. Beauchamp and J. Childress, Principles of Biomedical Ethics, 5th edn. (New York, 2001). H. T. Engelhardt, The Foundations of Bioethics (Oxford, 1986). K. W. M. Fulford, Donna L. Dickenson, and T. H. Murray (eds.), Healthcare Ethics and Human Values: An Introductory Text with Readings and Case Studies (Oxford, 2002). J. Glover, Causing Death and Saving Lives (Harmondsworth, 1977). F. Kamm, Morality, Mortality, vol. 2: Death and Whom to Save from It (New York, 1993). J. McMahan, The Ethics of Killing: Problems at the Margins of Life (New York, 2002). P. Singer, Rethinking Life and Death: The Collapse of our Traditional Ethics (New York, 1994). medicine, philosophy of. The philosophical, as distinct from the ethical and historical, problems in and around medicine are not largely discussed. There are fragments in 578 medical ethics Greek and Arabic writings, but in recent years there have been surprisingly few philosophical, as distinct from eth- ical, writings on medicine. Yet the philosophical problems of medicine are of great interest, because they include most of those discussed within the philosophy of science and the philosophy of social science, with the added twist that they occur in a distinctive context. Without claiming to cover every area of philosophical interest, I shall briefly mention four problems. First, there are problems of a kind familiar in the philoso- phy of science. For example, ‘evidence-based medicine’ has become a slogan in medical circles. But what is the nature of the evidence? The evidence often cited as the ‘gold standard’ is that of randomized control trials. Yet such trials are largely statistical correlations of treatments with percentage success rates using volunteer patients who may not be at all typical. Again, the evidence in diag- nosis, in identifying the patient’s medical problem, is quite different from that in control trials, being more like that used by a detective. And of course familiar problems of realism versus anti-realism arise if we ask ‘What is dis- ease?’ Secondly, much medical research is qualitative, involving questionnaire-based inquiries into concepts such as ‘quality of life’. The problem which arises here is over the attempt to quantify and produce measurement scales for such inquiries. These are problems in the philoso- phy of social science. Thirdly, there are problems of personal identity thrown up by genetics. How far are my personal traits determined by my genotype? Finally, there are problems of the acceptable limits of medicine. It used to be a set of techniques for repairing the human body when it broke down, but there are some who hold that it has the potential to make us immortal. The charm of the philosophy of medicine is that its problems are at the cut- ting edge of science, but these problems must be discussed from a humanistic perspective which prevents them from becoming dry abstractions. r.s.d. R. S. Downie and Jane Macnaughton, Clinical Judgement: Evidence in Practice (Oxford, 2001). Carl Elliott, A Philosophical Disease: Culture, Identity and Bioethics (New York, 1998). medieval philosophy. Histories of medieval philosophy tend to start with St Augustine (354–430), if not earlier; but Augustine was of the late Roman Empire, centuries before the Middle Ages, and is included in such books not because he was a medieval thinker but because he cast such a long shadow across medieval philosophy. He provided a role model in that he thought deeply, systematically, and in a philosophical way about Chris- tianity. He was familiar with the writings of the philoso- phers of Greece and Rome, particularly the Stoics and the Neoplatonic schools, and put that knowledge to work in the elucidation of fundamental concepts such as those of God, eternity, time, good and evil, and creation. The first great philosopher of the Middle Ages, St Anselm of Canterbury (1033–1109), was deeply influenced by him, and St Thomas Aquinas (1224/5–74) cited him far more often than he cited any other of the Church Fathers. Like St Augustine the medieval thinkers philosophized because they wished to understand Christianity. Indeed Anselm’s famous phrase fides quaerens intellectum (faith seeking understanding) is a perfect description of the phil- osophy written in the Christian West throughout the Middle Ages. A major part of their task of clarification involved demonstrating that Christianity is not incompat- ible with what can be demonstrated by reason. The doc- trine of *double truth, particularly associated with Averroës, which declares that a truth of faith can be incompatible with truths sanctioned by reason, made very little impact upon thinkers in the Christian West. For them it was crucially important to establish that Christianity was not incompatible with any proposition demonstrated by philosophy. For a proposition thus demonstrated must be true and anything incompatible with the truth is false. At the start of the period Aristotle’s works had been unknown except for a few treatises (fewer than was real- ized at the time since two treatises attributed to him were spurious, the Theology of Aristotle, which is really part of Plotinus’ Enneads, and the Book of Causes, which is an Arabic epitome of Proclus). As time passed more of Aris- totle’s writings became available, reaching the Christian West from the Muslim world, often accompanied by detailed and profound commentaries by Muslim thinkers such as al-Fa¯ra¯bı¯ (died c.950), Avicenna (980–1037), and Averroës (1126–98). These texts with their Arabic com- mentaries were promptly translated into Latin. Averroës’ interpretation of Aristotle was so influential that philoso- phers of the Christian West referred to him simply as ‘the Commentator’. And since Aristotle’s was the system to which every philosopher and theologian had to react (indeed he was referred to almost universally simply as philosophus—the Philosopher), the crucial question tackled was not whether Christianity and philosophy were compatible, but whether Christianity and Aristotelian phil- osophy were compatible. For the most part the answer given was affirmative. When, as rarely happened, it was not, then of course Aristotle’s position had to be rejected. Much the most important point of conflict concerned Aristotle’s analysis of motion in the Physics and Meta- physics, which led him to conclude that the world was eternal. Many medieval philosophers, believing that the world had a beginning in time, found it necessary to argue against Aristotle’s arguments. It should be noted that most did not attempt to prove that the world did have a beginning in time. A consensus formed round the opinion that the question whether it was eternal or had a temporal beginning was not philosophically demonstrable, and that the doctrine was to be accepted on faith, being the plain meaning of the first sentence of Genesis. Some indeed held that Aristotle did not think that he had demonstrated the eternity of the world but had merely presented the doctrine as a probable opinion. On that interpretation the standard interpretation of Genesis 1:1 was compatible with Aristotle’s teaching. Aquinas was one major figure medieval philosophy 579 . composed of swarms of elemen- tary particles. But these particles are organized into atoms, and these into molecules. The molecules in their turn make up the organs and other components of our bodies. You. prompting to action (e.g. excessively irascible) nor insufficiently sensitive to them, but responsive to the right extent, so at to choose to act on each motivation to the right degree, on the right. which those parts occur. Thus the truth or falsehood of the sentence ‘George Orwell wrote 1984’ is determined by the refer- ences of the individual words and the way they are put together. The overall